Subgroups — #26207880

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256363	C₂	1	1	yes	maximal, characteristic
2	#20256256	C₂	1	1	yes	maximal, characteristic, complemented
3	#20255728	C₂	1	2	yes	maximal, complemented
4	#20242980	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242922	C₂	1	2	yes	maximal, complemented
6	#20242792	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242748	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242517	C₂	1	2	yes	maximal, complemented
9	#20242463	C₂	1	2	yes	maximal, complemented
10	#20242274	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689833	C₂²	1	2	yes	—
13	#15689795	C₂²	1	2	yes	—
14	#15689528	C₂²	1	1	yes	characteristic
15	#15689525	—	2	1	no	—
16	#15689500	—	2	1	no	—
17	#15689500	C₂²	1	1	yes	characteristic
18	#15689496	C₂²	1	2	yes	—
19	#15689470	C₂²	1	2	yes	—
20	#15689438	—	2	2	no	—
21	#15689424	—	2	2	no	—
22	#15689418	C₂²	1	2	yes	—
23	#15689414	C₂²	1	2	yes	—
24	#15689372	—	2	2	no	—
25	#15689323	C₂²	1	2	yes	—
26	#15689319	—	2	2	no	—
27	#15689319	C₂²	1	2	yes	—
28	#15689284	—	2	2	no	—
29	C₄×D₃₇₆	—	2	1	no	—
30	C₄×D₃₇₆	—	2	1	no	—
31	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
32	#15689202	C₂²	1	2	yes	—
33	#15689123	—	2	2	no	—
34	#15688876	C₂²	1	1	yes	characteristic
35	#15688871	C₂²	1	2	yes	—
36	#15688866	C₂²	1	2	yes	—
37	#15688852	C₂²	1	2	yes	—
38	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#335	C₂²	1	1	yes	characteristic
42	C₄₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₄₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×#10625354	—	2	2	no	—
46	C₂×#10625325	—	4	2	no	—
47	C₂×#10625336	—	2	2	no	—
48	C₂×D₃₇₆	—	2	1	no	—
49	C₂×D₃₇₆	—	2	2	no	—
50	C₂×D₃₇₆	—	2	1	no	—
51	C₂×D₃₇₆	D₄	1	1	yes	characteristic
52	C₂×#10625324	—	2	2	no	—
53	#13094024	C₂³	1	2	yes	—
54	#13094021	—	2	2	no	—
55	#13094021	C₂³	1	2	yes	—
56	#13094016	—	2	2	no	—
57	#13094002	—	4	2	no	—
58	#13094000	—	2	2	no	—
59	#13093999	—	2	2	no	—
60	#13093999	—	4	2	no	—
61	C₄×D₁₈₈	—	2	2	no	—
62	C₄×D₁₈₈	—	2	2	no	—
63	C₄×D₁₈₈	C₂³	1	2	yes	—
64	C₄⋊C₄×D₄₇	—	2	2	no	—
65	#13093958	—	2	1	no	—
66	#13093958	—	2	1	no	—
67	#13093958	D₄	1	1	yes	characteristic
68	#13093957	—	2	2	no	—
69	#13093956	—	2	2	no	—
70	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
71	#13093948	—	2	2	no	—
72	#13093946	—	2	2	no	—
73	#13093946	—	2	2	no	—
74	#13093946	—	2	2	no	—
75	#13093946	—	2	2	no	—
76	#13093946	D₄	1	2	yes	—
77	#13093944	—	2	2	no	—
78	(C₂×D₄₇)⋊C₈	—	2	2	no	—
79	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
82	#107×C₄₇	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₄₇	—	2	1	no	—
84	C₄⋊C₈×C₄₇	—	2	1	no	—
85	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
87	#103×C₄₇	D₄	1	1	yes	characteristic
88	#103×C₄₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₄₇	—	2	1	no	—
90	C₄×C₈×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	#10625354	—	4	4	no	—
97	#10625354	—	8	2	no	—
98	C₂×D₁₈₈	—	2	2	no	—
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×D₁₈₈	—	4	2	no	—
102	C₂×D₁₈₈	—	4	2	no	—
103	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₄₇	—	2	2	no	—
105	C₂×C₄×D₄₇	—	4	2	no	—
106	C₂×C₄×D₄₇	—	4	2	no	—
107	C₂×Dic₉₄	—	2	2	no	—
108	Q₁₆×C₄₇	—	2	2	no	—
109	C₂×C₈×C₄₇	—	2	1	no	—
110	C₂×C₈×C₄₇	—	2	1	no	—
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₄₇	—	2	1	no	—
117	C₄⋊C₄×C₄₇	—	2	1	no	—
118	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	—	2	1	no	—
121	C₄²×C₄₇	—	2	1	no	—
122	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
123	#10625336	—	8	2	no	—
124	(C₂×D₄₇)⋊C₄	—	2	2	no	—
125	(C₂×D₄₇)⋊C₄	—	4	2	no	—
126	(C₂×D₄₇)⋊C₄	—	4	2	no	—
127	C₄⋊Dic₄₇	—	2	2	no	—
128	C₄⋊Dic₄₇	—	2	2	no	—
129	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
130	Dic₄₇⋊C₄	—	4	2	no	—
131	Dic₄₇⋊C₄	—	4	2	no	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	D₃₇₆	—	2	2	no	—
134	D₃₇₆	—	4	2	no	—
135	D₃₇₆	—	8	1	no	—
136	D₃₇₆	—	8	2	no	—
137	#10625325	—	8	4	no	—
138	#10625324	—	8	2	no	—
139	C₂²×D₄₇	—	2	2	no	—
140	C₂²×D₄₇	—	4	2	no	—
141	Q₈×C₄₇	—	2	1	no	—
142	Q₈×C₄₇	—	2	1	no	—
143	Q₈×C₄₇	—	2	1	no	—
144	Q₈×C₄₇	#135	1	2	yes	—
145	C₂×C₄×C₄₇	—	2	1	no	—
146	C₂×C₄×C₄₇	—	2	1	no	—
147	C₂×C₄×C₄₇	—	2	1	no	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	D₁₈₈	—	2	2	no	—
155	D₁₈₈	—	2	2	no	—
156	D₁₈₈	—	4	2	no	—
157	D₁₈₈	—	4	2	no	—
158	D₁₈₈	—	4	2	no	—
159	D₁₈₈	—	4	2	no	—
160	D₁₈₈	—	8	4	no	—
161	D₁₈₈	—	8	2	no	—
162	D₁₈₈	—	16	2	no	—
163	C₄×D₄₇	—	4	2	no	—
164	C₄×D₄₇	—	4	2	no	—
165	C₄×D₄₇	—	8	4	no	—
166	C₄×D₄₇	—	16	2	no	—
167	Dic₉₄	—	8	2	no	—
168	C₈×C₄₇	—	2	2	no	—
169	C₈×C₄₇	—	2	1	no	—
170	C₈×C₄₇	—	2	1	no	—
171	C₈×C₄₇	—	2	1	no	—
172	C₈×C₄₇	—	4	1	no	—
173	C₈×C₄₇	—	4	1	no	—
174	C₈×C₄₇	#137	1	2	yes	—
175	C₄₇⋊C₈	—	4	2	no	—
176	#19909	—	47	1	no	maximal, Sylow
177	C₂²×C₄₇	D₄²	1	1	yes	characteristic
178	C₂×D₄₇	—	4	2	no	—
179	C₂×D₄₇	—	4	2	no	—
180	C₂×D₄₇	—	4	2	no	—
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	8	2	no	—
184	C₄×C₄₇	—	2	2	no	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	4	1	no	—
187	C₄×C₄₇	—	4	1	no	—
188	C₄×C₄₇	—	4	1	no	—
189	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
190	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
191	C₄×C₄₇	#576	1	1	yes	characteristic
192	C₄×C₄₇	#577	1	1	yes	characteristic
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	2	no	—
195	#3391	—	47	1	no	—
196	#3380	—	47	1	no	—
197	#3333	—	47	2	no	—
198	#1706	—	47	1	no	—
199	#1697	—	47	2	no	—
200	#1676	—	47	1	no	—
201	#1668	—	47	1	no	—
202	#1631	—	47	2	no	—
203	#1622	—	47	2	no	—
204	#1589	—	47	1	no	—
205	C₈×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
206	C₂×C₄₇	#3278	1	1	yes	characteristic
207	C₂×C₄₇	#3280	1	1	yes	characteristic
208	C₂×C₄₇	#3292	1	1	yes	characteristic
209	D₄₇	—	8	2	no	—
210	D₄₇	—	16	2	no	—
211	#554	—	47	2	no	—
212	#550	—	47	2	no	—
213	#500	—	47	1	no	—
214	#499	—	94	1	no	—
215	#493	—	47	1	no	—
216	#493	—	94	1	no	—
217	#492	—	47	2	no	—
218	#486	—	47	2	no	—
219	#480	—	94	2	no	—
220	#478	—	94	2	no	—
221	#477	—	47	2	no	—
222	#476	—	47	2	no	—
223	#468	—	94	2	no	—
224	#460	—	47	2	no	—
225	#459	—	47	2	no	—
226	#459	—	94	2	no	—
227	#452	—	94	2	no	—
228	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
229	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
230	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
231	C₄×D₈	—	47	1	no	—
232	C₄×D₈	—	94	1	no	—
233	C₄×D₈	—	94	1	no	—
234	#436	—	47	2	no	—
235	#417	—	94	2	no	—
236	#335	—	47	1	no	—
237	#335	C₂×D₄₇	1	1	yes	characteristic
238	#333	—	47	2	no	—
239	#331	—	47	2	no	—
240	Q₈⋊C₈	C₂×D₄₇	1	1	yes	characteristic
241	Q₈⋊C₈	C₂×D₄₇	1	1	yes	characteristic
242	D₄⋊C₈	—	47	2	no	—
243	C₈²	C₂×D₄₇	1	1	yes	characteristic
244	C₄₇	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	94	2	no	—
246	C₂×Q₁₆	D₁₈₈	1	1	yes	characteristic
247	C₂×SD₁₆	—	94	2	no	—
248	C₂×SD₁₆	—	188	2	no	—
249	C₂×D₈	—	47	1	no	—
250	C₂×D₈	—	94	2	no	—
251	C₂×D₈	—	94	1	no	—
252	C₂×D₈	—	94	1	no	—
253	C₂×M₁₆	—	94	2	no	—
254	C₄⋊Q₈	—	47	2	no	—
255	#127	—	47	2	no	—
256	#127	—	94	2	no	—
257	#125	—	94	2	no	—
258	C₂²⋊Q₈	—	94	2	no	—
259	C₂²⋊Q₈	—	188	2	no	—
260	#121	—	94	2	no	—
261	#121	—	188	2	no	—
262	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
263	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
264	C₄×D₄	—	47	2	no	—
265	C₄×D₄	—	94	2	no	—
266	C₄×D₄	—	94	2	no	—
267	C₂×C₄⋊C₄	—	94	2	no	—
268	#107	—	47	1	no	—
269	#107	—	94	1	no	—
270	#107	—	94	1	no	—
271	#107	—	94	2	no	—
272	#107	D₁₈₈	1	1	yes	characteristic
273	#106	—	94	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	47	2	no	—
277	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
279	#103	—	94	2	no	—
280	#103	D₁₈₈	1	1	yes	characteristic
281	#103	D₁₈₈	1	1	yes	characteristic
282	#102	—	47	2	no	—
283	#102	—	94	2	no	—
284	#102	—	94	2	no	—
285	#102	—	94	2	no	—
286	#102	—	94	2	no	—
287	#102	—	94	2	no	—
288	C₂²⋊C₈	—	94	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
294	#41	—	376	2	no	—
295	#41	—	188	4	no	—
296	C₂×Q₈	—	94	2	no	—
297	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
298	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
299	C₂×D₄	—	47	2	no	—
300	C₂×D₄	—	94	2	no	—
301	C₂×D₄	—	94	2	no	—
302	C₂×D₄	—	94	2	no	—
303	C₂×D₄	—	188	2	no	—
304	C₂×D₄	—	188	2	no	—
305	C₂²×C₄	—	94	2	no	—
306	C₂²×C₄	—	188	2	no	—
307	C₂²×C₄	—	188	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	376	4	no	—
310	SD₁₆	—	376	2	no	—
311	D₈	—	94	2	no	—
312	D₈	—	376	1	no	—
313	D₈	—	376	2	no	—
314	D₈	—	188	2	no	—
315	M₁₆	—	376	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	94	2	no	—
320	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
321	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
322	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
323	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	47	2	no	—
327	C₄⋊C₄	—	94	2	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	—	188	2	no	—
330	C₄⋊C₄	—	188	2	no	—
331	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
332	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
333	C₂²⋊C₄	—	94	2	no	—
334	C₂²⋊C₄	—	188	2	no	—
335	C₂²⋊C₄	—	188	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	C₂³×D₄₇	1	1	yes	characteristic
339	C₂³	—	94	2	no	—
340	C₂³	—	188	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	376	2	no	—
345	Q₈	#13094046	1	2	yes	—
346	D₄	—	94	2	no	—
347	D₄	—	94	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	376	4	no	—
350	D₄	—	188	2	no	—
351	D₄	—	188	2	no	—
352	D₄	—	188	2	no	—
353	D₄	—	188	2	no	—
354	D₄	—	752	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	376	4	no	—
361	C₂×C₄	—	188	2	no	—
362	C₂×C₄	—	188	2	no	—
363	C₂×C₄	—	188	2	no	—
364	C₂×C₄	—	752	2	no	—
365	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
367	C₂×C₄	#13094070	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	188	2	no	—
375	C₈	#13094056	1	2	yes	—
376	C₂²	—	376	2	no	—
377	C₂²	—	376	2	no	—
378	C₂²	—	376	2	no	—
379	C₂²	—	188	2	no	—
380	C₂²	—	188	2	no	—
381	C₂²	—	188	2	no	—
382	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	376	2	no	—
389	C₄	—	188	2	no	—
390	C₄	C₂×#13094046	1	1	yes	characteristic
391	C₄	C₂×#13094056	1	1	yes	characteristic
392	C₄	#15689928	1	1	yes	characteristic
393	C₄	#15689930	1	1	yes	characteristic
394	C₂	—	376	2	no	—
395	C₂	—	752	2	no	—
396	C₂	#20255055	1	1	yes	characteristic
397	C₂	#20255075	1	1	yes	characteristic
398	C₂	#20255188	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20256363	C₂
hypercenter	C₈×Q₁₆	D₄₇
hypocenter	C₄₇	#19909
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20256256	C₂	semidirect
3. #20255728	C₂	semidirect
4. #20242980	C₂	semidirect
5. #20242922	C₂	semidirect
6. #20242792	C₂	semidirect
7. #20242748	C₂	semidirect
8. #20242517	C₂	semidirect
9. #20242463	C₂	semidirect
10. #20242274	C₂	semidirect
11. C₈×Q₁₆×C₄₇	C₂	semidirect
205. C₈×Q₁₆	D₄₇	semidirect
244. C₄₇	#19909	semidirect

Maximal subgroups

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

subgroup	index
#20256363	2
#20256256	2
#20255728	2
#20242980	2
#20242922	2
#20242792	2
#20242748	2
#20242517	2
#20242463	2
#20242274	2
C₈×Q₁₆×C₄₇	2
#19909	47