Subgroups — C₂₃×#19770 #19167396

Order 5888.

650 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3392	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3387	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3287	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1715	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1688	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1642	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#135	—	2	1	no	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#557	C₂²	1	1	yes	characteristic
19	C₂₃×#550	C₂²	1	1	yes	characteristic
20	C₂₃×#550	C₂²	1	1	yes	characteristic
21	C₂₃×#501	—	2	1	no	—
22	C₂₃×#498	C₂²	1	1	yes	characteristic
23	C₂₃×#493	C₂²	1	1	yes	characteristic
24	C₂₃×#492	—	2	1	no	—
25	C₂₃×#492	C₂²	1	1	yes	characteristic
26	C₂₃×#492	C₂²	1	1	yes	characteristic
27	C₂₃×#486	—	2	1	no	—
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#486	C₂²	1	1	yes	characteristic
30	C₂₃×#479	—	2	1	no	—
31	C₂₃×#477	C₂²	1	1	yes	characteristic
32	C₂₃×#476	—	2	1	no	—
33	C₂₃×#476	C₂²	1	1	yes	characteristic
34	C₂₃×#475	—	2	1	no	—
35	C₂₃×#475	C₂²	1	1	yes	characteristic
36	C₂₃×#466	—	2	1	no	—
37	C₂₃×#461	—	2	1	no	—
38	C₂₃×#461	C₂²	1	1	yes	characteristic
39	C₂₃×#460	—	2	1	no	—
40	C₂₃×#460	—	2	1	no	—
41	C₂₃×#460	C₂²	1	1	yes	characteristic
42	C₂₃×#460	C₂²	1	1	yes	characteristic
43	C₂₃×#460	C₂²	1	1	yes	characteristic
44	C₂₃×#460	C₂²	1	1	yes	characteristic
45	C₂₃×#449	—	2	1	no	—
46	C₂₃×#449	—	2	1	no	—
47	C₂₃×#446	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₂₃×#441	—	2	1	no	—
51	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₂₃	—	2	1	no	—
53	C₄×SD₁₆×C₂₃	—	2	1	no	—
54	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
55	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
56	C₂₃×#334	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#331	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂×#41×C₂₃	—	2	1	no	—
69	C₂₃×#135	—	4	4	no	—
70	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	4	1	no	—
79	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
80	C₂×D₈×C₂₃	—	2	1	no	—
81	C₂×D₈×C₂₃	—	2	1	no	—
82	C₂²×C₈×C₂₃	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	1	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	—	4	1	no	—
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×#127	—	2	1	no	—
90	C₂₃×#127	—	2	1	no	—
91	C₂₃×#127	C₂³	1	1	yes	characteristic
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#125	—	2	1	no	—
94	C₂₃×#125	—	4	1	no	—
95	C₂₃×#121	—	2	1	no	—
96	C₂₃×#121	—	2	1	no	—
97	C₂₃×#121	—	4	1	no	—
98	C₂₃×#121	—	4	1	no	—
99	C₄×Q₈×C₂₃	—	2	1	no	—
100	C₄×Q₈×C₂₃	—	2	1	no	—
101	C₄×Q₈×C₂₃	—	2	1	no	—
102	C₄×Q₈×C₂₃	—	2	1	no	—
103	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
104	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
105	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₄×D₄×C₂₃	—	2	1	no	—
107	C₄×D₄×C₂₃	—	2	1	no	—
108	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#107	—	2	1	no	—
110	C₂₃×#107	D₄	1	1	yes	characteristic
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×#106	—	2	1	no	—
113	C₂₃×#106	—	2	1	no	—
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	D₄	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	—	2	1	no	—
117	C₂₃×C₄⋊C₈	—	2	1	no	—
118	C₂₃×C₄⋊C₈	—	2	1	no	—
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	D₄	1	1	yes	characteristic
127	C₂₃×#103	D₄	1	1	yes	characteristic
128	C₂₃×#103	D₄	1	1	yes	characteristic
129	C₂₃×#102	—	2	1	no	—
130	C₂₃×#102	—	2	1	no	—
131	C₂₃×#102	—	2	1	no	—
132	C₂₃×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×#102	—	2	1	no	—
135	C₂₃×#102	—	2	1	no	—
136	C₂₃×#102	—	4	1	no	—
137	C₂₃×#102	D₄	1	1	yes	characteristic
138	C₂₃×C₂²⋊C₈	—	2	1	no	—
139	C₂₃×#97	—	2	1	no	—
140	C₄×C₈×C₂₃	—	2	1	no	—
141	C₄×C₈×C₂₃	—	2	1	no	—
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	#41×C₂₃	—	4	2	no	—
146	#41×C₂₃	—	4	2	no	—
147	#41×C₂₃	—	8	1	no	—
148	#41×C₂₃	—	8	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	4	1	no	—
152	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	—	4	1	no	—
162	C₂×D₄×C₂₃	—	4	1	no	—
163	C₂×D₄×C₂₃	—	4	1	no	—
164	C₂×D₄×C₂₃	—	4	1	no	—
165	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	4	1	no	—
168	C₂²×C₄×C₂₃	—	4	1	no	—
169	Q₁₆×C₂₃	SD₁₆	1	4	yes	complemented
170	SD₁₆×C₂₃	—	4	1	no	—
171	SD₁₆×C₂₃	—	4	2	no	—
172	SD₁₆×C₂₃	—	4	2	no	—
173	SD₁₆×C₂₃	—	4	2	no	—
174	SD₁₆×C₂₃	—	4	2	no	—
175	SD₁₆×C₂₃	—	8	1	no	—
176	SD₁₆×C₂₃	—	8	1	no	—
177	SD₁₆×C₂₃	—	8	1	no	—
178	SD₁₆×C₂₃	—	8	2	no	—
179	D₈×C₂₃	—	4	2	no	—
180	D₈×C₂₃	—	8	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	4	2	no	—
187	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	—	4	1	no	—
201	C₄⋊C₄×C₂₃	—	4	1	no	—
202	C₄⋊C₄×C₂₃	—	4	1	no	—
203	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	4	1	no	—
209	C₂²⋊C₄×C₂₃	—	4	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	4	1	no	—
214	C₄²×C₂₃	—	4	1	no	—
215	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
216	#19770	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂³×C₂₃	—	2	1	no	—
218	C₂³×C₂₃	—	4	1	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	2	1	no	—
222	Q₈×C₂₃	—	2	1	no	—
223	Q₈×C₂₃	—	2	1	no	—
224	Q₈×C₂₃	—	4	1	no	—
225	Q₈×C₂₃	—	4	1	no	—
226	Q₈×C₂₃	—	8	1	no	—
227	Q₈×C₂₃	—	16	1	no	—
228	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	—
229	Q₈×C₂₃	C₂×SD₁₆	1	2	yes	—
230	D₄×C₂₃	—	2	1	no	—
231	D₄×C₂₃	—	2	1	no	—
232	D₄×C₂₃	—	4	1	no	—
233	D₄×C₂₃	—	4	1	no	—
234	D₄×C₂₃	—	4	1	no	—
235	D₄×C₂₃	—	4	1	no	—
236	D₄×C₂₃	—	4	1	no	—
237	D₄×C₂₃	—	4	1	no	—
238	D₄×C₂₃	—	4	1	no	—
239	D₄×C₂₃	—	4	1	no	—
240	D₄×C₂₃	—	8	2	no	—
241	D₄×C₂₃	—	8	2	no	—
242	D₄×C₂₃	—	8	1	no	—
243	D₄×C₂₃	—	8	1	no	—
244	D₄×C₂₃	—	16	1	no	—
245	D₄×C₂₃	—	16	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	8	2	no	—
258	C₂×C₄×C₂₃	—	8	2	no	—
259	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
262	C₈×C₂₃	—	2	1	no	—
263	C₈×C₂₃	—	2	1	no	—
264	C₈×C₂₃	—	2	1	no	—
265	C₈×C₂₃	—	4	1	no	—
266	C₈×C₂₃	—	4	1	no	—
267	C₈×C₂₃	—	4	1	no	—
268	C₈×C₂₃	—	4	1	no	—
269	C₈×C₂₃	—	4	1	no	—
270	C₈×C₂₃	C₂×SD₁₆	1	2	yes	—
271	#3392	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#3387	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#3287	C₂×C₂₃	1	1	yes	characteristic
275	#1715	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1688	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1671	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1666	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1642	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₂²×C₂₃	—	4	1	no	—
287	C₂²×C₂₃	—	4	1	no	—
288	C₂²×C₂₃	—	4	1	no	—
289	C₂²×C₂₃	—	8	1	no	—
290	C₂²×C₂₃	—	8	1	no	—
291	C₂²×C₂₃	—	8	1	no	—
292	C₂²×C₂₃	—	8	1	no	—
293	C₂²×C₂₃	—	8	1	no	—
294	C₂²×C₂₃	—	8	1	no	—
295	C₂²×C₂₃	D₄²	1	1	yes	characteristic
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	2	no	—
298	C₄×C₂₃	—	2	1	no	—
299	C₄×C₂₃	—	4	1	no	—
300	C₄×C₂₃	—	4	1	no	—
301	C₄×C₂₃	—	4	1	no	—
302	C₄×C₂₃	—	4	1	no	—
303	C₄×C₂₃	—	4	1	no	—
304	C₄×C₂₃	—	8	1	no	—
305	C₄×C₂₃	—	8	1	no	—
306	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
307	C₄×C₂₃	#575	1	1	yes	characteristic
308	C₄×C₂₃	#577	1	1	yes	characteristic
309	C₄×C₂₃	#577	1	1	yes	characteristic
310	C₂×#135	—	2	1	no	—
311	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
312	#557	C₂²×C₂₃	1	1	yes	characteristic
313	#550	C₂²×C₂₃	1	1	yes	characteristic
314	#550	C₂²×C₂₃	1	1	yes	characteristic
315	#501	—	2	1	no	—
316	#498	C₂²×C₂₃	1	1	yes	characteristic
317	#493	C₂²×C₂₃	1	1	yes	characteristic
318	#492	—	2	1	no	—
319	#492	C₂²×C₂₃	1	1	yes	characteristic
320	#492	C₂²×C₂₃	1	1	yes	characteristic
321	#486	—	2	1	no	—
322	#486	C₂²×C₂₃	1	1	yes	characteristic
323	#486	C₂²×C₂₃	1	1	yes	characteristic
324	#479	—	2	1	no	—
325	#477	C₂²×C₂₃	1	1	yes	characteristic
326	#476	—	2	1	no	—
327	#476	C₂²×C₂₃	1	1	yes	characteristic
328	#475	—	2	1	no	—
329	#475	C₂²×C₂₃	1	1	yes	characteristic
330	#466	—	2	1	no	—
331	#461	—	2	1	no	—
332	#461	C₂²×C₂₃	1	1	yes	characteristic
333	#460	—	2	1	no	—
334	#460	—	2	1	no	—
335	#460	C₂²×C₂₃	1	1	yes	characteristic
336	#460	C₂²×C₂₃	1	1	yes	characteristic
337	#460	C₂²×C₂₃	1	1	yes	characteristic
338	#460	C₂²×C₂₃	1	1	yes	characteristic
339	#449	—	2	1	no	—
340	#449	—	2	1	no	—
341	#446	C₂²×C₂₃	1	1	yes	characteristic
342	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
344	#441	—	2	1	no	—
345	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
346	C₄×SD₁₆	—	2	1	no	—
347	C₄×SD₁₆	—	2	1	no	—
348	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
349	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
350	#334	C₂²×C₂₃	1	1	yes	characteristic
351	#334	C₂²×C₂₃	1	1	yes	characteristic
352	#333	C₂²×C₂₃	1	1	yes	characteristic
353	#332	C₂²×C₂₃	1	1	yes	characteristic
354	#331	C₂²×C₂₃	1	1	yes	characteristic
355	#331	C₂²×C₂₃	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
357	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
358	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
359	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
360	#322	C₂²×C₂₃	1	1	yes	characteristic
361	C₂×C₂₃	—	8	1	no	—
362	C₂×C₂₃	—	16	1	no	—
363	C₂×C₂₃	—	16	1	no	—
364	C₂×C₂₃	#3271	1	1	yes	characteristic
365	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
366	C₂×C₂₃	#3288	1	1	yes	characteristic
367	C₂×#41	—	2	1	no	—
368	C₂×#41	—	2	1	no	—
369	#135	—	4	4	no	—
370	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	—	2	1	no	—
374	C₂×SD₁₆	—	2	1	no	—
375	C₂×SD₁₆	—	2	1	no	—
376	C₂×SD₁₆	—	2	1	no	—
377	C₂×SD₁₆	—	2	1	no	—
378	C₂×SD₁₆	—	4	1	no	—
379	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
380	C₂×D₈	—	2	1	no	—
381	C₂×D₈	—	2	1	no	—
382	C₂²×C₈	—	2	1	no	—
383	C₄⋊Q₈	—	2	1	no	—
384	C₄⋊Q₈	—	2	1	no	—
385	C₄⋊Q₈	—	2	1	no	—
386	C₄⋊Q₈	—	4	1	no	—
387	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
389	#127	—	2	1	no	—
390	#127	—	2	1	no	—
391	#127	C₂³×C₂₃	1	1	yes	characteristic
392	#127	C₂³×C₂₃	1	1	yes	characteristic
393	#125	—	2	1	no	—
394	#125	—	4	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	4	1	no	—
398	#121	—	4	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
409	#107	—	2	1	no	—
410	#107	D₄×C₂₃	1	1	yes	characteristic
411	#106	—	2	1	no	—
412	#106	—	2	1	no	—
413	#106	—	2	1	no	—
414	#106	—	2	1	no	—
415	#106	D₄×C₂₃	1	1	yes	characteristic
416	C₄⋊C₈	—	2	1	no	—
417	C₄⋊C₈	—	2	1	no	—
418	C₄⋊C₈	—	2	1	no	—
419	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
423	#103	—	2	1	no	—
424	#103	—	2	1	no	—
425	#103	—	2	1	no	—
426	#103	D₄×C₂₃	1	1	yes	characteristic
427	#103	D₄×C₂₃	1	1	yes	characteristic
428	#103	D₄×C₂₃	1	1	yes	characteristic
429	#102	—	2	1	no	—
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	—	2	1	no	—
435	#102	—	2	1	no	—
436	#102	—	4	1	no	—
437	#102	D₄×C₂₃	1	1	yes	characteristic
438	C₂²⋊C₈	—	2	1	no	—
439	#97	—	2	1	no	—
440	C₄×C₈	—	2	1	no	—
441	C₄×C₈	—	2	1	no	—
442	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
443	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
444	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
445	C₂₃	#19770	1	1	yes	characteristic, direct product factor, Sylow
446	#41	—	4	2	no	—
447	#41	—	4	2	no	—
448	#41	—	8	1	no	—
449	#41	—	8	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	4	1	no	—
453	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	4	1	no	—
463	C₂×D₄	—	4	1	no	—
464	C₂×D₄	—	4	1	no	—
465	C₂×D₄	—	4	1	no	—
466	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	4	1	no	—
469	C₂²×C₄	—	4	1	no	—
470	Q₁₆	SD₁₆×C₂₃	1	4	yes	complemented
471	SD₁₆	—	4	1	no	—
472	SD₁₆	—	4	2	no	—
473	SD₁₆	—	4	2	no	—
474	SD₁₆	—	4	2	no	—
475	SD₁₆	—	4	2	no	—
476	SD₁₆	—	8	1	no	—
477	SD₁₆	—	8	1	no	—
478	SD₁₆	—	8	1	no	—
479	SD₁₆	—	8	2	no	—
480	D₈	—	4	2	no	—
481	D₈	—	8	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	4	2	no	—
488	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	—	4	1	no	—
503	C₄⋊C₄	—	4	1	no	—
504	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	4	1	no	—
515	C₄²	—	4	1	no	—
516	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
517	C₂³	—	2	1	no	—
518	C₂³	—	4	1	no	—
519	C₂³	—	4	1	no	—
520	Q₈	—	2	1	no	—
521	Q₈	—	2	1	no	—
522	Q₈	—	2	1	no	—
523	Q₈	—	2	1	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	8	1	no	—
527	Q₈	—	16	1	no	—
528	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	—
529	Q₈	C₂×SD₁₆×C₂₃	1	2	yes	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	1	no	—
535	D₄	—	4	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	1	no	—
540	D₄	—	8	2	no	—
541	D₄	—	8	2	no	—
542	D₄	—	8	1	no	—
543	D₄	—	8	1	no	—
544	D₄	—	16	1	no	—
545	D₄	—	16	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	8	2	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
560	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	C₂×SD₁₆×C₂₃	1	2	yes	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	8	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	—	8	1	no	—
579	C₂²	—	8	1	no	—
580	C₂²	D₄²×C₂₃	1	1	yes	characteristic
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	8	1	no	—
590	C₄	—	8	1	no	—
591	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
592	C₄	C₂₃×#575	1	1	yes	characteristic
593	C₄	C₂₃×#577	1	1	yes	characteristic
594	C₄	C₂₃×#577	1	1	yes	characteristic
595	C₂	—	8	1	no	—
596	C₂	—	16	1	no	—
597	C₂	—	16	1	no	—
598	C₂	C₂₃×#3271	1	1	yes	characteristic
599	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
600	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3287	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3392	C₂	semidirect
2. C₂₃×#3387	C₂	semidirect
3. C₂₃×#3362	C₂	semidirect
5. C₂₃×#1715	C₂	semidirect
6. C₂₃×#1688	C₂	semidirect
7. C₂₃×#1678	C₂	semidirect
8. C₂₃×#1671	C₂	semidirect
9. C₂₃×#1666	C₂	semidirect
10. C₂₃×#1642	C₂	semidirect
11. C₂₃×#1631	C₂	semidirect
12. C₂₃×#1626	C₂	semidirect
13. C₂₃×#1623	C₂	semidirect
14. C₂₃×#1582	C₂	semidirect
15. C₂₃×#1579	C₂	semidirect
169. Q₁₆×C₂₃	SD₁₆	semidirect
216. #19770	C₂₃	direct (N × H)
271. #3392	C₂×C₂₃	semidirect
272. #3387	C₂×C₂₃	semidirect
273. #3362	C₂×C₂₃	semidirect
275. #1715	C₂×C₂₃	semidirect
276. #1688	C₂×C₂₃	semidirect
277. #1678	C₂×C₂₃	semidirect
278. #1671	C₂×C₂₃	semidirect
279. #1666	C₂×C₂₃	semidirect
280. #1642	C₂×C₂₃	semidirect
281. #1631	C₂×C₂₃	semidirect
282. #1626	C₂×C₂₃	semidirect
283. #1623	C₂×C₂₃	semidirect
284. #1582	C₂×C₂₃	semidirect
285. #1579	C₂×C₂₃	semidirect
445. C₂₃	#19770	direct (N × H)
470. Q₁₆	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3392	2
C₂₃×#3387	2
C₂₃×#3362	2
C₂₃×#3287	2
C₂₃×#1715	2
C₂₃×#1688	2
C₂₃×#1678	2
C₂₃×#1671	2
C₂₃×#1666	2
C₂₃×#1642	2
C₂₃×#1631	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1582	2
C₂₃×#1579	2
#19770	23

Subgroups — C23×#19770 #19167396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19770 #19167396