Subgroups — #26207592

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255736	C₂	1	1	yes	maximal, characteristic
3	#20255035	C₂	1	2	yes	maximal, complemented
4	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242878	C₂	1	2	yes	maximal, complemented
6	#20242782	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242777	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242639	C₂	1	2	yes	maximal, complemented
9	#20242625	C₂	1	2	yes	maximal, complemented
10	#20242274	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094053	—	2	2	no	—
13	Q₈×D₁₈₈	C₂²	1	2	yes	—
14	#15689777	C₂²	1	2	yes	—
15	#15689528	—	2	1	no	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689528	C₂²	1	1	yes	characteristic
18	#15689528	C₂²	1	2	yes	—
19	#15689503	C₂²	1	2	yes	—
20	#15689437	—	2	2	no	—
21	#15689404	—	2	2	no	—
22	#15689404	C₂²	1	2	yes	—
23	#15689404	C₂²	1	2	yes	—
24	#15689382	—	2	2	no	—
25	#15689359	—	2	1	no	—
26	#15689333	C₂²	1	2	yes	—
27	#15689328	C₂²	1	2	yes	—
28	#15689286	—	2	2	no	—
29	#15689280	—	2	2	no	—
30	C₄×D₃₇₆	—	2	1	no	—
31	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
32	#15689202	C₂²	1	2	yes	—
33	C₂×#13093958	—	2	1	no	—
34	#15688876	C₂²	1	1	yes	characteristic
35	#15688868	C₂²	1	2	yes	—
36	#15688868	C₂²	1	2	yes	—
37	#15688852	C₂²	1	2	yes	—
38	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#334	C₂²	1	1	yes	characteristic
42	C₄₇×Q₈⋊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×#10625352	—	2	2	no	—
46	C₂×Q₈×D₄₇	—	2	2	no	—
47	#13094053	—	4	8	no	—
48	C₂×Dic₁₈₈	—	2	2	no	—
49	C₂×Dic₁₈₈	—	2	2	no	—
50	C₂×#10625337	—	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	—	2	2	no	—
55	#13094024	C₂³	1	2	yes	—
56	#13094024	C₂³	1	2	yes	—
57	#13094008	—	4	2	no	—
58	#13094004	—	2	2	no	—
59	#13094004	—	2	2	no	—
60	#13094004	—	4	2	no	—
61	#13093998	—	2	2	no	—
62	C₄×D₁₈₈	—	2	2	no	—
63	C₄×D₁₈₈	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₄₇	—	2	2	no	—
65	#13093958	—	2	1	no	—
66	#13093958	—	2	1	no	—
67	#13093958	—	2	1	no	—
68	#13093958	—	2	1	no	—
69	#13093958	—	2	2	no	—
70	#13093958	—	2	2	no	—
71	#13093958	—	4	2	no	—
72	#13093958	D₄	1	1	yes	characteristic
73	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
74	#13093947	—	2	2	no	—
75	#13093947	—	2	2	no	—
76	#13093946	—	2	2	no	—
77	#13093946	—	2	2	no	—
78	#13093946	D₄	1	2	yes	—
79	(C₂×D₄₇)⋊C₈	—	2	2	no	—
80	C₂×SD₁₆×C₄₇	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₄₇	—	2	1	no	—
82	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
84	#106×C₄₇	D₄	1	1	yes	characteristic
85	C₄⋊C₈×C₄₇	—	2	1	no	—
86	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
88	#103×C₄₇	D₄	1	1	yes	characteristic
89	#102×C₄₇	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₄₇	—	2	1	no	—
91	C₄×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×Q₈×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	C₂×C₂²⋊D₄₇	—	4	2	no	—
99	C₂²×Dic₄₇	—	4	2	no	—
100	Q₈×D₄₇	—	4	4	no	—
101	Q₈×D₄₇	—	8	2	no	—
102	#10625352	—	4	4	no	—
103	#10625352	—	8	2	no	—
104	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
105	C₂×C₄×D₄₇	—	2	2	no	—
106	C₂×C₄×D₄₇	—	4	2	no	—
107	C₂×Dic₉₄	—	2	2	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	4	2	no	—
110	SD₁₆×C₄₇	D₈	1	4	yes	complemented
111	C₂×C₈×C₄₇	—	2	1	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₂×C₈×C₄₇	—	2	1	no	—
114	C₂×C₈×C₄₇	—	4	2	no	—
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₇	—	2	1	no	—
120	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
122	C₂²⋊C₄×C₄₇	—	2	1	no	—
123	C₄²×C₄₇	—	2	1	no	—
124	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
125	#10625337	—	4	4	no	—
126	#10625335	—	4	4	no	—
127	(C₂×D₄₇)⋊C₄	—	2	2	no	—
128	(C₂×D₄₇)⋊C₄	—	4	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	—	4	2	no	—
136	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
137	C₂×C₄₇⋊C₈	—	2	2	no	—
138	Dic₁₈₈	—	4	4	no	—
139	Dic₁₈₈	—	8	2	no	—
140	D₃₇₆	—	2	2	no	—
141	#10625324	—	4	4	no	—
142	C₂³×C₄₇	—	2	1	no	—
143	C₂²×D₄₇	—	2	2	no	—
144	Q₈×C₄₇	—	2	1	no	—
145	Q₈×C₄₇	C₂×D₈	1	2	yes	—
146	D₄×C₄₇	—	2	1	no	—
147	D₄×C₄₇	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	2	1	no	—
150	C₂×C₄×C₄₇	—	2	1	no	—
151	C₂×C₄×C₄₇	—	4	1	no	—
152	C₂×C₄×C₄₇	—	4	1	no	—
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₄₇	—	8	4	no	—
157	C₂×Dic₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	C₂×Dic₄₇	—	4	2	no	—
160	C₂×Dic₄₇	—	8	4	no	—
161	D₁₈₈	—	2	2	no	—
162	D₁₈₈	—	2	2	no	—
163	C₄×D₄₇	—	4	2	no	—
164	C₄×D₄₇	—	4	2	no	—
165	C₄×D₄₇	—	8	4	no	—
166	Dic₉₄	—	4	2	no	—
167	Dic₉₄	—	4	2	no	—
168	Dic₉₄	—	4	2	no	—
169	Dic₉₄	—	4	2	no	—
170	Dic₉₄	—	8	4	no	—
171	C₈×C₄₇	—	2	2	no	—
172	C₈×C₄₇	—	2	1	no	—
173	C₈×C₄₇	—	2	1	no	—
174	C₈×C₄₇	—	4	1	no	—
175	C₈×C₄₇	—	4	1	no	—
176	C₈×C₄₇	C₂×D₈	1	2	yes	—
177	C₈×C₄₇	#137	1	2	yes	complemented
178	C₄₇⋊C₈	—	4	2	no	—
179	#19887	—	47	1	no	maximal, Sylow
180	C₂²×C₄₇	—	2	2	no	—
181	C₂²×C₄₇	—	4	1	no	—
182	C₂²×C₄₇	—	4	1	no	—
183	C₂²×C₄₇	D₄²	1	1	yes	characteristic
184	C₂×D₄₇	—	4	2	no	—
185	C₂×D₄₇	—	4	2	no	—
186	C₂×D₄₇	—	4	2	no	—
187	C₄×C₄₇	—	2	2	no	—
188	C₄×C₄₇	—	2	1	no	—
189	C₄×C₄₇	—	4	1	no	—
190	C₄×C₄₇	—	4	1	no	—
191	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₇	#576	1	1	yes	characteristic
194	C₄×C₄₇	#578	1	1	yes	characteristic
195	Dic₄₇	—	4	2	no	—
196	Dic₄₇	—	8	2	no	—
197	Dic₄₇	—	8	2	no	—
198	Q₈×D₈	—	47	1	no	—
199	#3334	—	47	1	no	—
200	#3276	—	47	2	no	—
201	#1713	—	47	1	no	—
202	#1689	—	47	2	no	—
203	#1674	—	47	1	no	—
204	#1673	—	47	1	no	—
205	#1649	—	47	2	no	—
206	#1647	—	47	2	no	—
207	#1589	—	47	1	no	—
208	C₈×SD₁₆	D₄₇	1	1	yes	characteristic, complemented
209	C₂×C₄₇	—	4	2	no	—
210	C₂×C₄₇	#3276	1	1	yes	characteristic
211	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
212	C₂×C₄₇	#3293	1	1	yes	characteristic
213	D₄₇	—	8	2	no	—
214	C₂×#137	—	94	2	no	—
215	D₄×Q₈	—	47	2	no	—
216	#547	—	47	2	no	—
217	#500	—	47	1	no	—
218	#500	—	47	1	no	—
219	#500	—	47	2	no	—
220	#500	—	94	1	no	—
221	#494	—	47	2	no	—
222	#480	—	94	2	no	—
223	#474	—	47	2	no	—
224	#474	—	47	2	no	—
225	#474	—	94	2	no	—
226	#470	—	94	2	no	—
227	#466	—	94	1	no	—
228	#462	—	47	2	no	—
229	#461	—	47	2	no	—
230	#452	—	94	2	no	—
231	#451	—	94	2	no	—
232	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
233	C₄×SD₁₆	C₂×D₄₇	1	1	yes	characteristic
234	C₄×D₈	—	47	1	no	—
235	C₄×D₈	—	94	1	no	—
236	#436	—	47	2	no	—
237	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
238	C₂×#107	—	94	1	no	—
239	#335	—	47	1	no	—
240	#334	C₂×D₄₇	1	1	yes	characteristic
241	#332	—	47	2	no	—
242	#332	—	47	2	no	—
243	Q₈⋊C₈	C₂×D₄₇	1	1	yes	characteristic
244	D₄⋊C₈	—	47	2	no	—
245	D₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
246	C₈²	C₂×D₄₇	1	1	yes	characteristic
247	C₄₇	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	94	2	no	—
249	C₂²×Q₈	—	94	2	no	—
250	#137	—	188	8	no	—
251	C₂×Q₁₆	—	94	2	no	—
252	C₂×Q₁₆	—	94	2	no	—
253	C₂×Q₁₆	—	94	2	no	—
254	C₂×SD₁₆	—	94	2	no	—
255	C₂×SD₁₆	D₁₈₈	1	1	yes	characteristic
256	C₂×D₈	—	47	1	no	—
257	C₂×M₁₆	—	94	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	47	2	no	—
260	C₄⋊Q₈	—	47	2	no	—
261	C₄⋊Q₈	—	94	2	no	—
262	#123	—	188	2	no	—
263	C₂²⋊Q₈	—	94	2	no	—
264	C₂²⋊Q₈	—	94	2	no	—
265	C₂²⋊Q₈	—	188	2	no	—
266	#121	—	94	2	no	—
267	C₄×Q₈	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×D₄	—	47	2	no	—
269	C₄×D₄	—	94	2	no	—
270	C₄×D₄	C₂²×D₄₇	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	94	2	no	—
272	#107	—	47	1	no	—
273	#107	—	94	2	no	—
274	#107	—	94	2	no	—
275	#107	—	94	1	no	—
276	#107	—	94	1	no	—
277	#107	—	94	1	no	—
278	#107	—	94	1	no	—
279	#107	—	188	2	no	—
280	#106	D₁₈₈	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	47	2	no	—
283	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
284	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
285	#103	—	94	2	no	—
286	#103	—	94	2	no	—
287	#103	D₁₈₈	1	1	yes	characteristic
288	#102	—	47	2	no	—
289	#102	—	94	2	no	—
290	#102	—	94	2	no	—
291	#102	D₁₈₈	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	94	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
298	#41	—	376	2	no	—
299	#41	—	188	4	no	—
300	C₂×Q₈	—	94	2	no	—
301	C₂×Q₈	—	94	2	no	—
302	C₂×Q₈	—	376	2	no	—
303	C₂×Q₈	—	188	2	no	—
304	C₂×Q₈	—	188	4	no	—
305	C₂×Q₈	C₂×D₁₈₈	1	1	yes	characteristic
306	C₂×D₄	—	47	2	no	—
307	C₂×D₄	—	188	2	no	—
308	C₂×D₄	C₂×D₁₈₈	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	94	2	no	—
311	C₂²×C₄	—	188	2	no	—
312	C₂²×C₄	—	188	2	no	—
313	Q₁₆	—	376	2	no	—
314	Q₁₆	—	188	4	no	—
315	Q₁₆	—	188	4	no	—
316	SD₁₆	—	188	4	no	—
317	SD₁₆	D₃₇₆	1	4	yes	complemented
318	D₈	—	94	2	no	—
319	M₁₆	—	188	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	94	2	no	—
325	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
326	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
327	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
328	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	47	2	no	—
331	C₄⋊C₄	—	94	2	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₄⋊C₄	—	94	2	no	—
334	C₄⋊C₄	—	94	2	no	—
335	C₄⋊C₄	—	188	2	no	—
336	C₄⋊C₄	—	188	2	no	—
337	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₁₈₈	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	94	2	no	—
341	C₂²⋊C₄	—	188	2	no	—
342	C₂²⋊C₄	—	188	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₄₇	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	94	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	376	4	no	—
349	Q₈	—	188	2	no	—
350	Q₈	—	188	2	no	—
351	Q₈	—	188	2	no	—
352	Q₈	—	188	2	no	—
353	Q₈	C₂×D₃₇₆	1	2	yes	—
354	D₄	—	2	1	no	—
355	D₄	—	94	2	no	—
356	D₄	—	94	2	no	—
357	D₄	—	376	4	no	—
358	D₄	C₂×D₃₇₆	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	94	2	no	—
365	C₂×C₄	—	376	4	no	—
366	C₂×C₄	—	376	4	no	—
367	C₂×C₄	—	188	2	no	—
368	C₂×C₄	—	188	2	no	—
369	C₂×C₄	—	188	2	no	—
370	C₂×C₄	—	188	2	no	—
371	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
373	C₂×C₄	#13094070	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	188	2	no	—
380	C₈	C₂×D₃₇₆	1	2	yes	—
381	C₈	#13094053	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	188	2	no	—
386	C₂²	—	188	2	no	—
387	C₂²	—	188	2	no	—
388	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	376	2	no	—
394	C₄	—	376	2	no	—
395	C₄	—	188	2	no	—
396	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
397	C₄	C₂×#13094053	1	1	yes	characteristic
398	C₄	#15689928	1	1	yes	characteristic
399	C₄	#15689937	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	376	2	no	—
402	C₂	#20255035	1	1	yes	characteristic
403	C₂	D₄×D₃₇₆	1	1	yes	characteristic
404	C₂	#20255193	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×SD₁₆×C₄₇	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20255736	C₂
hypercenter	C₈×SD₁₆	D₄₇
hypocenter	C₄₇	#19887
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₃₇₆	C₂	semidirect
3. #20255035	C₂	semidirect
4. #20243006	C₂	semidirect
5. #20242878	C₂	semidirect
6. #20242782	C₂	semidirect
7. #20242777	C₂	semidirect
8. #20242639	C₂	semidirect
9. #20242625	C₂	semidirect
10. #20242274	C₂	semidirect
11. C₈×SD₁₆×C₄₇	C₂	semidirect
110. SD₁₆×C₄₇	D₈	semidirect
177. C₈×C₄₇	#137	semidirect
208. C₈×SD₁₆	D₄₇	semidirect
247. C₄₇	#19887	semidirect
317. SD₁₆	D₃₇₆	semidirect
381. C₈	#13094053	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₃₇₆	2
#20255736	2
#20255035	2
#20243006	2
#20242878	2
#20242782	2
#20242777	2
#20242639	2
#20242625	2
#20242274	2
C₈×SD₁₆×C₄₇	2
#19887	47