Subgroups — #26207586

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	#20255735	C₂	1	1	yes	maximal, characteristic
3	#20255029	C₂	1	2	yes	maximal, complemented
4	#20243005	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242876	C₂	1	2	yes	maximal, complemented
6	#20242778	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242635	C₂	1	2	yes	maximal, complemented
8	#20242624	C₂	1	2	yes	maximal, complemented
9	#20242272	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10625336	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094056	—	2	2	no	—
13	Q₈×D₁₈₈	C₂²	1	2	yes	—
14	#15689776	C₂²	1	2	yes	—
15	#15689528	—	2	1	no	—
16	#15689528	C₂²	1	1	yes	characteristic
17	#15689526	C₂²	1	2	yes	—
18	#15689502	C₂²	1	2	yes	—
19	#15689438	—	2	2	no	—
20	#15689404	—	2	2	no	—
21	#15689404	C₂²	1	2	yes	—
22	#15689400	C₂²	1	2	yes	—
23	#15689380	—	2	2	no	—
24	#15689360	—	2	1	no	—
25	#15689329	C₂²	1	2	yes	—
26	#15689288	—	2	2	no	—
27	#15689277	—	2	2	no	—
28	C₄×#10625336	C₂²	1	1	yes	characteristic
29	C₄×D₃₇₆	—	2	1	no	—
30	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
31	#15689200	C₂²	1	2	yes	—
32	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
33	#107×D₄₇	—	2	1	no	—
34	#15688873	C₂²	1	1	yes	characteristic
35	#15688870	C₂²	1	2	yes	—
36	Q₈⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
37	#15688850	C₂²	1	1	yes	characteristic
38	#15688850	C₂²	1	2	yes	—
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₇×#500	C₂²	1	1	yes	characteristic
41	C₄₇×#462	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
43	C₄₇×#335	C₂²	1	1	yes	characteristic
44	C₄₇×#332	C₂²	1	2	yes	—
45	C₂×#10625354	—	2	2	no	—
46	C₂×Q₈×D₄₇	—	2	2	no	—
47	#13094056	—	4	8	no	—
48	C₂×#10625337	—	2	2	no	—
49	C₂×#10625337	—	2	2	no	—
50	C₂×#10625325	—	2	2	no	—
51	C₂×#10625336	D₄	1	1	yes	characteristic
52	C₂×D₃₇₆	D₄	1	1	yes	characteristic
53	C₂×#10625324	—	2	2	no	—
54	C₂×C₈×D₄₇	—	2	1	no	—
55	#13094024	—	2	2	no	—
56	#13094024	C₂³	1	2	yes	—
57	#13094009	—	4	2	no	—
58	#13094004	—	2	2	no	—
59	#13094004	—	4	2	no	—
60	#13094000	—	2	2	no	—
61	#13093999	—	2	2	no	—
62	C₄×D₁₈₈	—	2	2	no	—
63	C₄×D₁₈₈	C₂³	1	1	yes	characteristic
64	C₄×D₁₈₈	C₂³	1	2	yes	—
65	C₄⋊C₄×D₄₇	—	2	2	no	—
66	#13093958	—	2	1	no	—
67	#13093958	—	2	1	no	—
68	#13093958	—	2	2	no	—
69	#13093958	D₄	1	1	yes	characteristic
70	#13093957	—	2	2	no	—
71	#13093957	—	4	2	no	—
72	#13093955	D₄	1	1	yes	characteristic
73	C₄⋊(C₄₇⋊C₈)	C₂³	1	1	yes	characteristic
74	#13093949	—	2	2	no	—
75	#13093948	D₄	1	1	yes	characteristic
76	#13093946	—	2	2	no	—
77	#13093946	D₄	1	2	yes	—
78	#13093944	—	2	2	no	—
79	#13093944	D₄	1	1	yes	characteristic
80	(C₂×D₄₇)⋊C₈	—	2	1	no	—
81	(C₂×D₄₇)⋊C₈	—	2	2	no	—
82	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₄₇	—	2	2	no	—
85	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
86	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
87	#107×C₄₇	—	2	1	no	—
88	#107×C₄₇	—	2	1	no	—
89	C₄⋊C₈×C₄₇	—	2	1	no	—
90	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
91	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
92	#103×C₄₇	—	2	2	no	—
93	C₄×C₈×C₄₇	—	2	1	no	—
94	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₄₇	—	2	2	no	—
96	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	#10625354	—	4	4	no	—
98	#10625354	—	8	2	no	—
99	Q₈×D₄₇	—	4	4	no	—
100	Q₈×D₄₇	—	8	2	no	—
101	C₂×D₁₈₈	—	4	2	no	—
102	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
103	C₂×D₁₈₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₄₇	—	2	1	no	—
105	C₂×C₄×D₄₇	—	2	2	no	—
106	C₂×C₄×D₄₇	—	4	2	no	—
107	C₂×C₄×D₄₇	—	4	2	no	—
108	C₂×Dic₉₄	—	2	2	no	—
109	C₂×Dic₉₄	—	4	2	no	—
110	Q₁₆×C₄₇	—	4	4	no	—
111	C₂×C₈×C₄₇	—	2	2	no	—
112	C₂×C₈×C₄₇	—	2	1	no	—
113	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₇	—	2	1	no	—
116	C₄⋊C₄×C₄₇	—	2	2	no	—
117	C₄⋊C₄×C₄₇	—	2	2	no	—
118	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	—	2	1	no	—
120	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
121	#10625337	—	4	4	no	—
122	#10625337	—	8	2	no	—
123	#10625336	D₈	1	4	yes	complemented
124	(C₂×D₄₇)⋊C₄	—	2	2	no	—
125	(C₂×D₄₇)⋊C₄	—	2	1	no	—
126	(C₂×D₄₇)⋊C₄	—	4	2	no	—
127	(C₂×D₄₇)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₇	—	2	2	no	—
129	C₄⋊Dic₄₇	—	2	2	no	—
130	C₄⋊Dic₄₇	—	4	2	no	—
131	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
132	C₄⋊Dic₄₇	C₂×D₄	1	1	yes	characteristic
133	Dic₄₇⋊C₄	—	4	2	no	—
134	C₂×C₄₇⋊C₈	—	2	1	no	—
135	C₂×C₄₇⋊C₈	—	2	1	no	—
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	D₃₇₆	—	2	2	no	—
139	#10625325	—	4	4	no	—
140	#10625324	—	4	4	no	—
141	C₈×D₄₇	—	4	2	no	—
142	C₂²×D₄₇	—	2	1	no	—
143	C₂²×D₄₇	—	2	2	no	—
144	Q₈×C₄₇	—	2	1	no	—
145	Q₈×C₄₇	—	4	2	no	—
146	Q₈×C₄₇	—	4	2	no	—
147	Q₈×C₄₇	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₄₇	—	2	1	no	—
149	C₂×C₄×C₄₇	—	2	1	no	—
150	C₂×C₄×C₄₇	—	4	2	no	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₄₇	—	2	1	no	—
155	C₂×Dic₄₇	—	2	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	D₁₈₈	—	2	2	no	—
158	D₁₈₈	—	2	1	no	—
159	D₁₈₈	—	2	2	no	—
160	D₁₈₈	—	8	4	no	—
161	D₁₈₈	C₂×D₈	1	2	yes	—
162	C₄×D₄₇	—	4	2	no	—
163	C₄×D₄₇	—	4	1	no	—
164	C₄×D₄₇	—	4	2	no	—
165	C₄×D₄₇	—	4	1	no	—
166	C₄×D₄₇	—	8	4	no	—
167	C₄×D₄₇	—	8	4	no	—
168	Dic₉₄	—	4	2	no	—
169	Dic₉₄	—	4	2	no	—
170	Dic₉₄	—	8	4	no	—
171	C₈×C₄₇	—	2	1	no	—
172	C₈×C₄₇	—	4	2	no	—
173	C₈×C₄₇	—	4	1	no	—
174	C₈×C₄₇	#137	1	2	yes	complemented
175	C₄₇⋊C₈	—	2	2	no	—
176	C₄₇⋊C₈	—	2	1	no	—
177	C₄₇⋊C₈	—	4	1	no	—
178	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
179	#19887	—	47	1	no	maximal, Sylow
180	C₂²×C₄₇	D₄²	1	1	yes	characteristic
181	C₂×D₄₇	—	2	2	no	—
182	C₂×D₄₇	—	4	1	no	—
183	C₂×D₄₇	—	4	1	no	—
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	—	8	2	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	1	no	—
196	Dic₄₇	—	4	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	D₄₇	1	1	yes	characteristic, complemented
204	#1673	—	47	1	no	—
205	#1649	—	47	2	no	—
206	#1647	—	47	2	no	—
207	#1589	—	47	1	no	—
208	C₈×SD₁₆	—	47	1	no	—
209	C₂×C₄₇	#3276	1	1	yes	characteristic
210	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
211	C₂×C₄₇	#3293	1	1	yes	characteristic
212	D₄₇	—	4	2	no	—
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	2	no	—
218	#500	—	47	1	no	—
219	#500	—	94	1	no	—
220	#500	C₂×D₄₇	1	1	yes	characteristic
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	C₂×D₄₇	1	2	yes	—
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	—	47	1	no	—
234	C₄×D₈	—	47	1	no	—
235	C₄×D₈	—	94	1	no	—
236	#436	—	47	2	no	—
237	C₈×D₄	—	47	1	no	—
238	C₂×#107	—	94	1	no	—
239	#335	C₂×D₄₇	1	1	yes	characteristic
240	#334	—	47	1	no	—
241	#332	—	47	2	no	—
242	#332	C₂×D₄₇	1	2	yes	—
243	Q₈⋊C₈	—	47	1	no	—
244	D₄⋊C₈	—	47	1	no	—
245	D₄⋊C₈	—	47	2	no	—
246	C₈²	—	47	1	no	—
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	—	2	2	no	—
252	C₂×Q₁₆	—	94	2	no	—
253	C₂×Q₁₆	—	94	2	no	—
254	C₂×SD₁₆	—	47	1	no	—
255	C₂×SD₁₆	—	94	2	no	—
256	C₂×D₈	—	47	1	no	—
257	C₂×M₁₆	—	94	2	no	—
258	C₂²×C₈	—	94	1	no	—
259	C₄⋊Q₈	—	47	2	no	—
260	C₄⋊Q₈	—	94	2	no	—
261	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
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	1	no	—
269	C₄×D₄	—	47	2	no	—
270	C₄×D₄	—	94	2	no	—
271	C₂×C₄⋊C₄	—	94	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	47	1	no	—
275	#107	—	94	2	no	—
276	#107	—	94	2	no	—
277	#107	—	94	1	no	—
278	#107	—	94	1	no	—
279	#107	—	188	2	no	—
280	#106	—	47	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	47	1	no	—
283	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
284	C₄⋊C₈	C₂²×D₄₇	1	1	yes	characteristic
285	#103	—	2	2	no	—
286	#103	—	47	1	no	—
287	#103	—	94	2	no	—
288	#102	—	47	2	no	—
289	#102	—	47	1	no	—
290	#102	—	94	2	no	—
291	#102	—	94	2	no	—
292	C₂²⋊C₈	—	94	2	no	—
293	C₂²⋊C₈	—	94	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	47	1	no	—
296	C₄×C₈	—	47	1	no	—
297	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
298	#41	—	376	2	no	—
299	#41	—	188	4	no	—
300	C₂×Q₈	—	2	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	D₄×D₄₇	1	1	yes	characteristic
306	C₂×D₄	—	47	1	no	—
307	C₂×D₄	—	47	2	no	—
308	C₂×D₄	—	188	2	no	—
309	C₂²×C₄	—	94	1	no	—
310	C₂²×C₄	—	94	2	no	—
311	C₂²×C₄	—	188	2	no	—
312	C₂²×C₄	—	188	2	no	—
313	Q₁₆	—	4	4	no	—
314	Q₁₆	—	376	2	no	—
315	Q₁₆	—	188	4	no	—
316	SD₁₆	—	47	4	no	—
317	SD₁₆	—	188	4	no	—
318	D₈	—	94	2	no	—
319	M₁₆	—	188	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	47	1	no	—
323	C₂×C₈	—	47	1	no	—
324	C₂×C₈	—	94	1	no	—
325	C₂×C₈	—	94	1	no	—
326	C₂×C₈	—	188	2	no	—
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	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	47	2	no	—
333	C₄⋊C₄	—	47	1	no	—
334	C₄⋊C₄	—	94	2	no	—
335	C₄⋊C₄	—	94	2	no	—
336	C₄⋊C₄	—	188	2	no	—
337	C₄⋊C₄	—	188	2	no	—
338	C₄⋊C₄	D₄×D₄₇	1	1	yes	characteristic
339	C₂²⋊C₄	—	94	2	no	—
340	C₂²⋊C₄	—	94	1	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	—	94	1	no	—
346	C₂³	—	94	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	376	4	no	—
351	Q₈	—	188	2	no	—
352	Q₈	—	188	2	no	—
353	Q₈	D₈×D₄₇	1	2	yes	—
354	D₄	—	47	2	no	—
355	D₄	—	94	2	no	—
356	D₄	—	94	1	no	—
357	D₄	—	94	2	no	—
358	D₄	—	376	4	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	94	1	no	—
363	C₂×C₄	—	94	2	no	—
364	C₂×C₄	—	376	4	no	—
365	C₂×C₄	—	376	4	no	—
366	C₂×C₄	—	188	2	no	—
367	C₂×C₄	—	188	2	no	—
368	C₂×C₄	—	188	1	no	—
369	C₂×C₄	—	188	2	no	—
370	C₂×C₄	—	188	1	no	—
371	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
373	C₂×C₄	#13094070	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	1	no	—
377	C₈	—	47	2	no	—
378	C₈	—	94	2	no	—
379	C₈	—	94	1	no	—
380	C₈	—	188	1	no	—
381	C₈	#13094056	1	2	yes	complemented
382	C₂²	—	94	2	no	—
383	C₂²	—	188	2	no	—
384	C₂²	—	188	2	no	—
385	C₂²	—	188	2	no	—
386	C₂²	—	188	1	no	—
387	C₂²	—	188	1	no	—
388	C₂²	#15689757	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	376	2	no	—
394	C₄	—	188	1	no	—
395	C₄	—	188	2	no	—
396	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
397	C₄	C₂×#13094056	1	1	yes	characteristic
398	C₄	#15689925	1	1	yes	characteristic
399	C₄	#15689939	1	1	yes	characteristic
400	C₂	—	376	2	no	—
401	C₂	—	188	2	no	—
402	C₂	#20255029	1	1	yes	characteristic
403	C₂	#20255062	1	1	yes	characteristic
404	C₂	#20255191	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689757
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1674	C₂
socle	C₂²×C₄₇	D₄²
prime radical	#20255735	C₂
hypercenter	#1674	D₄₇
hypocenter	C₄₇	#19887
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
1. Q₈×D₃₇₆	C₂	semidirect
3. #20255029	C₂	semidirect
4. #20243005	C₂	semidirect
5. #20242876	C₂	semidirect
6. #20242778	C₂	semidirect
7. #20242635	C₂	semidirect
8. #20242624	C₂	semidirect
9. #20242272	C₂	semidirect
10. C₈×#10625336	C₂	semidirect
11. C₄₇×#1674	C₂	semidirect
123. #10625336	D₈	semidirect
174. C₈×C₄₇	#137	semidirect
203. #1674	D₄₇	semidirect
247. C₄₇	#19887	semidirect
381. C₈	#13094056	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
#20255735	2
#20255029	2
#20243005	2
#20242876	2
#20242778	2
#20242635	2
#20242624	2
#20242272	2
C₈×#10625336	2
C₄₇×#1674	2
#19887	47