Subgroups — #26370926

Order 12032.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20245553	C₂	1	2	yes	maximal, complemented
4	#20245546	C₂	1	2	yes	maximal, complemented
5	#20245541	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245536	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245531	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245526	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245172	C₂	1	2	yes	maximal, complemented
10	#20245096	C₂	1	2	yes	maximal
11	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689680	—	2	1	no	—
14	C₄×#10625350	—	2	1	no	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689525	—	2	1	no	—
18	#15689525	C₂²	1	2	yes	—
19	#15689521	—	2	1	no	—
20	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×#10625325	—	2	1	no	—
23	C₄×#10625325	—	2	1	no	—
24	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
25	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
26	#15689161	—	2	1	no	—
27	#15689161	C₂²	1	1	yes	characteristic
28	#15689161	C₂²	1	2	yes	—
29	#15689120	—	2	1	no	—
30	#15688951	C₂²	1	2	yes	—
31	#15688951	C₂²	1	2	yes	—
32	#15688949	C₂²	1	2	yes	—
33	#15688949	C₂²	1	2	yes	—
34	#15688928	C₂²	1	2	yes	—
35	#15688928	C₂²	1	2	yes	—
36	#15688925	C₂²	1	2	yes	—
37	#15688925	C₂²	1	2	yes	—
38	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
43	C₂×#10625350	—	2	1	no	—
44	#13094046	—	2	2	no	—
45	#13094046	Q₈	1	4	yes	complemented
46	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
47	C₂×#10625325	—	2	1	no	—
48	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
49	#13094024	—	2	1	no	—
50	#13094024	—	2	1	no	—
51	#13094016	—	2	2	no	—
52	#13094003	—	4	1	no	—
53	#13094003	—	4	1	no	—
54	C₄×Dic₉₄	—	2	1	no	—
55	C₄×Dic₉₄	—	2	1	no	—
56	C₄×Dic₉₄	—	4	1	no	—
57	C₄×Dic₉₄	—	4	1	no	—
58	C₄×D₁₈₈	—	2	1	no	—
59	C₄×D₁₈₈	—	2	1	no	—
60	C₄×C₂²⋊D₄₇	—	4	1	no	—
61	C₄×C₂²⋊D₄₇	—	4	1	no	—
62	#13093983	—	2	1	no	—
63	#13093983	—	2	2	no	—
64	#13093981	—	4	1	no	—
65	C₄²×D₄₇	—	4	1	no	—
66	#13093958	—	2	1	no	—
67	#13093958	—	2	1	no	—
68	#13093958	C₂³	1	1	yes	characteristic
69	#13093958	C₂³	1	1	yes	characteristic
70	#13093958	C₂³	1	2	yes	—
71	#13093958	C₂³	1	2	yes	—
72	#13093956	—	2	2	no	—
73	#13093956	—	2	2	no	—
74	#13093956	—	2	1	no	—
75	#13093949	—	2	1	no	—
76	#13093949	—	2	1	no	—
77	#13093946	—	2	1	no	—
78	#13093946	—	2	1	no	—
79	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
82	C₂×C₁₆×C₄₇	—	2	4	no	—
83	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
84	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
85	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
86	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	4	1	no	—
93	#10625350	—	2	4	no	—
94	#10625350	—	4	2	no	—
95	C₂×D₁₈₈	—	2	1	no	—
96	C₂×C₄×D₄₇	—	4	1	no	—
97	C₂×C₄×D₄₇	—	4	1	no	—
98	C₂×C₄×D₄₇	—	4	1	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	4	1	no	—
101	C₂×Dic₉₄	—	4	1	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₄²×C₄₇	D₈	1	1	yes	characteristic
110	C₄²×C₄₇	D₈	1	1	yes	characteristic
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₄₇	—	4	1	no	—
114	C₂²⋊Dic₄₇	—	4	2	no	—
115	(C₂×D₄₇)⋊C₄	—	4	1	no	—
116	(C₂×D₄₇)⋊C₄	—	4	1	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	4	1	no	—
122	C₄⋊Dic₄₇	—	4	1	no	—
123	Dic₄₇⋊C₄	—	4	1	no	—
124	Dic₄₇⋊C₄	—	4	1	no	—
125	Dic₄₇⋊C₄	—	4	2	no	—
126	Dic₄₇⋊C₄	—	4	2	no	—
127	C₄×Dic₄₇	—	4	1	no	—
128	C₄×Dic₄₇	—	4	1	no	—
129	C₄×Dic₄₇	—	4	1	no	—
130	C₄×Dic₄₇	—	4	2	no	—
131	Dic₁₈₈	—	2	1	no	—
132	Dic₁₈₈	C₂×Q₈	1	2	yes	—
133	D₃₇₆	—	2	1	no	—
134	D₃₇₆	C₂×Q₈	1	2	yes	—
135	#10625325	—	2	4	no	—
136	C₁₆×C₄₇	—	2	2	no	—
137	C₁₆×C₄₇	—	2	2	no	—
138	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₇	#137	1	2	yes	—
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₄₇	—	4	4	no	—
151	C₂×Dic₄₇	—	4	1	no	—
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	4	1	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	D₁₈₈	—	2	2	no	—
157	D₁₈₈	—	4	1	no	—
158	C₄×D₄₇	—	4	4	no	—
159	C₄×D₄₇	—	8	2	no	—
160	C₄×D₄₇	—	8	2	no	—
161	Dic₉₄	—	2	2	no	—
162	Dic₉₄	—	4	1	no	—
163	Dic₉₄	—	8	2	no	—
164	Dic₉₄	—	8	2	no	—
165	C₈×C₄₇	—	2	1	no	—
166	C₈×C₄₇	—	2	1	no	—
167	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
168	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
170	#33626	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	—	2	1	no	—
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	#474	1	2	yes	—
174	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₄₇	—	4	2	no	—
176	C₂×D₄₇	—	8	1	no	—
177	C₂×D₄₇	—	8	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	#474	1	2	yes	—
182	C₄×C₄₇	#508	1	2	yes	—
183	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	8	2	no	—
187	Dic₄₇	—	8	1	no	—
188	Dic₄₇	—	8	2	no	—
189	Dic₄₇	—	8	1	no	—
190	#3160	—	47	1	no	—
191	C₄×#135	—	47	1	no	—
192	#2229	—	47	2	no	—
193	#2228	—	47	2	no	—
194	#2227	—	47	1	no	—
195	#2226	—	47	1	no	—
196	#2225	—	47	1	no	—
197	#2224	—	47	1	no	—
198	#2158	—	47	2	no	—
199	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
200	#2138	—	47	2	no	—
201	C₂×C₄₇	—	2	2	no	—
202	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
203	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
204	C₂×C₄₇	#3414	1	1	yes	characteristic
205	D₄₇	—	8	2	no	—
206	C₂×#135	—	47	1	no	—
207	#533	—	94	1	no	—
208	C₄×#41	—	94	1	no	—
209	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
210	#500	—	47	1	no	—
211	#500	—	47	1	no	—
212	#499	—	47	2	no	—
213	#499	—	94	1	no	—
214	#498	—	94	1	no	—
215	C₄×Q₁₆	—	47	1	no	—
216	C₄×Q₁₆	—	47	1	no	—
217	C₄×SD₁₆	—	94	1	no	—
218	C₄×SD₁₆	—	94	1	no	—
219	C₄×D₈	—	47	1	no	—
220	C₄×D₈	—	47	1	no	—
221	#427	—	47	2	no	—
222	#427	—	47	1	no	—
223	#427	—	94	1	no	—
224	#416	—	94	1	no	—
225	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
226	#367	—	47	2	no	—
227	#367	—	47	2	no	—
228	#366	—	47	2	no	—
229	#366	—	47	2	no	—
230	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
231	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
232	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
233	#358	—	47	2	no	—
234	#358	—	47	2	no	—
235	#357	—	47	2	no	—
236	#357	—	47	2	no	—
237	C₄₇	#33626	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	94	1	no	—
239	#135	—	47	4	no	—
240	#135	—	94	2	no	—
241	C₂×Q₁₆	—	47	1	no	—
242	C₂×SD₁₆	—	94	1	no	—
243	C₂×D₈	—	47	1	no	—
244	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₄⋊Q₈	—	94	1	no	—
247	C₄⋊Q₈	—	94	1	no	—
248	#125	—	94	2	no	—
249	C₂²⋊Q₈	—	188	1	no	—
250	C₂²⋊Q₈	—	188	1	no	—
251	C₄×Q₈	—	94	1	no	—
252	C₄×Q₈	—	94	1	no	—
253	C₄×Q₈	—	188	1	no	—
254	C₄×Q₈	—	188	1	no	—
255	C₄×D₄	—	94	1	no	—
256	C₄×D₄	—	94	1	no	—
257	C₄×D₄	—	188	1	no	—
258	C₄×D₄	—	188	1	no	—
259	#117	—	94	1	no	—
260	#117	—	94	2	no	—
261	#117	—	188	1	no	—
262	C₂×C₄²	—	188	1	no	—
263	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
264	C₂×C₁₆	—	2	4	no	—
265	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
266	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
267	#107	—	47	2	no	—
268	#107	—	47	2	no	—
269	#107	—	47	1	no	—
270	#107	—	47	1	no	—
271	#107	—	94	1	no	—
272	#107	—	94	1	no	—
273	#106	—	94	1	no	—
274	#106	—	94	2	no	—
275	#106	—	94	2	no	—
276	#103	—	94	1	no	—
277	#103	—	94	1	no	—
278	#102	—	94	1	no	—
279	#102	—	94	1	no	—
280	C₄×C₈	D₁₈₈	1	1	yes	characteristic
281	C₄×C₈	D₁₈₈	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
284	#41	—	94	4	no	—
285	#41	—	188	2	no	—
286	C₂×Q₈	—	94	1	no	—
287	C₂×Q₈	—	188	1	no	—
288	C₂×Q₈	—	188	1	no	—
289	C₂×D₄	—	94	1	no	—
290	C₂×D₄	—	188	1	no	—
291	C₂²×C₄	—	188	1	no	—
292	C₂²×C₄	—	188	1	no	—
293	C₂²×C₄	—	188	1	no	—
294	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
295	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
297	Q₁₆	—	47	2	no	—
298	Q₁₆	—	94	1	no	—
299	SD₁₆	—	94	4	no	—
300	D₈	—	47	2	no	—
301	D₈	—	94	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
304	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
306	C₂×C₈	#10625352	1	2	yes	—
307	C₂×C₈	Q₈×D₄₇	1	2	yes	—
308	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
309	C₄⋊C₄	—	94	1	no	—
310	C₄⋊C₄	—	94	1	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	188	1	no	—
314	C₄⋊C₄	—	188	1	no	—
315	C₄⋊C₄	—	188	2	no	—
316	C₄⋊C₄	—	188	2	no	—
317	C₄⋊C₄	—	188	1	no	—
318	C₄⋊C₄	—	188	1	no	—
319	C₂²⋊C₄	—	188	1	no	—
320	C₂²⋊C₄	—	188	2	no	—
321	C₂²⋊C₄	—	188	1	no	—
322	C₂²⋊C₄	—	188	1	no	—
323	C₄²	—	188	1	no	—
324	C₄²	—	188	1	no	—
325	C₄²	—	188	1	no	—
326	C₄²	—	188	2	no	—
327	C₄²	D₃₇₆	1	1	yes	characteristic
328	C₄²	D₃₇₆	1	1	yes	characteristic
329	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
330	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	188	1	no	—
334	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
335	Q₈	—	94	2	no	—
336	Q₈	—	376	2	no	—
337	Q₈	—	376	2	no	—
338	Q₈	—	188	1	no	—
339	D₄	—	94	2	no	—
340	D₄	—	188	4	no	—
341	D₄	—	188	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	376	2	no	—
345	C₂×C₄	—	376	2	no	—
346	C₂×C₄	—	188	1	no	—
347	C₂×C₄	—	188	1	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	1	no	—
351	C₂×C₄	—	188	4	no	—
352	C₂×C₄	#13094004	1	2	yes	—
353	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
354	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
355	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
356	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
357	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
358	C₂×C₄	#13094057	1	2	yes	—
359	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#13094004	1	2	yes	—
363	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
364	C₈	C₂×#10625352	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	376	1	no	—
368	C₂²	—	376	1	no	—
369	C₂²	—	188	2	no	—
370	C₂²	#15689404	1	2	yes	—
371	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	376	1	no	—
376	C₄	—	376	2	no	—
377	C₄	—	376	1	no	—
378	C₄	—	376	2	no	—
379	C₄	—	188	2	no	—
380	C₄	#15689404	1	2	yes	—
381	C₄	#15689555	1	2	yes	—
382	C₄	C₂×#13094004	1	1	yes	characteristic
383	C₄	C₂×#13094057	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	376	2	no	—
386	C₂	C₂×#15689404	1	1	yes	characteristic
387	C₂	C₂×#15689555	1	1	yes	characteristic
388	C₂	#20256524	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#13094046	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#33626
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. #20253902	C₂	semidirect
3. #20245553	C₂	semidirect
4. #20245546	C₂	semidirect
5. #20245541	C₂	semidirect
6. #20245536	C₂	semidirect
7. #20245531	C₂	semidirect
8. #20245526	C₂	semidirect
9. #20245172	C₂	semidirect
11. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
45. #13094046	Q₈	semidirect
199. C₂×C₄⋊C₁₆	D₄₇	semidirect
237. C₄₇	#33626	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
C₄×#13094046	2
#20245553	2
#20245546	2
#20245541	2
#20245536	2
#20245531	2
#20245526	2
#20245172	2
#20245096	2
C₂×C₄₇×C₄⋊C₁₆	2
#33626	47