Subgroups — #26371454

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254047	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245596	C₂	1	2	yes	maximal, complemented
3	#20245545	C₂	1	2	yes	maximal, complemented
4	#20245433	C₂	1	2	yes	maximal, complemented
5	#20245317	C₂	1	2	yes	maximal
6	#20245252	C₂	1	2	yes	maximal, complemented
7	C₄×#13093968	C₂	1	2	yes	maximal, complemented
8	#20245195	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2948	C₂	1	1	yes	maximal, characteristic
10	#15689735	—	2	1	no	—
11	C₂×#13093968	C₂²	1	2	yes	—
12	C₂×#13093968	C₂²	1	2	yes	—
13	#15689525	C₂²	1	1	yes	characteristic, complemented
14	#15689503	C₂²	1	1	yes	characteristic, complemented
15	#15689492	—	2	1	no	—
16	#15689381	—	2	1	no	—
17	#15689364	C₂²	1	2	yes	—
18	#15689231	—	2	1	no	—
19	C₄×Dic₁₈₈	C₂²	1	2	yes	complemented
20	#15688950	C₂²	1	2	yes	complemented
21	#15688948	C₂²	1	2	yes	complemented
22	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
23	#15688926	C₂²	1	2	yes	complemented
24	#15688926	C₂²	1	2	yes	complemented
25	#15688924	C₂²	1	2	yes	—
26	#15688924	C₂²	1	2	yes	—
27	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
28	#15688899	C₂²	1	1	yes	characteristic, complemented
29	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#518	—	2	1	no	—
32	C₄₇×#442	—	2	1	no	—
33	C₄₇×#442	—	2	1	no	—
34	C₄×D₈×C₄₇	C₂²	1	2	yes	—
35	C₄×D₈×C₄₇	C₂²	1	2	yes	—
36	C₄₇×#427	C₂²	1	1	yes	characteristic
37	C₄₇×#418	—	2	1	no	—
38	C₄₇×#405	C₂²	1	1	yes	characteristic
39	C₂×Dic₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	C₂³	1	2	yes	—
41	#13094024	—	2	1	no	—
42	#13094017	—	4	2	no	—
43	#13094016	—	2	1	no	—
44	#13094014	—	4	2	no	—
45	#13094003	—	2	2	no	—
46	C₄×Dic₉₄	—	2	2	no	—
47	#13093968	—	2	4	no	—
48	#13093968	—	2	4	no	—
49	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	#13093958	C₂³	1	2	yes	—
52	#13093956	—	2	1	no	—
53	#13093949	—	2	2	no	—
54	#13093949	—	2	2	no	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	C₂×D₈×C₄₇	—	2	4	no	—
57	C₂×D₈×C₄₇	—	2	1	no	—
58	C₂×D₈×C₄₇	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₄×D₄×C₄₇	—	2	2	no	—
62	C₄×D₄×C₄₇	—	2	2	no	—
63	C₄×D₄×C₄₇	—	4	2	no	—
64	#117×C₄₇	—	2	1	no	—
65	#117×C₄₇	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₄₇	—	4	2	no	—
67	#107×C₄₇	C₂³	1	2	yes	—
68	#106×C₄₇	—	2	1	no	—
69	#102×C₄₇	—	2	2	no	—
70	#102×C₄₇	—	2	2	no	—
71	#97×C₄₇	D₄	1	1	yes	characteristic
72	#97×C₄₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂⁴×C₄₇	—	4	1	no	—
76	C₂×D₄×C₄₇	—	2	4	no	—
77	C₂×D₄×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂²×C₄×C₄₇	—	4	2	no	—
81	C₂²×C₄×C₄₇	—	4	2	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₉₄	—	2	2	no	—
84	D₈×C₄₇	—	2	4	no	—
85	D₈×C₄₇	—	2	2	no	—
86	D₈×C₄₇	#41	1	4	yes	—
87	M₁₆×C₄₇	—	4	1	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
95	C₂²⋊C₄×C₄₇	—	4	1	no	—
96	C₂²⋊C₄×C₄₇	—	4	2	no	—
97	C₂²⋊C₄×C₄₇	—	4	2	no	—
98	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
99	C₄²×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊Dic₄₇	—	4	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	Dic₄₇⋊C₄	—	4	2	no	—
106	Dic₄₇⋊C₄	—	4	2	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	C₄×Dic₄₇	—	4	2	no	—
109	Dic₁₈₈	—	2	2	no	—
110	Dic₁₈₈	—	2	2	no	—
111	Dic₁₈₈	—	4	2	no	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₄₇⋊C₁₆	—	2	2	no	—
114	C₂³×C₄₇	—	4	4	no	—
115	C₂³×C₄₇	—	4	2	no	—
116	C₂³×C₄₇	—	16	2	no	—
117	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
118	D₄×C₄₇	—	2	4	no	—
119	D₄×C₄₇	—	4	4	no	—
120	D₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	16	2	no	—
125	C₂×C₄×C₄₇	—	16	2	no	—
126	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	Dic₉₄	—	4	2	no	—
133	Dic₉₄	—	4	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	C₂×#41	1	2	yes	—
138	#33683	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	4	4	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	—	8	2	no	—
144	C₂²×C₄₇	—	16	4	no	—
145	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	8	2	no	—
150	C₄×C₄₇	#525	1	1	yes	characteristic
151	C₄×C₄₇	#575	1	1	yes	characteristic
152	Dic₄₇	—	8	2	no	—
153	Dic₄₇	—	8	2	no	—
154	#3177	—	47	1	no	—
155	#2948	D₄₇	1	1	yes	characteristic
156	#2237	—	47	2	no	—
157	#2228	—	47	2	no	—
158	#2208	—	47	2	no	—
159	#2187	—	47	2	no	—
160	#2175	—	47	2	no	—
161	C₄×SD₃₂	—	47	2	no	—
162	#2164	—	47	1	no	—
163	C₂×C₄₇	—	4	1	no	—
164	C₂×C₄₇	—	8	4	no	—
165	C₂×C₄₇	#3100	1	1	yes	characteristic
166	C₂×C₄₇	#3416	1	2	yes	—
167	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
168	#541	—	94	1	no	—
169	#518	—	2	1	no	—
170	C₂×SD₃₂	—	47	2	no	—
171	C₂×SD₃₂	—	47	2	no	—
172	#499	—	47	1	no	—
173	#494	—	47	1	no	—
174	#491	—	94	1	no	—
175	#470	—	94	1	no	—
176	#467	—	47	2	no	—
177	#442	—	2	1	no	—
178	#442	—	2	1	no	—
179	#441	—	94	1	no	—
180	C₄×Q₁₆	—	47	2	no	—
181	C₄×D₈	C₂×D₄₇	1	2	yes	—
182	C₄×D₈	C₂×D₄₇	1	2	yes	—
183	#427	C₂×D₄₇	1	1	yes	characteristic
184	#418	—	2	1	no	—
185	#405	C₂×D₄₇	1	1	yes	characteristic
186	#367	—	47	2	no	—
187	#366	—	47	2	no	—
188	C₄⋊C₁₆	—	47	2	no	—
189	#358	—	47	2	no	—
190	#358	—	47	2	no	—
191	#357	—	47	2	no	—
192	#357	—	47	2	no	—
193	C₂²⋊C₁₆	—	47	2	no	—
194	#346	—	47	1	no	—
195	C₄×C₁₆	—	47	1	no	—
196	C₄₇	#33683	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	2	1	no	—
198	C₂×Q₁₆	—	47	2	no	—
199	C₂×Q₁₆	—	94	1	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂²×D₄₇	1	2	yes	—
203	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	94	1	no	—
206	#126	—	188	2	no	—
207	#125	—	94	1	no	—
208	#125	—	188	2	no	—
209	C₂²⋊Q₈	—	94	2	no	—
210	C₄×Q₈	—	94	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	4	2	no	—
217	SD₃₂	—	94	4	no	—
218	SD₃₂	—	94	4	no	—
219	C₂×C₁₆	—	47	2	no	—
220	C₂×C₁₆	—	47	2	no	—
221	#107	—	47	2	no	—
222	#107	C₂²×D₄₇	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	94	1	no	—
225	#103	—	94	2	no	—
226	#103	—	94	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	#97	C₂²⋊D₄₇	1	1	yes	characteristic
230	#97	C₂²⋊D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	94	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	Q₁₆	—	94	2	no	—
243	Q₁₆	—	94	2	no	—
244	Q₁₆	—	188	2	no	—
245	D₈	—	2	4	no	—
246	D₈	—	2	2	no	—
247	D₈	#10625350	1	4	yes	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	188	2	no	—
257	C₄⋊C₄	—	188	2	no	—
258	C₄⋊C₄	—	188	2	no	—
259	C₄⋊C₄	—	188	2	no	—
260	C₄⋊C₄	#10625334	1	2	yes	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	188	2	no	—
265	C₂²⋊C₄	#10625334	1	2	yes	—
266	C₄²	—	4	1	no	—
267	C₄²	—	188	2	no	—
268	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₁₆	—	94	2	no	—
271	C₁₆	—	94	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	16	2	no	—
275	C₂³	C₂×#10625334	1	1	yes	characteristic
276	Q₈	—	188	2	no	—
277	Q₈	—	188	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	16	2	no	—
285	C₂×C₄	—	16	2	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625334	1	2	yes	—
290	C₂×C₄	C₂×#10625334	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₂×#10625350	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	C₂²×#10625334	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	376	2	no	—
308	C₄	—	376	2	no	—
309	C₄	#15689634	1	1	yes	characteristic
310	C₄	#15689921	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	8	4	no	—
313	C₂	#20253388	1	1	yes	characteristic
314	C₂	#20256535	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2948	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	#2948	D₄₇
hypocenter	C₄₇	#33683
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
1. #20254047	C₂	semidirect
2. #20245596	C₂	semidirect
3. #20245545	C₂	semidirect
4. #20245433	C₂	semidirect
6. #20245252	C₂	semidirect
7. C₄×#13093968	C₂	semidirect
8. #20245195	C₂	semidirect
13. #15689525	C₂²	semidirect
14. #15689503	C₂²	semidirect
19. C₄×Dic₁₈₈	C₂²	semidirect
20. #15688950	C₂²	semidirect
21. #15688948	C₂²	semidirect
22. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
23. #15688926	C₂²	semidirect
24. #15688926	C₂²	semidirect
28. #15688899	C₂²	semidirect
29. C₄×C₄₇⋊C₁₆	C₂²	semidirect
196. C₄₇	#33683	semidirect

Maximal subgroups

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

subgroup	index
#20254047	2
#20245596	2
#20245545	2
#20245433	2
#20245317	2
#20245252	2
C₄×#13093968	2
#20245195	2
C₄₇×#2948	2
#33683	47