Subgroups — #26370745

Order 12032.

690 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×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252969	C₂	1	2	yes	maximal, complemented
3	#20245501	C₂	1	4	yes	maximal, complemented
4	#20245496	C₂	1	2	yes	maximal, complemented
5	#20245491	C₂	1	2	yes	maximal, complemented
6	#20245172	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245096	C₂	1	2	yes	maximal, complemented
8	C₂²×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	2	yes	—
10	#15689634	—	2	2	no	—
11	#15689555	—	2	4	no	—
12	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×#13093969	C₂²	1	2	yes	complemented
14	C₂×D₇₅₂	C₂²	1	1	yes	characteristic, complemented
15	#15689364	C₂²	1	4	yes	—
16	#15689359	C₂²	1	4	yes	—
17	#15689352	—	2	4	no	—
18	#15689161	C₂²	1	2	yes	—
19	#15689120	—	2	2	no	—
20	#15688951	C₂²	1	2	yes	complemented
21	#15688949	C₂²	1	2	yes	complemented
22	#15688928	C₂²	1	4	yes	complemented
23	#15688925	C₂²	1	4	yes	complemented
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₄₇	—	2	4	no	—
26	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
27	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#10625350	—	2	2	no	—
31	#13094046	—	2	4	no	—
32	#13094046	—	2	4	no	—
33	C₂×Dic₁₈₈	C₂³	1	2	yes	—
34	C₂×#10625325	—	2	2	no	—
35	C₂×D₃₇₆	C₂³	1	2	yes	—
36	#13094007	—	4	2	no	—
37	#13094003	—	2	4	no	—
38	#13093998	—	2	4	no	—
39	#13093993	—	4	2	no	—
40	C₄×C₂²⋊D₄₇	—	4	4	no	—
41	#13093983	—	2	2	no	—
42	Dic₃₇₆	—	2	2	no	—
43	#13093969	—	2	4	no	—
44	D₇₅₂	—	2	2	no	—
45	#13093958	C₂³	1	4	yes	—
46	#13093956	—	2	2	no	—
47	#13093949	—	2	4	no	—
48	#13093946	—	2	4	no	—
49	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	D₄	1	2	yes	—
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	M₃₂×C₄₇	—	2	4	no	—
57	M₃₂×C₄₇	—	2	4	no	—
58	C₂×C₁₆×C₄₇	—	2	2	no	—
59	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
61	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
63	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	D₈	1	2	yes	—
67	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₇	—	4	4	no	—
71	C₂×C₂²⋊D₄₇	—	4	2	no	—
72	#10625350	—	4	4	no	—
73	#10625350	—	4	4	no	—
74	C₂×D₁₈₈	—	2	2	no	—
75	C₂×C₄×D₄₇	—	4	2	no	—
76	C₂×Dic₉₄	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₄₇	—	4	2	no	—
87	C₂²⋊Dic₄₇	—	4	4	no	—
88	(C₂×D₄₇)⋊C₄	—	4	4	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	Dic₄₇⋊C₄	—	4	4	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	Dic₁₈₈	—	2	2	no	—
93	Dic₁₈₈	—	2	2	no	—
94	D₃₇₆	—	2	2	no	—
95	D₃₇₆	—	2	2	no	—
96	#10625325	—	4	4	no	—
97	C₁₆×C₄₇	—	2	2	no	—
98	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
99	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	4	2	no	—
102	C₂³×C₄₇	C₂×D₈	1	2	yes	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	#121	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#135	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂²⋊D₄₇	—	8	8	no	—
115	C₂²⋊D₄₇	—	8	4	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	D₁₈₈	—	4	2	no	—
119	D₁₈₈	—	4	2	no	—
120	C₄×D₄₇	—	8	4	no	—
121	Dic₉₄	—	4	2	no	—
122	Dic₉₄	—	4	2	no	—
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	#121	1	2	yes	—
125	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
127	#33604	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	#466	1	2	yes	—
133	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₄₇	—	8	4	no	—
135	C₂×D₄₇	—	8	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	#466	1	2	yes	—
138	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
139	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
140	Dic₄₇	—	8	4	no	—
141	Dic₄₇	—	8	2	no	—
142	C₂×#508	—	47	1	no	—
143	C₂²×M₃₂	D₄₇	1	1	yes	characteristic, complemented
144	#3050	—	47	2	no	—
145	#2220	—	47	4	no	—
146	#2219	—	47	2	no	—
147	#2218	—	47	2	no	—
148	#2158	—	47	1	no	—
149	#2138	—	47	2	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	—	4	2	no	—
152	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
153	C₂×C₄₇	#3414	1	2	yes	—
154	D₄₇	—	16	2	no	—
155	C₂×#135	—	47	2	no	—
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#525	—	94	2	no	—
158	#508	—	94	4	no	—
159	C₂×Q₃₂	—	47	1	no	—
160	C₂×SD₃₂	—	47	2	no	—
161	C₂×D₁₆	—	47	1	no	—
162	C₂×M₃₂	—	2	4	no	—
163	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
164	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
165	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
166	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
167	#467	—	47	4	no	—
168	#466	—	47	4	no	—
169	#465	—	94	4	no	—
170	#427	—	47	2	no	—
171	#416	—	94	2	no	—
172	#367	—	47	2	no	—
173	#366	—	47	2	no	—
174	#358	—	47	4	no	—
175	#357	—	47	4	no	—
176	C₄₇	#33604	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	94	2	no	—
178	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
179	#135	—	94	4	no	—
180	#135	—	94	4	no	—
181	C₂×Q₁₆	—	47	2	no	—
182	C₂×SD₁₆	—	94	2	no	—
183	C₂×D₈	—	47	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
187	C₂²×C₈	D₁₈₈	1	2	yes	—
188	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	#123	—	188	2	no	—
191	C₂²⋊Q₈	—	94	4	no	—
192	#121	—	94	4	no	—
193	#120	—	188	2	no	—
194	C₄×D₄	—	188	4	no	—
195	#117	—	94	2	no	—
196	Q₃₂	—	94	2	no	—
197	SD₃₂	—	94	4	no	—
198	D₁₆	—	94	2	no	—
199	M₃₂	—	2	4	no	—
200	M₃₂	—	2	4	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	complemented
203	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
204	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
205	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
206	#107	—	47	4	no	—
207	#106	—	94	2	no	—
208	#103	—	94	4	no	—
209	#102	—	94	4	no	—
210	C₂⁴	D₃₇₆	1	1	yes	characteristic
211	#41	—	188	4	no	—
212	#41	—	188	4	no	—
213	C₂×Q₈	—	94	2	no	—
214	C₂×D₄	—	94	2	no	—
215	C₂×D₄	—	188	4	no	—
216	C₂×D₄	—	188	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	188	2	no	—
220	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
221	C₂²×C₄	D₃₇₆	1	2	yes	—
222	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
223	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
224	Q₁₆	—	94	2	no	—
225	Q₁₆	—	94	2	no	—
226	SD₁₆	—	188	4	no	—
227	D₈	—	94	2	no	—
228	D₈	—	94	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
234	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
235	C₂×C₈	#10625350	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
237	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	188	4	no	—
240	C₂²⋊C₄	—	188	2	no	—
241	C₂²⋊C₄	—	188	4	no	—
242	C₂²⋊C₄	—	188	4	no	—
243	C₄²	—	188	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	188	2	no	—
250	C₂³	C₂×D₃₇₆	1	2	yes	—
251	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
252	Q₈	—	188	2	no	—
253	Q₈	—	188	2	no	—
254	D₄	—	376	8	no	—
255	D₄	—	376	4	no	—
256	D₄	—	188	2	no	—
257	D₄	—	188	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	376	4	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	#13093998	1	2	yes	—
266	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
267	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
268	C₂×C₄	#13094046	1	2	yes	—
269	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	#13093998	1	2	yes	—
272	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
273	C₈	C₂×#10625350	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	376	4	no	—
279	C₂²	—	376	2	no	—
280	C₂²	#15689359	1	2	yes	—
281	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	376	4	no	—
284	C₄	—	376	2	no	—
285	C₄	#15689359	1	2	yes	—
286	C₄	C₂×#13093998	1	1	yes	characteristic
287	C₄	C₂×#13094046	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	752	2	no	—
291	C₂	C₂×#15689359	1	1	yes	characteristic
292	C₂	#20256524	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂²×M₃₂	D₄₇
hypocenter	C₄₇	#33604
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. #20252969	C₂	semidirect
3. #20245501	C₂	semidirect
4. #20245496	C₂	semidirect
5. #20245491	C₂	semidirect
6. #20245172	C₂	semidirect
7. #20245096	C₂	semidirect
8. C₂²×M₃₂×C₄₇	C₂	semidirect
12. C₂×Dic₃₇₆	C₂²	semidirect
13. C₂×#13093969	C₂²	semidirect
14. C₂×D₇₅₂	C₂²	semidirect
20. #15688951	C₂²	semidirect
21. #15688949	C₂²	semidirect
22. #15688928	C₂²	semidirect
23. #15688925	C₂²	semidirect
59. C₂×C₁₆×C₄₇	D₄	semidirect
143. C₂²×M₃₂	D₄₇	semidirect
176. C₄₇	#33604	semidirect
202. C₂×C₁₆	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
#20252969	2
#20245501	2
#20245496	2
#20245491	2
#20245172	2
#20245096	2
C₂²×M₃₂×C₄₇	2
#33604	47