Subgroups — #26106992

Order 12032.

498 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252344	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093950	C₂	1	1	yes	maximal, characteristic
3	#20243133	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241771	C₂	1	1	yes	maximal, characteristic
5	M₁₆×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241220	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10625350	C₂²	1	1	yes	characteristic
9	#15689247	C₄	1	2	yes	complemented
10	#15689188	C₄	1	2	yes	complemented
11	(C₂²⋊D₄₇)⋊C₈	—	2	2	no	—
12	#15689176	C₂²	1	1	yes	characteristic
13	#15689174	—	2	1	no	—
14	C₂×#13093950	C₄	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₄₇	—	2	1	no	—
17	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688884	C₄	1	2	yes	—
19	#15688877	—	2	2	no	—
20	#15688855	C₄	1	2	yes	complemented
21	#15688852	C₄	1	2	yes	complemented
22	#15688844	—	2	1	no	—
23	C₈×C₄₇⋊C₈	—	2	1	no	—
24	C₄₇×#431	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₂×C₈×D₄₇	—	2	2	no	—
30	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₇	—	2	1	no	—
35	C₄×C₂²⋊D₄₇	—	2	1	no	—
36	#13093983	C₂×C₄	1	1	yes	characteristic
37	#13093981	—	2	1	no	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
43	Dic₄₇⋊C₈	—	2	2	no	—
44	#13093950	C₈	1	8	yes	complemented
45	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
46	(C₂×D₄₇)⋊C₈	—	2	2	no	—
47	C₄×C₄₇⋊C₈	—	2	1	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
51	C₈×Dic₄₇	—	2	2	no	—
52	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	—	2	1	no	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₄₇	—	2	1	no	—
57	#97×C₄₇	—	2	1	no	—
58	C₄×C₈×C₄₇	—	2	1	no	—
59	C₄×C₈×C₄₇	—	2	1	no	—
60	C₄×C₈×C₄₇	—	2	1	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₇	—	2	1	no	—
66	C₂²×Dic₄₇	—	2	2	no	—
67	#10625350	—	2	2	no	—
68	#10625350	C₂×C₈	1	4	yes	—
69	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₄₇	—	2	1	no	—
71	C₂×C₄×D₄₇	—	2	1	no	—
72	C₂×C₄×D₄₇	—	2	1	no	—
73	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
74	M₁₆×C₄₇	—	2	2	no	—
75	M₁₆×C₄₇	C₂×C₈	1	4	yes	complemented
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	C₂²⋊Dic₄₇	—	2	1	no	—
85	(C₂×D₄₇)⋊C₄	—	2	1	no	—
86	(C₂×D₄₇)⋊C₄	—	2	1	no	—
87	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
88	C₄⋊Dic₄₇	C₄²	1	1	yes	characteristic
89	Dic₄₇⋊C₄	—	2	1	no	—
90	Dic₄₇⋊C₄	—	2	1	no	—
91	C₄×Dic₄₇	—	2	1	no	—
92	C₄×Dic₄₇	—	2	1	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	—	4	2	no	—
96	C₄×Dic₄₇	—	4	2	no	—
97	C₄×Dic₄₇	C₂×C₈	1	4	yes	complemented
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₈×D₄₇	—	4	4	no	—
103	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₄₇	#131	1	2	yes	—
115	C₂²⋊D₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	1	no	—
117	C₂×Dic₄₇	—	2	1	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	D₁₈₈	—	2	1	no	—
123	D₁₈₈	C₄×C₈	1	2	yes	—
124	C₄×D₄₇	—	2	4	no	—
125	C₄×D₄₇	—	4	2	no	—
126	C₄×D₄₇	—	4	2	no	—
127	Dic₉₄	—	2	1	no	—
128	Dic₉₄	C₄×C₈	1	2	yes	—
129	C₈×C₄₇	—	2	4	no	—
130	C₈×C₄₇	—	4	2	no	—
131	C₈×C₄₇	—	4	2	no	—
132	C₄₇⋊C₈	—	2	2	no	—
133	C₄₇⋊C₈	—	2	2	no	—
134	C₄₇⋊C₈	—	4	2	no	—
135	#8191	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₈×D₄	1	2	yes	—
140	C₂×D₄₇	—	2	2	no	—
141	C₂×D₄₇	—	4	1	no	—
142	C₂×D₄₇	—	4	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₄₇	#405	1	1	yes	characteristic
147	C₄×C₄₇	#421	1	2	yes	—
148	C₄×C₄₇	C₈×D₄	1	2	yes	—
149	C₄×C₄₇	#444	1	2	yes	—
150	Dic₄₇	—	2	2	no	—
151	Dic₄₇	—	4	4	no	—
152	Dic₄₇	—	4	1	no	—
153	Dic₄₇	—	4	1	no	—
154	C₈×#41	—	47	1	no	—
155	C₄×#104	—	47	1	no	—
156	#1744	—	47	1	no	—
157	#1476	—	47	1	no	—
158	C₈×M₁₆	—	47	1	no	—
159	#1278	D₄₇	1	1	yes	characteristic, complemented
160	#1276	—	47	1	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄₇	#1759	1	1	yes	characteristic
164	C₂×C₄₇	#1765	1	1	yes	characteristic
165	D₄₇	—	4	2	no	—
166	C₄×#41	—	47	1	no	—
167	C₈×Q₈	—	47	2	no	—
168	C₈×D₄	—	47	2	no	—
169	C₈×D₄	—	94	2	no	—
170	#431	—	47	1	no	—
171	#431	—	94	1	no	—
172	#431	C₂×D₄₇	1	1	yes	characteristic
173	C₂×#104	—	47	2	no	—
174	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₄×C₈	—	47	1	no	—
176	C₂×C₄×C₈	—	94	1	no	—
177	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₄³	—	47	1	no	—
179	#339	—	47	2	no	—
180	#336	—	94	2	no	—
181	Q₈⋊C₈	—	47	2	no	—
182	D₄⋊C₈	—	47	2	no	—
183	#322	—	94	1	no	—
184	C₈²	—	94	1	no	—
185	C₄₇	#8191	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	47	1	no	—
187	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	94	2	no	—
191	C₄×Q₈	—	47	1	no	—
192	C₄×Q₈	—	47	1	no	—
193	C₄×D₄	—	47	1	no	—
194	C₄×D₄	—	47	1	no	—
195	C₄×D₄	—	94	1	no	—
196	C₄×D₄	—	94	1	no	—
197	#117	—	47	1	no	—
198	#117	—	94	1	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	94	1	no	—
201	C₂×C₄²	—	94	1	no	—
202	C₂×C₄²	—	94	1	no	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	C₄⋊C₈	—	47	2	no	—
205	C₄⋊C₈	—	94	2	no	—
206	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
207	#104	—	47	8	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	94	2	no	—
210	C₂²⋊C₈	—	94	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	47	2	no	—
216	C₄×C₈	—	47	1	no	—
217	C₄×C₈	—	47	1	no	—
218	C₄×C₈	—	94	2	no	—
219	C₄×C₈	—	94	1	no	—
220	C₄×C₈	C₄×D₄₇	1	2	yes	—
221	#41	—	47	4	no	—
222	#41	—	94	2	no	—
223	C₂×Q₈	—	47	1	no	—
224	C₂×D₄	—	47	1	no	—
225	C₂×D₄	—	94	1	no	—
226	C₂²×C₄	—	94	1	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	1	no	—
229	C₂²×C₄	—	94	1	no	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
232	C₂²×C₄	#10625354	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	C₈×D₄₇	1	4	yes	complemented
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	47	2	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	188	4	no	—
243	C₂×C₈	—	188	2	no	—
244	C₄⋊C₄	—	47	1	no	—
245	C₄⋊C₄	—	47	1	no	—
246	C₄⋊C₄	—	94	1	no	—
247	C₄⋊C₄	—	94	1	no	—
248	C₂²⋊C₄	—	94	1	no	—
249	C₂²⋊C₄	—	94	1	no	—
250	C₂²⋊C₄	—	94	1	no	—
251	C₄²	—	47	4	no	—
252	C₄²	—	94	1	no	—
253	C₄²	—	94	1	no	—
254	C₄²	—	94	1	no	—
255	C₄²	—	94	2	no	—
256	C₄²	—	188	2	no	—
257	C₄²	—	188	2	no	—
258	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₄²	D₄×D₄₇	1	1	yes	characteristic
261	C₄²	#10625352	1	1	yes	characteristic
262	C₂³	—	94	1	no	—
263	C₂³	#13093986	1	1	yes	characteristic
264	Q₈	—	47	2	no	—
265	Q₈	—	94	1	no	—
266	D₄	—	47	2	no	—
267	D₄	—	94	4	no	—
268	D₄	—	94	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	94	1	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	1	no	—
274	C₂×C₄	—	94	1	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
282	C₂×C₄	#13093982	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#13093986	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
287	C₂×C₄	#13094032	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	94	2	no	—
292	C₈	—	94	2	no	—
293	C₈	—	188	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	94	2	no	—
297	C₂²	—	188	1	no	—
298	C₂²	—	188	1	no	—
299	C₂²	#15688970	1	1	yes	characteristic
300	C₂²	#15689188	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	94	2	no	—
304	C₄	—	188	4	no	—
305	C₄	—	188	1	no	—
306	C₄	—	188	1	no	—
307	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
308	C₄	#15689080	1	1	yes	characteristic
309	C₄	#15689138	1	2	yes	—
310	C₄	(C₂²⋊D₄₇)⋊C₈	1	2	yes	—
311	C₄	#15689244	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	188	2	no	—
314	C₂	#20243158	1	1	yes	characteristic
315	C₂	#20243197	1	1	yes	characteristic
316	C₂	#20243231	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1278	C₂
socle	C₂²×C₄₇	C₄×C₂²⋊C₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1278	D₄₇
hypocenter	C₄₇	#8191
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. #20243133	C₂	semidirect
5. M₁₆×C₄₇⋊C₈	C₂	semidirect
6. #20241220	C₂	semidirect
7. C₄₇×#1278	C₂	semidirect
9. #15689247	C₄	semidirect
10. #15689188	C₄	semidirect
20. #15688855	C₄	semidirect
21. #15688852	C₄	semidirect
44. #13093950	C₈	semidirect
75. M₁₆×C₄₇	C₂×C₈	semidirect
97. C₄×Dic₄₇	C₂×C₈	semidirect
159. #1278	D₄₇	semidirect
185. C₄₇	#8191	semidirect
234. M₁₆	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252344	2
C₄×#13093950	2
#20243133	2
#20241771	2
M₁₆×C₄₇⋊C₈	2
#20241220	2
C₄₇×#1278	2
#8191	47