Subgroups — #26204216

Order 12032.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254511	C₂	1	2	yes	maximal, complemented
2	#20253967	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	#20253847	C₂	1	2	yes	maximal, complemented
5	#20243059	C₂	1	4	yes	maximal, complemented
6	#20243006	C₂	1	2	yes	maximal, complemented
7	#20242980	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094057	—	2	4	no	—
10	#15689673	C₂²	1	2	yes	—
11	C₂×#13094024	C₂²	1	2	yes	—
12	#15689528	—	2	2	no	—
13	#15689528	C₂²	1	1	yes	characteristic
14	#15689528	C₂²	1	2	yes	—
15	#15689528	C₂²	1	4	yes	—
16	#15689528	C₂²	1	2	yes	—
17	#15689517	—	2	4	no	—
18	#15689503	C₂²	1	4	yes	—
19	#15689500	C₂²	1	1	yes	characteristic
20	#15689496	C₂²	1	4	yes	—
21	#15689470	C₂²	1	2	yes	—
22	#15689470	C₂²	1	4	yes	—
23	#15689437	—	2	4	no	—
24	#15689404	—	2	4	no	—
25	#15689359	—	2	2	no	—
26	C₄×D₃₇₆	—	2	2	no	—
27	C₂×#13093958	—	2	2	no	—
28	C₄₇×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#322	C₂²	1	2	yes	—
32	C₈²×C₄₇	C₂²	1	2	yes	—
33	C₂×#10625350	—	2	2	no	—
34	C₂²×Dic₉₄	—	2	2	no	—
35	#13094057	—	4	16	no	—
36	C₂×Dic₁₈₈	—	2	8	no	—
37	C₂×#10625325	—	2	8	no	—
38	C₂×D₃₇₆	—	2	2	no	—
39	#13094024	—	2	2	no	—
40	#13094024	—	2	2	no	—
41	#13094024	—	2	2	no	—
42	#13094024	C₂³	1	4	yes	—
43	#13094024	C₂³	1	2	yes	—
44	#13094021	C₂³	1	2	yes	—
45	#13094012	—	2	2	no	—
46	#13093998	—	2	2	no	—
47	#13093998	—	2	2	no	—
48	C₄×D₁₈₈	—	2	2	no	—
49	C₄×D₁₈₈	—	2	2	no	—
50	C₂×C₄⋊Dic₄₇	—	2	2	no	—
51	C₂×C₄⋊Dic₄₇	—	2	2	no	—
52	#13093958	—	2	4	no	—
53	#13093958	—	2	4	no	—
54	#13093958	—	2	2	no	—
55	#13093958	—	2	8	no	—
56	#13093946	—	2	4	no	—
57	#13093946	—	2	8	no	—
58	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
59	C₂²×C₈×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
62	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
63	#97×C₄₇	D₄	1	2	yes	—
64	C₄×C₈×C₄₇	D₄	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	2	yes	—
68	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₂²⋊D₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	2	no	—
74	#10625350	—	4	8	no	—
75	C₂×D₁₈₈	—	2	2	no	—
76	C₂×D₁₈₈	—	2	2	no	—
77	C₂×D₁₈₈	—	2	2	no	—
78	C₂×C₄×D₄₇	—	4	2	no	—
79	C₂×Dic₉₄	—	2	2	no	—
80	C₂×Dic₉₄	—	2	4	no	—
81	C₂×Dic₉₄	—	4	8	no	—
82	C₂×Dic₉₄	—	4	2	no	—
83	M₁₆×C₄₇	D₈	1	8	yes	complemented
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
93	(C₂×D₄₇)⋊C₄	—	4	2	no	—
94	(C₂×D₄₇)⋊C₄	—	4	2	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	4	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	Dic₁₈₈	—	4	16	no	—
102	D₃₇₆	—	4	4	no	—
103	#10625325	—	4	16	no	—
104	C₂³×C₄₇	#127	1	1	yes	characteristic
105	C₂²×D₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₄₇	—	8	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	8	4	no	—
119	D₁₈₈	—	4	4	no	—
120	D₁₈₈	—	4	4	no	—
121	D₁₈₈	—	4	4	no	—
122	C₄×D₄₇	—	8	4	no	—
123	Dic₉₄	—	4	8	no	—
124	Dic₉₄	—	4	4	no	—
125	Dic₉₄	—	8	4	no	—
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	—	2	4	no	—
128	C₈×C₄₇	C₂×D₈	1	8	yes	—
129	C₈×C₄₇	#137	1	4	yes	complemented
130	#19630	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	#493	1	2	yes	—
134	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
135	C₂×D₄₇	—	8	2	no	—
136	C₂×D₄₇	—	8	2	no	—
137	C₂×D₄₇	—	8	2	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	#493	1	2	yes	—
141	C₄×C₄₇	C₂²×D₈	1	2	yes	—
142	C₄×C₄₇	C₂×#137	1	2	yes	—
143	C₄×C₄₇	#578	1	2	yes	—
144	Dic₄₇	—	8	4	no	—
145	Dic₄₇	—	8	2	no	—
146	#3226	—	47	2	no	—
147	#3169	—	47	1	no	—
148	C₂×#500	—	47	1	no	—
149	#3152	—	47	2	no	—
150	#1722	—	47	4	no	—
151	#1713	—	47	2	no	—
152	#1706	—	47	2	no	—
153	C₈×M₁₆	D₄₇	1	1	yes	characteristic, complemented
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
156	C₂×C₄₇	#3152	1	1	yes	characteristic
157	C₂×C₄₇	#3156	1	1	yes	characteristic
158	D₄₇	—	16	2	no	—
159	C₂×#137	—	94	4	no	—
160	#532	—	47	2	no	—
161	C₂×C₄⋊Q₈	—	47	2	no	—
162	#500	—	47	2	no	—
163	#500	—	47	2	no	—
164	#500	—	47	1	no	—
165	#500	—	47	4	no	—
166	#500	—	94	2	no	—
167	#497	—	94	4	no	—
168	#494	—	47	4	no	—
169	#493	—	47	1	no	—
170	#492	—	47	4	no	—
171	#486	—	47	2	no	—
172	#486	—	47	4	no	—
173	#480	—	94	4	no	—
174	#474	—	94	4	no	—
175	#466	—	94	2	no	—
176	C₄×D₈	—	94	2	no	—
177	#431	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#107	—	94	2	no	—
179	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
181	#322	C₂×D₄₇	1	2	yes	—
182	C₈²	C₂×D₄₇	1	2	yes	—
183	C₄₇	#19630	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	94	2	no	—
185	C₂²×Q₈	—	94	2	no	—
186	#137	—	188	16	no	—
187	C₂×Q₁₆	—	94	8	no	—
188	C₂×SD₁₆	—	94	8	no	—
189	C₂×D₈	—	94	2	no	—
190	C₂×M₁₆	D₁₈₈	1	2	yes	—
191	C₂²×C₈	D₁₈₈	1	2	yes	—
192	C₄⋊Q₈	—	47	4	no	—
193	C₄⋊Q₈	—	47	2	no	—
194	C₄⋊Q₈	—	94	2	no	—
195	C₄⋊Q₈	—	94	2	no	—
196	C₄⋊Q₈	—	94	2	no	—
197	#127	—	47	2	no	—
198	#124	—	94	2	no	—
199	#121	—	94	2	no	—
200	#121	—	94	2	no	—
201	C₄×D₄	—	94	2	no	—
202	C₄×D₄	—	94	2	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	—	94	2	no	—
205	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
206	#107	—	94	4	no	—
207	#107	—	94	4	no	—
208	#107	—	94	2	no	—
209	#107	—	94	8	no	—
210	C₄⋊C₈	D₁₈₈	1	2	yes	—
211	#102	—	94	4	no	—
212	#102	—	94	8	no	—
213	C₂²⋊C₈	D₁₈₈	1	2	yes	—
214	#97	D₁₈₈	1	2	yes	—
215	C₄×C₈	D₁₈₈	1	2	yes	—
216	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₄₇	1	2	yes	—
218	C₄×C₈	C₂²×D₄₇	1	2	yes	—
219	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#41	—	188	8	no	—
221	C₂×Q₈	—	94	2	no	—
222	C₂×Q₈	—	94	4	no	—
223	C₂×Q₈	—	188	8	no	—
224	C₂×Q₈	—	188	2	no	—
225	C₂×D₄	—	94	2	no	—
226	C₂×D₄	—	94	2	no	—
227	C₂×D₄	—	94	2	no	—
228	C₂×D₄	—	188	2	no	—
229	C₂²×C₄	—	188	2	no	—
230	C₂²×C₄	—	188	2	no	—
231	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
234	Q₁₆	—	188	16	no	—
235	SD₁₆	—	188	16	no	—
236	D₈	—	188	4	no	—
237	M₁₆	D₃₇₆	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
240	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
241	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
242	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	94	4	no	—
246	C₄⋊C₄	—	94	2	no	—
247	C₄⋊C₄	—	188	2	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₂²⋊C₄	—	188	2	no	—
250	C₂²⋊C₄	—	188	2	no	—
251	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
252	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
253	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
254	C₄²	C₂³×D₄₇	1	1	yes	characteristic
255	C₂³	—	188	2	no	—
256	C₂³	#13094021	1	1	yes	characteristic
257	Q₈	—	376	4	no	—
258	Q₈	—	188	8	no	—
259	Q₈	—	188	4	no	—
260	D₄	—	376	4	no	—
261	D₄	—	188	4	no	—
262	D₄	—	188	4	no	—
263	D₄	—	188	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	376	4	no	—
267	C₂×C₄	—	376	4	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	#13094021	1	1	yes	characteristic
271	C₂×C₄	#13094021	1	1	yes	characteristic
272	C₂×C₄	#13094021	1	1	yes	characteristic
273	C₂×C₄	C₂×D₃₇₆	1	4	yes	—
274	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₃₇₆	1	8	yes	—
280	C₈	#13094057	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	376	2	no	—
284	C₂²	—	376	2	no	—
285	C₂²	—	376	2	no	—
286	C₂²	#15689500	1	2	yes	—
287	C₂²	C₂×#13094021	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	376	2	no	—
291	C₄	—	376	4	no	—
292	C₄	#15689500	1	2	yes	—
293	C₄	C₂²×D₃₇₆	1	2	yes	—
294	C₄	C₂×#13094057	1	2	yes	—
295	C₄	#15689940	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	752	2	no	—
298	C₂	C₂×#15689500	1	1	yes	characteristic
299	C₂	#20253847	1	1	yes	characteristic
300	C₂	#20253882	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13094021
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#127
prime radical	#20253967	C₂
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#19630
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20254511	C₂	semidirect
3. C₂×#15689528	C₂	semidirect
4. #20253847	C₂	semidirect
5. #20243059	C₂	semidirect
6. #20243006	C₂	semidirect
7. #20242980	C₂	semidirect
8. C₈×M₁₆×C₄₇	C₂	semidirect
83. M₁₆×C₄₇	D₈	semidirect
129. C₈×C₄₇	#137	semidirect
153. C₈×M₁₆	D₄₇	semidirect
183. C₄₇	#19630	semidirect
237. M₁₆	D₃₇₆	semidirect
280. C₈	#13094057	semidirect

Maximal subgroups

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

subgroup	index
#20254511	2
#20253967	2
C₂×#15689528	2
#20253847	2
#20243059	2
#20243006	2
#20242980	2
C₈×M₁₆×C₄₇	2
#19630	47