Subgroups — C₂₃×#25446 #19173072

Order 5888.

876 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3154	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3069	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3068	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3067	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#471	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2049	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2048	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2009	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2008	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1915	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1914	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#575	C₂²	1	1	yes	characteristic
17	C₂₃×#575	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
23	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#568	C₂²	1	1	yes	characteristic
25	C₂₃×#568	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
27	C₂₃×#536	C₂²	1	1	yes	characteristic
28	C₂₃×#535	C₂²	1	1	yes	characteristic
29	C₂₃×#497	—	2	1	no	—
30	C₂₃×#496	—	2	1	no	—
31	C₂₃×#495	—	2	1	no	—
32	C₂₃×#492	—	2	1	no	—
33	C₂₃×#473	—	2	1	no	—
34	C₂₃×#473	—	2	2	no	—
35	C₂₃×#472	—	2	1	no	—
36	C₂₃×#472	—	2	2	no	—
37	C₂₃×#471	—	2	1	no	—
38	C₂₃×#471	—	2	1	no	—
39	C₂₃×#471	—	2	2	no	—
40	C₂₃×#471	—	2	2	no	—
41	C₂₃×#471	—	2	2	no	—
42	C₂₃×#471	—	2	2	no	—
43	C₂₃×#444	—	2	2	no	—
44	C₂₃×#443	—	2	2	no	—
45	C₂₃×#430	C₂²	1	1	yes	characteristic
46	C₂₃×#430	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
49	C₂₃×#421	C₂²	1	1	yes	characteristic
50	C₂₃×#421	C₂²	1	1	yes	characteristic
51	C₂₃×#419	C₂²	1	1	yes	characteristic
52	C₂₃×#418	C₂²	1	1	yes	characteristic
53	C₂₃×#413	C₂²	1	1	yes	characteristic
54	C₂₃×#413	C₂²	1	1	yes	characteristic
55	C₂₃×#413	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
58	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
59	C₂×C₂₃×#100	C₂²	1	1	yes	characteristic
60	C₂₃×#410	C₂²	1	1	yes	characteristic
61	C₂₃×#405	C₂²	1	1	yes	characteristic
62	C₂₃×#405	C₂²	1	1	yes	characteristic
63	C₂₃×#344	C₂²	1	1	yes	characteristic
64	C₂₃×#344	C₂²	1	1	yes	characteristic
65	C₂₃×#344	C₂²	1	1	yes	characteristic
66	C₂₃×#344	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×#137	—	2	1	no	—
73	C₂₃×#137	—	2	1	no	—
74	C₂₃×#137	—	2	1	no	—
75	C₂₃×#137	—	2	1	no	—
76	C₂₃×#137	—	2	2	no	—
77	C₂₃×#137	—	2	2	no	—
78	C₂₃×#137	—	4	1	no	—
79	C₂₃×#137	—	4	1	no	—
80	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
81	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
82	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
83	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
84	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
85	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
86	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
87	C₂₃×AGL(1,ℤ/8)	—	4	1	no	—
88	C₂₃×#135	—	2	1	no	—
89	C₂₃×#135	—	2	1	no	—
90	C₂₃×#135	—	2	1	no	—
91	C₂₃×#135	—	2	1	no	—
92	C₂₃×#135	—	4	1	no	—
93	C₂₃×#135	—	4	1	no	—
94	C₂×Q₁₆×C₂₃	—	2	1	no	—
95	C₂×Q₁₆×C₂₃	—	2	1	no	—
96	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
97	C₂×SD₁₆×C₂₃	—	2	1	no	—
98	C₂×SD₁₆×C₂₃	—	2	1	no	—
99	C₂×SD₁₆×C₂₃	—	2	1	no	—
100	C₂×SD₁₆×C₂₃	—	2	1	no	—
101	C₂×SD₁₆×C₂₃	—	2	1	no	—
102	C₂×SD₁₆×C₂₃	—	2	1	no	—
103	C₂×SD₁₆×C₂₃	—	2	1	no	—
104	C₂×SD₁₆×C₂₃	—	2	1	no	—
105	C₂×SD₁₆×C₂₃	—	2	1	no	—
106	C₂×SD₁₆×C₂₃	—	2	1	no	—
107	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
108	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
109	C₂×D₈×C₂₃	—	2	1	no	—
110	C₂×D₈×C₂₃	—	2	1	no	—
111	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
112	C₂₃×#131	—	2	2	no	—
113	C₂₃×#131	—	2	2	no	—
114	C₂₃×#131	—	2	2	no	—
115	C₂₃×#131	—	2	2	no	—
116	C₂₃×#131	—	2	1	no	—
117	C₂₃×#131	—	2	1	no	—
118	C₂₃×#131	—	2	1	no	—
119	C₂₃×#131	—	2	1	no	—
120	C₂×M₁₆×C₂₃	—	2	1	no	—
121	C₂×M₁₆×C₂₃	—	4	1	no	—
122	C₂×M₁₆×C₂₃	—	4	1	no	—
123	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
124	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
130	C₂²×C₈×C₂₃	—	2	1	no	—
131	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
132	C₂₃×C₄⋊Q₈	—	2	1	no	—
133	C₂₃×#127	—	2	1	no	—
134	C₂₃×#124	—	2	1	no	—
135	C₂₃×#124	—	2	1	no	—
136	C₂₃×#123	—	2	2	no	—
137	C₂₃×C₂²⋊Q₈	—	4	1	no	—
138	C₂₃×#121	—	4	1	no	—
139	C₂₃×#120	—	2	2	no	—
140	C₂₃×#117	C₂³	1	1	yes	characteristic
141	C₂₃×#117	C₂³	1	1	yes	characteristic
142	C₂₃×#108	—	2	2	no	—
143	C₂₃×#108	—	2	2	no	—
144	C₂₃×#108	—	2	1	no	—
145	C₂₃×#108	—	2	1	no	—
146	C₂₃×#108	D₄	1	2	yes	complemented
147	C₂₃×#108	D₄	1	2	yes	complemented
148	C₂₃×#104	—	2	2	no	—
149	C₂₃×#104	—	2	2	no	—
150	C₂₃×#103	—	4	1	no	—
151	C₂₃×#102	—	4	1	no	—
152	C₂₃×#101	—	2	1	no	—
153	C₂₃×#101	—	2	1	no	—
154	C₂₃×#101	—	2	2	no	—
155	C₂₃×#101	—	2	2	no	—
156	C₂₃×#101	D₄	1	2	yes	complemented
157	C₂₃×#100	—	2	1	no	—
158	C₂₃×#100	—	2	1	no	—
159	C₂₃×#100	—	2	2	no	—
160	C₂₃×#100	—	2	2	no	—
161	C₂₃×#100	D₄	1	2	yes	complemented
162	C₂₃×#99	—	2	2	no	—
163	C₂₃×#97	—	2	1	no	—
164	C₂₃×#97	—	2	1	no	—
165	C₄×C₈×C₂₃	—	2	1	no	—
166	C₄×C₈×C₂₃	—	2	1	no	—
167	C₂⁴×C₂₃	—	2	1	no	—
168	#41×C₂₃	—	2	1	no	—
169	#41×C₂₃	—	2	1	no	—
170	#41×C₂₃	—	2	1	no	—
171	#41×C₂₃	—	2	1	no	—
172	#41×C₂₃	—	2	1	no	—
173	#41×C₂₃	—	2	1	no	—
174	#41×C₂₃	—	2	1	no	—
175	#41×C₂₃	—	2	1	no	—
176	#41×C₂₃	—	2	1	no	—
177	#41×C₂₃	—	2	1	no	—
178	#41×C₂₃	—	2	1	no	—
179	#41×C₂₃	—	2	1	no	—
180	C₂×Q₈×C₂₃	—	2	1	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	1	no	—
183	C₂×Q₈×C₂₃	—	2	1	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	2	1	no	—
186	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×D₄×C₂₃	—	2	1	no	—
190	C₂×D₄×C₂₃	—	2	1	no	—
191	C₂×D₄×C₂₃	—	2	1	no	—
192	C₂×D₄×C₂₃	—	2	1	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	—	2	1	no	—
196	C₂×D₄×C₂₃	—	2	1	no	—
197	C₂×D₄×C₂₃	—	2	1	no	—
198	C₂×D₄×C₂₃	—	4	2	no	—
199	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₂₃	—	2	1	no	—
203	C₂²×C₄×C₂₃	—	2	1	no	—
204	C₂²×C₄×C₂₃	—	2	1	no	—
205	C₂²×C₄×C₂₃	—	2	1	no	—
206	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₂₃	—	2	1	no	—
208	Q₁₆×C₂₃	—	2	1	no	—
209	Q₁₆×C₂₃	—	4	2	no	—
210	Q₁₆×C₂₃	—	4	2	no	—
211	Q₁₆×C₂₃	—	4	1	no	—
212	Q₁₆×C₂₃	—	4	1	no	—
213	SD₁₆×C₂₃	—	2	1	no	—
214	SD₁₆×C₂₃	—	2	1	no	—
215	SD₁₆×C₂₃	—	2	1	no	—
216	SD₁₆×C₂₃	—	2	1	no	—
217	SD₁₆×C₂₃	—	4	2	no	—
218	SD₁₆×C₂₃	—	4	2	no	—
219	SD₁₆×C₂₃	—	4	2	no	—
220	SD₁₆×C₂₃	—	4	2	no	—
221	SD₁₆×C₂₃	—	4	1	no	—
222	SD₁₆×C₂₃	—	4	1	no	—
223	SD₁₆×C₂₃	—	4	1	no	—
224	SD₁₆×C₂₃	—	4	1	no	—
225	SD₁₆×C₂₃	—	4	1	no	—
226	SD₁₆×C₂₃	—	4	1	no	—
227	SD₁₆×C₂₃	—	4	1	no	—
228	SD₁₆×C₂₃	—	4	1	no	—
229	D₈×C₂₃	—	2	1	no	—
230	D₈×C₂₃	—	2	1	no	—
231	D₈×C₂₃	—	4	2	no	—
232	D₈×C₂₃	—	4	2	no	—
233	D₈×C₂₃	—	4	1	no	—
234	D₈×C₂₃	—	4	1	no	—
235	M₁₆×C₂₃	—	2	1	no	—
236	M₁₆×C₂₃	—	2	1	no	—
237	M₁₆×C₂₃	—	2	1	no	—
238	M₁₆×C₂₃	—	2	1	no	—
239	M₁₆×C₂₃	—	2	1	no	—
240	M₁₆×C₂₃	—	2	1	no	—
241	M₁₆×C₂₃	—	2	1	no	—
242	M₁₆×C₂₃	—	2	1	no	—
243	M₁₆×C₂₃	—	2	1	no	—
244	M₁₆×C₂₃	—	2	1	no	—
245	M₁₆×C₂₃	—	4	2	no	—
246	M₁₆×C₂₃	—	4	2	no	—
247	M₁₆×C₂₃	—	4	1	no	—
248	M₁₆×C₂₃	—	4	1	no	—
249	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
250	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
251	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
252	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
253	C₂×C₈×C₂₃	—	2	1	no	—
254	C₂×C₈×C₂₃	—	2	1	no	—
255	C₂×C₈×C₂₃	—	2	1	no	—
256	C₂×C₈×C₂₃	—	2	1	no	—
257	C₂×C₈×C₂₃	—	2	1	no	—
258	C₂×C₈×C₂₃	—	2	1	no	—
259	C₂×C₈×C₂₃	—	2	1	no	—
260	C₂×C₈×C₂₃	—	4	2	no	—
261	C₂×C₈×C₂₃	—	4	2	no	—
262	C₂×C₈×C₂₃	—	4	1	no	—
263	C₂×C₈×C₂₃	—	4	1	no	—
264	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
265	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
266	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
268	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
269	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
270	C₄⋊C₄×C₂₃	—	2	1	no	—
271	C₄⋊C₄×C₂₃	—	2	1	no	—
272	C₄⋊C₄×C₂₃	—	4	2	no	—
273	C₂²⋊C₄×C₂₃	—	2	1	no	—
274	C₂²⋊C₄×C₂₃	—	4	2	no	—
275	C₂²⋊C₄×C₂₃	—	4	2	no	—
276	C₂²⋊C₄×C₂₃	#41	1	2	yes	—
277	C₄²×C₂₃	—	2	1	no	—
278	C₄²×C₂₃	—	2	1	no	—
279	#25446	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂³×C₂₃	—	2	1	no	—
281	C₂³×C₂₃	—	2	1	no	—
282	C₂³×C₂₃	—	2	1	no	—
283	C₂³×C₂₃	—	4	1	no	—
284	C₂³×C₂₃	—	4	1	no	—
285	C₂³×C₂₃	—	4	1	no	—
286	C₂³×C₂₃	—	16	1	no	—
287	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
288	Q₈×C₂₃	—	2	1	no	—
289	Q₈×C₂₃	—	2	1	no	—
290	Q₈×C₂₃	—	2	1	no	—
291	Q₈×C₂₃	—	2	1	no	—
292	Q₈×C₂₃	—	2	1	no	—
293	Q₈×C₂₃	—	2	1	no	—
294	Q₈×C₂₃	—	4	1	no	—
295	Q₈×C₂₃	—	4	1	no	—
296	Q₈×C₂₃	—	4	1	no	—
297	Q₈×C₂₃	—	4	1	no	—
298	D₄×C₂₃	—	2	1	no	—
299	D₄×C₂₃	—	2	1	no	—
300	D₄×C₂₃	—	2	1	no	—
301	D₄×C₂₃	—	2	1	no	—
302	D₄×C₂₃	—	2	1	no	—
303	D₄×C₂₃	—	2	1	no	—
304	D₄×C₂₃	—	4	1	no	—
305	D₄×C₂₃	—	4	1	no	—
306	D₄×C₂₃	—	4	1	no	—
307	D₄×C₂₃	—	4	1	no	—
308	D₄×C₂₃	—	4	1	no	—
309	D₄×C₂₃	—	4	1	no	—
310	D₄×C₂₃	—	4	1	no	—
311	D₄×C₂₃	—	4	1	no	—
312	D₄×C₂₃	—	4	1	no	—
313	D₄×C₂₃	—	4	1	no	—
314	D₄×C₂₃	—	16	2	no	—
315	C₂×C₄×C₂₃	—	2	1	no	—
316	C₂×C₄×C₂₃	—	2	2	no	—
317	C₂×C₄×C₂₃	—	2	1	no	—
318	C₂×C₄×C₂₃	—	2	1	no	—
319	C₂×C₄×C₂₃	—	4	1	no	—
320	C₂×C₄×C₂₃	—	4	1	no	—
321	C₂×C₄×C₂₃	—	4	1	no	—
322	C₂×C₄×C₂₃	—	4	1	no	—
323	C₂×C₄×C₂₃	—	4	1	no	—
324	C₂×C₄×C₂₃	—	4	1	no	—
325	C₂×C₄×C₂₃	—	4	1	no	—
326	C₂×C₄×C₂₃	—	4	1	no	—
327	C₂×C₄×C₂₃	—	4	1	no	—
328	C₂×C₄×C₂₃	—	4	1	no	—
329	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
332	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
333	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
334	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
335	C₈×C₂₃	—	2	2	no	—
336	C₈×C₂₃	—	2	2	no	—
337	C₈×C₂₃	—	2	2	no	—
338	C₈×C₂₃	—	2	2	no	—
339	C₈×C₂₃	—	4	1	no	—
340	C₈×C₂₃	—	4	1	no	—
341	C₈×C₂₃	—	4	1	no	—
342	C₈×C₂₃	#121	1	1	yes	characteristic
343	C₈×C₂₃	#121	1	1	yes	characteristic
344	C₈×C₂₃	#121	1	1	yes	characteristic
345	C₈×C₂₃	#121	1	1	yes	characteristic
346	#3154	C₂×C₂₃	1	1	yes	characteristic, complemented
347	#3069	C₂×C₂₃	1	1	yes	characteristic, complemented
348	#3068	C₂×C₂₃	1	1	yes	characteristic, complemented
349	#3067	C₂×C₂₃	1	1	yes	characteristic, complemented
350	C₂×#471	C₂×C₂₃	1	1	yes	characteristic, complemented
351	#2957	C₂×C₂₃	1	1	yes	characteristic, complemented
352	#2049	C₂×C₂₃	1	1	yes	characteristic, complemented
353	#2048	C₂×C₂₃	1	1	yes	characteristic, complemented
354	#2009	C₂×C₂₃	1	1	yes	characteristic, complemented
355	#2008	C₂×C₂₃	1	1	yes	characteristic, complemented
356	#1977	C₂×C₂₃	1	1	yes	characteristic, complemented
357	#1953	C₂×C₂₃	1	1	yes	characteristic, complemented
358	#1915	C₂×C₂₃	1	1	yes	characteristic, complemented
359	#1914	C₂×C₂₃	1	1	yes	characteristic, complemented
360	#1757	C₂×C₂₃	1	1	yes	characteristic, complemented
361	C₂²×C₂₃	—	4	1	no	—
362	C₂²×C₂₃	—	4	1	no	—
363	C₂²×C₂₃	—	4	1	no	—
364	C₂²×C₂₃	—	4	1	no	—
365	C₂²×C₂₃	—	8	1	no	—
366	C₂²×C₂₃	—	8	1	no	—
367	C₂²×C₂₃	—	8	1	no	—
368	C₂²×C₂₃	—	8	2	no	—
369	C₂²×C₂₃	—	8	1	no	—
370	C₂²×C₂₃	—	16	1	no	—
371	C₂²×C₂₃	—	16	1	no	—
372	C₂²×C₂₃	#532	1	1	yes	characteristic
373	C₂²×C₂₃	#546	1	1	yes	characteristic
374	C₂²×C₂₃	#550	1	1	yes	characteristic
375	C₄×C₂₃	—	4	2	no	—
376	C₄×C₂₃	—	4	1	no	—
377	C₄×C₂₃	—	4	1	no	—
378	C₄×C₂₃	—	4	1	no	—
379	C₄×C₂₃	—	8	1	no	—
380	C₄×C₂₃	—	8	1	no	—
381	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
382	C₄×C₂₃	#540	1	1	yes	characteristic
383	C₄×C₂₃	D₄²	1	1	yes	characteristic
384	C₄×C₂₃	D₄²	1	1	yes	characteristic
385	#575	C₂²×C₂₃	1	1	yes	characteristic
386	#575	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
389	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
392	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
393	#568	C₂²×C₂₃	1	1	yes	characteristic
394	#568	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
396	#536	C₂²×C₂₃	1	1	yes	characteristic
397	#535	C₂²×C₂₃	1	1	yes	characteristic
398	#497	—	2	1	no	—
399	#496	—	2	1	no	—
400	#495	—	2	1	no	—
401	#492	—	2	1	no	—
402	#473	—	2	1	no	—
403	#473	—	2	2	no	—
404	#472	—	2	1	no	—
405	#472	—	2	2	no	—
406	#471	—	2	1	no	—
407	#471	—	2	1	no	—
408	#471	—	2	2	no	—
409	#471	—	2	2	no	—
410	#471	—	2	2	no	—
411	#471	—	2	2	no	—
412	#444	—	2	2	no	—
413	#443	—	2	2	no	—
414	#430	C₂²×C₂₃	1	1	yes	characteristic
415	#430	C₂²×C₂₃	1	1	yes	characteristic
416	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
417	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
418	#421	C₂²×C₂₃	1	1	yes	characteristic
419	#421	C₂²×C₂₃	1	1	yes	characteristic
420	#419	C₂²×C₂₃	1	1	yes	characteristic
421	#418	C₂²×C₂₃	1	1	yes	characteristic
422	#413	C₂²×C₂₃	1	1	yes	characteristic
423	#413	C₂²×C₂₃	1	1	yes	characteristic
424	#413	C₂²×C₂₃	1	1	yes	characteristic
425	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
426	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
427	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
428	C₂×#100	C₂²×C₂₃	1	1	yes	characteristic
429	#410	C₂²×C₂₃	1	1	yes	characteristic
430	#405	C₂²×C₂₃	1	1	yes	characteristic
431	#405	C₂²×C₂₃	1	1	yes	characteristic
432	#344	C₂²×C₂₃	1	1	yes	characteristic
433	#344	C₂²×C₂₃	1	1	yes	characteristic
434	#344	C₂²×C₂₃	1	1	yes	characteristic
435	#344	C₂²×C₂₃	1	1	yes	characteristic
436	C₂×C₂₃	—	2	1	no	—
437	C₂×C₂₃	—	2	1	no	—
438	C₂×C₂₃	—	2	1	no	—
439	C₂×C₂₃	—	8	1	no	—
440	C₂×C₂₃	—	8	1	no	—
441	C₂×C₂₃	—	8	1	no	—
442	C₂×C₂₃	—	16	1	no	—
443	C₂×C₂₃	#2428	1	1	yes	characteristic
444	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
447	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
449	#137	—	2	1	no	—
450	#137	—	2	1	no	—
451	#137	—	2	1	no	—
452	#137	—	2	1	no	—
453	#137	—	2	2	no	—
454	#137	—	2	2	no	—
455	#137	—	4	1	no	—
456	#137	—	4	1	no	—
457	AGL(1,ℤ/8)	—	2	1	no	—
458	AGL(1,ℤ/8)	—	2	1	no	—
459	AGL(1,ℤ/8)	—	2	1	no	—
460	AGL(1,ℤ/8)	—	2	1	no	—
461	AGL(1,ℤ/8)	—	2	2	no	—
462	AGL(1,ℤ/8)	—	2	2	no	—
463	AGL(1,ℤ/8)	—	4	1	no	—
464	AGL(1,ℤ/8)	—	4	1	no	—
465	#135	—	2	1	no	—
466	#135	—	2	1	no	—
467	#135	—	2	1	no	—
468	#135	—	2	1	no	—
469	#135	—	4	1	no	—
470	#135	—	4	1	no	—
471	C₂×Q₁₆	—	2	1	no	—
472	C₂×Q₁₆	—	2	1	no	—
473	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
474	C₂×SD₁₆	—	2	1	no	—
475	C₂×SD₁₆	—	2	1	no	—
476	C₂×SD₁₆	—	2	1	no	—
477	C₂×SD₁₆	—	2	1	no	—
478	C₂×SD₁₆	—	2	1	no	—
479	C₂×SD₁₆	—	2	1	no	—
480	C₂×SD₁₆	—	2	1	no	—
481	C₂×SD₁₆	—	2	1	no	—
482	C₂×SD₁₆	—	2	1	no	—
483	C₂×SD₁₆	—	2	1	no	—
484	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
485	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
486	C₂×D₈	—	2	1	no	—
487	C₂×D₈	—	2	1	no	—
488	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
489	#131	—	2	2	no	—
490	#131	—	2	2	no	—
491	#131	—	2	2	no	—
492	#131	—	2	2	no	—
493	#131	—	2	1	no	—
494	#131	—	2	1	no	—
495	#131	—	2	1	no	—
496	#131	—	2	1	no	—
497	C₂×M₁₆	—	2	1	no	—
498	C₂×M₁₆	—	4	1	no	—
499	C₂×M₁₆	—	4	1	no	—
500	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
501	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
502	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
503	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
504	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
505	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
506	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
507	C₂²×C₈	—	2	1	no	—
508	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
509	C₄⋊Q₈	—	2	1	no	—
510	#127	—	2	1	no	—
511	#124	—	2	1	no	—
512	#124	—	2	1	no	—
513	#123	—	2	2	no	—
514	C₂²⋊Q₈	—	4	1	no	—
515	#121	—	4	1	no	—
516	#120	—	2	2	no	—
517	#117	C₂³×C₂₃	1	1	yes	characteristic
518	#117	C₂³×C₂₃	1	1	yes	characteristic
519	#108	—	2	2	no	—
520	#108	—	2	2	no	—
521	#108	—	2	1	no	—
522	#108	—	2	1	no	—
523	#108	D₄×C₂₃	1	2	yes	complemented
524	#108	D₄×C₂₃	1	2	yes	complemented
525	#104	—	2	2	no	—
526	#104	—	2	2	no	—
527	#103	—	4	1	no	—
528	#102	—	4	1	no	—
529	#101	—	2	1	no	—
530	#101	—	2	1	no	—
531	#101	—	2	2	no	—
532	#101	—	2	2	no	—
533	#101	D₄×C₂₃	1	2	yes	complemented
534	#100	—	2	1	no	—
535	#100	—	2	1	no	—
536	#100	—	2	2	no	—
537	#100	—	2	2	no	—
538	#100	D₄×C₂₃	1	2	yes	complemented
539	#99	—	2	2	no	—
540	#97	—	2	1	no	—
541	#97	—	2	1	no	—
542	C₄×C₈	—	2	1	no	—
543	C₄×C₈	—	2	1	no	—
544	C₂₃	#25446	1	1	yes	characteristic, direct product factor, Sylow
545	C₂⁴	—	2	1	no	—
546	#41	—	2	1	no	—
547	#41	—	2	1	no	—
548	#41	—	2	1	no	—
549	#41	—	2	1	no	—
550	#41	—	2	1	no	—
551	#41	—	2	1	no	—
552	#41	—	2	1	no	—
553	#41	—	2	1	no	—
554	#41	—	2	1	no	—
555	#41	—	2	1	no	—
556	#41	—	2	1	no	—
557	#41	—	2	1	no	—
558	C₂×Q₈	—	2	1	no	—
559	C₂×Q₈	—	2	1	no	—
560	C₂×Q₈	—	2	1	no	—
561	C₂×Q₈	—	2	1	no	—
562	C₂×Q₈	—	2	1	no	—
563	C₂×Q₈	—	2	1	no	—
564	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
565	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂×D₄	—	2	1	no	—
568	C₂×D₄	—	2	1	no	—
569	C₂×D₄	—	2	1	no	—
570	C₂×D₄	—	2	1	no	—
571	C₂×D₄	—	2	1	no	—
572	C₂×D₄	—	2	1	no	—
573	C₂×D₄	—	2	1	no	—
574	C₂×D₄	—	2	1	no	—
575	C₂×D₄	—	2	1	no	—
576	C₂×D₄	—	4	2	no	—
577	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
578	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
579	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
580	C₂²×C₄	—	2	1	no	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	1	no	—
583	C₂²×C₄	—	2	1	no	—
584	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
585	Q₁₆	—	2	1	no	—
586	Q₁₆	—	2	1	no	—
587	Q₁₆	—	4	2	no	—
588	Q₁₆	—	4	2	no	—
589	Q₁₆	—	4	1	no	—
590	Q₁₆	—	4	1	no	—
591	SD₁₆	—	2	1	no	—
592	SD₁₆	—	2	1	no	—
593	SD₁₆	—	2	1	no	—
594	SD₁₆	—	2	1	no	—
595	SD₁₆	—	4	2	no	—
596	SD₁₆	—	4	2	no	—
597	SD₁₆	—	4	2	no	—
598	SD₁₆	—	4	2	no	—
599	SD₁₆	—	4	1	no	—
600	SD₁₆	—	4	1	no	—
601	SD₁₆	—	4	1	no	—
602	SD₁₆	—	4	1	no	—
603	SD₁₆	—	4	1	no	—
604	SD₁₆	—	4	1	no	—
605	SD₁₆	—	4	1	no	—
606	SD₁₆	—	4	1	no	—
607	D₈	—	2	1	no	—
608	D₈	—	2	1	no	—
609	D₈	—	4	2	no	—
610	D₈	—	4	2	no	—
611	D₈	—	4	1	no	—
612	D₈	—	4	1	no	—
613	M₁₆	—	2	1	no	—
614	M₁₆	—	2	1	no	—
615	M₁₆	—	2	1	no	—
616	M₁₆	—	2	1	no	—
617	M₁₆	—	2	1	no	—
618	M₁₆	—	2	1	no	—
619	M₁₆	—	2	1	no	—
620	M₁₆	—	2	1	no	—
621	M₁₆	—	2	1	no	—
622	M₁₆	—	2	1	no	—
623	M₁₆	—	4	2	no	—
624	M₁₆	—	4	2	no	—
625	M₁₆	—	4	1	no	—
626	M₁₆	—	4	1	no	—
627	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
628	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
629	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
630	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
631	C₂×C₈	—	2	1	no	—
632	C₂×C₈	—	2	1	no	—
633	C₂×C₈	—	2	1	no	—
634	C₂×C₈	—	2	1	no	—
635	C₂×C₈	—	2	1	no	—
636	C₂×C₈	—	2	1	no	—
637	C₂×C₈	—	2	1	no	—
638	C₂×C₈	—	4	2	no	—
639	C₂×C₈	—	4	2	no	—
640	C₂×C₈	—	4	1	no	—
641	C₂×C₈	—	4	1	no	—
642	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
643	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
644	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
645	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
646	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
647	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
648	C₄⋊C₄	—	2	1	no	—
649	C₄⋊C₄	—	2	1	no	—
650	C₄⋊C₄	—	4	2	no	—
651	C₂²⋊C₄	—	2	1	no	—
652	C₂²⋊C₄	—	4	2	no	—
653	C₂²⋊C₄	—	4	2	no	—
654	C₂²⋊C₄	#41×C₂₃	1	2	yes	—
655	C₄²	—	2	1	no	—
656	C₄²	—	2	1	no	—
657	C₂³	—	2	1	no	—
658	C₂³	—	2	1	no	—
659	C₂³	—	2	1	no	—
660	C₂³	—	4	1	no	—
661	C₂³	—	4	1	no	—
662	C₂³	—	4	1	no	—
663	C₂³	—	16	1	no	—
664	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
665	Q₈	—	2	1	no	—
666	Q₈	—	2	1	no	—
667	Q₈	—	2	1	no	—
668	Q₈	—	2	1	no	—
669	Q₈	—	2	1	no	—
670	Q₈	—	2	1	no	—
671	Q₈	—	4	1	no	—
672	Q₈	—	4	1	no	—
673	Q₈	—	4	1	no	—
674	Q₈	—	4	1	no	—
675	D₄	—	2	1	no	—
676	D₄	—	2	1	no	—
677	D₄	—	2	1	no	—
678	D₄	—	2	1	no	—
679	D₄	—	2	1	no	—
680	D₄	—	2	1	no	—
681	D₄	—	4	1	no	—
682	D₄	—	4	1	no	—
683	D₄	—	4	1	no	—
684	D₄	—	4	1	no	—
685	D₄	—	4	1	no	—
686	D₄	—	4	1	no	—
687	D₄	—	4	1	no	—
688	D₄	—	4	1	no	—
689	D₄	—	4	1	no	—
690	D₄	—	4	1	no	—
691	D₄	—	16	2	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	2	1	no	—
695	C₂×C₄	—	2	1	no	—
696	C₂×C₄	—	4	1	no	—
697	C₂×C₄	—	4	1	no	—
698	C₂×C₄	—	4	1	no	—
699	C₂×C₄	—	4	1	no	—
700	C₂×C₄	—	4	1	no	—
701	C₂×C₄	—	4	1	no	—
702	C₂×C₄	—	4	1	no	—
703	C₂×C₄	—	4	1	no	—
704	C₂×C₄	—	4	1	no	—
705	C₂×C₄	—	4	1	no	—
706	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
707	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
708	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
709	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
710	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
711	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
712	C₈	—	2	2	no	—
713	C₈	—	2	2	no	—
714	C₈	—	2	2	no	—
715	C₈	—	2	2	no	—
716	C₈	—	4	1	no	—
717	C₈	—	4	1	no	—
718	C₈	—	4	1	no	—
719	C₈	C₂₃×#121	1	1	yes	characteristic
720	C₈	C₂₃×#121	1	1	yes	characteristic
721	C₈	C₂₃×#121	1	1	yes	characteristic
722	C₈	C₂₃×#121	1	1	yes	characteristic
723	C₂²	—	4	1	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	1	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	8	1	no	—
728	C₂²	—	8	1	no	—
729	C₂²	—	8	1	no	—
730	C₂²	—	8	2	no	—
731	C₂²	—	8	1	no	—
732	C₂²	—	16	1	no	—
733	C₂²	—	16	1	no	—
734	C₂²	C₂₃×#532	1	1	yes	characteristic
735	C₂²	C₂₃×#546	1	1	yes	characteristic
736	C₂²	C₂₃×#550	1	1	yes	characteristic
737	C₄	—	4	2	no	—
738	C₄	—	4	1	no	—
739	C₄	—	4	1	no	—
740	C₄	—	4	1	no	—
741	C₄	—	8	1	no	—
742	C₄	—	8	1	no	—
743	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
744	C₄	C₂₃×#540	1	1	yes	characteristic
745	C₄	D₄²×C₂₃	1	1	yes	characteristic
746	C₄	D₄²×C₂₃	1	1	yes	characteristic
747	C₂	—	2	1	no	—
748	C₂	—	2	1	no	—
749	C₂	—	2	1	no	—
750	C₂	—	8	1	no	—
751	C₂	—	8	1	no	—
752	C₂	—	8	1	no	—
753	C₂	—	16	1	no	—
754	C₂	C₂₃×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3154	C₂	semidirect
2. C₂₃×#3069	C₂	semidirect
3. C₂₃×#3068	C₂	semidirect
4. C₂₃×#3067	C₂	semidirect
5. C₂×C₂₃×#471	C₂	semidirect
6. C₂₃×#2957	C₂	semidirect
7. C₂₃×#2049	C₂	semidirect
8. C₂₃×#2048	C₂	semidirect
9. C₂₃×#2009	C₂	semidirect
10. C₂₃×#2008	C₂	semidirect
11. C₂₃×#1977	C₂	semidirect
12. C₂₃×#1953	C₂	semidirect
13. C₂₃×#1915	C₂	semidirect
14. C₂₃×#1914	C₂	semidirect
15. C₂₃×#1757	C₂	semidirect
146. C₂₃×#108	D₄	semidirect
147. C₂₃×#108	D₄	semidirect
156. C₂₃×#101	D₄	semidirect
161. C₂₃×#100	D₄	semidirect
279. #25446	C₂₃	direct (N × H)
346. #3154	C₂×C₂₃	semidirect
347. #3069	C₂×C₂₃	semidirect
348. #3068	C₂×C₂₃	semidirect
349. #3067	C₂×C₂₃	semidirect
350. C₂×#471	C₂×C₂₃	semidirect
351. #2957	C₂×C₂₃	semidirect
352. #2049	C₂×C₂₃	semidirect
353. #2048	C₂×C₂₃	semidirect
354. #2009	C₂×C₂₃	semidirect
355. #2008	C₂×C₂₃	semidirect
356. #1977	C₂×C₂₃	semidirect
357. #1953	C₂×C₂₃	semidirect
358. #1915	C₂×C₂₃	semidirect
359. #1914	C₂×C₂₃	semidirect
360. #1757	C₂×C₂₃	semidirect
523. #108	D₄×C₂₃	semidirect
524. #108	D₄×C₂₃	semidirect
533. #101	D₄×C₂₃	semidirect
538. #100	D₄×C₂₃	semidirect
544. C₂₃	#25446	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3154	2
C₂₃×#3069	2
C₂₃×#3068	2
C₂₃×#3067	2
C₂×C₂₃×#471	2
C₂₃×#2957	2
C₂₃×#2049	2
C₂₃×#2048	2
C₂₃×#2009	2
C₂₃×#2008	2
C₂₃×#1977	2
C₂₃×#1953	2
C₂₃×#1915	2
C₂₃×#1914	2
C₂₃×#1757	2
#25446	23

Subgroups — C23×#25446 #19173072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25446 #19173072