Subgroups — #27213224

Order 13504.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₂₁₁⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#16809946	—	2	4	no	—
7	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₂₁₁	C₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
13	C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
14	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂₁₁⋊C₈	—	2	2	no	—
16	C₂₁₁⋊C₈	—	2	1	no	—
17	C₂₁₁⋊C₈	D₄	1	2	yes	complemented
18	C₂³×C₂₁₁	C₂×C₄	1	2	yes	—
19	D₄×C₂₁₁	—	2	2	no	—
20	C₂×C₄×C₂₁₁	—	2	2	no	—
21	C₂×C₄×C₂₁₁	—	2	2	no	—
22	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂²×C₂₁₁	—	2	2	no	—
27	C₂²×C₂₁₁	—	2	2	no	—
28	C₂²×C₂₁₁	—	2	2	no	—
29	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₁₁	—	2	2	no	—
31	C₄×C₂₁₁	—	2	1	no	—
32	C₄×C₂₁₁	M₁₆	1	2	yes	complemented
33	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₁₁	#41	1	1	yes	characteristic
35	C₂×C₂₁₁	—	4	2	no	—
36	C₂×C₂₁₁	C₄×D₄	1	1	yes	characteristic
37	C₂×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₂₁₁	#131	1	1	yes	characteristic
39	C₂₁₁	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	211	1	no	maximal, Sylow
41	C₂×M₁₆	—	211	2	no	—
42	C₄×D₄	D₂₁₁	1	1	yes	characteristic
43	C₄⋊C₈	—	211	1	no	—
44	C₂²⋊C₈	—	211	2	no	—
45	C₄×C₈	—	211	1	no	—
46	C₂×D₄	Dic₂₁₁	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
48	M₁₆	—	422	4	no	—
49	C₂×C₈	—	211	1	no	—
50	C₂×C₈	—	211	2	no	—
51	C₂×C₈	—	211	1	no	—
52	C₄⋊C₄	Dic₂₁₁	1	1	yes	characteristic
53	C₂²⋊C₄	Dic₂₁₁	1	2	yes	—
54	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂³	C₂×Dic₂₁₁	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
61	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₈	—	422	2	no	—
64	C₈	—	422	1	no	—
65	C₈	—	211	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	#16809946	1	2	yes	complemented
73	C₄	D₄×D₂₁₁	1	1	yes	characteristic
74	C₄	#16809961	1	1	yes	characteristic
75	C₂	—	4	2	no	—
76	C₂	D₄×Dic₂₁₁	1	1	yes	characteristic
77	C₂	C₂×#16809946	1	1	yes	characteristic
78	C₂	#20269984	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁₁
commutator, socle	C₂²×C₂₁₁	C₂²×C₄
Fitting	C₄×D₄×C₂₁₁	C₂
prime radical	C₂×D₄×C₂₁₁	C₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	C₄×D₄	D₂₁₁
hypocenter	C₂₁₁	#436
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16809946	C₂	semidirect
2. C₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
3. C₂²⋊(C₂₁₁⋊C₈)	C₂	semidirect
4. C₄×C₂₁₁⋊C₈	C₂	semidirect
17. C₂₁₁⋊C₈	D₄	semidirect
32. C₄×C₂₁₁	M₁₆	semidirect
39. C₂₁₁	#436	semidirect
72. C₄	#16809946	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
C₄⋊(C₂₁₁⋊C₈)	2
C₂²⋊(C₂₁₁⋊C₈)	2
C₄×C₂₁₁⋊C₈	2
C₄×D₄×C₂₁₁	2
#436	211