Subgroups — C₂₃×#21559 #19169185

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3067	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2955	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1932	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1906	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1757	C₂	1	2	yes	maximal, complemented
12	C₂₃×#575	C₂²	1	2	yes	—
13	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#131	C₂²	1	2	yes	—
16	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#473	C₄	1	2	yes	complemented
19	C₂₃×#472	C₄	1	2	yes	complemented
20	C₂₃×#471	C₄	1	4	yes	complemented
21	C₂₃×#444	—	2	1	no	—
22	C₂₃×#444	—	2	1	no	—
23	C₂₃×#444	—	2	1	no	—
24	C₂₃×#444	—	2	1	no	—
25	C₂₃×#444	—	2	4	no	—
26	C₂₃×#443	—	2	4	no	—
27	C₂₃×#430	C₂²	1	2	yes	—
28	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
30	C₂₃×#427	C₂²	1	1	yes	characteristic
31	C₂₃×#421	C₂²	1	2	yes	—
32	C₂₃×#421	C₂²	1	2	yes	—
33	C₂×C₂₃×#104	C₂²	1	2	yes	—
34	C₂₃×#413	C₂²	1	2	yes	—
35	C₂₃×#410	C₂²	1	2	yes	—
36	C₂₃×#405	—	2	1	no	—
37	C₂₃×#405	—	2	1	no	—
38	C₂₃×#405	C₂²	1	1	yes	characteristic
39	C₂₃×#405	C₂²	1	2	yes	—
40	C₂₃×#405	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₂₃	—	2	1	no	—
42	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#97	—	2	1	no	—
44	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
45	C₂₃×#344	C₂²	1	2	yes	—
46	C₂₃×#344	C₂²	1	2	yes	—
47	C₂₃×#339	C₂²	1	2	yes	—
48	C₂×#41×C₂₃	C₂³	1	2	yes	—
49	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#137	—	2	2	no	—
51	C₂₃×#137	C₂×C₄	1	4	yes	—
52	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
53	C₂₃×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	—
54	C₂₃×#135	—	2	4	no	—
55	C₂₃×#135	—	2	1	no	—
56	C₂₃×#135	—	2	1	no	—
57	C₂₃×#135	—	2	1	no	—
58	C₂₃×#135	—	2	1	no	—
59	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
61	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂₃×#131	—	2	1	no	—
63	C₂₃×#131	—	2	1	no	—
64	C₂₃×#131	—	2	2	no	—
65	C₂₃×#131	—	2	2	no	—
66	C₂₃×#131	—	2	2	no	—
67	C₂₃×#131	—	2	2	no	—
68	C₂₃×#131	—	2	2	no	—
69	C₂₃×#131	C₂×C₄	1	2	yes	—
70	C₂₃×#131	C₂×C₄	1	2	yes	—
71	C₂₃×#131	D₄	1	4	yes	—
72	C₂×M₁₆×C₂₃	—	2	1	no	—
73	C₂×M₁₆×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	2	no	—
75	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
76	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	—	2	1	no	—
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂²×C₈×C₂₃	—	2	2	no	—
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
84	C₄×Q₈×C₂₃	—	2	1	no	—
85	C₄×D₄×C₂₃	—	2	1	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₂₃×#117	—	2	1	no	—
88	C₂₃×#117	C₂³	1	2	yes	—
89	C₂₃×#117	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#108	—	2	4	no	—
93	C₂₃×#108	—	2	1	no	—
94	C₂₃×#108	—	2	1	no	—
95	C₂₃×#108	C₂×C₄	1	2	yes	complemented
96	C₂₃×#108	C₂×C₄	1	2	yes	complemented
97	C₂₃×#107	—	2	1	no	—
98	C₂₃×#106	—	2	1	no	—
99	C₂₃×#104	—	2	4	no	—
100	C₂₃×#104	—	2	4	no	—
101	C₂₃×#104	—	2	2	no	—
102	C₂₃×#104	—	2	2	no	—
103	C₂₃×#101	C₂×C₄	1	4	yes	complemented
104	C₂₃×#100	C₂×C₄	1	4	yes	complemented
105	C₂₃×#99	—	2	4	no	—
106	C₂₃×#97	—	2	1	no	—
107	C₂₃×#97	—	2	1	no	—
108	C₂₃×#97	—	2	2	no	—
109	C₂₃×#97	—	2	1	no	—
110	C₂₃×#97	D₄	1	1	yes	characteristic
111	C₂₃×#97	D₄	1	1	yes	characteristic
112	C₂₃×#97	D₄	1	1	yes	characteristic
113	C₂₃×#97	D₄	1	1	yes	characteristic
114	C₄×C₈×C₂₃	—	2	1	no	—
115	C₄×C₈×C₂₃	—	2	1	no	—
116	C₄×C₈×C₂₃	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	1	no	—
118	#41×C₂₃	—	2	1	no	—
119	#41×C₂₃	—	2	1	no	—
120	#41×C₂₃	—	2	2	no	—
121	#41×C₂₃	—	2	2	no	—
122	#41×C₂₃	—	2	2	no	—
123	#41×C₂₃	—	2	2	no	—
124	#41×C₂₃	C₂²×C₄	1	2	yes	—
125	#41×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
127	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₂₃	—	2	4	no	—
138	Q₁₆×C₂₃	—	2	1	no	—
139	Q₁₆×C₂₃	—	2	1	no	—
140	SD₁₆×C₂₃	—	2	4	no	—
141	SD₁₆×C₂₃	—	2	4	no	—
142	SD₁₆×C₂₃	—	2	2	no	—
143	SD₁₆×C₂₃	—	2	2	no	—
144	D₈×C₂₃	—	2	4	no	—
145	D₈×C₂₃	—	2	1	no	—
146	D₈×C₂₃	—	2	1	no	—
147	M₁₆×C₂₃	—	2	1	no	—
148	M₁₆×C₂₃	—	2	1	no	—
149	M₁₆×C₂₃	—	2	2	no	—
150	M₁₆×C₂₃	—	2	2	no	—
151	M₁₆×C₂₃	—	2	2	no	—
152	M₁₆×C₂₃	—	2	4	no	—
153	M₁₆×C₂₃	—	2	2	no	—
154	M₁₆×C₂₃	—	2	2	no	—
155	M₁₆×C₂₃	—	4	2	no	—
156	M₁₆×C₂₃	—	4	2	no	—
157	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
158	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
159	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₂₃	—	2	2	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	2	no	—
164	C₂×C₈×C₂₃	—	2	2	no	—
165	C₂×C₈×C₂₃	—	2	4	no	—
166	C₂×C₈×C₂₃	—	2	2	no	—
167	C₂×C₈×C₂₃	—	4	2	no	—
168	C₂×C₈×C₂₃	—	4	2	no	—
169	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
177	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
178	C₂×C₈×C₂₃	#41	1	2	yes	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₂²⋊C₄×C₂₃	—	2	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	#41	1	2	yes	—
185	C₄²×C₂₃	—	2	2	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	2	2	no	—
189	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
191	#21559	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₂₃	—	2	1	no	—
193	C₂³×C₂₃	—	2	2	no	—
194	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
195	Q₈×C₂₃	—	2	1	no	—
196	Q₈×C₂₃	—	2	2	no	—
197	Q₈×C₂₃	—	2	2	no	—
198	Q₈×C₂₃	C₄×D₄	1	2	yes	—
199	D₄×C₂₃	—	2	2	no	—
200	D₄×C₂₃	—	2	2	no	—
201	D₄×C₂₃	—	2	2	no	—
202	D₄×C₂₃	—	2	2	no	—
203	D₄×C₂₃	—	2	1	no	—
204	D₄×C₂₃	—	4	2	no	—
205	D₄×C₂₃	—	4	2	no	—
206	D₄×C₂₃	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₂₃	—	2	1	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	4	2	no	—
215	C₂×C₄×C₂₃	—	4	2	no	—
216	C₂×C₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	8	1	no	—
218	C₂×C₄×C₂₃	—	8	2	no	—
219	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
225	C₈×C₂₃	—	2	2	no	—
226	C₈×C₂₃	—	2	4	no	—
227	C₈×C₂₃	—	2	2	no	—
228	C₈×C₂₃	—	4	2	no	—
229	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
230	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
231	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
232	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
233	C₈×C₂₃	#121	1	2	yes	—
234	C₈×C₂₃	#121	1	2	yes	—
235	#3067	C₂×C₂₃	1	1	yes	characteristic
236	#2955	C₂×C₂₃	1	2	yes	complemented
237	C₂×#444	C₂×C₂₃	1	1	yes	characteristic, complemented
238	#2875	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#1976	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1953	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#1932	C₂×C₂₃	1	2	yes	complemented
242	#1906	C₂×C₂₃	1	2	yes	complemented
243	#1814	C₂×C₂₃	1	1	yes	characteristic, complemented
244	#1773	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#1757	C₂×C₂₃	1	2	yes	complemented
246	C₂²×C₂₃	—	2	2	no	—
247	C₂²×C₂₃	—	4	1	no	—
248	C₂²×C₂₃	—	4	2	no	—
249	C₂²×C₂₃	—	4	2	no	—
250	C₂²×C₂₃	—	8	2	no	—
251	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
252	C₂²×C₂₃	#525	1	1	yes	characteristic
253	C₂²×C₂₃	#532	1	1	yes	characteristic
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	4	2	no	—
256	C₄×C₂₃	—	4	1	no	—
257	C₄×C₂₃	—	4	1	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	—	8	2	no	—
260	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
261	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
262	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
263	C₄×C₂₃	#529	1	1	yes	characteristic
264	#575	C₂²×C₂₃	1	2	yes	—
265	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#131	C₂²×C₂₃	1	2	yes	—
267	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
268	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
269	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
270	#473	C₄×C₂₃	1	2	yes	complemented
271	#472	C₄×C₂₃	1	2	yes	complemented
272	#471	C₄×C₂₃	1	4	yes	complemented
273	#444	—	2	4	no	—
274	#444	—	2	1	no	—
275	#444	—	2	1	no	—
276	#444	—	2	1	no	—
277	#444	—	2	1	no	—
278	#443	—	2	4	no	—
279	#430	C₂²×C₂₃	1	2	yes	—
280	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
282	#427	C₂²×C₂₃	1	1	yes	characteristic
283	#421	C₂²×C₂₃	1	2	yes	—
284	#421	C₂²×C₂₃	1	2	yes	—
285	C₂×#104	C₂²×C₂₃	1	2	yes	—
286	#413	C₂²×C₂₃	1	2	yes	—
287	#410	C₂²×C₂₃	1	2	yes	—
288	#405	—	2	1	no	—
289	#405	—	2	1	no	—
290	#405	C₂²×C₂₃	1	1	yes	characteristic
291	#405	C₂²×C₂₃	1	1	yes	characteristic
292	#405	C₂²×C₂₃	1	2	yes	—
293	C₄×M₁₆	—	2	1	no	—
294	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×#97	—	2	1	no	—
296	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
297	#344	C₂²×C₂₃	1	2	yes	—
298	#344	C₂²×C₂₃	1	2	yes	—
299	#339	C₂²×C₂₃	1	2	yes	—
300	C₂×C₂₃	—	2	1	no	—
301	C₂×C₂₃	—	2	1	no	—
302	C₂×C₂₃	—	2	1	no	—
303	C₂×C₂₃	—	4	2	no	—
304	C₂×C₂₃	—	8	2	no	—
305	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
306	C₂×#41	C₂³×C₂₃	1	2	yes	—
307	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
308	#137	—	2	2	no	—
309	#137	C₂×C₄×C₂₃	1	4	yes	—
310	AGL(1,ℤ/8)	—	2	2	no	—
311	AGL(1,ℤ/8)	C₂×C₄×C₂₃	1	4	yes	—
312	#135	—	2	4	no	—
313	#135	—	2	1	no	—
314	#135	—	2	1	no	—
315	#135	—	2	1	no	—
316	#135	—	2	1	no	—
317	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
318	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
319	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
320	#131	—	2	1	no	—
321	#131	—	2	1	no	—
322	#131	—	2	2	no	—
323	#131	—	2	2	no	—
324	#131	—	2	2	no	—
325	#131	—	2	2	no	—
326	#131	—	2	2	no	—
327	#131	C₂×C₄×C₂₃	1	2	yes	—
328	#131	C₂×C₄×C₂₃	1	2	yes	—
329	#131	D₄×C₂₃	1	4	yes	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	2	2	no	—
333	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
334	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
336	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
337	C₂²×C₈	—	2	1	no	—
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	—	2	2	no	—
340	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
342	C₄×Q₈	—	2	1	no	—
343	C₄×D₄	—	2	1	no	—
344	C₄×D₄	—	2	2	no	—
345	#117	—	2	1	no	—
346	#117	C₂³×C₂₃	1	2	yes	—
347	#117	C₂³×C₂₃	1	1	yes	characteristic
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	#108	—	2	4	no	—
351	#108	—	2	1	no	—
352	#108	—	2	1	no	—
353	#108	C₂×C₄×C₂₃	1	2	yes	complemented
354	#108	C₂×C₄×C₂₃	1	2	yes	complemented
355	#107	—	2	1	no	—
356	#106	—	2	1	no	—
357	#104	—	2	4	no	—
358	#104	—	2	4	no	—
359	#104	—	2	2	no	—
360	#104	—	2	2	no	—
361	#101	C₂×C₄×C₂₃	1	4	yes	complemented
362	#100	C₂×C₄×C₂₃	1	4	yes	complemented
363	#99	—	2	4	no	—
364	#97	—	2	1	no	—
365	#97	—	2	1	no	—
366	#97	—	2	2	no	—
367	#97	—	2	1	no	—
368	#97	D₄×C₂₃	1	1	yes	characteristic
369	#97	D₄×C₂₃	1	1	yes	characteristic
370	#97	D₄×C₂₃	1	1	yes	characteristic
371	#97	D₄×C₂₃	1	1	yes	characteristic
372	C₄×C₈	—	2	1	no	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	—	2	2	no	—
375	C₄×C₈	—	2	1	no	—
376	C₂₃	#21559	1	1	yes	characteristic, direct product factor, Sylow
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	#41	—	2	2	no	—
382	#41	—	2	2	no	—
383	#41	C₂²×C₄×C₂₃	1	2	yes	—
384	#41	C₂×D₄×C₂₃	1	2	yes	—
385	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
386	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
390	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
396	Q₁₆	—	2	4	no	—
397	Q₁₆	—	2	1	no	—
398	Q₁₆	—	2	1	no	—
399	SD₁₆	—	2	4	no	—
400	SD₁₆	—	2	4	no	—
401	SD₁₆	—	2	2	no	—
402	SD₁₆	—	2	2	no	—
403	D₈	—	2	4	no	—
404	D₈	—	2	1	no	—
405	D₈	—	2	1	no	—
406	M₁₆	—	2	1	no	—
407	M₁₆	—	2	1	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	2	2	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	4	2	no	—
415	M₁₆	—	4	2	no	—
416	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
417	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
418	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	4	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	—	4	2	no	—
428	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
429	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
433	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
436	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
437	C₂×C₈	#41×C₂₃	1	2	yes	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	#41×C₂₃	1	2	yes	—
444	C₄²	—	2	2	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	—	2	2	no	—
448	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₂³	—	2	1	no	—
451	C₂³	—	2	2	no	—
452	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
453	Q₈	—	2	1	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	C₄×D₄×C₂₃	1	2	yes	—
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	2	2	no	—
460	D₄	—	2	2	no	—
461	D₄	—	2	1	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	2	no	—
464	D₄	C₄×D₄×C₂₃	1	2	yes	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	8	1	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
481	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	2	4	no	—
485	C₈	—	2	2	no	—
486	C₈	—	4	2	no	—
487	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
488	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
489	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
490	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
491	C₈	C₂₃×#121	1	2	yes	—
492	C₈	C₂₃×#121	1	2	yes	—
493	C₂²	—	2	2	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
499	C₂²	C₂₃×#525	1	1	yes	characteristic
500	C₂²	C₂₃×#532	1	1	yes	characteristic
501	C₄	—	2	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
508	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
509	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
510	C₄	C₂₃×#529	1	1	yes	characteristic
511	C₂	—	2	1	no	—
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	4	2	no	—
515	C₂	—	8	2	no	—
516	C₂	C₄×C₂₃×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	C₄×#121
prime radical	C₂₃×#3067	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2955	C₂	semidirect
3. C₂×C₂₃×#444	C₂	semidirect
4. C₂₃×#2875	C₂	semidirect
5. C₂₃×#1976	C₂	semidirect
6. C₂₃×#1953	C₂	semidirect
7. C₂₃×#1932	C₂	semidirect
8. C₂₃×#1906	C₂	semidirect
9. C₂₃×#1814	C₂	semidirect
10. C₂₃×#1773	C₂	semidirect
11. C₂₃×#1757	C₂	semidirect
18. C₂₃×#473	C₄	semidirect
19. C₂₃×#472	C₄	semidirect
20. C₂₃×#471	C₄	semidirect
59. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
60. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
61. C₂×D₈×C₂₃	C₂×C₄	semidirect
95. C₂₃×#108	C₂×C₄	semidirect
96. C₂₃×#108	C₂×C₄	semidirect
103. C₂₃×#101	C₂×C₄	semidirect
104. C₂₃×#100	C₂×C₄	semidirect
191. #21559	C₂₃	direct (N × H)
236. #2955	C₂×C₂₃	semidirect
237. C₂×#444	C₂×C₂₃	semidirect
238. #2875	C₂×C₂₃	semidirect
239. #1976	C₂×C₂₃	semidirect
240. #1953	C₂×C₂₃	semidirect
241. #1932	C₂×C₂₃	semidirect
242. #1906	C₂×C₂₃	semidirect
243. #1814	C₂×C₂₃	semidirect
244. #1773	C₂×C₂₃	semidirect
245. #1757	C₂×C₂₃	semidirect
270. #473	C₄×C₂₃	semidirect
271. #472	C₄×C₂₃	semidirect
272. #471	C₄×C₂₃	semidirect
317. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
318. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
319. C₂×D₈	C₂×C₄×C₂₃	semidirect
353. #108	C₂×C₄×C₂₃	semidirect
354. #108	C₂×C₄×C₂₃	semidirect
361. #101	C₂×C₄×C₂₃	semidirect
362. #100	C₂×C₄×C₂₃	semidirect
376. C₂₃	#21559	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×#3067	2
C₂₃×#2955	2
C₂×C₂₃×#444	2
C₂₃×#2875	2
C₂₃×#1976	2
C₂₃×#1953	2
C₂₃×#1932	2
C₂₃×#1906	2
C₂₃×#1814	2
C₂₃×#1773	2
C₂₃×#1757	2
#21559	23

Subgroups — C23×#21559 #19169185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21559 #19169185