Subgroups — #26224430

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253711	C₂	1	1	yes	maximal, characteristic
2	#20252318	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252099	C₂	1	1	yes	maximal, characteristic
4	#20244364	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244215	C₂	1	1	yes	maximal, characteristic
6	#20244090	C₂	1	4	yes	maximal, complemented
7	#20243224	C₂	1	4	yes	maximal, complemented
8	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094032	C₂²	1	2	yes	—
11	#15689680	C₂²	1	1	yes	characteristic
12	#15689673	C₂²	1	1	yes	characteristic
13	#15689478	C₄	1	4	yes	complemented
14	#15689474	C₄	1	2	yes	complemented
15	#15689470	C₄	1	2	yes	complemented
16	#15689250	—	2	2	no	—
17	#15689247	—	2	2	no	—
18	#15689200	—	2	2	no	—
19	#15689188	—	2	2	no	—
20	#15689151	C₂²	1	2	yes	—
21	C₂×#13093950	C₂²	1	4	yes	—
22	#15689120	C₂²	1	4	yes	—
23	#15689091	C₂²	1	2	yes	—
24	#15689081	C₂²	1	2	yes	—
25	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
27	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688884	C₂²	1	4	yes	—
30	C₄₇×#423	C₂²	1	1	yes	characteristic
31	C₄₇×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#341	C₂²	1	4	yes	—
34	C₂×#10625350	C₂³	1	2	yes	—
35	#13094032	—	2	8	no	—
36	C₂×#10625324	—	2	2	no	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₄₇	—	2	2	no	—
40	#13094024	C₂×C₄	1	1	yes	characteristic
41	#13094024	C₂×C₄	1	2	yes	—
42	#13094021	C₂×C₄	1	1	yes	characteristic
43	#13094016	C₂×C₄	1	2	yes	—
44	#13094012	C₂×C₄	1	2	yes	—
45	#13094003	—	2	2	no	—
46	#13093998	—	2	2	no	—
47	C₄×Dic₉₄	—	2	2	no	—
48	C₄×D₁₈₈	—	2	2	no	—
49	#13093983	C₂³	1	2	yes	—
50	C₂×C₄×Dic₄₇	—	2	2	no	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	Dic₄₇⋊C₈	—	2	4	no	—
53	#13093950	—	2	8	no	—
54	#13093949	C₂×C₄	1	8	yes	—
55	#13093946	C₂×C₄	1	8	yes	—
56	(C₂×D₄₇)⋊C₈	—	2	4	no	—
57	#13093934	—	2	2	no	—
58	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
60	C₈×Dic₄₇	—	2	4	no	—
61	C₈×Dic₄₇	—	2	2	no	—
62	C₈×Dic₄₇	—	2	4	no	—
63	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
64	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	—	2	2	no	—
68	C₄⋊C₈×C₄₇	—	2	2	no	—
69	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	#10625350	—	2	4	no	—
76	#10625350	—	2	4	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
79	C₂×C₄×D₄₇	—	2	2	no	—
80	C₂×Dic₉₄	—	2	2	no	—
81	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
82	M₁₆×C₄₇	—	2	4	no	—
83	M₁₆×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²⋊Dic₄₇	—	2	2	no	—
93	(C₂×D₄₇)⋊C₄	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
97	Dic₄₇⋊C₄	—	2	4	no	—
98	C₄×Dic₄₇	—	2	8	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	#41	1	4	yes	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	#41	1	4	yes	—
107	#10625324	—	4	4	no	—
108	C₈×D₄₇	—	4	4	no	—
109	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₂²×D₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂²⋊D₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	D₁₈₈	—	2	4	no	—
124	D₁₈₈	—	4	4	no	—
125	C₄×D₄₇	—	4	2	no	—
126	C₄×D₄₇	—	4	2	no	—
127	Dic₉₄	—	2	4	no	—
128	Dic₉₄	—	4	4	no	—
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	#124	1	2	yes	—
134	C₄₇⋊C₈	#124	1	2	yes	—
135	#21658	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄₇	#529	1	2	yes	—
139	C₂×D₄₇	—	4	4	no	—
140	C₂×D₄₇	—	4	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	#443	1	4	yes	—
143	C₄×C₄₇	C₄×#41	1	2	yes	—
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	#532	1	1	yes	characteristic
146	Dic₄₇	—	2	8	no	—
147	Dic₄₇	—	4	4	no	—
148	Dic₄₇	—	4	2	no	—
149	#3131	—	47	1	no	—
150	#2960	—	47	1	no	—
151	#2931	—	47	1	no	—
152	#1994	—	47	1	no	—
153	#1962	—	47	1	no	—
154	#1937	—	47	4	no	—
155	#1837	D₄₇	1	1	yes	characteristic, complemented
156	#1764	—	47	4	no	—
157	C₄²×C₈	—	47	1	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#443	1	2	yes	—
161	D₄₇	—	8	2	no	—
162	C₂×#131	—	47	2	no	—
163	#533	—	47	1	no	—
164	#532	—	47	1	no	—
165	#488	—	47	4	no	—
166	#487	—	47	2	no	—
167	#486	—	47	2	no	—
168	#446	—	94	2	no	—
169	C₈×Q₈	—	94	2	no	—
170	#436	—	94	2	no	—
171	C₈×D₄	—	94	2	no	—
172	#424	—	47	2	no	—
173	#423	C₂×D₄₇	1	1	yes	characteristic
174	#423	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#104	—	47	4	no	—
176	#416	—	47	4	no	—
177	#408	—	47	2	no	—
178	#405	—	47	2	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	C₂×C₄×C₈	—	47	2	no	—
181	C₂×C₄×C₈	—	47	2	no	—
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₄³	—	47	1	no	—
184	#341	C₂×D₄₇	1	4	yes	—
185	#339	—	47	4	no	—
186	C₄₇	#21658	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	47	2	no	—
188	#131	—	94	8	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
191	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
196	C₄⋊Q₈	—	47	2	no	—
197	C₄⋊Q₈	—	47	1	no	—
198	#127	—	47	1	no	—
199	#125	—	47	2	no	—
200	#124	—	47	2	no	—
201	C₂²⋊Q₈	—	94	2	no	—
202	#121	—	94	2	no	—
203	C₄×Q₈	—	94	2	no	—
204	C₄×D₄	—	94	2	no	—
205	#117	—	47	2	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	94	4	no	—
212	#104	—	94	8	no	—
213	#103	—	47	8	no	—
214	#102	—	47	8	no	—
215	C₂²⋊C₈	—	94	4	no	—
216	#97	—	94	2	no	—
217	C₄×C₈	—	47	2	no	—
218	C₄×C₈	—	47	2	no	—
219	C₄×C₈	—	94	4	no	—
220	C₄×C₈	—	94	4	no	—
221	C₄×C₈	—	94	2	no	—
222	C₄×C₈	C₄×D₄₇	1	2	yes	—
223	C₄×C₈	C₄×D₄₇	1	2	yes	—
224	#41	—	94	4	no	—
225	#41	—	94	4	no	—
226	C₂×Q₈	—	47	2	no	—
227	C₂×Q₈	—	94	2	no	—
228	C₂×D₄	—	47	2	no	—
229	C₂×D₄	—	94	2	no	—
230	C₂×D₄	—	94	2	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	#10625354	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	188	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	47	1	no	—
243	C₂×C₈	—	47	1	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	47	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	188	4	no	—
248	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
249	C₄⋊C₄	—	47	4	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₂²⋊C₄	—	94	2	no	—
254	C₂²⋊C₄	—	94	4	no	—
255	C₄²	—	47	4	no	—
256	C₄²	—	94	8	no	—
257	C₄²	—	94	2	no	—
258	C₄²	—	94	2	no	—
259	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₄²	D₄×D₄₇	1	2	yes	—
262	C₂³	—	94	2	no	—
263	C₂³	C₂×#10625354	1	1	yes	characteristic
264	Q₈	—	94	4	no	—
265	Q₈	—	188	4	no	—
266	D₄	—	94	4	no	—
267	D₄	—	188	4	no	—
268	D₄	—	188	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	94	8	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	2	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	#13093986	1	2	yes	—
277	C₂×C₄	#13093986	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₂×C₄	#41×D₄₇	1	2	yes	—
281	C₂×C₄	#41×D₄₇	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	47	2	no	—
286	C₈	—	47	2	no	—
287	C₈	—	94	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	188	4	no	—
290	C₂²	—	188	2	no	—
291	C₂²	C₂×#13093986	1	1	yes	characteristic
292	C₂²	#15689658	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	94	8	no	—
295	C₄	—	188	4	no	—
296	C₄	—	188	2	no	—
297	C₄	#15689239	1	4	yes	—
298	C₄	#15689591	1	2	yes	—
299	C₄	#124×D₄₇	1	1	yes	characteristic
300	C₄	#15689672	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	376	2	no	—
303	C₂	#20245927	1	1	yes	characteristic
304	C₂	C₂×#15689239	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1837	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#21658
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
2. #20252318	C₂	semidirect
4. #20244364	C₂	semidirect
6. #20244090	C₂	semidirect
7. #20243224	C₂	semidirect
8. C₄×C₈×Dic₄₇	C₂	semidirect
9. C₄₇×#1837	C₂	semidirect
13. #15689478	C₄	semidirect
14. #15689474	C₄	semidirect
15. #15689470	C₄	semidirect
155. #1837	D₄₇	semidirect
186. C₄₇	#21658	semidirect

Maximal subgroups

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

subgroup	index
#20253711	2
#20252318	2
#20252099	2
#20244364	2
#20244215	2
#20244090	2
#20243224	2
C₄×C₈×Dic₄₇	2
C₄₇×#1837	2
#21658	47