Subgroups — #26207795

Order 12032.

672 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20255736	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255695	C₂	1	2	yes	maximal, complemented
3	#20255110	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242985	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242936	C₂	1	2	yes	maximal, complemented
6	#20242777	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242753	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242675	C₂	1	2	yes	maximal, complemented
9	#20242674	C₂	1	2	yes	maximal, complemented
10	#20242274	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	—	2	1	no	—
13	#15689778	C₂²	1	2	yes	—
14	#15689777	C₂²	1	2	yes	—
15	#15689528	C₂²	1	1	yes	characteristic
16	#15689521	C₂²	1	2	yes	—
17	#15689503	C₂²	1	1	yes	characteristic
18	#15689474	C₂²	1	2	yes	—
19	#15689460	—	2	2	no	—
20	#15689437	—	2	2	no	—
21	#15689404	C₂²	1	2	yes	—
22	#15689403	C₂²	1	2	yes	—
23	#15689368	—	2	2	no	—
24	#15689364	—	2	1	no	—
25	#15689359	—	2	1	no	—
26	#15689329	C₂²	1	2	yes	—
27	#15689328	C₂²	1	2	yes	—
28	#15689287	—	2	2	no	—
29	#15689265	—	2	2	no	—
30	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
31	#15689202	C₂²	1	2	yes	—
32	C₂×#13093958	—	2	1	no	—
33	#15689121	—	2	2	no	—
34	#15688876	C₂²	1	1	yes	characteristic
35	#15688870	C₂²	1	2	yes	—
36	#15688868	C₂²	1	2	yes	—
37	#15688852	C₂²	1	2	yes	—
38	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#335	C₂²	1	1	yes	characteristic
42	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×#10625350	—	2	2	no	—
46	C₂×#10625352	—	2	2	no	—
47	#13094046	—	4	4	no	—
48	C₂×Dic₁₈₈	—	2	1	no	—
49	C₂×Dic₁₈₈	—	2	1	no	—
50	C₂×#10625325	—	2	2	no	—
51	C₂×#10625335	—	2	2	no	—
52	C₂×D₃₇₆	D₄	1	1	yes	characteristic
53	C₂×#10625324	—	2	2	no	—
54	#13094024	C₂³	1	2	yes	—
55	#13094024	C₂³	1	2	yes	—
56	#13094008	—	4	2	no	—
57	#13094006	—	4	2	no	—
58	#13094004	—	2	2	no	—
59	#13094003	—	2	2	no	—
60	#13093998	—	2	2	no	—
61	#13093995	—	2	2	no	—
62	C₄×D₁₈₈	C₂³	1	2	yes	—
63	C₂×C₄⋊Dic₄₇	—	2	2	no	—
64	C₄⋊C₄×D₄₇	—	2	2	no	—
65	#13093958	—	2	1	no	—
66	#13093958	—	2	1	no	—
67	#13093958	—	2	2	no	—
68	#13093958	D₄	1	1	yes	characteristic
69	#13093957	—	2	2	no	—
70	#13093956	—	4	2	no	—
71	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
72	#13093949	—	2	2	no	—
73	#13093949	—	2	2	no	—
74	#13093949	—	2	2	no	—
75	#13093947	—	2	2	no	—
76	#13093946	—	2	2	no	—
77	#13093946	D₄	1	2	yes	—
78	#13093945	—	2	2	no	—
79	(C₂×D₄₇)⋊C₈	—	2	2	no	—
80	C₂×D₈×C₄₇	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₄₇	—	2	1	no	—
82	C₂²×C₈×C₄₇	—	2	1	no	—
83	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
85	#107×C₄₇	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
88	#102×C₄₇	D₄	1	1	yes	characteristic
89	#102×C₄₇	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₄₇	—	2	1	no	—
91	C₂²⋊C₈×C₄₇	—	2	1	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	—	2	1	no	—
98	C₂²×C₄×C₄₇	—	2	1	no	—
99	C₂×C₂²⋊D₄₇	—	4	2	no	—
100	C₂×C₂²⋊D₄₇	—	4	2	no	—
101	C₂²×Dic₄₇	—	4	2	no	—
102	#10625352	—	4	4	no	—
103	#10625352	—	8	2	no	—
104	#10625350	—	4	4	no	—
105	#10625350	—	8	2	no	—
106	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₄₇	—	2	2	no	—
108	C₂×C₄×D₄₇	—	4	2	no	—
109	C₂×Dic₉₄	—	2	2	no	—
110	C₂×Dic₉₄	—	2	2	no	—
111	D₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₂×C₈×C₄₇	—	2	1	no	—
114	C₂×C₈×C₄₇	—	2	1	no	—
115	C₂×C₈×C₄₇	—	4	2	no	—
116	C₂×C₈×C₄₇	—	4	2	no	—
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
123	C₂²⋊C₄×C₄₇	—	2	1	no	—
124	C₂²⋊C₄×C₄₇	—	2	1	no	—
125	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊Dic₄₇	—	4	2	no	—
127	#10625335	—	8	2	no	—
128	(C₂×D₄₇)⋊C₄	—	2	2	no	—
129	(C₂×D₄₇)⋊C₄	—	4	2	no	—
130	C₄⋊Dic₄₇	—	2	2	no	—
131	C₄⋊Dic₄₇	—	2	2	no	—
132	C₄⋊Dic₄₇	—	2	2	no	—
133	C₄⋊Dic₄₇	—	2	2	no	—
134	C₄⋊Dic₄₇	—	4	2	no	—
135	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
136	Dic₄₇⋊C₄	—	4	2	no	—
137	C₂×C₄₇⋊C₈	—	2	2	no	—
138	Dic₁₈₈	—	4	2	no	—
139	Dic₁₈₈	—	8	1	no	—
140	D₃₇₆	—	2	2	no	—
141	#10625325	—	4	4	no	—
142	#10625324	—	8	2	no	—
143	C₂³×C₄₇	—	2	1	no	—
144	C₂³×C₄₇	—	2	1	no	—
145	C₂²×D₄₇	—	2	2	no	—
146	D₄×C₄₇	—	2	1	no	—
147	D₄×C₄₇	—	2	1	no	—
148	D₄×C₄₇	—	2	1	no	—
149	D₄×C₄₇	#135	1	2	yes	complemented
150	C₂×C₄×C₄₇	—	2	1	no	—
151	C₂×C₄×C₄₇	—	2	1	no	—
152	C₂×C₄×C₄₇	—	4	1	no	—
153	C₂×C₄×C₄₇	—	4	1	no	—
154	C₂×C₄×C₄₇	—	4	1	no	—
155	C₂×C₄×C₄₇	—	4	1	no	—
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²⋊D₄₇	—	8	4	no	—
160	C₂²⋊D₄₇	—	8	4	no	—
161	C₂×Dic₄₇	—	2	2	no	—
162	C₂×Dic₄₇	—	4	2	no	—
163	C₂×Dic₄₇	—	4	2	no	—
164	C₂×Dic₄₇	—	8	4	no	—
165	D₁₈₈	—	2	2	no	—
166	D₁₈₈	—	2	2	no	—
167	C₄×D₄₇	—	4	2	no	—
168	C₄×D₄₇	—	4	2	no	—
169	C₄×D₄₇	—	8	4	no	—
170	Dic₉₄	—	4	2	no	—
171	Dic₉₄	—	4	2	no	—
172	Dic₉₄	—	4	2	no	—
173	Dic₉₄	—	4	2	no	—
174	C₈×C₄₇	—	2	2	no	—
175	C₈×C₄₇	—	2	1	no	—
176	C₈×C₄₇	—	2	1	no	—
177	C₈×C₄₇	—	2	1	no	—
178	C₈×C₄₇	—	4	1	no	—
179	C₈×C₄₇	—	4	1	no	—
180	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
181	C₄₇⋊C₈	—	4	2	no	—
182	#19902	—	47	1	no	maximal, Sylow
183	C₂²×C₄₇	—	2	2	no	—
184	C₂²×C₄₇	—	4	1	no	—
185	C₂²×C₄₇	—	4	1	no	—
186	C₂²×C₄₇	—	4	1	no	—
187	C₂²×C₄₇	—	4	1	no	—
188	C₂²×C₄₇	—	4	1	no	—
189	C₂²×C₄₇	D₄²	1	1	yes	characteristic
190	C₂×D₄₇	—	4	2	no	—
191	C₂×D₄₇	—	4	2	no	—
192	C₂×D₄₇	—	4	2	no	—
193	C₄×C₄₇	—	2	1	no	—
194	C₄×C₄₇	—	4	1	no	—
195	C₄×C₄₇	—	4	1	no	—
196	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
197	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₄₇	#576	1	1	yes	characteristic
199	C₄×C₄₇	#577	1	1	yes	characteristic
200	Dic₄₇	—	4	2	no	—
201	Dic₄₇	—	8	2	no	—
202	Dic₄₇	—	8	2	no	—
203	#3334	—	47	1	no	—
204	#3330	—	47	2	no	—
205	#3283	—	47	1	no	—
206	#1707	—	47	1	no	—
207	#1699	—	47	2	no	—
208	#1673	—	47	1	no	—
209	#1669	—	47	1	no	—
210	#1654	—	47	2	no	—
211	#1654	—	47	2	no	—
212	#1589	—	47	1	no	—
213	C₈×D₈	D₄₇	1	1	yes	characteristic, complemented
214	C₂×C₄₇	—	4	2	no	—
215	C₂×C₄₇	—	8	1	no	—
216	C₂×C₄₇	#3272	1	1	yes	characteristic
217	C₂×C₄₇	#3280	1	1	yes	characteristic
218	C₂×C₄₇	#3293	1	1	yes	characteristic
219	D₄₇	—	8	2	no	—
220	C₂×#135	—	94	1	no	—
221	#547	—	47	2	no	—
222	#547	—	47	2	no	—
223	#500	—	47	1	no	—
224	#498	—	47	2	no	—
225	#494	—	47	1	no	—
226	#487	—	47	2	no	—
227	#484	—	94	2	no	—
228	#480	—	94	2	no	—
229	#474	—	47	2	no	—
230	#474	—	47	2	no	—
231	#468	—	94	2	no	—
232	#467	—	94	1	no	—
233	#466	—	94	1	no	—
234	#461	—	47	2	no	—
235	#461	—	47	2	no	—
236	#452	—	94	2	no	—
237	#449	—	94	2	no	—
238	C₄×D₈	—	47	1	no	—
239	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
240	#436	—	47	2	no	—
241	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
242	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
243	C₂×#107	—	94	1	no	—
244	#417	—	94	2	no	—
245	#335	—	47	1	no	—
246	#335	C₂×D₄₇	1	1	yes	characteristic
247	#332	—	47	2	no	—
248	#332	—	47	2	no	—
249	D₄⋊C₈	—	47	2	no	—
250	D₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
251	D₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
252	C₈²	C₂×D₄₇	1	1	yes	characteristic
253	C₄₇	#19902	1	1	yes	characteristic, complemented, Sylow
254	C₂×#41	—	94	2	no	—
255	C₂×#41	—	94	2	no	—
256	#135	—	188	4	no	—
257	C₂×Q₁₆	—	94	1	no	—
258	C₂×Q₁₆	—	94	1	no	—
259	C₂×SD₁₆	—	94	2	no	—
260	C₂×SD₁₆	—	94	2	no	—
261	C₂×D₈	—	47	1	no	—
262	C₂×D₈	D₁₈₈	1	1	yes	characteristic
263	C₂×M₁₆	—	94	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₄⋊Q₈	—	47	2	no	—
267	C₄⋊Q₈	—	47	2	no	—
268	#123	—	188	2	no	—
269	#123	—	188	2	no	—
270	C₂²⋊Q₈	—	94	2	no	—
271	C₂²⋊Q₈	—	94	2	no	—
272	#121	—	94	2	no	—
273	#121	—	94	2	no	—
274	C₄×D₄	—	47	2	no	—
275	C₄×D₄	C₂²×D₄₇	1	1	yes	characteristic
276	C₄×D₄	C₂²×D₄₇	1	1	yes	characteristic
277	C₂×C₄⋊C₄	—	94	2	no	—
278	C₂×C₄⋊C₄	—	94	2	no	—
279	#107	—	47	1	no	—
280	#107	—	94	2	no	—
281	#107	—	94	1	no	—
282	#107	—	94	1	no	—
283	#107	—	94	2	no	—
284	#107	D₁₈₈	1	1	yes	characteristic
285	#106	—	188	2	no	—
286	C₄⋊C₈	—	47	2	no	—
287	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
289	#103	—	94	2	no	—
290	#103	—	94	2	no	—
291	#103	—	94	2	no	—
292	#103	—	94	2	no	—
293	#102	—	47	2	no	—
294	#102	—	94	2	no	—
295	#102	—	94	2	no	—
296	#102	D₁₈₈	1	1	yes	characteristic
297	#102	D₁₈₈	1	1	yes	characteristic
298	C₂²⋊C₈	—	2	1	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	—	94	2	no	—
301	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
302	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
304	#41	—	376	2	no	—
305	#41	—	376	2	no	—
306	#41	—	188	4	no	—
307	#41	—	188	4	no	—
308	C₂×Q₈	—	94	2	no	—
309	C₂×Q₈	—	94	2	no	—
310	C₂×D₄	—	47	2	no	—
311	C₂×D₄	—	188	2	no	—
312	C₂×D₄	—	188	2	no	—
313	C₂×D₄	C₂×D₁₈₈	1	1	yes	characteristic
314	C₂×D₄	C₂×D₁₈₈	1	1	yes	characteristic
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	94	2	no	—
318	C₂²×C₄	—	188	2	no	—
319	C₂²×C₄	—	188	2	no	—
320	Q₁₆	—	376	1	no	—
321	Q₁₆	—	188	2	no	—
322	SD₁₆	—	376	2	no	—
323	SD₁₆	—	188	4	no	—
324	D₈	—	2	2	no	—
325	D₈	—	94	2	no	—
326	M₁₆	—	376	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	—	94	2	no	—
333	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
334	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
335	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
336	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
337	C₄⋊C₄	—	47	2	no	—
338	C₄⋊C₄	—	94	2	no	—
339	C₄⋊C₄	—	94	2	no	—
340	C₄⋊C₄	—	94	2	no	—
341	C₄⋊C₄	—	94	2	no	—
342	C₄⋊C₄	—	188	2	no	—
343	C₄⋊C₄	—	188	2	no	—
344	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
345	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	94	2	no	—
349	C₂²⋊C₄	—	188	2	no	—
350	C₂²⋊C₄	—	188	2	no	—
351	C₄²	C₂³×D₄₇	1	1	yes	characteristic
352	C₂³	—	2	1	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	94	2	no	—
355	Q₈	—	188	2	no	—
356	Q₈	—	188	2	no	—
357	Q₈	—	188	2	no	—
358	Q₈	—	188	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	2	1	no	—
361	D₄	—	2	1	no	—
362	D₄	—	94	2	no	—
363	D₄	—	94	2	no	—
364	D₄	—	376	4	no	—
365	D₄	—	376	4	no	—
366	D₄	#13094046	1	2	yes	complemented
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	94	2	no	—
374	C₂×C₄	—	376	4	no	—
375	C₂×C₄	—	376	4	no	—
376	C₂×C₄	—	188	2	no	—
377	C₂×C₄	—	188	2	no	—
378	C₂×C₄	—	188	2	no	—
379	C₂×C₄	—	188	2	no	—
380	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
382	C₂×C₄	#13094070	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	1	no	—
387	C₈	—	4	1	no	—
388	C₈	—	4	1	no	—
389	C₈	—	188	2	no	—
390	C₈	#13094048	1	2	yes	—
391	C₂²	—	2	2	no	—
392	C₂²	—	4	1	no	—
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	188	2	no	—
398	C₂²	—	188	2	no	—
399	C₂²	—	188	2	no	—
400	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	1	no	—
404	C₄	—	376	2	no	—
405	C₄	—	376	2	no	—
406	C₄	—	188	2	no	—
407	C₄	C₂×#13094046	1	1	yes	characteristic
408	C₄	C₂×#13094048	1	1	yes	characteristic
409	C₄	#15689928	1	1	yes	characteristic
410	C₄	#15689930	1	1	yes	characteristic
411	C₂	—	4	2	no	—
412	C₂	—	8	1	no	—
413	C₂	—	376	2	no	—
414	C₂	#20254989	1	1	yes	characteristic
415	C₂	#20255075	1	1	yes	characteristic
416	C₂	#20255193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×D₈×C₄₇	C₂
socle	C₂²×C₄₇	D₄²
hypercenter	C₈×D₈	D₄₇
hypocenter	C₄₇	#19902
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
1. #20255736	C₂	semidirect
2. #20255695	C₂	semidirect
3. #20255110	C₂	semidirect
4. #20242985	C₂	semidirect
5. #20242936	C₂	semidirect
6. #20242777	C₂	semidirect
7. #20242753	C₂	semidirect
8. #20242675	C₂	semidirect
9. #20242674	C₂	semidirect
10. #20242274	C₂	semidirect
11. C₈×D₈×C₄₇	C₂	semidirect
149. D₄×C₄₇	#135	semidirect
213. C₈×D₈	D₄₇	semidirect
253. C₄₇	#19902	semidirect
366. D₄	#13094046	semidirect

Maximal subgroups

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

subgroup	index
#20255736	2
#20255695	2
#20255110	2
#20242985	2
#20242936	2
#20242777	2
#20242753	2
#20242675	2
#20242674	2
#20242274	2
C₈×D₈×C₄₇	2
#19902	47