Subgroups — #15682357

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67290	C₂	1	1	yes	maximal, characteristic
2	#13092546	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092545	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092493	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092492	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092489	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092488	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092485	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092469	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092467	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092467	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092369	C₂	1	1	yes	maximal, characteristic, complemented
13	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#67290	C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	D₄×Dic₂₃	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×#67267	C₂²	1	1	yes	characteristic
26	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×#67262	C₂²	1	1	yes	characteristic
28	C₂×#67262	C₂²	1	1	yes	characteristic
29	#10625063	C₄	1	1	yes	characteristic
30	#10625058	C₄	1	1	yes	characteristic
31	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
32	(C₂²⋊D₂₃)⋊C₄	C₄	1	1	yes	characteristic
33	#10625048	C₄	1	1	yes	characteristic
34	#10625048	C₄	1	1	yes	characteristic
35	C₄²×D₂₃	C₂²	1	1	yes	characteristic
36	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
43	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₂₃)⋊C₈	C₂²	1	1	yes	characteristic
45	#10624997	C₂²	1	1	yes	characteristic
46	#10624997	C₂²	1	1	yes	characteristic
47	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
54	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
55	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
56	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
57	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
58	#10624985	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
68	#67290	—	2	4	no	—
69	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
73	M₁₆×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₂₃)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
88	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
89	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
90	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
91	Dic₂₃⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
101	#67267	—	2	2	no	—
102	C₂×C₂₃⋊C₈	—	2	2	no	—
103	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
107	#67262	—	2	2	no	—
108	#67262	—	2	2	no	—
109	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₂₃	—	2	2	no	—
123	C₂²⋊D₂₃	—	2	2	no	—
124	C₂×Dic₂₃	—	2	2	no	—
125	C₂×Dic₂₃	—	2	2	no	—
126	C₂×Dic₂₃	—	2	2	no	—
127	C₂×Dic₂₃	M₁₆	1	4	yes	complemented
128	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₂₃	—	2	1	no	—
136	C₄×D₂₃	—	2	1	no	—
137	C₄×D₂₃	—	2	2	no	—
138	C₄×D₂₃	—	2	2	no	—
139	Dic₄₆	—	2	2	no	—
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₈×C₂₃	#41	1	2	yes	complemented
144	C₂₃⋊C₈	—	2	1	no	—
145	C₂₃⋊C₈	—	2	1	no	—
146	C₂₃⋊C₈	—	2	1	no	—
147	C₂₃⋊C₈	#41	1	2	yes	complemented
148	#2965	—	23	1	no	maximal, Sylow
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	2	1	no	—
152	C₂²×C₂₃	—	2	1	no	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
155	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×D₂₃	—	2	1	no	—
157	C₂×D₂₃	—	2	1	no	—
158	C₂×D₂₃	—	2	1	no	—
159	C₄×C₂₃	—	2	1	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
165	Dic₂₃	—	2	1	no	—
166	Dic₂₃	—	2	1	no	—
167	Dic₂₃	—	2	1	no	—
168	Dic₂₃	—	2	1	no	—
169	Dic₂₃	C₂×M₁₆	1	2	yes	—
170	Dic₂₃	C₂×M₁₆	1	2	yes	—
171	Dic₂₃	#131	1	2	yes	—
172	C₄×#41	—	23	1	no	—
173	#446	—	23	1	no	—
174	#446	—	23	1	no	—
175	#436	—	23	1	no	—
176	#436	—	23	1	no	—
177	#435	—	23	1	no	—
178	#435	—	23	1	no	—
179	#435	—	23	1	no	—
180	#435	D₂₃	1	1	yes	characteristic, complemented
181	#431	—	23	1	no	—
182	#431	—	23	1	no	—
183	#431	—	23	1	no	—
184	C₄×M₁₆	—	23	1	no	—
185	C₄×M₁₆	—	23	1	no	—
186	C₂×C₄×C₈	—	23	1	no	—
187	C₂×C₂₃	—	2	2	no	—
188	C₂×C₂₃	—	4	1	no	—
189	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
190	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
192	D₂₃	—	4	1	no	—
193	C₂×#41	—	23	1	no	—
194	C₂×M₁₆	—	23	1	no	—
195	C₂×M₁₆	—	23	1	no	—
196	C₂×M₁₆	—	23	1	no	—
197	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
198	C₂²×C₈	—	23	1	no	—
199	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
200	C₄×Q₈	—	23	1	no	—
201	C₄×Q₈	—	23	1	no	—
202	C₄×D₄	—	23	1	no	—
203	C₄×D₄	—	23	1	no	—
204	C₄×D₄	—	23	1	no	—
205	C₄×D₄	—	23	1	no	—
206	C₄×D₄	—	23	1	no	—
207	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
208	#117	—	23	1	no	—
209	#117	—	23	1	no	—
210	#117	—	23	1	no	—
211	C₂×C₄²	—	23	1	no	—
212	C₂×C₄²	—	23	1	no	—
213	C₂×C₄²	—	23	1	no	—
214	C₄⋊C₈	—	23	1	no	—
215	C₄⋊C₈	—	23	1	no	—
216	C₄⋊C₈	—	23	1	no	—
217	C₄⋊C₈	—	23	1	no	—
218	C₄⋊C₈	—	23	1	no	—
219	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
220	C₂²⋊C₈	—	23	1	no	—
221	C₂²⋊C₈	—	23	1	no	—
222	C₂²⋊C₈	—	23	1	no	—
223	C₂²⋊C₈	—	23	1	no	—
224	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
226	#97	—	23	1	no	—
227	#97	—	23	1	no	—
228	#97	—	23	1	no	—
229	#97	C₂×D₂₃	1	1	yes	characteristic
230	C₄×C₈	—	23	1	no	—
231	C₄×C₈	—	23	1	no	—
232	C₄×C₈	—	23	1	no	—
233	C₄×C₈	—	23	1	no	—
234	C₄×C₈	—	23	1	no	—
235	C₄×C₈	—	23	1	no	—
236	C₂₃	#2965	1	1	yes	characteristic, complemented, Sylow
237	#41	—	46	4	no	—
238	C₂×Q₈	—	23	1	no	—
239	C₂×D₄	—	23	1	no	—
240	C₂×D₄	—	23	1	no	—
241	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
242	C₂²×C₄	—	23	1	no	—
243	C₂²×C₄	—	23	1	no	—
244	C₂²×C₄	—	23	1	no	—
245	C₂²×C₄	—	23	1	no	—
246	C₂²×C₄	—	23	1	no	—
247	C₂²×C₄	—	23	1	no	—
248	C₂²×C₄	—	23	1	no	—
249	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	46	2	no	—
253	M₁₆	—	46	2	no	—
254	M₁₆	—	46	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	23	1	no	—
257	C₂×C₈	—	23	1	no	—
258	C₂×C₈	—	23	1	no	—
259	C₂×C₈	—	23	1	no	—
260	C₂×C₈	—	46	2	no	—
261	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
265	C₄⋊C₄	—	23	1	no	—
266	C₄⋊C₄	—	23	1	no	—
267	C₄⋊C₄	—	23	1	no	—
268	C₄⋊C₄	—	23	1	no	—
269	C₄⋊C₄	—	23	1	no	—
270	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
271	C₂²⋊C₄	—	23	1	no	—
272	C₂²⋊C₄	—	23	1	no	—
273	C₂²⋊C₄	—	23	1	no	—
274	C₂²⋊C₄	—	23	1	no	—
275	C₂²⋊C₄	C₄×D₂₃	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₂₃	1	1	yes	characteristic
277	C₄²	—	23	1	no	—
278	C₄²	—	23	1	no	—
279	C₄²	—	23	1	no	—
280	C₄²	—	23	1	no	—
281	C₄²	—	23	1	no	—
282	C₄²	—	23	1	no	—
283	C₄²	—	23	1	no	—
284	C₄²	—	23	1	no	—
285	C₄²	—	23	1	no	—
286	C₄²	C₂²×D₂₃	1	1	yes	characteristic
287	C₂³	—	23	1	no	—
288	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
290	Q₈	—	46	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	46	2	no	—
293	D₄	—	46	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	23	4	no	—
299	C₂×C₄	—	23	1	no	—
300	C₂×C₄	—	23	1	no	—
301	C₂×C₄	—	23	1	no	—
302	C₂×C₄	—	23	1	no	—
303	C₂×C₄	—	23	1	no	—
304	C₂×C₄	—	23	1	no	—
305	C₂×C₄	—	23	1	no	—
306	C₂×C₄	—	46	2	no	—
307	C₂×C₄	—	46	2	no	—
308	C₂×C₄	—	46	2	no	—
309	C₂×C₄	—	46	2	no	—
310	C₂×C₄	—	46	2	no	—
311	C₂×C₄	—	46	1	no	—
312	C₂×C₄	—	46	1	no	—
313	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	23	2	no	—
322	C₈	—	46	1	no	—
323	C₈	—	46	1	no	—
324	C₈	—	46	1	no	—
325	C₈	#67290	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	46	1	no	—
332	C₂²	—	46	1	no	—
333	C₂²	—	46	1	no	—
334	C₂²	M₁₆×D₂₃	1	2	yes	—
335	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	23	2	no	—
341	C₄	—	23	2	no	—
342	C₄	—	23	2	no	—
343	C₄	—	46	1	no	—
344	C₄	—	46	1	no	—
345	C₄	—	46	1	no	—
346	C₄	—	46	1	no	—
347	C₄	C₂×#67290	1	1	yes	characteristic
348	C₄	#41×D₂₃	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	92	1	no	—
352	C₂	C₄×#67290	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
354	C₂	#131×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#435	C₂
prime radical	C₂×#67290	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#435	D₂₃
hypocenter	C₂₃	#2965
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #13092546	C₂	semidirect
3. #13092545	C₂	semidirect
4. #13092493	C₂	semidirect
5. #13092492	C₂	semidirect
6. #13092489	C₂	semidirect
7. #13092488	C₂	semidirect
8. #13092485	C₂	semidirect
9. #13092469	C₂	semidirect
10. #13092467	C₂	semidirect
11. #13092467	C₂	semidirect
12. #13092369	C₂	semidirect
13. M₁₆×Dic₂₃	C₂	semidirect
14. C₂×C₈×Dic₂₃	C₂	semidirect
15. C₂₃×#435	C₂	semidirect
127. C₂×Dic₂₃	M₁₆	semidirect
143. C₈×C₂₃	#41	semidirect
147. C₂₃⋊C₈	#41	semidirect
180. #435	D₂₃	semidirect
236. C₂₃	#2965	semidirect
325. C₈	#67290	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67290	2
#13092546	2
#13092545	2
#13092493	2
#13092492	2
#13092489	2
#13092488	2
#13092485	2
#13092469	2
#13092467	2
#13092369	2
M₁₆×Dic₂₃	2
C₂×C₈×Dic₂₃	2
C₂₃×#435	2
#2965	23