Subgroups — #26374006

Order 12032.

508 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256377	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245627	C₂	1	2	yes	maximal, complemented
3	#20245617	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245537	C₂	1	2	yes	maximal, complemented
5	#20245522	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245443	C₂	1	2	yes	maximal
7	#20245433	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245297	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245252	C₂	1	2	yes	maximal, complemented
10	C₄×#13093965	C₂	1	1	yes	maximal, characteristic
11	Q₈×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689846	—	2	1	no	—
13	C₂×#13093968	C₂²	1	2	yes	—
14	C₂×#13093965	C₂²	1	1	yes	characteristic
15	#15689507	C₂²	1	2	yes	—
16	#15689503	C₂²	1	1	yes	characteristic
17	#15689346	—	2	2	no	—
18	#15689329	—	2	1	no	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	—
20	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
21	#15688948	C₂²	1	2	yes	—
22	#15688948	C₂²	1	1	yes	characteristic
23	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
25	#15688926	C₂²	1	2	yes	—
26	#15688926	C₂²	1	2	yes	—
27	#15688926	C₂²	1	2	yes	—
28	#15688924	C₂²	1	1	yes	characteristic
29	#15688923	C₂²	1	2	yes	—
30	#15688923	C₂²	1	1	yes	characteristic
31	#15688899	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	D₄×Q₈×C₄₇	—	2	1	no	—
34	C₄₇×#500	C₂²	1	1	yes	characteristic
35	C₄₇×#500	C₂²	1	2	yes	—
36	C₄₇×#474	—	2	1	no	—
37	C₄₇×#474	—	2	2	no	—
38	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₄₇	C₂²	1	2	yes	—
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₈₈	—	2	1	no	—
42	C₂×Dic₁₈₈	C₂³	1	2	yes	—
43	C₂×#10625325	—	2	2	no	—
44	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
45	#13094024	—	2	1	no	—
46	#13094019	—	4	2	no	—
47	#13094012	—	2	2	no	—
48	#13094000	—	4	1	no	—
49	C₄×Dic₉₄	—	2	2	no	—
50	C₄×D₁₈₈	—	2	1	no	—
51	#13093982	—	4	1	no	—
52	#13093968	—	2	4	no	—
53	#13093965	—	2	2	no	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	#13093958	C₂³	1	1	yes	characteristic
58	#13093949	—	2	2	no	—
59	#13093946	—	2	1	no	—
60	C₂²×Q₈×C₄₇	—	4	1	no	—
61	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₄₇	—	2	1	no	—
63	C₄⋊Q₈×C₄₇	—	2	2	no	—
64	C₂²⋊Q₈×C₄₇	—	4	1	no	—
65	C₂²⋊Q₈×C₄₇	—	4	2	no	—
66	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
67	C₄×D₄×C₄₇	—	2	1	no	—
68	C₄×D₄×C₄₇	—	2	2	no	—
69	#107×C₄₇	—	2	1	no	—
70	#107×C₄₇	—	2	2	no	—
71	#107×C₄₇	C₂³	1	2	yes	—
72	#107×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
75	#102×C₄₇	—	2	1	no	—
76	#102×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	C₂³	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₄₇	—	4	2	no	—
80	C₂×Q₈×C₄₇	—	4	1	no	—
81	C₂×Q₈×C₄₇	—	8	4	no	—
82	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
83	C₂×D₄×C₄₇	—	2	1	no	—
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	—	4	1	no	—
86	C₂×D₁₈₈	—	2	1	no	—
87	C₂×C₄×D₄₇	—	4	1	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	D₈×C₄₇	—	2	1	no	—
90	D₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₇	—	2	2	no	—
95	C₄⋊C₄×C₄₇	—	2	1	no	—
96	C₄⋊C₄×C₄₇	—	4	2	no	—
97	C₄⋊C₄×C₄₇	—	4	1	no	—
98	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
99	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
100	C₂²⋊C₄×C₄₇	—	4	2	no	—
101	C₂²⋊C₄×C₄₇	—	4	1	no	—
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	(C₂×D₄₇)⋊C₄	—	4	2	no	—
105	(C₂×D₄₇)⋊C₄	—	4	1	no	—
106	(C₂×D₄₇)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₇	—	2	1	no	—
108	C₄⋊Dic₄₇	—	4	1	no	—
109	C₄⋊Dic₄₇	—	4	1	no	—
110	Dic₄₇⋊C₄	—	4	2	no	—
111	C₄×Dic₄₇	—	4	2	no	—
112	Dic₁₈₈	—	2	2	no	—
113	Dic₁₈₈	—	2	2	no	—
114	Dic₁₈₈	—	8	1	no	—
115	D₃₇₆	—	2	1	no	—
116	D₃₇₆	—	2	1	no	—
117	#10625325	—	8	2	no	—
118	C₄₇⋊C₁₆	—	2	1	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₄₇⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₇	—	4	1	no	—
122	C₂²×D₄₇	—	4	1	no	—
123	Q₈×C₄₇	—	2	2	no	—
124	Q₈×C₄₇	—	8	4	no	—
125	Q₈×C₄₇	—	8	2	no	—
126	D₄×C₄₇	—	2	2	no	—
127	D₄×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	—	8	4	no	—
131	C₂×C₄×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₄₇	—	4	1	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	D₁₈₈	—	4	1	no	—
140	D₁₈₈	—	4	1	no	—
141	C₄×D₄₇	—	8	2	no	—
142	Dic₉₄	—	4	2	no	—
143	Dic₉₄	—	4	2	no	—
144	C₈×C₄₇	—	2	2	no	—
145	C₈×C₄₇	—	2	1	no	—
146	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
148	#33934	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	4	2	no	—
150	C₂²×C₄₇	—	8	1	no	—
151	C₂²×C₄₇	—	8	1	no	—
152	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₄₇	—	8	1	no	—
154	C₂×D₄₇	—	8	1	no	—
155	C₂×D₄₇	—	8	1	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	—	8	1	no	—
161	C₄×C₄₇	#509	1	2	yes	—
162	C₄×C₄₇	#548	1	1	yes	characteristic
163	C₄×C₄₇	#577	1	1	yes	characteristic
164	Dic₄₇	—	8	2	no	—
165	Dic₄₇	—	8	1	no	—
166	#3393	—	47	1	no	—
167	Q₈×D₈	D₄₇	1	1	yes	characteristic, complemented
168	#2243	—	47	2	no	—
169	#2241	—	47	1	no	—
170	#2227	—	47	2	no	—
171	#2224	—	47	1	no	—
172	#2210	—	47	2	no	—
173	#2208	—	47	1	no	—
174	#2183	—	47	1	no	—
175	#2175	—	47	2	no	—
176	C₄×D₁₆	—	47	1	no	—
177	C₂×C₄₇	—	8	2	no	—
178	C₂×C₄₇	#3358	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
180	C₂×C₄₇	#3417	1	1	yes	characteristic
181	D₄₇	—	16	1	no	—
182	#556	—	94	1	no	—
183	D₄×Q₈	—	2	1	no	—
184	C₂×SD₃₂	—	47	2	no	—
185	C₂×D₁₆	—	47	1	no	—
186	#500	C₂×D₄₇	1	1	yes	characteristic
187	#500	C₂×D₄₇	1	2	yes	—
188	#495	—	47	2	no	—
189	#494	—	47	1	no	—
190	#474	—	2	1	no	—
191	#474	—	2	2	no	—
192	#464	—	94	2	no	—
193	#461	—	94	1	no	—
194	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
195	C₄×Q₁₆	—	47	2	no	—
196	C₄×D₈	—	47	1	no	—
197	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
198	C₄×D₈	C₂×D₄₇	1	2	yes	—
199	#366	—	47	1	no	—
200	#366	—	47	2	no	—
201	C₄⋊C₁₆	—	47	1	no	—
202	C₄⋊C₁₆	—	47	2	no	—
203	#358	—	47	2	no	—
204	#358	—	47	2	no	—
205	#358	—	47	2	no	—
206	#357	—	47	1	no	—
207	#357	—	47	1	no	—
208	#357	—	47	2	no	—
209	#346	—	47	2	no	—
210	C₄×C₁₆	—	47	1	no	—
211	C₄₇	#33934	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	—	4	1	no	—
213	C₂×Q₁₆	—	47	2	no	—
214	C₂×Q₁₆	—	94	1	no	—
215	C₂×SD₁₆	—	94	2	no	—
216	C₂×D₈	—	47	1	no	—
217	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	94	1	no	—
221	#126	—	188	2	no	—
222	#124	—	94	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₂²⋊Q₈	—	4	2	no	—
225	C₂²⋊Q₈	—	188	1	no	—
226	C₄×Q₈	—	94	2	no	—
227	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	94	1	no	—
231	#117	—	188	1	no	—
232	SD₃₂	—	94	4	no	—
233	D₁₆	—	94	2	no	—
234	C₂×C₁₆	—	47	1	no	—
235	C₂×C₁₆	—	47	1	no	—
236	C₂×C₁₆	—	47	2	no	—
237	#107	—	2	1	no	—
238	#107	—	2	2	no	—
239	#107	—	47	1	no	—
240	#107	C₂²×D₄₇	1	1	yes	characteristic
241	#107	C₂²×D₄₇	1	2	yes	—
242	C₄⋊C₈	C₂²⋊D₄₇	1	1	yes	characteristic
243	C₄⋊C₈	C₂²⋊D₄₇	1	2	yes	—
244	#103	—	94	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	#102	—	94	1	no	—
248	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₄₇	1	2	yes	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	8	4	no	—
253	C₂×Q₈	—	94	2	no	—
254	C₂×Q₈	#10625334	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	94	1	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	188	1	no	—
260	Q₁₆	—	94	2	no	—
261	Q₁₆	—	94	2	no	—
262	Q₁₆	—	376	1	no	—
263	SD₁₆	—	376	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	94	1	no	—
266	D₈	—	94	1	no	—
267	D₈	#10625354	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
270	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	94	1	no	—
276	C₄⋊C₄	—	188	2	no	—
277	C₄⋊C₄	—	188	1	no	—
278	C₄⋊C₄	—	188	1	no	—
279	C₄⋊C₄	#10625334	1	1	yes	characteristic
280	C₄⋊C₄	#10625334	1	2	yes	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	188	2	no	—
284	C₂²⋊C₄	—	188	1	no	—
285	C₂²⋊C₄	—	188	2	no	—
286	C₄²	—	188	2	no	—
287	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
289	C₁₆	—	94	1	no	—
290	C₁₆	—	94	2	no	—
291	C₁₆	—	94	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	188	1	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	8	4	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	188	2	no	—
298	Q₈	—	188	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	1	no	—
301	D₄	—	188	1	no	—
302	D₄	—	188	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	376	2	no	—
308	C₂×C₄	—	188	1	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	C₂×#10625334	1	2	yes	—
311	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
312	C₂×C₄	C₂×#10625334	1	2	yes	—
313	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	C₂×#10625354	1	1	yes	characteristic
318	C₈	#13094072	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	376	1	no	—
323	C₂²	—	376	1	no	—
324	C₂²	—	376	1	no	—
325	C₂²	C₂²×#10625334	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	—	376	2	no	—
332	C₄	—	376	1	no	—
333	C₄	#15689559	1	2	yes	—
334	C₄	#15689784	1	1	yes	characteristic
335	C₄	#15689931	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	752	1	no	—
338	C₂	#20256023	1	1	yes	characteristic
339	C₂	C₂×#15689559	1	1	yes	characteristic
340	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	Q₈×D₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#13093965	C₂²
hypercenter	Q₈×D₈	D₄₇
hypocenter	C₄₇	#33934
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
1. #20256377	C₂	semidirect
2. #20245627	C₂	semidirect
3. #20245617	C₂	semidirect
4. #20245537	C₂	semidirect
5. #20245522	C₂	semidirect
7. #20245433	C₂	semidirect
8. #20245297	C₂	semidirect
9. #20245252	C₂	semidirect
11. Q₈×D₈×C₄₇	C₂	semidirect
167. Q₈×D₈	D₄₇	semidirect
211. C₄₇	#33934	semidirect

Maximal subgroups

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

subgroup	index
#20256377	2
#20245627	2
#20245617	2
#20245537	2
#20245522	2
#20245443	2
#20245433	2
#20245297	2
#20245252	2
C₄×#13093965	2
Q₈×D₈×C₄₇	2
#33934	47