Subgroups — #26204263

Order 12032.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254326	C₂	1	2	yes	maximal, complemented
2	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
3	#20253764	C₂	1	2	yes	maximal, complemented
4	#20243030	C₂	1	4	yes	maximal, complemented
5	#20242995	C₂	1	2	yes	maximal, complemented
6	#20242981	C₂	1	2	yes	maximal, complemented
7	C₈×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	—	2	2	no	—
10	C₂×#13094021	C₂²	1	2	yes	—
11	#15689500	—	2	2	no	—
12	#15689500	C₂²	1	2	yes	—
13	#15689500	C₂²	1	2	yes	—
14	#15689497	C₂²	1	4	yes	complemented
15	#15689489	—	2	4	no	—
16	#15689471	C₂²	1	4	yes	complemented
17	#15689471	C₂²	1	4	yes	complemented
18	#15689471	C₂²	1	4	yes	complemented
19	#15689323	—	2	4	no	—
20	#15689273	—	2	4	no	—
21	#15689176	C₂²	1	1	yes	characteristic
22	#15689127	—	2	4	no	—
23	C₄×#10625329	C₂²	1	1	yes	characteristic
24	#15688845	C₂²	1	2	yes	complemented
25	C₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
26	C₄₇×#533	C₂²	1	2	yes	—
27	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
28	C₄₇×#494	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#487	C₂²	1	2	yes	complemented
30	C₄₇×#467	—	2	2	no	—
31	C₄×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×D₁₈₈	—	2	2	no	—
34	C₂²×D₁₈₈	—	2	2	no	—
35	C₂²×D₁₈₈	—	2	2	no	—
36	C₂×#10625336	—	2	8	no	—
37	C₂×D₃₇₆	—	2	4	no	—
38	C₂×D₃₇₆	—	2	4	no	—
39	C₂×D₃₇₆	—	4	8	no	—
40	C₂×#10625329	D₄	1	2	yes	—
41	#13094021	—	2	2	no	—
42	#13094021	—	2	2	no	—
43	#13094021	—	2	2	no	—
44	#13094021	C₂³	1	4	yes	—
45	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
46	#13093944	—	2	8	no	—
47	#13093944	—	2	8	no	—
48	#13093944	—	2	8	no	—
49	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
50	#13093935	D₄	1	2	yes	—
51	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
52	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
53	C₂×Q₁₆×C₄₇	—	2	2	no	—
54	C₂²×C₈×C₄₇	D₄	1	2	yes	—
55	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
57	#125×C₄₇	—	2	2	no	—
58	C₂²⋊Q₈×C₄₇	—	2	2	no	—
59	C₂²⋊Q₈×C₄₇	—	2	2	no	—
60	C₄×Q₈×C₄₇	—	2	2	no	—
61	C₄×Q₈×C₄₇	—	2	2	no	—
62	#117×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#107×C₄₇	—	2	2	no	—
65	#103×C₄₇	—	2	4	no	—
66	C₄×C₈×C₄₇	D₄	1	2	yes	—
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂³×D₄₇	—	4	2	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×D₁₈₈	—	2	4	no	—
76	C₂×D₁₈₈	—	2	4	no	—
77	C₂×D₁₈₈	—	2	4	no	—
78	C₂×D₁₈₈	—	4	8	no	—
79	C₂×D₁₈₈	—	4	8	no	—
80	C₂×D₁₈₈	—	4	2	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	4	2	no	—
83	C₂×D₁₈₈	—	8	4	no	—
84	Q₁₆×C₄₇	—	4	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	2	no	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	4	2	no	—
93	C₄⋊C₄×C₄₇	—	4	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	—	4	2	no	—
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
100	#10625336	—	8	8	no	—
101	#10625329	—	2	4	no	—
102	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
103	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
104	D₃₇₆	—	4	8	no	—
105	D₃₇₆	—	4	8	no	—
106	C₂³×C₄₇	#127	1	1	yes	characteristic
107	C₂²×D₄₇	—	4	4	no	—
108	C₂²×D₄₇	—	8	4	no	—
109	C₂²×D₄₇	—	8	4	no	—
110	C₂²×D₄₇	—	8	4	no	—
111	Q₈×C₄₇	—	4	4	no	—
112	Q₈×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#135	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	D₁₈₈	—	4	8	no	—
125	D₁₈₈	—	4	8	no	—
126	D₁₈₈	—	4	8	no	—
127	D₁₈₈	—	8	4	no	—
128	D₁₈₈	—	8	4	no	—
129	D₁₈₈	—	8	4	no	—
130	C₈×C₄₇	—	2	2	no	—
131	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	4	no	—
134	#19635	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	#495	1	2	yes	—
138	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
139	C₂×D₄₇	—	8	4	no	—
140	C₂×D₄₇	—	8	4	no	—
141	C₂×D₄₇	—	8	4	no	—
142	C₂×D₄₇	—	16	8	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	—	8	2	no	—
147	C₄×C₄₇	#495	1	2	yes	—
148	C₄×C₄₇	C₂×#135	1	2	yes	—
149	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₄×C₄₇	#576	1	2	yes	—
151	#3208	—	47	2	no	—
152	#3170	D₄₇	1	1	yes	characteristic, complemented
153	C₂×#493	—	47	1	no	—
154	#3138	—	47	2	no	—
155	#1717	—	47	4	no	—
156	#1709	—	47	2	no	—
157	#1706	—	47	2	no	—
158	C₈×M₁₆	—	47	1	no	—
159	C₂×C₄₇	—	2	2	no	—
160	C₂×C₄₇	C₂×#495	1	1	yes	characteristic
161	C₂×C₄₇	#3151	1	1	yes	characteristic
162	C₂×C₄₇	#3155	1	1	yes	characteristic
163	D₄₇	—	16	4	no	—
164	C₂²×D₈	—	94	2	no	—
165	#533	C₂×D₄₇	1	2	yes	—
166	C₂×#127	—	47	2	no	—
167	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
168	#494	C₂×D₄₇	1	1	yes	characteristic, complemented
169	#493	—	47	2	no	—
170	#493	—	47	2	no	—
171	#493	—	94	2	no	—
172	#492	—	47	4	no	—
173	#490	—	94	4	no	—
174	#487	C₂×D₄₇	1	2	yes	complemented
175	#486	—	47	4	no	—
176	#486	—	47	4	no	—
177	#486	—	47	4	no	—
178	#467	—	2	2	no	—
179	#460	—	94	4	no	—
180	#450	—	94	4	no	—
181	C₄×Q₁₆	—	2	2	no	—
182	#431	—	47	1	no	—
183	#418	—	94	4	no	—
184	C₄×M₁₆	—	47	1	no	—
185	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
186	#322	—	47	2	no	—
187	C₈²	—	47	2	no	—
188	C₄₇	#19635	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	—	94	2	no	—
190	C₂²×D₄	—	94	2	no	—
191	C₂²×D₄	—	94	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	94	8	no	—
194	C₂×D₈	—	94	4	no	—
195	C₂×D₈	—	94	4	no	—
196	C₂×D₈	—	188	8	no	—
197	C₂×M₁₆	—	47	2	no	—
198	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
199	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
200	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
201	#127	—	47	4	no	—
202	#127	—	94	2	no	—
203	#127	—	94	2	no	—
204	#127	—	94	2	no	—
205	#125	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	#117	—	2	2	no	—
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	#107	—	2	2	no	—
213	C₄⋊C₈	—	47	2	no	—
214	#103	—	2	4	no	—
215	#102	—	94	8	no	—
216	#102	—	94	8	no	—
217	#102	—	94	8	no	—
218	C₂²⋊C₈	—	47	2	no	—
219	#97	—	47	2	no	—
220	C₄×C₈	—	47	2	no	—
221	C₄×C₈	—	47	2	no	—
222	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
223	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂⁴	—	188	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×D₄	—	94	4	no	—
229	C₂×D₄	—	94	4	no	—
230	C₂×D₄	—	94	4	no	—
231	C₂×D₄	—	376	4	no	—
232	C₂×D₄	—	188	8	no	—
233	C₂×D₄	—	188	2	no	—
234	C₂×D₄	—	188	2	no	—
235	C₂×D₄	—	188	2	no	—
236	C₂×D₄	—	188	8	no	—
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
240	Q₁₆	—	4	4	no	—
241	SD₁₆	—	376	8	no	—
242	D₈	—	188	8	no	—
243	D₈	—	188	8	no	—
244	M₁₆	—	94	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	D₄×D₄₇	1	1	yes	characteristic
259	C₄²	D₄×D₄₇	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₄²	C₂³×D₄₇	1	1	yes	characteristic
262	C₂³	—	376	4	no	—
263	C₂³	—	376	4	no	—
264	C₂³	—	376	4	no	—
265	C₂³	—	188	4	no	—
266	C₂³	#13094020	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	D₄	—	376	4	no	—
270	D₄	—	376	4	no	—
271	D₄	—	376	4	no	—
272	D₄	—	188	8	no	—
273	D₄	—	188	8	no	—
274	D₄	—	188	8	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	#13094020	1	1	yes	characteristic
280	C₂×C₄	#13094020	1	1	yes	characteristic
281	C₂×C₄	#13094020	1	1	yes	characteristic
282	C₂×C₄	#13094047	1	4	yes	—
283	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	94	4	no	—
288	C₈	—	94	4	no	—
289	C₈	#13094049	1	4	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	376	4	no	—
293	C₂²	—	376	4	no	—
294	C₂²	—	376	4	no	—
295	C₂²	—	752	8	no	—
296	C₂²	#15689508	1	2	yes	—
297	C₂²	C₂×#13094020	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#15689508	1	2	yes	—
303	C₄	C₂×#13094047	1	2	yes	—
304	C₄	C₂×#13094049	1	2	yes	—
305	C₄	#15689925	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	752	4	no	—
308	C₂	C₂×#15689508	1	1	yes	characteristic
309	C₂	#20253841	1	1	yes	characteristic
310	C₂	#20253875	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3170	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	C₂×#15689500	C₂
hypercenter	#3170	D₄₇
hypocenter	C₄₇	#19635
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
1. #20254326	C₂	semidirect
3. #20253764	C₂	semidirect
4. #20243030	C₂	semidirect
5. #20242995	C₂	semidirect
6. #20242981	C₂	semidirect
7. C₈×#10625329	C₂	semidirect
8. C₄₇×#3170	C₂	semidirect
14. #15689497	C₂²	semidirect
16. #15689471	C₂²	semidirect
17. #15689471	C₂²	semidirect
18. #15689471	C₂²	semidirect
24. #15688845	C₂²	semidirect
25. C₈×C₄₇⋊C₈	C₂²	semidirect
27. C₄₇×#500	C₂²	semidirect
28. C₄₇×#494	C₂²	semidirect
29. C₄₇×#487	C₂²	semidirect
152. #3170	D₄₇	semidirect
167. #500	C₂×D₄₇	semidirect
168. #494	C₂×D₄₇	semidirect
174. #487	C₂×D₄₇	semidirect
188. C₄₇	#19635	semidirect

Maximal subgroups

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

subgroup	index
#20254326	2
C₂×#15689500	2
#20253764	2
#20243030	2
#20242995	2
#20242981	2
C₈×#10625329	2
C₄₇×#3170	2
#19635	47