Subgroups — C₂₃×#18749 #19166375

Order 5888.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3077	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1644	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1632	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1478	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#554	—	2	2	no	—
11	C₂₃×#554	—	2	2	no	—
12	C₂₃×#554	—	2	2	no	—
13	C₂₃×#554	C₂²	1	4	yes	—
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#125	—	2	1	no	—
17	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	—	2	1	no	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₂₃×#494	C₂²	1	2	yes	complemented
24	C₂₃×#487	C₂²	1	2	yes	complemented
25	C₂₃×#484	—	2	1	no	—
26	C₂₃×#478	—	2	2	no	—
27	C₂₃×#477	C₂²	1	2	yes	complemented
28	C₂₃×#476	C₂²	1	2	yes	—
29	C₂₃×#474	C₂²	1	2	yes	—
30	C₂₃×#462	—	2	2	no	—
31	C₂₃×#462	C₂²	1	2	yes	complemented
32	C₂₃×#451	—	2	2	no	—
33	C₂₃×#451	—	2	1	no	—
34	C₄×Q₁₆×C₂₃	—	2	2	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#428	—	2	1	no	—
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#418	—	2	1	no	—
40	C₂₃×#333	C₂²	1	4	yes	complemented
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
45	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	2	no	—
47	C₂³×C₄×C₂₃	—	2	1	no	—
48	C₂×Q₁₆×C₂₃	—	2	2	no	—
49	C₂×Q₁₆×C₂₃	—	2	4	no	—
50	C₂×M₁₆×C₂₃	—	2	1	no	—
51	C₂₃×C₄⋊Q₈	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	—	2	1	no	—
53	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
54	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₂₃×#125	—	2	4	no	—
58	C₂₃×#125	—	2	4	no	—
59	C₂₃×#125	—	2	1	no	—
60	C₂₃×C₂²⋊Q₈	—	2	4	no	—
61	C₂₃×C₂²⋊Q₈	—	2	8	no	—
62	C₂₃×C₂²⋊Q₈	—	2	4	no	—
63	C₂₃×C₂²⋊Q₈	—	2	8	no	—
64	C₂₃×C₂²⋊Q₈	—	2	1	no	—
65	C₂₃×C₂²⋊Q₈	—	2	1	no	—
66	C₄×Q₈×C₂₃	—	2	1	no	—
67	C₄×Q₈×C₂₃	—	2	1	no	—
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	—	2	1	no	—
70	C₄×Q₈×C₂₃	D₄	1	2	yes	—
71	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
72	C₄×D₄×C₂₃	—	2	4	no	—
73	C₄×D₄×C₂₃	—	2	4	no	—
74	C₄×D₄×C₂₃	—	2	1	no	—
75	C₄×D₄×C₂₃	—	2	1	no	—
76	C₄×D₄×C₂₃	D₄	1	2	yes	—
77	C₄×D₄×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#117	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#107	—	2	2	no	—
93	C₂₃×#107	—	2	2	no	—
94	C₂₃×#106	—	2	2	no	—
95	C₂₃×C₄⋊C₈	—	2	1	no	—
96	C₂₃×C₄⋊C₈	—	2	1	no	—
97	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
98	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
99	C₂₃×#103	—	2	2	no	—
100	C₂₃×#103	—	2	4	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#102	—	2	2	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×C₂²⋊C₈	—	2	1	no	—
105	C₂₃×C₂²⋊C₈	—	2	1	no	—
106	C₄×C₈×C₂₃	—	2	1	no	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₂⁴×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	1	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	—	4	2	no	—
114	C₂×Q₈×C₂₃	—	8	2	no	—
115	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₂₃	—	2	1	no	—
118	C₂×D₄×C₂₃	—	2	2	no	—
119	C₂×D₄×C₂₃	Q₁₆	1	4	yes	complemented
120	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	8	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	4	4	no	—
129	C₂²×C₄×C₂₃	—	8	4	no	—
130	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	Q₁₆×C₂₃	—	4	4	no	—
134	Q₁₆×C₂₃	—	8	4	no	—
135	M₁₆×C₂₃	—	4	2	no	—
136	C₂×C₈×C₂₃	—	2	2	no	—
137	C₂×C₈×C₂₃	—	2	2	no	—
138	C₂×C₈×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	2	8	no	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	2	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	4	1	no	—
154	C₄⋊C₄×C₂₃	—	4	1	no	—
155	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	2	no	—
159	C₂²⋊C₄×C₂₃	—	2	2	no	—
160	C₂²⋊C₄×C₂₃	—	4	1	no	—
161	C₄²×C₂₃	—	2	2	no	—
162	C₄²×C₂₃	—	2	2	no	—
163	C₄²×C₂₃	—	4	1	no	—
164	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
168	#18749	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₂₃	—	2	2	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	4	2	no	—
172	C₂³×C₂₃	—	4	2	no	—
173	C₂³×C₂₃	#120	1	1	yes	characteristic
174	Q₈×C₂₃	—	2	2	no	—
175	Q₈×C₂₃	—	2	2	no	—
176	Q₈×C₂₃	—	4	4	no	—
177	Q₈×C₂₃	—	4	2	no	—
178	Q₈×C₂₃	—	4	2	no	—
179	Q₈×C₂₃	—	4	2	no	—
180	Q₈×C₂₃	—	8	2	no	—
181	D₄×C₂₃	—	2	2	no	—
182	D₄×C₂₃	—	2	4	no	—
183	D₄×C₂₃	C₂×Q₁₆	1	4	yes	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	4	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	2	no	—
193	C₂×C₄×C₂₃	—	4	4	no	—
194	C₂×C₄×C₂₃	—	4	2	no	—
195	C₂×C₄×C₂₃	—	4	4	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	8	8	no	—
199	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
203	C₂×C₄×C₂₃	#137	1	2	yes	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	C₈×C₂₃	—	4	2	no	—
208	C₈×C₂₃	—	4	2	no	—
209	C₈×C₂₃	—	4	2	no	—
210	C₂×#554	C₂×C₂₃	1	1	yes	characteristic
211	#3209	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#3104	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#3077	C₂×C₂₃	1	1	yes	characteristic
214	#1644	C₂×C₂₃	1	4	yes	complemented
215	#1632	C₂×C₂₃	1	4	yes	complemented
216	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#1478	C₂×C₂₃	1	1	yes	characteristic
219	C₂²×C₂₃	—	2	1	no	—
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	2	4	no	—
222	C₂²×C₂₃	—	4	8	no	—
223	C₂²×C₂₃	—	4	2	no	—
224	C₂²×C₂₃	—	4	2	no	—
225	C₂²×C₂₃	#451	1	2	yes	—
226	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	2	1	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	4	2	no	—
232	C₄×C₂₃	—	4	2	no	—
233	C₄×C₂₃	—	4	4	no	—
234	C₄×C₂₃	—	8	1	no	—
235	C₄×C₂₃	—	8	1	no	—
236	C₄×C₂₃	#451	1	2	yes	—
237	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
238	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
239	C₄×C₂₃	#575	1	2	yes	—
240	#554	—	2	2	no	—
241	#554	—	2	2	no	—
242	#554	—	2	2	no	—
243	#554	C₂²×C₂₃	1	4	yes	—
244	#533	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×#125	—	2	1	no	—
247	C₂×C₂²⋊Q₈	—	2	1	no	—
248	C₂×C₂²⋊Q₈	—	2	2	no	—
249	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×C₄×D₄	—	2	1	no	—
251	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
252	C₂²×C₄⋊C₄	—	2	1	no	—
253	#494	C₂²×C₂₃	1	2	yes	complemented
254	#487	C₂²×C₂₃	1	2	yes	complemented
255	#484	—	2	1	no	—
256	#478	—	2	2	no	—
257	#477	C₂²×C₂₃	1	2	yes	complemented
258	#476	C₂²×C₂₃	1	2	yes	—
259	#474	C₂²×C₂₃	1	2	yes	—
260	#462	—	2	2	no	—
261	#462	C₂²×C₂₃	1	2	yes	complemented
262	#451	—	2	1	no	—
263	#451	—	2	2	no	—
264	C₄×Q₁₆	—	2	2	no	—
265	#431	C₂²×C₂₃	1	1	yes	characteristic
266	#431	C₂²×C₂₃	1	1	yes	characteristic
267	#428	—	2	1	no	—
268	#423	C₂²×C₂₃	1	1	yes	characteristic
269	#418	—	2	1	no	—
270	#333	C₂²×C₂₃	1	4	yes	complemented
271	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
272	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
273	C₂×C₂₃	—	2	2	no	—
274	C₂×C₂₃	—	4	4	no	—
275	C₂×C₂₃	C₂×#451	1	1	yes	characteristic
276	C₂×C₂₃	#3005	1	1	yes	characteristic
277	C₂×C₂₃	#3019	1	1	yes	characteristic
278	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
279	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
280	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
281	C₂³×C₄	—	2	2	no	—
282	C₂³×C₄	—	2	1	no	—
283	C₂×Q₁₆	—	2	2	no	—
284	C₂×Q₁₆	—	2	4	no	—
285	C₂×M₁₆	—	2	1	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
289	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
290	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
291	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
292	#125	—	2	4	no	—
293	#125	—	2	4	no	—
294	#125	—	2	1	no	—
295	C₂²⋊Q₈	—	2	4	no	—
296	C₂²⋊Q₈	—	2	8	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	C₂²⋊Q₈	—	2	8	no	—
299	C₂²⋊Q₈	—	2	1	no	—
300	C₂²⋊Q₈	—	2	1	no	—
301	C₄×Q₈	—	2	1	no	—
302	C₄×Q₈	—	2	1	no	—
303	C₄×Q₈	—	2	1	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×Q₈	D₄×C₂₃	1	2	yes	—
306	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	D₄×C₂₃	1	2	yes	—
312	C₄×D₄	C₂³×C₂₃	1	2	yes	—
313	#117	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	4	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	4	no	—
321	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
327	#107	—	2	2	no	—
328	#107	—	2	2	no	—
329	#106	—	2	2	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
333	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
334	#103	—	2	2	no	—
335	#103	—	2	4	no	—
336	#103	—	2	2	no	—
337	#102	—	2	2	no	—
338	#102	—	2	2	no	—
339	C₂²⋊C₈	—	2	1	no	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	C₂³×C₂₃	1	2	yes	—
343	C₂₃	#18749	1	1	yes	characteristic, direct product factor, Sylow
344	C₂⁴	—	2	1	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	8	2	no	—
351	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
352	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	Q₁₆×C₂₃	1	4	yes	complemented
356	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	—	8	4	no	—
366	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
369	Q₁₆	—	4	4	no	—
370	Q₁₆	—	8	4	no	—
371	M₁₆	—	4	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₄⋊C₄	—	2	8	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	4	1	no	—
390	C₄⋊C₄	—	4	1	no	—
391	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
392	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	4	1	no	—
400	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
402	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
403	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
404	C₂³	—	2	4	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	C₂₃×#120	1	1	yes	characteristic
409	Q₈	—	2	2	no	—
410	Q₈	—	2	2	no	—
411	Q₈	—	4	4	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	Q₈	—	8	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	4	no	—
418	D₄	C₂×Q₁₆×C₂₃	1	4	yes	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	8	8	no	—
434	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
435	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
436	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
437	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
438	C₂×C₄	C₂₃×#137	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
442	C₈	—	4	2	no	—
443	C₈	—	4	2	no	—
444	C₈	—	4	2	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	8	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	C₂₃×#451	1	2	yes	—
452	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	4	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	1	no	—
462	C₄	C₂₃×#451	1	2	yes	—
463	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
464	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
465	C₄	C₂₃×#575	1	2	yes	—
466	C₂	—	2	2	no	—
467	C₂	—	4	4	no	—
468	C₂	C₂×C₂₃×#451	1	1	yes	characteristic
469	C₂	C₂₃×#3005	1	1	yes	characteristic
470	C₂	C₂₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
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
2. C₂₃×#3209	C₂	semidirect
3. C₂₃×#3104	C₂	semidirect
5. C₂₃×#1644	C₂	semidirect
6. C₂₃×#1632	C₂	semidirect
7. C₂₃×#1548	C₂	semidirect
8. C₂₃×#1487	C₂	semidirect
23. C₂₃×#494	C₂²	semidirect
24. C₂₃×#487	C₂²	semidirect
27. C₂₃×#477	C₂²	semidirect
31. C₂₃×#462	C₂²	semidirect
40. C₂₃×#333	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
119. C₂×D₄×C₂₃	Q₁₆	semidirect
168. #18749	C₂₃	direct (N × H)
211. #3209	C₂×C₂₃	semidirect
212. #3104	C₂×C₂₃	semidirect
214. #1644	C₂×C₂₃	semidirect
215. #1632	C₂×C₂₃	semidirect
216. #1548	C₂×C₂₃	semidirect
217. #1487	C₂×C₂₃	semidirect
253. #494	C₂²×C₂₃	semidirect
254. #487	C₂²×C₂₃	semidirect
257. #477	C₂²×C₂₃	semidirect
261. #462	C₂²×C₂₃	semidirect
270. #333	C₂²×C₂₃	semidirect
271. Q₈⋊C₈	C₂²×C₂₃	semidirect
343. C₂₃	#18749	direct (N × H)
355. C₂×D₄	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#554	2
C₂₃×#3209	2
C₂₃×#3104	2
C₂₃×#3077	2
C₂₃×#1644	2
C₂₃×#1632	2
C₂₃×#1548	2
C₂₃×#1487	2
C₂₃×#1478	2
#18749	23

Subgroups — C23×#18749 #19166375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18749 #19166375