Subgroups — #41011344

Order 27008.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212872	C₂	1	2	yes	maximal
2	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
4	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
5	C₂×C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₁₁	C₄	1	1	yes	characteristic
8	C₂×C₂₁₁⋊C₈	—	4	2	no	—
9	C₂²×C₄×C₂₁₁	—	2	1	no	—
10	C₂²×C₄×C₂₁₁	—	2	1	no	—
11	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
13	C₄⋊C₄×C₂₁₁	—	4	1	no	—
14	C₄⋊C₄×C₂₁₁	—	4	1	no	—
15	C₄⋊C₄×C₂₁₁	—	4	4	no	—
16	C₄²×C₂₁₁	—	2	1	no	—
17	C₄²×C₂₁₁	C₈	1	2	yes	complemented
18	C₂₁₁⋊C₈	—	8	2	no	—
19	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₁₁	—	2	2	no	—
21	C₂×C₄×C₂₁₁	—	2	2	no	—
22	C₂×C₄×C₂₁₁	—	2	2	no	—
23	C₂×C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄×C₂₁₁	—	2	1	no	—
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	—	4	2	no	—
27	C₂×C₄×C₂₁₁	—	4	2	no	—
28	C₂×C₄×C₂₁₁	—	4	1	no	—
29	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
31	C₂²×C₂₁₁	—	2	1	no	—
32	C₂²×C₂₁₁	—	2	1	no	—
33	C₂²×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
34	C₂²×C₂₁₁	#99	1	1	yes	characteristic
35	C₂²×C₂₁₁	#101	1	1	yes	characteristic
36	C₄×C₂₁₁	—	2	4	no	—
37	C₄×C₂₁₁	—	2	1	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	4	2	no	—
40	C₄×C₂₁₁	—	8	1	no	—
41	C₂×C₂₁₁	—	2	1	no	—
42	C₂×C₂₁₁	—	2	1	no	—
43	C₂×C₂₁₁	#324	1	1	yes	characteristic
44	C₂×C₂₁₁	#353	1	1	yes	characteristic
45	C₂×C₂₁₁	#356	1	1	yes	characteristic
46	C₂₁₁	#1325	1	1	yes	characteristic, complemented, Sylow
47	#1325	—	211	1	no	maximal, Sylow
48	#384	D₂₁₁	1	1	yes	characteristic
49	#324	—	211	2	no	—
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	Dic₂₁₁	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
53	C₂×C₄²	Dic₂₁₁	1	1	yes	characteristic
54	C₂²⋊C₈	—	422	2	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
59	C₂×C₈	—	844	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄²	—	2	1	no	—
64	C₄²	C₂₁₁⋊C₈	1	2	yes	complemented
65	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#16809946	1	1	yes	characteristic
77	C₈	—	1688	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
81	C₂²	#20269892	1	1	yes	characteristic
82	C₂²	#20269896	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	8	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#27212872	1	1	yes	characteristic
91	C₂	#27212939	1	1	yes	characteristic
92	C₂	#27212945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269892
commutator	C₂×C₄×C₂₁₁	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₂₁₁
Fitting	#384×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#99
prime radical	C₂³×C₂₁₁	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₂₁₁	C₂²
hypercenter	#384	D₂₁₁
hypocenter	C₂₁₁	#1325
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #384

Complements

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

N	H ≅ G/N	type
17. C₄²×C₂₁₁	C₈	semidirect
46. C₂₁₁	#1325	semidirect
64. C₄²	C₂₁₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#27212872	2
#384×C₂₁₁	2
#1325	211