Subgroups — #26215846

Order 12032.

1646 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	2	yes	maximal, complemented
2	#20245771	C₂	1	2	yes	maximal
3	#20243417	C₂	1	4	yes	maximal, complemented
4	#20243413	C₂	1	4	yes	maximal, complemented
5	#20243105	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₂²	1	4	yes	—
8	#15689161	—	2	8	no	—
9	C₂×#13093958	C₂²	1	4	yes	complemented
10	C₂×#13093956	C₂²	1	4	yes	complemented
11	#15688986	—	2	2	no	—
12	#15688983	C₂²	1	8	yes	—
13	#15688979	—	2	2	no	—
14	C₄×C₂²⋊Dic₄₇	—	2	4	no	—
15	#15688886	C₂²	1	8	yes	complemented
16	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₇	—	2	8	no	—
18	C₂³×C₈×C₄₇	C₄	1	8	yes	complemented
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	#13093983	—	2	8	no	—
24	#13093983	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
26	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
27	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄×Dic₄₇	—	2	4	no	—
29	#13093958	—	2	8	no	—
30	#13093956	—	2	8	no	—
31	C₄₇⋊#95	—	2	8	no	—
32	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	—	2	4	no	—
34	C₂³×C₄×C₄₇	—	2	4	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	4	no	—
42	C₂²×C₈×C₄₇	—	2	32	no	—
43	C₂²×C₈×C₄₇	—	2	8	no	—
44	C₂²×C₈×C₄₇	—	2	8	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₄₇	D₄	1	8	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂⁴×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	—	2	8	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	16	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂×C₈×C₄₇	—	2	8	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	16	no	—
80	C₂×C₈×C₄₇	—	2	16	no	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	complemented
82	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
87	C₂²⋊Dic₄₇	—	4	8	no	—
88	C₂²⋊Dic₄₇	—	4	16	no	—
89	C₄⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	4	8	no	—
92	C₂³×C₄₇	—	2	32	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	8	no	—
97	C₂³×C₄₇	—	2	16	no	—
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂³×C₄₇	#121	1	4	yes	—
101	C₂³×C₄₇	#135	1	8	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	—	2	16	no	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	#117	1	4	yes	—
113	C₂×C₄×C₄₇	#121	1	4	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
115	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
119	C₂×Dic₄₇	—	4	8	no	—
120	C₂×Dic₄₇	—	4	16	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
124	#20658	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	32	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	—	2	16	no	—
131	C₂²×C₄₇	#380	1	4	yes	—
132	C₂²×C₄₇	#427	1	8	yes	complemented
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×#121	1	2	yes	—
135	C₂²×C₄₇	C₂×#135	1	4	yes	—
136	C₄×C₄₇	—	2	4	no	—
137	C₄×C₄₇	#380	1	4	yes	—
138	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
140	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
141	Dic₄₇	—	8	8	no	—
142	C₂⁴×C₈	D₄₇	1	1	yes	characteristic
143	C₂×#427	—	47	2	no	—
144	#2280	—	47	2	no	—
145	#1808	—	47	4	no	—
146	#1807	—	47	4	no	—
147	#1735	—	47	2	no	—
148	C₂×C₄₇	—	2	4	no	—
149	C₂×C₄₇	—	2	8	no	—
150	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
151	C₂×C₄₇	C₂×#427	1	2	yes	—
152	C₂×C₄₇	#3050	1	4	yes	—
153	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
154	C₂³×C₈	—	2	8	no	—
155	C₂³×C₈	Dic₄₇	1	8	yes	complemented
156	C₂³×C₈	C₂×D₄₇	1	2	yes	—
157	C₂³×C₈	C₂×D₄₇	1	2	yes	—
158	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×#117	—	47	4	no	—
161	#427	—	94	8	no	—
162	C₂×#107	—	47	4	no	—
163	C₂×#106	—	47	4	no	—
164	#381	—	94	2	no	—
165	#380	—	47	8	no	—
166	#379	—	94	2	no	—
167	C₄×C₂²⋊C₄	—	94	4	no	—
168	#340	—	47	8	no	—
169	C₄₇	#20658	1	1	yes	characteristic, complemented, Sylow
170	C₂⁵	D₁₈₈	1	1	yes	characteristic
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	Dic₉₄	1	4	yes	—
174	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
175	C₂³×C₄	D₁₈₈	1	2	yes	—
176	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
177	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
178	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	32	no	—
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	—	2	8	no	—
183	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
184	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
185	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
186	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
187	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
188	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	#117	—	94	8	no	—
192	#117	—	94	8	no	—
193	C₂×C₄⋊C₄	—	47	8	no	—
194	C₂×C₂²⋊C₄	—	94	4	no	—
195	C₂×C₂²⋊C₄	—	94	8	no	—
196	C₂×C₄²	—	94	4	no	—
197	#107	—	94	8	no	—
198	#106	—	94	8	no	—
199	#95	—	94	8	no	—
200	C₂⁴	—	2	4	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
203	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂⁴	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	94	8	no	—
213	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
214	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
215	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
216	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
217	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
218	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂²×C₄	#10625350	1	4	yes	—
220	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	16	no	—
227	C₂×C₈	—	2	16	no	—
228	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	complemented
229	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
230	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
231	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
233	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
234	C₄⋊C₄	—	94	8	no	—
235	C₄⋊C₄	—	94	8	no	—
236	C₂²⋊C₄	—	188	8	no	—
237	C₂²⋊C₄	—	188	16	no	—
238	C₄²	—	188	8	no	—
239	C₂³	—	2	32	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	16	no	—
245	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
246	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
247	C₂³	#13093998	1	4	yes	—
248	C₂³	#13094046	1	8	yes	—
249	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	188	8	no	—
257	C₂×C₄	—	188	16	no	—
258	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
261	C₂×C₄	#13093983	1	4	yes	—
262	C₂×C₄	#13093998	1	4	yes	—
263	C₂×C₄	#13094003	1	8	yes	—
264	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
266	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
267	C₂×C₄	C₂×#10625350	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
270	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
271	C₂²	—	2	32	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	16	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#15688983	1	4	yes	—
278	C₂²	#15689161	1	8	yes	complemented
279	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
280	C₂²	C₂×#13093998	1	2	yes	—
281	C₂²	C₂×#13094046	1	4	yes	—
282	C₄	—	2	4	no	—
283	C₄	—	376	8	no	—
284	C₄	#15688983	1	4	yes	—
285	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
286	C₄	C₂×#13093983	1	1	yes	characteristic
287	C₄	C₂×#13094003	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	2	8	no	—
290	C₂	C₂×#15688983	1	1	yes	characteristic
291	C₂	C₂×#15689161	1	2	yes	—
292	C₂	#20252969	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂⁴×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂⁴×C₈	D₄₇
hypocenter	C₄₇	#20658
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689161	C₂	semidirect
3. #20243417	C₂	semidirect
4. #20243413	C₂	semidirect
5. #20243105	C₂	semidirect
9. C₂×#13093958	C₂²	semidirect
10. C₂×#13093956	C₂²	semidirect
15. #15688886	C₂²	semidirect
18. C₂³×C₈×C₄₇	C₄	semidirect
81. C₂×C₈×C₄₇	C₂²⋊C₄	semidirect
132. C₂²×C₄₇	#427	semidirect
155. C₂³×C₈	Dic₄₇	semidirect
169. C₄₇	#20658	semidirect
228. C₂×C₈	C₂²⋊Dic₄₇	semidirect
278. C₂²	#15689161	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
#20245771	2
#20243417	2
#20243413	2
#20243105	2
C₂⁴×C₈×C₄₇	2
#20658	47