Subgroups — C₂₃×#61469 #19209095

Order 5888.

1866 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3483	C₂	1	1	yes	maximal, characteristic
2	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3392	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3363	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3357	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3333	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3329	C₂	1	3	yes	maximal, complemented
10	C₂₃×#3278	C₂	1	3	yes	maximal, complemented
11	C₂₃×#3182	C₂	1	3	yes	maximal, complemented
12	C₂₃×#3116	C₂	1	3	yes	maximal, complemented
13	C₂₃×#3112	C₂	1	3	yes	maximal, complemented
14	C₂₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#137	C₂	1	3	yes	maximal, complemented
16	C₂×C₂₃×#137	C₂²	1	3	yes	—
17	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
18	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#557	—	2	1	no	—
20	C₂₃×#557	C₂²	1	1	yes	characteristic
21	C₂₃×#555	—	2	3	no	—
22	C₂₃×#555	C₂²	1	3	yes	—
23	C₂₃×#554	—	2	3	no	—
24	C₂₃×#554	C₂²	1	3	yes	—
25	C₂₃×#550	—	2	1	no	—
26	C₂₃×#550	C₂²	1	3	yes	—
27	C₂₃×#550	C₂²	1	1	yes	characteristic
28	D₄×Q₈×C₂₃	C₂²	1	3	yes	—
29	C₂₃×#533	—	2	3	no	—
30	C₂₃×#533	C₂²	1	3	yes	—
31	C₂₃×#532	—	2	3	no	—
32	C₂₃×#532	C₂²	1	3	yes	—
33	C₄×#41×C₂₃	—	2	1	no	—
34	C₄×#41×C₂₃	—	2	1	no	—
35	C₄×#41×C₂₃	C₂²	1	3	yes	—
36	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
38	C₂₃×#500	C₂²	1	3	yes	—
39	C₂₃×#492	C₂²	1	3	yes	—
40	C₂₃×#487	C₂²	1	3	yes	—
41	C₂₃×#486	C₂²	1	3	yes	—
42	C₂₃×#485	C₂²	1	6	yes	—
43	C₂₃×#480	C₂²	1	6	yes	—
44	C₂₃×#477	C₂²	1	3	yes	—
45	C₂₃×#477	C₂²	1	3	yes	—
46	C₂₃×#477	C₂²	1	3	yes	—
47	C₂₃×#476	C₂²	1	3	yes	—
48	C₂₃×#468	C₂²	1	6	yes	—
49	C₂₃×#468	C₂²	1	6	yes	—
50	C₂₃×#462	C₂²	1	3	yes	—
51	C₂₃×#461	C₂²	1	3	yes	—
52	C₂₃×#460	C₂²	1	3	yes	—
53	C₂₃×#460	C₂²	1	3	yes	—
54	C₂₃×#452	C₂²	1	6	yes	—
55	C₂₃×#448	C₂²	1	6	yes	—
56	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
57	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
58	C₂₃×#441	C₂²	1	1	yes	characteristic
59	C₂₃×#441	C₂²	1	3	yes	—
60	C₂₃×#440	C₂²	1	3	yes	—
61	C₂₃×#440	C₂²	1	6	yes	—
62	C₂₃×#440	C₂²	1	3	yes	—
63	C₄×Q₁₆×C₂₃	C₂²	1	3	yes	—
64	C₄×Q₁₆×C₂₃	C₂²	1	3	yes	—
65	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
66	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
67	C₂₃×#436	C₂²	1	3	yes	—
68	C₂₃×#436	C₂²	1	3	yes	—
69	C₂₃×#432	C₂²	1	3	yes	—
70	C₂₃×#428	C₂²	1	3	yes	—
71	C₂₃×#419	C₂²	1	3	yes	—
72	C₂₃×#417	C₂²	1	3	yes	—
73	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
74	C₂×#41×C₂₃	—	2	1	no	—
75	C₂×#41×C₂₃	—	2	3	no	—
76	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
77	C₂×#41×C₂₃	C₂³	1	3	yes	—
78	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
79	C₂₃×#137	—	2	12	no	—
80	C₂×Q₁₆×C₂₃	C₂³	1	3	yes	—
81	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
83	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
84	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
85	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
86	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
87	C₂₃×C₄⋊Q₈	—	2	3	no	—
88	C₂₃×C₄⋊Q₈	—	2	3	no	—
89	C₂₃×C₄⋊Q₈	—	2	3	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
91	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
92	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
93	C₂₃×#127	—	2	3	no	—
94	C₂₃×#127	C₂³	1	3	yes	—
95	C₂₃×#125	—	2	3	no	—
96	C₂₃×#125	—	2	3	no	—
97	C₂₃×#125	—	2	3	no	—
98	C₂₃×#124	—	2	6	no	—
99	C₂₃×#124	—	2	3	no	—
100	C₂₃×C₂²⋊Q₈	—	2	3	no	—
101	C₂₃×C₂²⋊Q₈	—	2	6	no	—
102	C₂₃×C₂²⋊Q₈	—	2	6	no	—
103	C₂₃×C₂²⋊Q₈	C₂³	1	6	yes	—
104	C₂₃×#121	—	2	3	no	—
105	C₂₃×#121	—	2	3	no	—
106	C₂₃×#121	—	2	3	no	—
107	C₂₃×#121	—	2	6	no	—
108	C₂₃×#121	C₂³	1	6	yes	—
109	C₄×Q₈×C₂₃	—	2	1	no	—
110	C₄×Q₈×C₂₃	—	2	1	no	—
111	C₄×Q₈×C₂₃	—	2	1	no	—
112	C₄×Q₈×C₂₃	—	2	3	no	—
113	C₄×Q₈×C₂₃	—	2	3	no	—
114	C₄×Q₈×C₂₃	—	2	3	no	—
115	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
116	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
117	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
118	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
119	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
121	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
122	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
123	C₄×D₄×C₂₃	—	2	3	no	—
124	C₄×D₄×C₂₃	—	2	3	no	—
125	C₄×D₄×C₂₃	—	2	3	no	—
126	C₄×D₄×C₂₃	—	2	3	no	—
127	C₄×D₄×C₂₃	—	2	3	no	—
128	C₄×D₄×C₂₃	—	2	3	no	—
129	C₄×D₄×C₂₃	D₄	1	3	yes	—
130	C₄×D₄×C₂₃	C₂³	1	3	yes	—
131	C₄×D₄×C₂₃	C₂³	1	3	yes	—
132	C₄×D₄×C₂₃	C₂³	1	3	yes	—
133	C₂₃×#117	—	2	3	no	—
134	C₂₃×#117	—	2	3	no	—
135	C₂₃×#117	—	2	3	no	—
136	C₂₃×#117	D₄	1	3	yes	—
137	C₂₃×#117	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
139	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
140	C₂×C₄²×C₂₃	—	2	3	no	—
141	C₂×C₄²×C₂₃	—	2	3	no	—
142	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
143	C₂₃×#107	C₂³	1	3	yes	—
144	C₂₃×#107	C₂³	1	3	yes	—
145	C₂₃×#107	C₂³	1	6	yes	—
146	C₂₃×#106	C₂³	1	1	yes	characteristic
147	C₂₃×#106	C₂³	1	3	yes	—
148	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
149	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
150	C₂₃×#103	C₂³	1	1	yes	characteristic
151	C₂₃×#103	C₂³	1	3	yes	—
152	C₂₃×#103	C₂³	1	1	yes	characteristic
153	C₂₃×#103	C₂³	1	3	yes	—
154	C₂₃×#103	C₂³	1	3	yes	—
155	C₂₃×#103	C₂³	1	3	yes	—
156	C₂₃×#103	C₂³	1	6	yes	—
157	C₂₃×#102	C₂³	1	3	yes	—
158	C₂₃×#102	C₂³	1	3	yes	—
159	C₂₃×#102	C₂³	1	6	yes	—
160	C₂₃×C₂²⋊C₈	C₂³	1	6	yes	—
161	C₂₃×#97	C₂³	1	1	yes	characteristic
162	C₂₃×#97	C₂³	1	3	yes	—
163	C₄×C₈×C₂₃	C₂³	1	3	yes	—
164	C₄×C₈×C₂₃	C₂³	1	3	yes	—
165	#41×C₂₃	—	2	4	no	—
166	#41×C₂₃	—	2	6	no	—
167	#41×C₂₃	—	4	3	no	—
168	#41×C₂₃	—	4	4	no	—
169	#41×C₂₃	—	8	6	no	—
170	C₂×Q₈×C₂₃	—	2	3	no	—
171	C₂×Q₈×C₂₃	—	2	3	no	—
172	C₂×Q₈×C₂₃	—	2	3	no	—
173	C₂×Q₈×C₂₃	—	2	6	no	—
174	C₂×Q₈×C₂₃	—	4	3	no	—
175	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
177	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
179	C₂×D₄×C₂₃	—	2	3	no	—
180	C₂×D₄×C₂₃	—	2	6	no	—
181	C₂×D₄×C₂₃	—	2	3	no	—
182	C₂×D₄×C₂₃	—	2	3	no	—
183	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
184	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
185	C₂²×C₄×C₂₃	—	2	3	no	—
186	C₂²×C₄×C₂₃	—	2	3	no	—
187	C₂²×C₄×C₂₃	—	2	3	no	—
188	C₂²×C₄×C₂₃	—	2	3	no	—
189	C₂²×C₄×C₂₃	—	2	3	no	—
190	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
191	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
193	Q₁₆×C₂₃	—	2	6	no	—
194	Q₁₆×C₂₃	#41	1	4	yes	complemented
195	SD₁₆×C₂₃	—	2	6	no	—
196	SD₁₆×C₂₃	—	2	6	no	—
197	SD₁₆×C₂₃	—	4	6	no	—
198	M₁₆×C₂₃	—	2	6	no	—
199	M₁₆×C₂₃	—	4	3	no	—
200	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
201	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
202	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
203	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	3	no	—
206	C₄⋊C₄×C₂₃	—	2	3	no	—
207	C₄⋊C₄×C₂₃	—	2	3	no	—
208	C₄⋊C₄×C₂₃	—	2	3	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	2	3	no	—
211	C₄⋊C₄×C₂₃	—	2	3	no	—
212	C₄⋊C₄×C₂₃	—	2	3	no	—
213	C₄⋊C₄×C₂₃	—	2	3	no	—
214	C₄⋊C₄×C₂₃	—	2	3	no	—
215	C₄⋊C₄×C₂₃	—	2	6	no	—
216	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
217	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
218	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
219	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
220	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
222	C₂²⋊C₄×C₂₃	—	2	6	no	—
223	C₂²⋊C₄×C₂₃	—	2	3	no	—
224	C₂²⋊C₄×C₂₃	—	2	6	no	—
225	C₂²⋊C₄×C₂₃	—	2	3	no	—
226	C₂²⋊C₄×C₂₃	—	2	3	no	—
227	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
228	C₄²×C₂₃	—	2	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	2	3	no	—
231	C₄²×C₂₃	—	2	3	no	—
232	C₄²×C₂₃	—	2	3	no	—
233	C₄²×C₂₃	—	2	3	no	—
234	C₄²×C₂₃	—	2	3	no	—
235	C₄²×C₂₃	C₂×D₄	1	3	yes	—
236	C₄²×C₂₃	C₂×D₄	1	3	yes	—
237	C₄²×C₂₃	C₂⁴	1	3	yes	—
238	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
239	#61469	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₂₃	—	2	3	no	—
241	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
242	Q₈×C₂₃	—	2	1	no	—
243	Q₈×C₂₃	—	2	1	no	—
244	Q₈×C₂₃	—	2	3	no	—
245	Q₈×C₂₃	—	2	3	no	—
246	Q₈×C₂₃	—	4	6	no	—
247	Q₈×C₂₃	—	8	3	no	—
248	Q₈×C₂₃	—	8	3	no	—
249	Q₈×C₂₃	#137	1	4	yes	complemented
250	Q₈×C₂₃	C₂×#41	1	2	yes	—
251	Q₈×C₂₃	C₂×#41	1	2	yes	—
252	D₄×C₂₃	—	2	6	no	—
253	D₄×C₂₃	—	2	3	no	—
254	D₄×C₂₃	—	2	3	no	—
255	D₄×C₂₃	—	4	6	no	—
256	D₄×C₂₃	—	4	6	no	—
257	D₄×C₂₃	—	8	3	no	—
258	D₄×C₂₃	—	8	6	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	6	no	—
262	C₂×C₄×C₂₃	—	2	3	no	—
263	C₂×C₄×C₂₃	—	2	3	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	3	no	—
267	C₂×C₄×C₂₃	—	2	3	no	—
268	C₂×C₄×C₂₃	—	2	3	no	—
269	C₂×C₄×C₂₃	—	4	3	no	—
270	C₂×C₄×C₂₃	—	4	6	no	—
271	C₂×C₄×C₂₃	—	4	6	no	—
272	C₂×C₄×C₂₃	—	4	6	no	—
273	C₂×C₄×C₂₃	—	4	6	no	—
274	C₂×C₄×C₂₃	—	8	3	no	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
278	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
279	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
280	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
281	C₈×C₂₃	—	2	3	no	—
282	C₈×C₂₃	—	2	1	no	—
283	C₈×C₂₃	—	2	3	no	—
284	C₈×C₂₃	C₂×#41	1	2	yes	—
285	#3483	C₂×C₂₃	1	1	yes	characteristic
286	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#3392	C₂×C₂₃	1	1	yes	characteristic, complemented
288	Q₈×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#3363	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#3357	C₂×C₂₃	1	3	yes	complemented
292	#3333	C₂×C₂₃	1	3	yes	complemented
293	#3329	C₂×C₂₃	1	3	yes	complemented
294	#3278	C₂×C₂₃	1	3	yes	complemented
295	#3182	C₂×C₂₃	1	3	yes	complemented
296	#3116	C₂×C₂₃	1	3	yes	complemented
297	#3112	C₂×C₂₃	1	3	yes	complemented
298	#2971	C₂×C₂₃	1	1	yes	characteristic, complemented
299	C₄×#137	C₂×C₂₃	1	3	yes	complemented
300	C₂²×C₂₃	—	2	3	no	—
301	C₂²×C₂₃	—	2	3	no	—
302	C₂²×C₂₃	—	2	3	no	—
303	C₂²×C₂₃	—	4	3	no	—
304	C₂²×C₂₃	—	4	3	no	—
305	C₂²×C₂₃	—	4	3	no	—
306	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₂₃	—	2	1	no	—
308	C₄×C₂₃	—	2	3	no	—
309	C₄×C₂₃	—	2	3	no	—
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	2	no	—
312	C₄×C₂₃	—	2	1	no	—
313	C₄×C₂₃	—	4	3	no	—
314	C₄×C₂₃	—	4	1	no	—
315	C₄×C₂₃	—	4	3	no	—
316	C₄×C₂₃	—	4	3	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	C₂×#137	1	6	yes	—
319	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
320	C₄×C₂₃	#585	1	1	yes	characteristic
321	C₂×#137	C₂²×C₂₃	1	3	yes	—
322	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
323	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
324	#557	—	2	1	no	—
325	#557	C₂²×C₂₃	1	1	yes	characteristic
326	#555	—	2	3	no	—
327	#555	C₂²×C₂₃	1	3	yes	—
328	#554	—	2	3	no	—
329	#554	C₂²×C₂₃	1	3	yes	—
330	#550	—	2	1	no	—
331	#550	C₂²×C₂₃	1	3	yes	—
332	#550	C₂²×C₂₃	1	1	yes	characteristic
333	D₄×Q₈	C₂²×C₂₃	1	3	yes	—
334	#533	—	2	3	no	—
335	#533	C₂²×C₂₃	1	3	yes	—
336	#532	—	2	3	no	—
337	#532	C₂²×C₂₃	1	3	yes	—
338	C₄×#41	—	2	1	no	—
339	C₄×#41	—	2	1	no	—
340	C₄×#41	C₂²×C₂₃	1	3	yes	—
341	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
343	#500	C₂²×C₂₃	1	3	yes	—
344	#492	C₂²×C₂₃	1	3	yes	—
345	#487	C₂²×C₂₃	1	3	yes	—
346	#486	C₂²×C₂₃	1	3	yes	—
347	#485	C₂²×C₂₃	1	6	yes	—
348	#480	C₂²×C₂₃	1	6	yes	—
349	#477	C₂²×C₂₃	1	3	yes	—
350	#477	C₂²×C₂₃	1	3	yes	—
351	#477	C₂²×C₂₃	1	3	yes	—
352	#476	C₂²×C₂₃	1	3	yes	—
353	#468	C₂²×C₂₃	1	6	yes	—
354	#468	C₂²×C₂₃	1	6	yes	—
355	#462	C₂²×C₂₃	1	3	yes	—
356	#461	C₂²×C₂₃	1	3	yes	—
357	#460	C₂²×C₂₃	1	3	yes	—
358	#460	C₂²×C₂₃	1	3	yes	—
359	#452	C₂²×C₂₃	1	6	yes	—
360	#448	C₂²×C₂₃	1	6	yes	—
361	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
362	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
363	#441	C₂²×C₂₃	1	3	yes	—
364	#441	C₂²×C₂₃	1	1	yes	characteristic
365	#440	C₂²×C₂₃	1	3	yes	—
366	#440	C₂²×C₂₃	1	6	yes	—
367	#440	C₂²×C₂₃	1	3	yes	—
368	C₄×Q₁₆	C₂²×C₂₃	1	3	yes	—
369	C₄×Q₁₆	C₂²×C₂₃	1	3	yes	—
370	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
371	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
372	#436	C₂²×C₂₃	1	3	yes	—
373	#436	C₂²×C₂₃	1	3	yes	—
374	#432	C₂²×C₂₃	1	3	yes	—
375	#428	C₂²×C₂₃	1	3	yes	—
376	#419	C₂²×C₂₃	1	3	yes	—
377	#417	C₂²×C₂₃	1	3	yes	—
378	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
379	C₂×C₂₃	—	4	3	no	—
380	C₂×C₂₃	—	8	3	no	—
381	C₂×C₂₃	D₄×#41	1	1	yes	characteristic
382	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
383	C₂×C₂₃	#3585	1	1	yes	characteristic
384	C₂×#41	—	2	1	no	—
385	C₂×#41	—	2	3	no	—
386	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
387	C₂×#41	C₂³×C₂₃	1	3	yes	—
388	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
389	#137	—	2	12	no	—
390	C₂×Q₁₆	C₂³×C₂₃	1	3	yes	—
391	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
393	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
394	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
395	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
396	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
397	C₄⋊Q₈	—	2	3	no	—
398	C₄⋊Q₈	—	2	3	no	—
399	C₄⋊Q₈	—	2	3	no	—
400	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
401	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
402	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
403	#127	—	2	3	no	—
404	#127	C₂³×C₂₃	1	3	yes	—
405	#125	—	2	3	no	—
406	#125	—	2	3	no	—
407	#125	—	2	3	no	—
408	#124	—	2	6	no	—
409	#124	—	2	3	no	—
410	C₂²⋊Q₈	—	2	3	no	—
411	C₂²⋊Q₈	—	2	6	no	—
412	C₂²⋊Q₈	—	2	6	no	—
413	C₂²⋊Q₈	C₂³×C₂₃	1	6	yes	—
414	#121	—	2	3	no	—
415	#121	—	2	3	no	—
416	#121	—	2	3	no	—
417	#121	—	2	6	no	—
418	#121	C₂³×C₂₃	1	6	yes	—
419	C₄×Q₈	—	2	1	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×Q₈	—	2	3	no	—
423	C₄×Q₈	—	2	3	no	—
424	C₄×Q₈	—	2	3	no	—
425	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
426	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
428	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
429	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
430	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
432	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₄×D₄	—	2	3	no	—
434	C₄×D₄	—	2	3	no	—
435	C₄×D₄	—	2	3	no	—
436	C₄×D₄	—	2	3	no	—
437	C₄×D₄	—	2	3	no	—
438	C₄×D₄	—	2	3	no	—
439	C₄×D₄	D₄×C₂₃	1	3	yes	—
440	C₄×D₄	C₂³×C₂₃	1	3	yes	—
441	C₄×D₄	C₂³×C₂₃	1	3	yes	—
442	C₄×D₄	C₂³×C₂₃	1	3	yes	—
443	#117	—	2	3	no	—
444	#117	—	2	3	no	—
445	#117	—	2	3	no	—
446	#117	D₄×C₂₃	1	3	yes	—
447	#117	C₂³×C₂₃	1	3	yes	—
448	C₂×C₄⋊C₄	—	2	3	no	—
449	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
450	C₂×C₄²	—	2	3	no	—
451	C₂×C₄²	—	2	3	no	—
452	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
453	#107	C₂³×C₂₃	1	3	yes	—
454	#107	C₂³×C₂₃	1	3	yes	—
455	#107	C₂³×C₂₃	1	6	yes	—
456	#106	C₂³×C₂₃	1	1	yes	characteristic
457	#106	C₂³×C₂₃	1	3	yes	—
458	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
459	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
460	#103	C₂³×C₂₃	1	1	yes	characteristic
461	#103	C₂³×C₂₃	1	3	yes	—
462	#103	C₂³×C₂₃	1	1	yes	characteristic
463	#103	C₂³×C₂₃	1	3	yes	—
464	#103	C₂³×C₂₃	1	3	yes	—
465	#103	C₂³×C₂₃	1	3	yes	—
466	#103	C₂³×C₂₃	1	6	yes	—
467	#102	C₂³×C₂₃	1	3	yes	—
468	#102	C₂³×C₂₃	1	3	yes	—
469	#102	C₂³×C₂₃	1	6	yes	—
470	C₂²⋊C₈	C₂³×C₂₃	1	6	yes	—
471	#97	C₂³×C₂₃	1	1	yes	characteristic
472	#97	C₂³×C₂₃	1	3	yes	—
473	C₄×C₈	C₂³×C₂₃	1	3	yes	—
474	C₄×C₈	C₂³×C₂₃	1	3	yes	—
475	C₂₃	#61469	1	1	yes	characteristic, direct product factor, Sylow
476	#41	—	2	4	no	—
477	#41	—	2	6	no	—
478	#41	—	4	3	no	—
479	#41	—	4	4	no	—
480	#41	—	8	6	no	—
481	C₂×Q₈	—	2	3	no	—
482	C₂×Q₈	—	2	3	no	—
483	C₂×Q₈	—	2	3	no	—
484	C₂×Q₈	—	2	6	no	—
485	C₂×Q₈	—	4	3	no	—
486	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
488	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
489	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
490	C₂×D₄	—	2	3	no	—
491	C₂×D₄	—	2	6	no	—
492	C₂×D₄	—	2	3	no	—
493	C₂×D₄	—	2	3	no	—
494	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
495	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
496	C₂²×C₄	—	2	3	no	—
497	C₂²×C₄	—	2	3	no	—
498	C₂²×C₄	—	2	3	no	—
499	C₂²×C₄	—	2	3	no	—
500	C₂²×C₄	—	2	3	no	—
501	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
502	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
503	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
504	Q₁₆	—	2	6	no	—
505	Q₁₆	#41×C₂₃	1	4	yes	complemented
506	SD₁₆	—	2	6	no	—
507	SD₁₆	—	2	6	no	—
508	SD₁₆	—	4	6	no	—
509	M₁₆	—	2	6	no	—
510	M₁₆	—	4	3	no	—
511	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
512	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
513	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
514	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	3	no	—
517	C₄⋊C₄	—	2	3	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	3	no	—
522	C₄⋊C₄	—	2	3	no	—
523	C₄⋊C₄	—	2	3	no	—
524	C₄⋊C₄	—	2	3	no	—
525	C₄⋊C₄	—	2	3	no	—
526	C₄⋊C₄	—	2	6	no	—
527	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
528	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
529	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
530	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
531	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
532	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
533	C₂²⋊C₄	—	2	6	no	—
534	C₂²⋊C₄	—	2	3	no	—
535	C₂²⋊C₄	—	2	6	no	—
536	C₂²⋊C₄	—	2	3	no	—
537	C₂²⋊C₄	—	2	3	no	—
538	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	3	no	—
542	C₄²	—	2	3	no	—
543	C₄²	—	2	3	no	—
544	C₄²	—	2	3	no	—
545	C₄²	—	2	3	no	—
546	C₄²	C₂×D₄×C₂₃	1	3	yes	—
547	C₄²	C₂×D₄×C₂₃	1	3	yes	—
548	C₄²	C₂⁴×C₂₃	1	3	yes	—
549	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
550	C₂³	—	2	3	no	—
551	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
552	Q₈	—	2	1	no	—
553	Q₈	—	2	1	no	—
554	Q₈	—	2	3	no	—
555	Q₈	—	2	3	no	—
556	Q₈	—	4	6	no	—
557	Q₈	—	8	3	no	—
558	Q₈	—	8	3	no	—
559	Q₈	C₂₃×#137	1	4	yes	complemented
560	Q₈	C₂×#41×C₂₃	1	2	yes	—
561	Q₈	C₂×#41×C₂₃	1	2	yes	—
562	D₄	—	2	6	no	—
563	D₄	—	2	3	no	—
564	D₄	—	2	3	no	—
565	D₄	—	4	6	no	—
566	D₄	—	4	6	no	—
567	D₄	—	8	3	no	—
568	D₄	—	8	6	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	6	no	—
572	C₂×C₄	—	2	3	no	—
573	C₂×C₄	—	2	3	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	3	no	—
577	C₂×C₄	—	2	3	no	—
578	C₂×C₄	—	2	3	no	—
579	C₂×C₄	—	4	3	no	—
580	C₂×C₄	—	4	6	no	—
581	C₂×C₄	—	4	6	no	—
582	C₂×C₄	—	4	6	no	—
583	C₂×C₄	—	4	6	no	—
584	C₂×C₄	—	8	3	no	—
585	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
586	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
588	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
589	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
590	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
591	C₈	—	2	3	no	—
592	C₈	—	2	1	no	—
593	C₈	—	2	3	no	—
594	C₈	C₂×#41×C₂₃	1	2	yes	—
595	C₂²	—	2	3	no	—
596	C₂²	—	2	3	no	—
597	C₂²	—	2	3	no	—
598	C₂²	—	4	3	no	—
599	C₂²	—	4	3	no	—
600	C₂²	—	4	3	no	—
601	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
602	C₄	—	2	3	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	3	no	—
608	C₄	—	4	3	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	3	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	3	no	—
613	C₄	C₂×C₂₃×#137	1	6	yes	—
614	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
615	C₄	C₂₃×#585	1	1	yes	characteristic
616	C₂	—	4	3	no	—
617	C₂	—	8	3	no	—
618	C₂	D₄×#41×C₂₃	1	1	yes	characteristic
619	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
620	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3483	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×#41×C₂₃	C₂	semidirect
3. C₂₃×#3392	C₂	semidirect
4. Q₈×Q₁₆×C₂₃	C₂	semidirect
5. C₂₃×#3363	C₂	semidirect
6. C₂₃×#3362	C₂	semidirect
7. C₂₃×#3357	C₂	semidirect
8. C₂₃×#3333	C₂	semidirect
9. C₂₃×#3329	C₂	semidirect
10. C₂₃×#3278	C₂	semidirect
11. C₂₃×#3182	C₂	semidirect
12. C₂₃×#3116	C₂	semidirect
13. C₂₃×#3112	C₂	semidirect
14. C₂₃×#2971	C₂	semidirect
15. C₄×C₂₃×#137	C₂	semidirect
194. Q₁₆×C₂₃	#41	semidirect
239. #61469	C₂₃	direct (N × H)
249. Q₈×C₂₃	#137	semidirect
286. Q₈×#41	C₂×C₂₃	semidirect
287. #3392	C₂×C₂₃	semidirect
288. Q₈×Q₁₆	C₂×C₂₃	semidirect
289. #3363	C₂×C₂₃	semidirect
290. #3362	C₂×C₂₃	semidirect
291. #3357	C₂×C₂₃	semidirect
292. #3333	C₂×C₂₃	semidirect
293. #3329	C₂×C₂₃	semidirect
294. #3278	C₂×C₂₃	semidirect
295. #3182	C₂×C₂₃	semidirect
296. #3116	C₂×C₂₃	semidirect
297. #3112	C₂×C₂₃	semidirect
298. #2971	C₂×C₂₃	semidirect
299. C₄×#137	C₂×C₂₃	semidirect
475. C₂₃	#61469	direct (N × H)
505. Q₁₆	#41×C₂₃	semidirect
559. Q₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3483	2
Q₈×#41×C₂₃	2
C₂₃×#3392	2
Q₈×Q₁₆×C₂₃	2
C₂₃×#3363	2
C₂₃×#3362	2
C₂₃×#3357	2
C₂₃×#3333	2
C₂₃×#3329	2
C₂₃×#3278	2
C₂₃×#3182	2
C₂₃×#3116	2
C₂₃×#3112	2
C₂₃×#2971	2
C₄×C₂₃×#137	2
#61469	23

Subgroups — C23×#61469 #19209095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61469 #19209095