Subgroups — #15682613

Order 2944.

764 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67288	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10625121	C₂	1	2	yes	maximal, complemented
3	#13092577	C₂	1	2	yes	maximal, complemented
4	#13092572	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092567	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092560	C₂	1	2	yes	maximal, complemented
7	#13092555	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092550	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092411	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₂₃	—	2	1	no	—
15	C₂×#67288	—	2	1	no	—
16	C₂×#67288	—	2	1	no	—
17	C₂×#67288	—	2	8	no	—
18	C₂×#67288	C₂²	1	2	yes	—
19	C₂×#67288	C₂²	1	2	yes	—
20	C₂²×D₉₂	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₂₃	—	2	1	no	—
22	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
23	C₂₃×#123	—	2	1	no	—
24	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#121	C₂²	1	1	yes	characteristic
27	C₂₃×#121	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#120	—	2	1	no	—
29	C₄×D₄×C₂₃	—	2	1	no	—
30	C₄×D₄×C₂₃	—	2	1	no	—
31	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
32	#10625121	—	2	8	no	—
33	C₂×#67275	C₂²	1	2	yes	complemented
34	C₂×#67274	C₂²	1	2	yes	complemented
35	C₂×#67273	C₂²	1	2	yes	complemented
36	C₂×#67272	C₂²	1	2	yes	complemented
37	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	complemented
38	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
39	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
40	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	complemented
41	#10624992	C₂²	1	2	yes	complemented
42	#10624992	C₂²	1	2	yes	complemented
43	#10624991	C₂²	1	2	yes	complemented
44	#10624991	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×D₂₃	—	2	1	no	—
47	#41×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₂₃	—	2	8	no	—
62	C₂×C₂²⋊D₂₃	—	2	2	no	—
63	C₂×C₂²⋊D₂₃	—	2	2	no	—
64	C₂²×Dic₂₃	—	2	1	no	—
65	#67288	—	2	8	no	—
66	#67288	—	2	8	no	—
67	#67288	—	2	4	no	—
68	#67288	—	2	4	no	—
69	#67288	—	2	4	no	—
70	#67288	D₄	1	8	yes	complemented
71	C₂×D₉₂	—	2	1	no	—
72	C₂×D₉₂	—	2	4	no	—
73	C₂×D₉₂	C₂³	1	2	yes	—
74	C₂×C₄×D₂₃	—	2	8	no	—
75	C₂×C₄×D₂₃	—	2	2	no	—
76	C₂×C₄×D₂₃	—	2	2	no	—
77	C₂×Dic₄₆	—	2	1	no	—
78	C₂×Dic₄₆	—	2	4	no	—
79	C₂×Dic₄₆	C₂³	1	2	yes	—
80	C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₂₃	—	2	1	no	—
85	C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂²⋊C₄×C₂₃	—	2	1	no	—
89	C₄²×C₂₃	—	2	1	no	—
90	#67275	—	2	4	no	—
91	#67274	—	2	4	no	—
92	#67273	—	2	4	no	—
93	#67272	—	2	4	no	—
94	C₂×C₂₃⋊C₈	—	2	4	no	—
95	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
96	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
97	C₂³×C₂₃	—	2	2	no	—
98	C₂³×C₂₃	—	2	2	no	—
99	C₂³×C₂₃	—	2	1	no	—
100	C₂³×C₂₃	—	4	1	no	—
101	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×D₂₃	—	2	4	no	—
105	C₂²×D₂₃	—	2	2	no	—
106	C₂²×D₂₃	—	4	2	no	—
107	C₂²×D₂₃	—	4	2	no	—
108	Q₈×C₂₃	—	2	2	no	—
109	D₄×C₂₃	—	2	2	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	1	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	1	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
123	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₂₃	—	2	8	no	—
128	C₂²⋊D₂₃	—	2	8	no	—
129	C₂×Dic₂₃	—	2	4	no	—
130	C₂×Dic₂₃	—	2	2	no	—
131	D₉₂	—	2	4	no	—
132	D₉₂	—	2	2	no	—
133	D₉₂	C₂×D₄	1	4	yes	—
134	C₄×D₂₃	—	2	8	no	—
135	C₄×D₂₃	—	2	8	no	—
136	Dic₄₆	—	2	4	no	—
137	Dic₄₆	—	2	2	no	—
138	Dic₄₆	C₂×D₄	1	4	yes	—
139	C₂₃⋊C₈	—	2	2	no	—
140	C₂₃⋊C₈	—	2	2	no	—
141	#3002	—	23	1	no	maximal, Sylow
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	2	1	no	—
148	C₂²×C₂₃	—	2	1	no	—
149	C₂²×C₂₃	—	4	4	no	—
150	C₂²×C₂₃	—	4	1	no	—
151	C₂²×C₂₃	—	4	1	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	#120	1	2	yes	—
154	C₂²×C₂₃	#135	1	2	yes	complemented
155	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₂₃	—	2	4	no	—
157	C₂×D₂₃	—	4	8	no	—
158	C₂×D₂₃	—	4	2	no	—
159	C₂×D₂₃	—	4	2	no	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	4	1	no	—
163	C₄×C₂₃	—	4	1	no	—
164	C₄×C₂₃	—	4	1	no	—
165	C₄×C₂₃	#120	1	2	yes	—
166	C₄×C₂₃	C₂²×D₄	1	2	yes	—
167	Dic₂₃	—	2	4	no	—
168	C₂²×#41	—	23	1	no	—
169	C₂×#135	—	23	2	no	—
170	#525	D₂₃	1	1	yes	characteristic, complemented
171	#452	—	23	2	no	—
172	#451	—	23	1	no	—
173	#450	—	23	1	no	—
174	#449	—	23	2	no	—
175	#448	—	23	1	no	—
176	#447	—	23	1	no	—
177	#416	—	23	2	no	—
178	C₂×C₂²⋊C₈	—	23	1	no	—
179	C₂×C₂₃	—	2	2	no	—
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	4	1	no	—
182	C₂×C₂₃	—	8	1	no	—
183	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
184	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
185	C₂×C₂₃	#575	1	1	yes	characteristic
186	D₂₃	—	4	4	no	—
187	C₂×#41	—	23	2	no	—
188	C₂×#41	—	23	2	no	—
189	C₂×#41	—	46	8	no	—
190	C₂×#41	—	46	1	no	—
191	C₂×#41	—	46	1	no	—
192	C₂×#41	C₂×D₂₃	1	1	yes	characteristic, complemented
193	C₂²×Q₈	—	23	1	no	—
194	C₂²×D₄	—	23	1	no	—
195	C₂²×D₄	—	46	1	no	—
196	C₂³×C₄	—	46	1	no	—
197	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
198	#135	—	46	8	no	—
199	C₂×Q₁₆	—	23	2	no	—
200	C₂×SD₁₆	—	23	2	no	—
201	C₂×SD₁₆	—	23	2	no	—
202	C₂×D₈	—	23	2	no	—
203	C₂²×C₈	—	23	2	no	—
204	#123	—	2	1	no	—
205	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
206	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
207	#121	C₂×D₂₃	1	1	yes	characteristic
208	#121	C₂×D₂₃	1	1	yes	characteristic, complemented
209	#120	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	C₂×D₂₃	1	1	yes	characteristic, complemented
213	#103	—	23	2	no	—
214	#103	—	23	2	no	—
215	#102	—	23	2	no	—
216	#102	—	23	2	no	—
217	C₂²⋊C₈	—	23	1	no	—
218	C₂²⋊C₈	—	23	1	no	—
219	C₂²⋊C₈	—	23	2	no	—
220	C₂₃	#3002	1	1	yes	characteristic, complemented, Sylow
221	C₂⁴	—	46	1	no	—
222	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
223	#41	—	2	4	no	—
224	#41	—	23	8	no	—
225	#41	—	46	4	no	—
226	#41	—	46	4	no	—
227	#41	—	46	8	no	—
228	#41	—	46	8	no	—
229	#41	—	46	4	no	—
230	C₂×Q₈	—	23	2	no	—
231	C₂×Q₈	—	46	4	no	—
232	C₂×Q₈	—	46	1	no	—
233	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	23	2	no	—
238	C₂×D₄	—	46	4	no	—
239	C₂×D₄	—	46	8	no	—
240	C₂×D₄	—	46	2	no	—
241	C₂×D₄	—	46	2	no	—
242	C₂×D₄	—	46	1	no	—
243	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	46	8	no	—
247	C₂²×C₄	—	46	1	no	—
248	C₂²×C₄	—	46	2	no	—
249	C₂²×C₄	—	46	2	no	—
250	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
256	Q₁₆	—	46	4	no	—
257	SD₁₆	—	46	4	no	—
258	SD₁₆	—	46	4	no	—
259	D₈	—	46	4	no	—
260	C₂×C₈	—	23	2	no	—
261	C₂×C₈	—	23	2	no	—
262	C₂×C₈	—	46	4	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	46	4	no	—
278	C₂³	—	46	2	no	—
279	C₂³	—	92	2	no	—
280	C₂³	—	92	2	no	—
281	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
283	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	23	4	no	—
286	Q₈	—	46	4	no	—
287	Q₈	—	46	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	23	4	no	—
293	D₄	—	46	4	no	—
294	D₄	—	46	8	no	—
295	D₄	—	46	8	no	—
296	D₄	—	46	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	46	8	no	—
307	C₂×C₄	—	46	4	no	—
308	C₂×C₄	—	46	2	no	—
309	C₂×C₄	—	46	8	no	—
310	C₂×C₄	D₄×D₂₃	1	4	yes	—
311	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
314	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
315	C₈	—	46	2	no	—
316	C₈	—	46	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	46	4	no	—
329	C₂²	—	92	8	no	—
330	C₂²	—	92	2	no	—
331	C₂²	—	92	2	no	—
332	C₂²	#10625108	1	2	yes	—
333	C₂²	#10625121	1	2	yes	complemented
334	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	46	4	no	—
341	C₄	#10625108	1	2	yes	—
342	C₄	C₂×D₄×D₂₃	1	2	yes	—
343	C₂	—	2	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	—	8	1	no	—
347	C₂	—	92	4	no	—
348	C₂	C₂×#10625108	1	1	yes	characteristic
349	C₂	C₂×#10625121	1	1	yes	characteristic
350	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#525	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	#525	D₂₃
hypocenter	C₂₃	#3002
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. C₂×#10625121	C₂	semidirect
3. #13092577	C₂	semidirect
4. #13092572	C₂	semidirect
5. #13092567	C₂	semidirect
6. #13092560	C₂	semidirect
7. #13092555	C₂	semidirect
8. #13092550	C₂	semidirect
9. #13092411	C₂	semidirect
10. C₂×C₂²⋊(C₂₃⋊C₈)	C₂	semidirect
11. C₂₃×#525	C₂	semidirect
12. C₂×#41×C₂₃	C₂²	semidirect
25. C₂₃×C₂²⋊Q₈	C₂²	semidirect
27. C₂₃×#121	C₂²	semidirect
31. C₂₃×#117	C₂²	semidirect
33. C₂×#67275	C₂²	semidirect
34. C₂×#67274	C₂²	semidirect
35. C₂×#67273	C₂²	semidirect
36. C₂×#67272	C₂²	semidirect
37. C₂²×C₂₃⋊C₈	C₂²	semidirect
40. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
41. #10624992	C₂²	semidirect
42. #10624992	C₂²	semidirect
43. #10624991	C₂²	semidirect
44. #10624991	C₂²	semidirect
70. #67288	D₄	semidirect
154. C₂²×C₂₃	#135	semidirect
170. #525	D₂₃	semidirect
192. C₂×#41	C₂×D₂₃	semidirect
206. C₂²⋊Q₈	C₂×D₂₃	semidirect
208. #121	C₂×D₂₃	semidirect
212. #117	C₂×D₂₃	semidirect
220. C₂₃	#3002	semidirect
333. C₂²	#10625121	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67288	2
C₂×#10625121	2
#13092577	2
#13092572	2
#13092567	2
#13092560	2
#13092555	2
#13092550	2
#13092411	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
C₂₃×#525	2
#3002	23