Subgroups — #41012063

Order 27008.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213199	C₂	1	2	yes	maximal
2	#27212892	C₂	1	4	yes	maximal, complemented
3	#533×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₂₁₁⋊C₈)	C₂²	1	4	yes	—
5	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
6	C₄×C₂₁₁⋊C₈	—	2	2	no	—
7	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
8	#125×C₂₁₁	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₂₁₁	—	2	2	no	—
10	C₄×Q₈×C₂₁₁	—	2	2	no	—
11	#117×C₂₁₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₂₁₁⋊C₈	—	2	4	no	—
14	C₂×Q₈×C₂₁₁	—	2	2	no	—
15	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
17	C₄⋊C₄×C₂₁₁	—	2	4	no	—
18	C₄⋊C₄×C₂₁₁	—	2	2	no	—
19	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
20	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
21	C₄²×C₂₁₁	—	2	2	no	—
22	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₁₁	D₄	1	2	yes	—
24	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂₁₁⋊C₈	—	4	4	no	—
26	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₂₁₁	—	4	4	no	—
28	C₂×C₄×C₂₁₁	—	2	2	no	—
29	C₂×C₄×C₂₁₁	—	2	4	no	—
30	C₂×C₄×C₂₁₁	—	2	4	no	—
31	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
35	C₂²×C₂₁₁	—	2	1	no	—
36	C₂²×C₂₁₁	—	2	2	no	—
37	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	2	2	no	—
40	C₄×C₂₁₁	—	4	4	no	—
41	C₄×C₂₁₁	#101	1	2	yes	—
42	C₄×C₂₁₁	#135	1	4	yes	—
43	C₂×C₂₁₁	—	4	1	no	—
44	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
45	C₂×C₂₁₁	#416	1	2	yes	—
46	C₂₁₁	#1544	1	1	yes	characteristic, complemented, Sylow
47	#1544	—	211	1	no	maximal, Sylow
48	#533	D₂₁₁	1	1	yes	characteristic
49	#431	—	211	2	no	—
50	#332	—	211	4	no	—
51	C₄⋊Q₈	C₂×D₂₁₁	1	2	yes	—
52	#125	Dic₂₁₁	1	2	yes	—
53	C₂²⋊Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	#117	Dic₂₁₁	1	2	yes	—
56	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
57	C₄⋊C₈	—	211	4	no	—
58	C₂²⋊C₈	—	422	2	no	—
59	C₄×C₈	—	422	2	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
62	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
63	C₂×C₈	—	422	4	no	—
64	C₄⋊C₄	—	2	4	no	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
67	C₂²⋊C₄	—	2	2	no	—
68	C₄²	—	2	2	no	—
69	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
70	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
71	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
78	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
79	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
80	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
81	C₈	—	844	4	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₄	#20269896	1	2	yes	—
89	C₄	#20269997	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₂×#20269896	1	1	yes	characteristic
92	C₂	#27213142	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#533×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#533	D₂₁₁
hypocenter	C₂₁₁	#1544
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #27212892	C₂	semidirect
46. C₂₁₁	#1544	semidirect

Maximal subgroups

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

subgroup	index
#27213199	2
#27212892	2
#533×C₂₁₁	2
#1544	211