Subgroups — C₂₃×#25198 #19172824

Order 5888.

958 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3069	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3069	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3020	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#456	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2889	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2049	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2005	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1966	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1932	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1915	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1902	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1861	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂₃×#575	C₂²	1	1	yes	characteristic
18	C₂₃×#575	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#137	C₂²	1	1	yes	characteristic
22	C₂₃×#568	C₂²	1	1	yes	characteristic
23	C₂₃×#568	C₂²	1	1	yes	characteristic
24	C₂₃×#568	C₂²	1	1	yes	characteristic
25	C₂₃×#536	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂₃×#473	—	2	2	no	—
28	C₂₃×#473	—	2	2	no	—
29	C₂₃×#471	—	2	2	no	—
30	C₂₃×#471	—	2	2	no	—
31	C₂₃×#456	—	2	2	no	—
32	C₂₃×#456	—	2	2	no	—
33	C₂₃×#456	—	2	2	no	—
34	C₂₃×#454	—	2	2	no	—
35	C₂₃×#444	—	2	2	no	—
36	C₂₃×#443	—	2	2	no	—
37	C₂₃×#430	C₂²	1	1	yes	characteristic
38	C₂₃×#430	C₂²	1	1	yes	characteristic
39	C₂₃×#430	C₂²	1	1	yes	characteristic
40	C₂₃×#423	C₂²	1	1	yes	characteristic
41	C₂₃×#421	C₂²	1	1	yes	characteristic
42	C₂₃×#421	C₂²	1	1	yes	characteristic
43	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
44	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
46	C₂₃×#419	C₂²	1	1	yes	characteristic
47	C₂₃×#413	—	2	2	no	—
48	C₂₃×#413	C₂²	1	1	yes	characteristic
49	C₂₃×#413	C₂²	1	1	yes	characteristic
50	C₂₃×#413	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#101	—	2	2	no	—
52	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
56	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
57	C₂×C₂₃×#101	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂₃×#344	C₂²	1	1	yes	characteristic
60	C₂₃×#344	C₂²	1	1	yes	characteristic
61	C₂₃×#339	C₂²	1	1	yes	characteristic
62	C₂₃×#338	C₂²	1	1	yes	characteristic
63	C₂₃×2₋¹⁺⁴	—	2	2	no	—
64	C₂₃×2₋¹⁺⁴	—	2	2	no	—
65	C₂₃×2₋¹⁺⁴	—	4	1	no	—
66	C₂₃×2₋¹⁺⁴	—	4	1	no	—
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	—	2	1	no	—
69	C₂×#41×C₂₃	—	2	2	no	—
70	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₂₃	—	2	1	no	—
74	C₂²×Q₈×C₂₃	—	2	1	no	—
75	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×#137	—	2	1	no	—
78	C₂₃×#137	—	2	2	no	—
79	C₂₃×#137	—	2	2	no	—
80	C₂₃×#137	—	2	1	no	—
81	C₂₃×#137	—	2	1	no	—
82	C₂₃×#137	—	2	2	no	—
83	C₂₃×#137	—	2	2	no	—
84	C₂₃×#137	—	4	1	no	—
85	C₂₃×#137	—	4	1	no	—
86	C₂₃×#137	D₄	1	2	yes	—
87	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
88	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
89	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
90	C₂₃×AGL(1,ℤ/8)	D₄	1	2	yes	—
91	C₂₃×#135	—	2	2	no	—
92	C₂₃×#135	—	4	1	no	—
93	C₂×Q₁₆×C₂₃	—	2	1	no	—
94	C₂×Q₁₆×C₂₃	—	2	1	no	—
95	C₂×Q₁₆×C₂₃	—	2	1	no	—
96	C₂×SD₁₆×C₂₃	—	2	1	no	—
97	C₂×SD₁₆×C₂₃	—	2	1	no	—
98	C₂×SD₁₆×C₂₃	—	2	1	no	—
99	C₂₃×#131	—	2	2	no	—
100	C₂₃×#131	—	2	2	no	—
101	C₂₃×#131	—	2	2	no	—
102	C₂₃×#131	—	2	2	no	—
103	C₂₃×#131	—	4	1	no	—
104	C₂₃×#131	—	4	1	no	—
105	C₂×M₁₆×C₂₃	—	2	2	no	—
106	C₂×M₁₆×C₂₃	—	2	2	no	—
107	C₂×M₁₆×C₂₃	—	4	1	no	—
108	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
115	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊Q₈	—	2	1	no	—
117	C₂₃×C₄⋊Q₈	—	2	1	no	—
118	C₂₃×C₄⋊Q₈	—	2	1	no	—
119	C₂₃×C₄⋊Q₈	—	4	1	no	—
120	C₂₃×#124	—	2	1	no	—
121	C₂₃×#123	—	2	2	no	—
122	C₂₃×C₂²⋊Q₈	—	4	1	no	—
123	C₂₃×#117	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂₃×#108	—	2	2	no	—
127	C₂₃×#108	—	2	2	no	—
128	C₂₃×#108	—	2	2	no	—
129	C₂₃×C₄⋊C₈	—	4	1	no	—
130	C₂₃×#104	—	2	2	no	—
131	C₂₃×#104	—	2	2	no	—
132	C₂₃×#104	—	2	1	no	—
133	C₂₃×#104	—	2	1	no	—
134	C₂₃×#104	—	2	1	no	—
135	C₂₃×#104	—	2	1	no	—
136	C₂₃×#104	D₄	1	2	yes	—
137	C₂₃×#104	D₄	1	2	yes	—
138	C₂₃×#103	—	4	1	no	—
139	C₂₃×#101	—	2	1	no	—
140	C₂₃×#101	—	2	1	no	—
141	C₂₃×#101	—	2	2	no	—
142	C₂₃×#101	—	2	2	no	—
143	C₂₃×#101	—	2	2	no	—
144	C₂₃×#101	—	2	2	no	—
145	C₂₃×#101	—	2	2	no	—
146	C₂₃×#101	—	2	2	no	—
147	C₂₃×#101	D₄	1	2	yes	—
148	C₂₃×#100	—	2	1	no	—
149	C₂₃×#100	—	2	1	no	—
150	C₂₃×#100	D₄	1	2	yes	—
151	C₂₃×#97	—	2	1	no	—
152	C₄×C₈×C₂₃	—	2	1	no	—
153	#41×C₂₃	—	2	1	no	—
154	#41×C₂₃	—	2	1	no	—
155	#41×C₂₃	—	2	1	no	—
156	#41×C₂₃	—	2	1	no	—
157	#41×C₂₃	—	2	1	no	—
158	#41×C₂₃	—	2	1	no	—
159	#41×C₂₃	—	2	1	no	—
160	#41×C₂₃	—	2	1	no	—
161	#41×C₂₃	—	2	1	no	—
162	#41×C₂₃	—	2	1	no	—
163	#41×C₂₃	—	4	2	no	—
164	#41×C₂₃	—	4	2	no	—
165	#41×C₂₃	—	4	4	no	—
166	#41×C₂₃	—	4	4	no	—
167	#41×C₂₃	—	4	2	no	—
168	#41×C₂₃	—	4	2	no	—
169	#41×C₂₃	C₂×D₄	1	2	yes	—
170	#41×C₂₃	C₂×D₄	1	2	yes	—
171	C₂×Q₈×C₂₃	—	2	2	no	—
172	C₂×Q₈×C₂₃	—	2	1	no	—
173	C₂×Q₈×C₂₃	—	2	1	no	—
174	C₂×Q₈×C₂₃	—	2	1	no	—
175	C₂×Q₈×C₂₃	—	2	1	no	—
176	C₂×Q₈×C₂₃	—	2	1	no	—
177	C₂×Q₈×C₂₃	—	2	1	no	—
178	C₂×Q₈×C₂₃	—	2	1	no	—
179	C₂×Q₈×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	—	4	2	no	—
185	C₂×Q₈×C₂₃	—	4	2	no	—
186	C₂×Q₈×C₂₃	—	4	2	no	—
187	C₂×Q₈×C₂₃	—	4	2	no	—
188	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
191	C₂×D₄×C₂₃	—	2	1	no	—
192	C₂×D₄×C₂₃	—	2	2	no	—
193	C₂×D₄×C₂₃	—	2	1	no	—
194	C₂×D₄×C₂₃	—	2	1	no	—
195	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	2	no	—
200	C₂²×C₄×C₂₃	—	2	2	no	—
201	C₂²×C₄×C₂₃	—	2	1	no	—
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	#41	1	2	yes	—
206	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₂₃	—	2	2	no	—
208	Q₁₆×C₂₃	—	4	2	no	—
209	Q₁₆×C₂₃	—	4	2	no	—
210	Q₁₆×C₂₃	—	4	1	no	—
211	Q₁₆×C₂₃	—	4	2	no	—
212	SD₁₆×C₂₃	—	2	2	no	—
213	SD₁₆×C₂₃	—	2	2	no	—
214	SD₁₆×C₂₃	—	4	2	no	—
215	SD₁₆×C₂₃	—	4	2	no	—
216	SD₁₆×C₂₃	—	4	1	no	—
217	SD₁₆×C₂₃	—	4	1	no	—
218	SD₁₆×C₂₃	—	4	2	no	—
219	D₈×C₂₃	—	2	2	no	—
220	D₈×C₂₃	—	4	1	no	—
221	M₁₆×C₂₃	—	2	1	no	—
222	M₁₆×C₂₃	—	2	1	no	—
223	M₁₆×C₂₃	—	2	1	no	—
224	M₁₆×C₂₃	—	2	1	no	—
225	M₁₆×C₂₃	—	2	1	no	—
226	M₁₆×C₂₃	—	2	1	no	—
227	M₁₆×C₂₃	—	2	1	no	—
228	M₁₆×C₂₃	—	2	1	no	—
229	M₁₆×C₂₃	—	2	1	no	—
230	M₁₆×C₂₃	—	2	1	no	—
231	M₁₆×C₂₃	—	2	1	no	—
232	M₁₆×C₂₃	—	2	1	no	—
233	M₁₆×C₂₃	—	4	2	no	—
234	M₁₆×C₂₃	—	4	2	no	—
235	M₁₆×C₂₃	—	4	2	no	—
236	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
237	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
238	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
239	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
240	C₂×C₈×C₂₃	—	2	1	no	—
241	C₂×C₈×C₂₃	—	2	1	no	—
242	C₂×C₈×C₂₃	—	2	1	no	—
243	C₂×C₈×C₂₃	—	2	1	no	—
244	C₂×C₈×C₂₃	—	2	1	no	—
245	C₂×C₈×C₂₃	—	2	1	no	—
246	C₂×C₈×C₂₃	—	2	1	no	—
247	C₂×C₈×C₂₃	—	2	1	no	—
248	C₂×C₈×C₂₃	—	4	2	no	—
249	C₂×C₈×C₂₃	—	4	2	no	—
250	C₂×C₈×C₂₃	—	4	2	no	—
251	C₄⋊C₄×C₂₃	—	2	1	no	—
252	C₄⋊C₄×C₂₃	—	4	2	no	—
253	C₄⋊C₄×C₂₃	—	4	2	no	—
254	C₄⋊C₄×C₂₃	—	4	2	no	—
255	C₂²⋊C₄×C₂₃	—	2	1	no	—
256	C₂²⋊C₄×C₂₃	—	4	2	no	—
257	C₄²×C₂₃	—	2	1	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
260	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
261	#25198	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
262	C₂³×C₂₃	—	2	1	no	—
263	C₂³×C₂₃	—	2	1	no	—
264	C₂³×C₂₃	—	2	1	no	—
265	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
266	Q₈×C₂₃	—	2	1	no	—
267	Q₈×C₂₃	—	2	1	no	—
268	Q₈×C₂₃	—	2	1	no	—
269	Q₈×C₂₃	—	2	1	no	—
270	Q₈×C₂₃	—	2	1	no	—
271	Q₈×C₂₃	—	4	4	no	—
272	Q₈×C₂₃	—	4	1	no	—
273	Q₈×C₂₃	—	4	1	no	—
274	Q₈×C₂₃	—	4	1	no	—
275	Q₈×C₂₃	—	4	1	no	—
276	Q₈×C₂₃	—	4	1	no	—
277	Q₈×C₂₃	—	4	1	no	—
278	Q₈×C₂₃	—	4	1	no	—
279	Q₈×C₂₃	—	4	1	no	—
280	Q₈×C₂₃	#121	1	2	yes	—
281	D₄×C₂₃	—	2	2	no	—
282	D₄×C₂₃	—	2	2	no	—
283	D₄×C₂₃	—	2	1	no	—
284	D₄×C₂₃	—	2	1	no	—
285	D₄×C₂₃	—	2	1	no	—
286	D₄×C₂₃	—	2	1	no	—
287	D₄×C₂₃	—	2	1	no	—
288	D₄×C₂₃	—	4	4	no	—
289	D₄×C₂₃	—	4	1	no	—
290	D₄×C₂₃	—	4	1	no	—
291	D₄×C₂₃	—	4	1	no	—
292	D₄×C₂₃	—	4	1	no	—
293	D₄×C₂₃	#121	1	2	yes	—
294	C₂×C₄×C₂₃	—	2	1	no	—
295	C₂×C₄×C₂₃	—	2	1	no	—
296	C₂×C₄×C₂₃	—	2	2	no	—
297	C₂×C₄×C₂₃	—	2	2	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	2	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	2	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	2	no	—
304	C₂×C₄×C₂₃	—	4	2	no	—
305	C₂×C₄×C₂₃	—	4	4	no	—
306	C₂×C₄×C₂₃	—	4	4	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	—	4	1	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	1	no	—
315	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
319	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
320	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
321	C₈×C₂₃	—	2	2	no	—
322	C₈×C₂₃	—	2	2	no	—
323	C₈×C₂₃	—	2	2	no	—
324	C₈×C₂₃	—	2	2	no	—
325	C₈×C₂₃	—	4	1	no	—
326	C₈×C₂₃	—	4	1	no	—
327	C₈×C₂₃	—	4	1	no	—
328	C₈×C₂₃	—	4	1	no	—
329	#3069	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#3069	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#3020	C₂×C₂₃	1	1	yes	characteristic
332	C₂×#456	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#2956	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#2889	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#2070	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#2050	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#2049	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#2005	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1966	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1932	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1915	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1902	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1861	C₂×C₂₃	1	1	yes	characteristic, complemented
344	C₂²×C₂₃	—	2	2	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	—	4	1	no	—
347	C₂²×C₂₃	—	4	1	no	—
348	C₂²×C₂₃	—	8	1	no	—
349	C₂²×C₂₃	—	8	1	no	—
350	C₂²×C₂₃	—	8	1	no	—
351	C₂²×C₂₃	#525	1	1	yes	characteristic
352	C₂²×C₂₃	#546	1	1	yes	characteristic
353	C₂²×C₂₃	#546	1	1	yes	characteristic
354	C₄×C₂₃	—	2	2	no	—
355	C₄×C₂₃	—	2	2	no	—
356	C₄×C₂₃	—	4	1	no	—
357	C₄×C₂₃	—	4	2	no	—
358	C₄×C₂₃	—	4	1	no	—
359	C₄×C₂₃	—	4	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	—	8	1	no	—
362	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
363	C₄×C₂₃	#538	1	1	yes	characteristic
364	C₄×C₂₃	D₄²	1	1	yes	characteristic
365	C₄×C₂₃	D₄²	1	1	yes	characteristic
366	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
367	#575	C₂²×C₂₃	1	1	yes	characteristic
368	#575	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×#137	C₂²×C₂₃	1	1	yes	characteristic
372	#568	C₂²×C₂₃	1	1	yes	characteristic
373	#568	C₂²×C₂₃	1	1	yes	characteristic
374	#568	C₂²×C₂₃	1	1	yes	characteristic
375	#536	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
377	#473	—	2	2	no	—
378	#473	—	2	2	no	—
379	#471	—	2	2	no	—
380	#471	—	2	2	no	—
381	#456	—	2	2	no	—
382	#456	—	2	2	no	—
383	#456	—	2	2	no	—
384	#454	—	2	2	no	—
385	#444	—	2	2	no	—
386	#443	—	2	2	no	—
387	#430	C₂²×C₂₃	1	1	yes	characteristic
388	#430	C₂²×C₂₃	1	1	yes	characteristic
389	#430	C₂²×C₂₃	1	1	yes	characteristic
390	#423	C₂²×C₂₃	1	1	yes	characteristic
391	#421	C₂²×C₂₃	1	1	yes	characteristic
392	#421	C₂²×C₂₃	1	1	yes	characteristic
393	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
394	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
395	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
396	#419	C₂²×C₂₃	1	1	yes	characteristic
397	#413	—	2	2	no	—
398	#413	C₂²×C₂₃	1	1	yes	characteristic
399	#413	C₂²×C₂₃	1	1	yes	characteristic
400	#413	C₂²×C₂₃	1	1	yes	characteristic
401	C₂×#101	—	2	2	no	—
402	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
404	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
405	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×#101	C₂²×C₂₃	1	1	yes	characteristic
408	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
409	#344	C₂²×C₂₃	1	1	yes	characteristic
410	#344	C₂²×C₂₃	1	1	yes	characteristic
411	#339	C₂²×C₂₃	1	1	yes	characteristic
412	#338	C₂²×C₂₃	1	1	yes	characteristic
413	C₂×C₂₃	—	2	1	no	—
414	C₂×C₂₃	—	2	1	no	—
415	C₂×C₂₃	—	2	1	no	—
416	C₂×C₂₃	—	4	2	no	—
417	C₂×C₂₃	—	8	1	no	—
418	C₂×C₂₃	—	8	1	no	—
419	C₂×C₂₃	#2424	1	1	yes	characteristic
420	2₋¹⁺⁴	—	2	2	no	—
421	2₋¹⁺⁴	—	2	2	no	—
422	2₋¹⁺⁴	—	4	1	no	—
423	2₋¹⁺⁴	—	4	1	no	—
424	C₂×#41	—	2	1	no	—
425	C₂×#41	—	2	1	no	—
426	C₂×#41	—	2	2	no	—
427	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
430	C₂²×Q₈	—	2	1	no	—
431	C₂²×Q₈	—	2	1	no	—
432	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
434	#137	—	2	1	no	—
435	#137	—	2	2	no	—
436	#137	—	2	2	no	—
437	#137	—	2	1	no	—
438	#137	—	2	1	no	—
439	#137	—	2	2	no	—
440	#137	—	2	2	no	—
441	#137	—	4	1	no	—
442	#137	—	4	1	no	—
443	#137	D₄×C₂₃	1	2	yes	—
444	AGL(1,ℤ/8)	—	2	1	no	—
445	AGL(1,ℤ/8)	—	2	1	no	—
446	AGL(1,ℤ/8)	—	2	1	no	—
447	AGL(1,ℤ/8)	D₄×C₂₃	1	2	yes	—
448	#135	—	2	2	no	—
449	#135	—	4	1	no	—
450	C₂×Q₁₆	—	2	1	no	—
451	C₂×Q₁₆	—	2	1	no	—
452	C₂×Q₁₆	—	2	1	no	—
453	C₂×SD₁₆	—	2	1	no	—
454	C₂×SD₁₆	—	2	1	no	—
455	C₂×SD₁₆	—	2	1	no	—
456	#131	—	2	2	no	—
457	#131	—	2	2	no	—
458	#131	—	2	2	no	—
459	#131	—	2	2	no	—
460	#131	—	4	1	no	—
461	#131	—	4	1	no	—
462	C₂×M₁₆	—	2	2	no	—
463	C₂×M₁₆	—	2	2	no	—
464	C₂×M₁₆	—	4	1	no	—
465	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
466	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
467	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
468	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
470	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
471	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
472	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
473	C₄⋊Q₈	—	2	1	no	—
474	C₄⋊Q₈	—	2	1	no	—
475	C₄⋊Q₈	—	2	1	no	—
476	C₄⋊Q₈	—	4	1	no	—
477	#124	—	2	1	no	—
478	#123	—	2	2	no	—
479	C₂²⋊Q₈	—	4	1	no	—
480	#117	C₂³×C₂₃	1	1	yes	characteristic
481	C₂×C₄⋊C₄	—	2	2	no	—
482	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
483	#108	—	2	2	no	—
484	#108	—	2	2	no	—
485	#108	—	2	2	no	—
486	C₄⋊C₈	—	4	1	no	—
487	#104	—	2	2	no	—
488	#104	—	2	2	no	—
489	#104	—	2	1	no	—
490	#104	—	2	1	no	—
491	#104	—	2	1	no	—
492	#104	—	2	1	no	—
493	#104	D₄×C₂₃	1	2	yes	—
494	#104	D₄×C₂₃	1	2	yes	—
495	#103	—	4	1	no	—
496	#101	—	2	1	no	—
497	#101	—	2	1	no	—
498	#101	—	2	2	no	—
499	#101	—	2	2	no	—
500	#101	—	2	2	no	—
501	#101	—	2	2	no	—
502	#101	—	2	2	no	—
503	#101	—	2	2	no	—
504	#101	D₄×C₂₃	1	2	yes	—
505	#100	—	2	1	no	—
506	#100	—	2	1	no	—
507	#100	D₄×C₂₃	1	2	yes	—
508	#97	—	2	1	no	—
509	C₄×C₈	—	2	1	no	—
510	C₂₃	#25198	1	1	yes	characteristic, direct product factor, Sylow
511	#41	—	2	1	no	—
512	#41	—	2	1	no	—
513	#41	—	2	1	no	—
514	#41	—	2	1	no	—
515	#41	—	2	1	no	—
516	#41	—	2	1	no	—
517	#41	—	2	1	no	—
518	#41	—	2	1	no	—
519	#41	—	2	1	no	—
520	#41	—	2	1	no	—
521	#41	—	4	2	no	—
522	#41	—	4	2	no	—
523	#41	—	4	4	no	—
524	#41	—	4	4	no	—
525	#41	—	4	2	no	—
526	#41	—	4	2	no	—
527	#41	C₂×D₄×C₂₃	1	2	yes	—
528	#41	C₂×D₄×C₂₃	1	2	yes	—
529	C₂×Q₈	—	2	2	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	1	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	1	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	2	1	no	—
542	C₂×Q₈	—	4	2	no	—
543	C₂×Q₈	—	4	2	no	—
544	C₂×Q₈	—	4	2	no	—
545	C₂×Q₈	—	4	2	no	—
546	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	1	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
554	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	2	2	no	—
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	2	1	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	1	no	—
563	C₂²×C₄	#41×C₂₃	1	2	yes	—
564	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
565	Q₁₆	—	2	2	no	—
566	Q₁₆	—	4	2	no	—
567	Q₁₆	—	4	2	no	—
568	Q₁₆	—	4	1	no	—
569	Q₁₆	—	4	2	no	—
570	SD₁₆	—	2	2	no	—
571	SD₁₆	—	2	2	no	—
572	SD₁₆	—	4	2	no	—
573	SD₁₆	—	4	2	no	—
574	SD₁₆	—	4	1	no	—
575	SD₁₆	—	4	1	no	—
576	SD₁₆	—	4	2	no	—
577	D₈	—	2	2	no	—
578	D₈	—	4	1	no	—
579	M₁₆	—	2	1	no	—
580	M₁₆	—	2	1	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	2	1	no	—
587	M₁₆	—	2	1	no	—
588	M₁₆	—	2	1	no	—
589	M₁₆	—	2	1	no	—
590	M₁₆	—	2	1	no	—
591	M₁₆	—	4	2	no	—
592	M₁₆	—	4	2	no	—
593	M₁₆	—	4	2	no	—
594	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
595	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
596	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
597	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	2	1	no	—
603	C₂×C₈	—	2	1	no	—
604	C₂×C₈	—	2	1	no	—
605	C₂×C₈	—	2	1	no	—
606	C₂×C₈	—	4	2	no	—
607	C₂×C₈	—	4	2	no	—
608	C₂×C₈	—	4	2	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	4	2	no	—
611	C₄⋊C₄	—	4	2	no	—
612	C₄⋊C₄	—	4	2	no	—
613	C₂²⋊C₄	—	2	1	no	—
614	C₂²⋊C₄	—	4	2	no	—
615	C₄²	—	2	1	no	—
616	C₄²	—	2	1	no	—
617	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
618	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
623	Q₈	—	2	1	no	—
624	Q₈	—	2	1	no	—
625	Q₈	—	2	1	no	—
626	Q₈	—	2	1	no	—
627	Q₈	—	2	1	no	—
628	Q₈	—	4	4	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	C₂₃×#121	1	2	yes	—
638	D₄	—	2	2	no	—
639	D₄	—	2	2	no	—
640	D₄	—	2	1	no	—
641	D₄	—	2	1	no	—
642	D₄	—	2	1	no	—
643	D₄	—	2	1	no	—
644	D₄	—	2	1	no	—
645	D₄	—	4	4	no	—
646	D₄	—	4	1	no	—
647	D₄	—	4	1	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	1	no	—
650	D₄	C₂₃×#121	1	2	yes	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	2	no	—
663	C₂×C₄	—	4	4	no	—
664	C₂×C₄	—	4	1	no	—
665	C₂×C₄	—	4	1	no	—
666	C₂×C₄	—	4	1	no	—
667	C₂×C₄	—	4	1	no	—
668	C₂×C₄	—	4	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
674	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
675	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
676	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
677	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
678	C₈	—	2	2	no	—
679	C₈	—	2	2	no	—
680	C₈	—	2	2	no	—
681	C₈	—	2	2	no	—
682	C₈	—	4	1	no	—
683	C₈	—	4	1	no	—
684	C₈	—	4	1	no	—
685	C₈	—	4	1	no	—
686	C₂²	—	2	2	no	—
687	C₂²	—	4	1	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	1	no	—
690	C₂²	—	8	1	no	—
691	C₂²	—	8	1	no	—
692	C₂²	—	8	1	no	—
693	C₂²	C₂₃×#525	1	1	yes	characteristic
694	C₂²	C₂₃×#546	1	1	yes	characteristic
695	C₂²	C₂₃×#546	1	1	yes	characteristic
696	C₄	—	2	2	no	—
697	C₄	—	2	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	1	no	—
701	C₄	—	4	1	no	—
702	C₄	—	8	1	no	—
703	C₄	—	8	1	no	—
704	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
705	C₄	C₂₃×#538	1	1	yes	characteristic
706	C₄	D₄²×C₂₃	1	1	yes	characteristic
707	C₄	D₄²×C₂₃	1	1	yes	characteristic
708	C₂	—	2	1	no	—
709	C₂	—	2	1	no	—
710	C₂	—	2	1	no	—
711	C₂	—	4	2	no	—
712	C₂	—	8	1	no	—
713	C₂	—	8	1	no	—
714	C₂	C₂₃×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3020	C₂
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×#3069	C₂	semidirect
2. C₂₃×#3069	C₂	semidirect
4. C₂×C₂₃×#456	C₂	semidirect
5. C₂₃×#2956	C₂	semidirect
6. C₂₃×#2889	C₂	semidirect
7. C₂₃×#2070	C₂	semidirect
8. C₂₃×#2050	C₂	semidirect
9. C₂₃×#2049	C₂	semidirect
10. C₂₃×#2005	C₂	semidirect
11. C₂₃×#1966	C₂	semidirect
12. C₂₃×#1932	C₂	semidirect
13. C₂₃×#1915	C₂	semidirect
14. C₂₃×#1902	C₂	semidirect
15. C₂₃×#1861	C₂	semidirect
261. #25198	C₂₃	direct (N × H)
329. #3069	C₂×C₂₃	semidirect
330. #3069	C₂×C₂₃	semidirect
332. C₂×#456	C₂×C₂₃	semidirect
333. #2956	C₂×C₂₃	semidirect
334. #2889	C₂×C₂₃	semidirect
335. #2070	C₂×C₂₃	semidirect
336. #2050	C₂×C₂₃	semidirect
337. #2049	C₂×C₂₃	semidirect
338. #2005	C₂×C₂₃	semidirect
339. #1966	C₂×C₂₃	semidirect
340. #1932	C₂×C₂₃	semidirect
341. #1915	C₂×C₂₃	semidirect
342. #1902	C₂×C₂₃	semidirect
343. #1861	C₂×C₂₃	semidirect
510. C₂₃	#25198	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×#3069	2
C₂₃×#3020	2
C₂×C₂₃×#456	2
C₂₃×#2956	2
C₂₃×#2889	2
C₂₃×#2070	2
C₂₃×#2050	2
C₂₃×#2049	2
C₂₃×#2005	2
C₂₃×#1966	2
C₂₃×#1932	2
C₂₃×#1915	2
C₂₃×#1902	2
C₂₃×#1861	2
#25198	23

Subgroups — C23×#25198 #19172824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25198 #19172824