Subgroups — C₂₃×#18797 #19166423

Order 5888.

924 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3210	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3112	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3106	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1636	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1633	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1632	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1554	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
16	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#557	—	2	1	no	—
18	C₂₃×#557	C₂²	1	1	yes	characteristic
19	C₂₃×#554	—	2	2	no	—
20	C₂₃×#554	—	2	1	no	—
21	C₂₃×#554	—	2	1	no	—
22	C₂₃×#554	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₂₃	—	2	1	no	—
24	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#533	—	2	2	no	—
26	C₂₃×#533	—	2	1	no	—
27	C₂₃×#533	—	2	1	no	—
28	C₂₃×#533	C₂²	1	1	yes	characteristic
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₄×#41×C₂₃	—	2	1	no	—
31	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
33	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#494	C₂²	1	1	yes	characteristic
35	C₂₃×#492	C₂²	1	1	yes	characteristic
36	C₂₃×#487	C₂²	1	1	yes	characteristic
37	C₂₃×#486	C₂²	1	1	yes	characteristic
38	C₂₃×#482	—	2	1	no	—
39	C₂₃×#480	—	2	1	no	—
40	C₂₃×#477	C₂²	1	1	yes	characteristic
41	C₂₃×#477	C₂²	1	1	yes	characteristic
42	C₂₃×#477	C₂²	1	1	yes	characteristic
43	C₂₃×#476	C₂²	1	1	yes	characteristic
44	C₂₃×#464	—	2	2	no	—
45	C₂₃×#460	C₂²	1	1	yes	characteristic
46	C₂₃×#460	C₂²	1	1	yes	characteristic
47	C₂₃×#460	C₂²	1	1	yes	characteristic
48	C₂₃×#459	C₂²	1	1	yes	characteristic
49	C₂₃×#452	—	2	2	no	—
50	C₂₃×#449	—	2	1	no	—
51	C₂₃×#448	—	2	1	no	—
52	C₂₃×#440	—	2	1	no	—
53	C₂₃×#440	—	2	1	no	—
54	C₄×SD₁₆×C₂₃	—	2	1	no	—
55	C₄×D₈×C₂₃	—	2	1	no	—
56	C₂₃×#432	C₂²	1	1	yes	characteristic
57	C₂₃×#431	C₂²	1	1	yes	characteristic
58	C₂₃×#431	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
73	C₂²×Q₈×C₂₃	—	2	1	no	—
74	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
75	C₂×Q₁₆×C₂₃	—	2	2	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	2	2	no	—
80	C₂×D₈×C₂₃	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	—	2	2	no	—
82	C₂₃×C₄⋊Q₈	—	2	1	no	—
83	C₂₃×C₄⋊Q₈	—	2	2	no	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	—	2	1	no	—
87	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×#127	C₂³	1	1	yes	characteristic
91	C₂₃×#125	—	2	2	no	—
92	C₂₃×#125	—	2	2	no	—
93	C₂₃×#125	—	2	2	no	—
94	C₂₃×#125	—	2	1	no	—
95	C₂₃×#125	—	2	1	no	—
96	C₂₃×#125	—	2	1	no	—
97	C₂₃×#124	—	2	1	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	—	2	2	no	—
100	C₂₃×C₂²⋊Q₈	—	2	2	no	—
101	C₂₃×C₂²⋊Q₈	—	2	2	no	—
102	C₂₃×C₂²⋊Q₈	—	2	2	no	—
103	C₂₃×C₂²⋊Q₈	—	2	1	no	—
104	C₂₃×C₂²⋊Q₈	—	2	2	no	—
105	C₂₃×C₂²⋊Q₈	—	2	1	no	—
106	C₂₃×C₂²⋊Q₈	—	2	1	no	—
107	C₂₃×C₂²⋊Q₈	—	2	2	no	—
108	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
109	C₂₃×#121	—	2	1	no	—
110	C₂₃×#121	—	2	1	no	—
111	C₄×Q₈×C₂₃	—	2	2	no	—
112	C₄×Q₈×C₂₃	—	2	2	no	—
113	C₄×Q₈×C₂₃	—	2	2	no	—
114	C₄×Q₈×C₂₃	—	2	1	no	—
115	C₄×Q₈×C₂₃	—	2	1	no	—
116	C₄×Q₈×C₂₃	—	2	1	no	—
117	C₄×Q₈×C₂₃	—	2	1	no	—
118	C₄×Q₈×C₂₃	—	2	1	no	—
119	C₄×Q₈×C₂₃	—	2	1	no	—
120	C₄×Q₈×C₂₃	—	2	1	no	—
121	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
122	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
124	C₄×Q₈×C₂₃	D₄	1	2	yes	—
125	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
127	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
128	C₄×D₄×C₂₃	—	2	2	no	—
129	C₄×D₄×C₂₃	—	2	2	no	—
130	C₄×D₄×C₂₃	—	2	1	no	—
131	C₄×D₄×C₂₃	—	2	1	no	—
132	C₄×D₄×C₂₃	—	2	1	no	—
133	C₄×D₄×C₂₃	—	2	1	no	—
134	C₄×D₄×C₂₃	—	2	1	no	—
135	C₄×D₄×C₂₃	—	2	1	no	—
136	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
137	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
138	C₂₃×#117	—	2	2	no	—
139	C₂₃×#117	—	2	1	no	—
140	C₂₃×#117	—	2	1	no	—
141	C₂₃×#117	D₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
144	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
146	C₂×C₄²×C₂₃	—	2	2	no	—
147	C₂×C₄²×C₂₃	—	2	1	no	—
148	C₂×C₄²×C₂₃	—	2	1	no	—
149	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
150	C₂₃×#107	—	2	1	no	—
151	C₂₃×#107	—	2	1	no	—
152	C₂₃×#107	—	2	1	no	—
153	C₂₃×#106	—	2	1	no	—
154	C₂₃×C₄⋊C₈	—	2	1	no	—
155	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
156	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₂₃×#103	—	2	1	no	—
160	C₂₃×#103	—	2	1	no	—
161	C₂₃×#103	—	2	1	no	—
162	C₂₃×#103	—	2	2	no	—
163	C₂₃×#102	—	2	1	no	—
164	C₂₃×#102	—	2	1	no	—
165	C₂₃×#102	—	2	1	no	—
166	C₂₃×#102	—	2	1	no	—
167	C₂₃×#102	—	2	1	no	—
168	C₂₃×#102	—	2	2	no	—
169	C₂₃×C₂²⋊C₈	—	2	1	no	—
170	C₂₃×C₂²⋊C₈	—	2	1	no	—
171	C₂₃×C₂²⋊C₈	—	2	1	no	—
172	C₂₃×#97	—	2	1	no	—
173	C₄×C₈×C₂₃	—	2	1	no	—
174	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
176	#41×C₂₃	—	2	2	no	—
177	#41×C₂₃	—	2	2	no	—
178	#41×C₂₃	—	8	2	no	—
179	C₂×Q₈×C₂₃	—	2	2	no	—
180	C₂×Q₈×C₂₃	—	2	2	no	—
181	C₂×Q₈×C₂₃	—	2	1	no	—
182	C₂×Q₈×C₂₃	—	2	1	no	—
183	C₂×Q₈×C₂₃	—	2	2	no	—
184	C₂×Q₈×C₂₃	—	2	1	no	—
185	C₂×Q₈×C₂₃	—	4	1	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	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
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	—	4	1	no	—
197	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₂₃	—	2	1	no	—
199	C₂²×C₄×C₂₃	—	2	2	no	—
200	C₂²×C₄×C₂₃	—	2	1	no	—
201	C₂²×C₄×C₂₃	—	2	2	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	—	4	1	no	—
206	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
209	Q₁₆×C₂₃	—	8	2	no	—
210	SD₁₆×C₂₃	—	4	2	no	—
211	SD₁₆×C₂₃	—	4	2	no	—
212	SD₁₆×C₂₃	—	4	2	no	—
213	SD₁₆×C₂₃	—	8	2	no	—
214	D₈×C₂₃	—	4	2	no	—
215	C₂×C₈×C₂₃	—	2	1	no	—
216	C₂×C₈×C₂₃	—	2	1	no	—
217	C₂×C₈×C₂₃	—	2	1	no	—
218	C₂×C₈×C₂₃	—	2	1	no	—
219	C₂×C₈×C₂₃	—	2	1	no	—
220	C₂×C₈×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	2	2	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	—	2	2	no	—
225	C₄⋊C₄×C₂₃	—	2	2	no	—
226	C₄⋊C₄×C₂₃	—	2	2	no	—
227	C₄⋊C₄×C₂₃	—	2	2	no	—
228	C₄⋊C₄×C₂₃	—	2	2	no	—
229	C₄⋊C₄×C₂₃	—	2	1	no	—
230	C₄⋊C₄×C₂₃	—	2	2	no	—
231	C₄⋊C₄×C₂₃	—	2	1	no	—
232	C₄⋊C₄×C₂₃	—	2	1	no	—
233	C₄⋊C₄×C₂₃	—	2	1	no	—
234	C₄⋊C₄×C₂₃	—	2	2	no	—
235	C₄⋊C₄×C₂₃	—	2	1	no	—
236	C₄⋊C₄×C₂₃	—	2	1	no	—
237	C₄⋊C₄×C₂₃	—	2	1	no	—
238	C₄⋊C₄×C₂₃	—	2	1	no	—
239	C₄⋊C₄×C₂₃	—	2	1	no	—
240	C₄⋊C₄×C₂₃	—	2	1	no	—
241	C₄⋊C₄×C₂₃	—	2	1	no	—
242	C₄⋊C₄×C₂₃	—	2	1	no	—
243	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
245	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
246	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
247	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
248	C₂²⋊C₄×C₂₃	—	2	2	no	—
249	C₂²⋊C₄×C₂₃	—	2	2	no	—
250	C₂²⋊C₄×C₂₃	—	2	2	no	—
251	C₂²⋊C₄×C₂₃	—	2	1	no	—
252	C₂²⋊C₄×C₂₃	—	2	1	no	—
253	C₂²⋊C₄×C₂₃	—	2	1	no	—
254	C₂²⋊C₄×C₂₃	—	4	1	no	—
255	C₂²⋊C₄×C₂₃	—	4	1	no	—
256	C₄²×C₂₃	—	2	2	no	—
257	C₄²×C₂₃	—	2	2	no	—
258	C₄²×C₂₃	—	2	1	no	—
259	C₄²×C₂₃	—	2	2	no	—
260	C₄²×C₂₃	—	2	1	no	—
261	C₄²×C₂₃	—	2	1	no	—
262	C₄²×C₂₃	—	2	1	no	—
263	C₄²×C₂₃	—	2	1	no	—
264	C₄²×C₂₃	—	2	1	no	—
265	C₄²×C₂₃	—	2	1	no	—
266	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
268	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
269	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
270	#18797	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂³×C₂₃	—	2	1	no	—
272	C₂³×C₂₃	—	4	1	no	—
273	C₂³×C₂₃	#120	1	1	yes	characteristic
274	Q₈×C₂₃	—	2	1	no	—
275	Q₈×C₂₃	—	2	1	no	—
276	Q₈×C₂₃	—	2	1	no	—
277	Q₈×C₂₃	—	2	1	no	—
278	Q₈×C₂₃	—	4	2	no	—
279	Q₈×C₂₃	—	4	2	no	—
280	Q₈×C₂₃	—	4	4	no	—
281	Q₈×C₂₃	—	4	2	no	—
282	Q₈×C₂₃	—	4	1	no	—
283	Q₈×C₂₃	—	4	1	no	—
284	Q₈×C₂₃	—	8	2	no	—
285	Q₈×C₂₃	#135	1	2	yes	—
286	Q₈×C₂₃	#137	1	2	yes	—
287	D₄×C₂₃	—	2	4	no	—
288	D₄×C₂₃	—	2	1	no	—
289	D₄×C₂₃	—	4	2	no	—
290	D₄×C₂₃	—	4	1	no	—
291	D₄×C₂₃	—	4	1	no	—
292	D₄×C₂₃	—	4	1	no	—
293	D₄×C₂₃	—	4	1	no	—
294	D₄×C₂₃	—	8	2	no	—
295	D₄×C₂₃	#137	1	2	yes	—
296	C₂×C₄×C₂₃	—	2	4	no	—
297	C₂×C₄×C₂₃	—	2	1	no	—
298	C₂×C₄×C₂₃	—	2	1	no	—
299	C₂×C₄×C₂₃	—	2	1	no	—
300	C₂×C₄×C₂₃	—	2	1	no	—
301	C₂×C₄×C₂₃	—	2	1	no	—
302	C₂×C₄×C₂₃	—	2	1	no	—
303	C₂×C₄×C₂₃	—	2	1	no	—
304	C₂×C₄×C₂₃	—	2	1	no	—
305	C₂×C₄×C₂₃	—	2	2	no	—
306	C₂×C₄×C₂₃	—	2	2	no	—
307	C₂×C₄×C₂₃	—	2	1	no	—
308	C₂×C₄×C₂₃	—	2	1	no	—
309	C₂×C₄×C₂₃	—	2	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	2	no	—
312	C₂×C₄×C₂₃	—	4	4	no	—
313	C₂×C₄×C₂₃	—	4	1	no	—
314	C₂×C₄×C₂₃	—	4	1	no	—
315	C₂×C₄×C₂₃	—	4	1	no	—
316	C₂×C₄×C₂₃	—	8	1	no	—
317	C₂×C₄×C₂₃	—	8	2	no	—
318	C₂×C₄×C₂₃	—	8	1	no	—
319	C₂×C₄×C₂₃	—	8	2	no	—
320	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
321	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
322	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
323	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
326	C₈×C₂₃	—	4	1	no	—
327	C₈×C₂₃	—	4	1	no	—
328	C₈×C₂₃	—	4	1	no	—
329	C₈×C₂₃	—	4	1	no	—
330	C₈×C₂₃	—	4	1	no	—
331	C₈×C₂₃	—	4	1	no	—
332	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#3210	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#3112	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#3106	C₂×C₂₃	1	1	yes	characteristic
336	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1636	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1633	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
340	#1627	C₂×C₂₃	1	1	yes	characteristic, complemented
341	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
342	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
343	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
344	#1554	C₂×C₂₃	1	1	yes	characteristic, complemented
345	#1488	C₂×C₂₃	1	1	yes	characteristic, complemented
346	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
347	C₂²×C₂₃	—	2	1	no	—
348	C₂²×C₂₃	—	2	1	no	—
349	C₂²×C₂₃	—	2	2	no	—
350	C₂²×C₂₃	—	2	1	no	—
351	C₂²×C₂₃	—	4	1	no	—
352	C₂²×C₂₃	—	4	1	no	—
353	C₂²×C₂₃	—	8	1	no	—
354	C₂²×C₂₃	—	8	1	no	—
355	C₂²×C₂₃	—	8	1	no	—
356	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
357	C₄×C₂₃	—	2	1	no	—
358	C₄×C₂₃	—	2	2	no	—
359	C₄×C₂₃	—	2	1	no	—
360	C₄×C₂₃	—	2	2	no	—
361	C₄×C₂₃	—	2	1	no	—
362	C₄×C₂₃	—	2	1	no	—
363	C₄×C₂₃	—	4	1	no	—
364	C₄×C₂₃	—	4	2	no	—
365	C₄×C₂₃	—	4	1	no	—
366	C₄×C₂₃	—	4	1	no	—
367	C₄×C₂₃	—	4	2	no	—
368	C₄×C₂₃	—	4	1	no	—
369	C₄×C₂₃	—	4	1	no	—
370	C₄×C₂₃	—	8	1	no	—
371	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
372	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
373	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
374	C₄×C₂₃	#575	1	1	yes	characteristic
375	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
376	#557	—	2	1	no	—
377	#557	C₂²×C₂₃	1	1	yes	characteristic
378	#554	—	2	2	no	—
379	#554	—	2	1	no	—
380	#554	—	2	1	no	—
381	#554	C₂²×C₂₃	1	1	yes	characteristic
382	D₄×Q₈	—	2	1	no	—
383	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
384	#533	—	2	2	no	—
385	#533	—	2	1	no	—
386	#533	—	2	1	no	—
387	#533	C₂²×C₂₃	1	1	yes	characteristic
388	#532	C₂²×C₂₃	1	1	yes	characteristic
389	C₄×#41	—	2	1	no	—
390	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₄×Q₈	—	2	1	no	—
392	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
393	#494	C₂²×C₂₃	1	1	yes	characteristic
394	#492	C₂²×C₂₃	1	1	yes	characteristic
395	#487	C₂²×C₂₃	1	1	yes	characteristic
396	#486	C₂²×C₂₃	1	1	yes	characteristic
397	#482	—	2	1	no	—
398	#480	—	2	1	no	—
399	#477	C₂²×C₂₃	1	1	yes	characteristic
400	#477	C₂²×C₂₃	1	1	yes	characteristic
401	#477	C₂²×C₂₃	1	1	yes	characteristic
402	#476	C₂²×C₂₃	1	1	yes	characteristic
403	#464	—	2	2	no	—
404	#460	C₂²×C₂₃	1	1	yes	characteristic
405	#460	C₂²×C₂₃	1	1	yes	characteristic
406	#460	C₂²×C₂₃	1	1	yes	characteristic
407	#459	C₂²×C₂₃	1	1	yes	characteristic
408	#452	—	2	2	no	—
409	#449	—	2	1	no	—
410	#448	—	2	1	no	—
411	#440	—	2	1	no	—
412	#440	—	2	1	no	—
413	C₄×SD₁₆	—	2	1	no	—
414	C₄×D₈	—	2	1	no	—
415	#432	C₂²×C₂₃	1	1	yes	characteristic
416	#431	C₂²×C₂₃	1	1	yes	characteristic
417	#431	C₂²×C₂₃	1	1	yes	characteristic
418	#333	C₂²×C₂₃	1	1	yes	characteristic
419	#333	C₂²×C₂₃	1	1	yes	characteristic
420	#332	C₂²×C₂₃	1	1	yes	characteristic
421	#332	C₂²×C₂₃	1	1	yes	characteristic
422	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
423	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
424	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
425	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
426	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
427	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
428	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
429	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
430	C₂×C₂₃	—	4	1	no	—
431	C₂×C₂₃	—	4	2	no	—
432	C₂×C₂₃	—	16	1	no	—
433	C₂×C₂₃	#3002	1	1	yes	characteristic
434	C₂×C₂₃	#3004	1	1	yes	characteristic
435	C₂×C₂₃	#3021	1	1	yes	characteristic
436	C₂×#41	—	2	1	no	—
437	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
438	C₂²×Q₈	—	2	1	no	—
439	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
440	C₂×Q₁₆	—	2	2	no	—
441	C₂×SD₁₆	—	2	1	no	—
442	C₂×SD₁₆	—	2	1	no	—
443	C₂×SD₁₆	—	2	1	no	—
444	C₂×SD₁₆	—	2	2	no	—
445	C₂×D₈	—	2	1	no	—
446	C₄⋊Q₈	—	2	2	no	—
447	C₄⋊Q₈	—	2	1	no	—
448	C₄⋊Q₈	—	2	2	no	—
449	C₄⋊Q₈	—	2	1	no	—
450	C₄⋊Q₈	—	2	1	no	—
451	C₄⋊Q₈	—	2	1	no	—
452	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
453	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
454	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
455	#127	C₂³×C₂₃	1	1	yes	characteristic
456	#125	—	2	2	no	—
457	#125	—	2	2	no	—
458	#125	—	2	2	no	—
459	#125	—	2	1	no	—
460	#125	—	2	1	no	—
461	#125	—	2	1	no	—
462	#124	—	2	1	no	—
463	C₂²⋊Q₈	—	2	2	no	—
464	C₂²⋊Q₈	—	2	2	no	—
465	C₂²⋊Q₈	—	2	2	no	—
466	C₂²⋊Q₈	—	2	2	no	—
467	C₂²⋊Q₈	—	2	2	no	—
468	C₂²⋊Q₈	—	2	1	no	—
469	C₂²⋊Q₈	—	2	2	no	—
470	C₂²⋊Q₈	—	2	1	no	—
471	C₂²⋊Q₈	—	2	1	no	—
472	C₂²⋊Q₈	—	2	2	no	—
473	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
474	#121	—	2	1	no	—
475	#121	—	2	1	no	—
476	C₄×Q₈	—	2	2	no	—
477	C₄×Q₈	—	2	2	no	—
478	C₄×Q₈	—	2	2	no	—
479	C₄×Q₈	—	2	1	no	—
480	C₄×Q₈	—	2	1	no	—
481	C₄×Q₈	—	2	1	no	—
482	C₄×Q₈	—	2	1	no	—
483	C₄×Q₈	—	2	1	no	—
484	C₄×Q₈	—	2	1	no	—
485	C₄×Q₈	—	2	1	no	—
486	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
487	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
488	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
489	C₄×Q₈	D₄×C₂₃	1	2	yes	—
490	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
491	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
492	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
493	C₄×D₄	—	2	2	no	—
494	C₄×D₄	—	2	2	no	—
495	C₄×D₄	—	2	1	no	—
496	C₄×D₄	—	2	1	no	—
497	C₄×D₄	—	2	1	no	—
498	C₄×D₄	—	2	1	no	—
499	C₄×D₄	—	2	1	no	—
500	C₄×D₄	—	2	1	no	—
501	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
502	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
503	#117	—	2	2	no	—
504	#117	—	2	1	no	—
505	#117	—	2	1	no	—
506	#117	D₄×C₂₃	1	1	yes	characteristic
507	C₂×C₄⋊C₄	—	2	2	no	—
508	C₂×C₄⋊C₄	—	2	1	no	—
509	C₂×C₄⋊C₄	—	2	1	no	—
510	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
511	C₂×C₄²	—	2	2	no	—
512	C₂×C₄²	—	2	1	no	—
513	C₂×C₄²	—	2	1	no	—
514	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
515	#107	—	2	1	no	—
516	#107	—	2	1	no	—
517	#107	—	2	1	no	—
518	#106	—	2	1	no	—
519	C₄⋊C₈	—	2	1	no	—
520	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
521	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
522	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
523	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
524	#103	—	2	1	no	—
525	#103	—	2	1	no	—
526	#103	—	2	1	no	—
527	#103	—	2	2	no	—
528	#102	—	2	1	no	—
529	#102	—	2	1	no	—
530	#102	—	2	1	no	—
531	#102	—	2	1	no	—
532	#102	—	2	1	no	—
533	#102	—	2	2	no	—
534	C₂²⋊C₈	—	2	1	no	—
535	C₂²⋊C₈	—	2	1	no	—
536	C₂²⋊C₈	—	2	1	no	—
537	#97	—	2	1	no	—
538	C₄×C₈	—	2	1	no	—
539	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
540	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
541	C₂₃	#18797	1	1	yes	characteristic, direct product factor, Sylow
542	#41	—	2	2	no	—
543	#41	—	2	2	no	—
544	#41	—	8	2	no	—
545	C₂×Q₈	—	2	2	no	—
546	C₂×Q₈	—	2	2	no	—
547	C₂×Q₈	—	2	1	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	2	2	no	—
550	C₂×Q₈	—	2	1	no	—
551	C₂×Q₈	—	4	1	no	—
552	C₂×Q₈	—	4	2	no	—
553	C₂×Q₈	—	4	2	no	—
554	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
555	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
558	C₂×D₄	—	2	1	no	—
559	C₂×D₄	—	2	1	no	—
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	4	1	no	—
563	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	2	no	—
566	C₂²×C₄	—	2	1	no	—
567	C₂²×C₄	—	2	2	no	—
568	C₂²×C₄	—	2	1	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	4	1	no	—
572	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
573	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
574	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
575	Q₁₆	—	8	2	no	—
576	SD₁₆	—	4	2	no	—
577	SD₁₆	—	4	2	no	—
578	SD₁₆	—	4	2	no	—
579	SD₁₆	—	8	2	no	—
580	D₈	—	4	2	no	—
581	C₂×C₈	—	2	1	no	—
582	C₂×C₈	—	2	1	no	—
583	C₂×C₈	—	2	1	no	—
584	C₂×C₈	—	2	1	no	—
585	C₂×C₈	—	2	1	no	—
586	C₂×C₈	—	2	1	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	2	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	1	no	—
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	1	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	—	2	1	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
610	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
611	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
612	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
613	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
614	C₂²⋊C₄	—	2	2	no	—
615	C₂²⋊C₄	—	2	2	no	—
616	C₂²⋊C₄	—	2	2	no	—
617	C₂²⋊C₄	—	2	1	no	—
618	C₂²⋊C₄	—	2	1	no	—
619	C₂²⋊C₄	—	2	1	no	—
620	C₂²⋊C₄	—	4	1	no	—
621	C₂²⋊C₄	—	4	1	no	—
622	C₄²	—	2	2	no	—
623	C₄²	—	2	2	no	—
624	C₄²	—	2	1	no	—
625	C₄²	—	2	2	no	—
626	C₄²	—	2	1	no	—
627	C₄²	—	2	1	no	—
628	C₄²	—	2	1	no	—
629	C₄²	—	2	1	no	—
630	C₄²	—	2	1	no	—
631	C₄²	—	2	1	no	—
632	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
633	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
634	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
635	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
636	C₂³	—	2	1	no	—
637	C₂³	—	4	1	no	—
638	C₂³	C₂₃×#120	1	1	yes	characteristic
639	Q₈	—	2	1	no	—
640	Q₈	—	2	1	no	—
641	Q₈	—	2	1	no	—
642	Q₈	—	2	1	no	—
643	Q₈	—	4	2	no	—
644	Q₈	—	4	2	no	—
645	Q₈	—	4	4	no	—
646	Q₈	—	4	2	no	—
647	Q₈	—	4	1	no	—
648	Q₈	—	4	1	no	—
649	Q₈	—	8	2	no	—
650	Q₈	C₂₃×#135	1	2	yes	—
651	Q₈	C₂₃×#137	1	2	yes	—
652	D₄	—	2	4	no	—
653	D₄	—	2	1	no	—
654	D₄	—	4	2	no	—
655	D₄	—	4	1	no	—
656	D₄	—	4	1	no	—
657	D₄	—	4	1	no	—
658	D₄	—	4	1	no	—
659	D₄	—	8	2	no	—
660	D₄	C₂₃×#137	1	2	yes	—
661	C₂×C₄	—	2	4	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	2	no	—
677	C₂×C₄	—	4	4	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	8	1	no	—
682	C₂×C₄	—	8	2	no	—
683	C₂×C₄	—	8	1	no	—
684	C₂×C₄	—	8	2	no	—
685	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
686	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
687	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
688	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
689	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
690	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
691	C₈	—	4	1	no	—
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₈	—	4	1	no	—
697	C₂²	—	2	1	no	—
698	C₂²	—	2	1	no	—
699	C₂²	—	2	2	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	8	1	no	—
704	C₂²	—	8	1	no	—
705	C₂²	—	8	1	no	—
706	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
707	C₄	—	2	1	no	—
708	C₄	—	2	2	no	—
709	C₄	—	2	1	no	—
710	C₄	—	2	2	no	—
711	C₄	—	2	1	no	—
712	C₄	—	2	1	no	—
713	C₄	—	4	1	no	—
714	C₄	—	4	2	no	—
715	C₄	—	4	1	no	—
716	C₄	—	4	1	no	—
717	C₄	—	4	2	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	8	1	no	—
721	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
722	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
723	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
724	C₄	C₂₃×#575	1	1	yes	characteristic
725	C₂	—	4	1	no	—
726	C₂	—	4	2	no	—
727	C₂	—	16	1	no	—
728	C₂	C₂₃×#3002	1	1	yes	characteristic
729	C₂	C₂₃×#3004	1	1	yes	characteristic
730	C₂	C₂₃×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3106	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₂₃	C₂	semidirect
2. C₂₃×#3210	C₂	semidirect
3. C₂₃×#3112	C₂	semidirect
5. C₂₃×#1637	C₂	semidirect
6. C₂₃×#1636	C₂	semidirect
7. C₂₃×#1633	C₂	semidirect
8. C₂₃×#1632	C₂	semidirect
9. C₂₃×#1627	C₂	semidirect
10. C₂₃×#1626	C₂	semidirect
11. C₂₃×#1623	C₂	semidirect
12. C₂₃×#1622	C₂	semidirect
13. C₂₃×#1554	C₂	semidirect
14. C₂₃×#1488	C₂	semidirect
15. C₂₃×#1479	C₂	semidirect
270. #18797	C₂₃	direct (N × H)
332. Q₈×#41	C₂×C₂₃	semidirect
333. #3210	C₂×C₂₃	semidirect
334. #3112	C₂×C₂₃	semidirect
336. #1637	C₂×C₂₃	semidirect
337. #1636	C₂×C₂₃	semidirect
338. #1633	C₂×C₂₃	semidirect
339. #1632	C₂×C₂₃	semidirect
340. #1627	C₂×C₂₃	semidirect
341. #1626	C₂×C₂₃	semidirect
342. #1623	C₂×C₂₃	semidirect
343. #1622	C₂×C₂₃	semidirect
344. #1554	C₂×C₂₃	semidirect
345. #1488	C₂×C₂₃	semidirect
346. #1479	C₂×C₂₃	semidirect
541. C₂₃	#18797	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41×C₂₃	2
C₂₃×#3210	2
C₂₃×#3112	2
C₂₃×#3106	2
C₂₃×#1637	2
C₂₃×#1636	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1627	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1622	2
C₂₃×#1554	2
C₂₃×#1488	2
C₂₃×#1479	2
#18797	23

Subgroups — C23×#18797 #19166423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18797 #19166423