Subgroups — #27213113

Order 13504.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₂₁₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#16809946	—	2	2	no	—
6	C₂×C₂₁₁⋊C₈	—	2	2	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	#41×C₂₁₁	—	2	2	no	—
10	#41×C₂₁₁	—	2	2	no	—
11	C₂×Q₈×C₂₁₁	C₄	1	1	yes	characteristic
12	C₂×D₄×C₂₁₁	C₄	1	2	yes	—
13	C₂×D₄×C₂₁₁	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂₁₁⋊C₈	—	2	2	no	—
17	C₂₁₁⋊C₈	—	2	2	no	—
18	C₂³×C₂₁₁	C₂×C₄	1	2	yes	—
19	C₂³×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	Q₈×C₂₁₁	—	2	2	no	—
21	D₄×C₂₁₁	—	2	4	no	—
22	D₄×C₂₁₁	—	2	2	no	—
23	C₂×C₄×C₂₁₁	—	2	2	no	—
24	C₂×C₄×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
28	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₂₁₁	—	2	2	no	—
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	—	2	4	no	—
33	C₂²×C₂₁₁	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₁₁	—	2	2	no	—
36	C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
37	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₂₁₁	—	2	2	no	—
40	C₂×C₂₁₁	—	4	2	no	—
41	C₂×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₂₁₁	#131	1	2	yes	—
43	C₂₁₁	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	211	1	no	maximal, Sylow
45	C₂×#41	D₂₁₁	1	1	yes	characteristic
46	C₂×M₁₆	—	211	1	no	—
47	C₂²×C₈	—	211	1	no	—
48	C₂²⋊C₈	—	211	4	no	—
49	#41	—	2	2	no	—
50	#41	—	2	2	no	—
51	C₂×Q₈	Dic₂₁₁	1	1	yes	characteristic
52	C₂×D₄	Dic₂₁₁	1	2	yes	—
53	C₂×D₄	Dic₂₁₁	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
55	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
56	M₁₆	—	422	2	no	—
57	C₂×C₈	—	422	2	no	—
58	C₂×C₈	—	211	2	no	—
59	C₂×C₈	—	211	2	no	—
60	C₂³	C₂×Dic₂₁₁	1	2	yes	—
61	C₂³	C₂×Dic₂₁₁	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
68	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
70	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
71	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₈	—	422	2	no	—
73	C₈	—	422	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	C₂²⋊Dic₂₁₁	1	2	yes	—
78	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
81	C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	C₂	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
86	C₂	#20269984	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁₁
commutator, socle	C₂²×C₂₁₁	C₂²×C₄
Fitting, prime radical	C₂×#41×C₂₁₁	C₂
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	C₂×#41	D₂₁₁
hypocenter	C₂₁₁	#408
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16809946	C₂	semidirect
2. C₂²×C₂₁₁⋊C₈	C₂	semidirect
3. C₂²⋊(C₂₁₁⋊C₈)	C₂	semidirect
43. C₂₁₁	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809946	2
C₂²×C₂₁₁⋊C₈	2
C₂²⋊(C₂₁₁⋊C₈)	2
C₂×#41×C₂₁₁	2
#408	211