Subgroups — #26224435

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	#20252322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252101	C₂	1	1	yes	maximal, characteristic
4	#20244365	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244216	C₂	1	1	yes	maximal, characteristic
6	#20244092	C₂	1	4	yes	maximal, complemented
7	#20243526	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243225	C₂	1	4	yes	maximal, complemented
9	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	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	#15689253	—	2	2	no	—
17	C₈×Dic₉₄	—	2	2	no	—
18	#15689202	—	2	2	no	—
19	C₈×D₁₈₈	—	2	2	no	—
20	#15689149	C₂²	1	2	yes	—
21	#15689147	C₂²	1	1	yes	characteristic
22	#15689147	C₂²	1	1	yes	characteristic
23	C₂×#13093952	C₂²	1	4	yes	—
24	#15689120	C₂²	1	4	yes	—
25	#15689089	C₂²	1	2	yes	—
26	#15689078	C₂²	1	2	yes	—
27	#15688888	C₂²	1	4	yes	—
28	#15688883	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄³×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625350	C₂³	1	2	yes	—
35	#13094030	—	2	8	no	—
36	C₂×#10625329	C₂³	1	2	yes	—
37	C₂×#10625329	C₂³	1	2	yes	—
38	C₂×#10625324	—	2	2	no	—
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	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
52	Dic₄₇⋊C₈	—	2	4	no	—
53	#13093952	—	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	C₈×Dic₄₇	—	2	2	no	—
59	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	—	2	2	no	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₄₇	—	2	4	no	—
66	C₄×C₈×C₄₇	—	2	4	no	—
67	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₄₇	D₄	1	2	yes	—
70	C₄×C₈×C₄₇	D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₇	—	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	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	#41	1	4	yes	—
90	C₄²×C₄₇	—	2	8	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	2	yes	—
95	C₄²×C₄₇	#41	1	4	yes	—
96	C₂²⋊Dic₄₇	—	2	2	no	—
97	(C₂×D₄₇)⋊C₄	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
101	Dic₄₇⋊C₄	—	2	4	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	#10625329	—	2	4	no	—
104	#10625329	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
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	8	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂²⋊D₄₇	—	4	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	—	2	2	no	—
131	C₈×C₄₇	#124	1	2	yes	—
132	C₈×C₄₇	#124	1	2	yes	—
133	C₄₇⋊C₈	—	4	2	no	—
134	C₄₇⋊C₈	—	4	2	no	—
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	8	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	#443	1	4	yes	—
144	C₄×C₄₇	C₄×#41	1	2	yes	—
145	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
146	C₄×C₄₇	#532	1	1	yes	characteristic
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	—	47	1	no	—
156	#1764	—	47	4	no	—
157	C₄²×C₈	D₄₇	1	1	yes	characteristic, complemented
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	—	47	1	no	—
174	#423	—	47	1	no	—
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	C₂×D₄₇	1	2	yes	—
180	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
181	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₄³	C₂×D₄₇	1	1	yes	characteristic
184	#341	—	47	4	no	—
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	—	47	2	no	—
190	C₂×M₁₆	—	47	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
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	—	2	2	no	—
207	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
209	C₄⋊C₈	—	94	4	no	—
210	C₄⋊C₈	—	94	2	no	—
211	C₄⋊C₈	—	94	2	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	—	2	4	no	—
218	C₄×C₈	—	2	4	no	—
219	C₄×C₈	—	94	2	no	—
220	C₄×C₈	C₄×D₄₇	1	2	yes	—
221	C₄×C₈	C₄×D₄₇	1	2	yes	—
222	C₄×C₈	D₁₈₈	1	2	yes	—
223	C₄×C₈	D₁₈₈	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	—	2	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	#10625350	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	M₁₆	—	94	4	no	—
236	M₁₆	—	94	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	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	188	4	no	—
244	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
245	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
246	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
248	C₂×C₈	#10625350	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	—	2	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	94	2	no	—
258	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
260	C₄²	C₂×D₁₈₈	1	2	yes	—
261	C₄²	#10625350	1	4	yes	—
262	C₂³	—	94	2	no	—
263	C₂³	C₂×#10625350	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	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	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	C₄×D₁₈₈	1	2	yes	—
277	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
280	C₂×C₄	C₂×#10625350	1	2	yes	—
281	C₂×C₄	C₂×#10625350	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	188	2	no	—
285	C₈	—	188	2	no	—
286	C₈	#13094012	1	2	yes	—
287	C₈	#13094012	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	188	4	no	—
290	C₂²	—	188	2	no	—
291	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
292	C₂²	#15689657	1	2	yes	—
293	C₄	—	2	8	no	—
294	C₄	—	2	2	no	—
295	C₄	—	188	4	no	—
296	C₄	—	188	2	no	—
297	C₄	#15689241	1	4	yes	—
298	C₄	C₄×#10625350	1	2	yes	—
299	C₄	C₂×#13094012	1	1	yes	characteristic
300	C₄	#15689673	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	376	2	no	—
303	C₂	C₄×#13094012	1	1	yes	characteristic
304	C₂	C₂×#15689241	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#21658
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. #20252322	C₂	semidirect
4. #20244365	C₂	semidirect
6. #20244092	C₂	semidirect
7. #20243526	C₂	semidirect
8. #20243225	C₂	semidirect
9. C₄²×C₈×C₄₇	C₂	semidirect
13. #15689478	C₄	semidirect
14. #15689474	C₄	semidirect
15. #15689470	C₄	semidirect
157. C₄²×C₈	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
#20252322	2
#20252101	2
#20244365	2
#20244216	2
#20244092	2
#20243526	2
#20243225	2
C₄²×C₈×C₄₇	2
#21658	47