Subgroups — #27213170

Order 13504.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	2	yes	maximal
2	C₄⋊(C₂₁₁⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#16809946	—	2	4	no	—
5	C₂×C₂₁₁⋊C₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₂₁₁	C₄	1	2	yes	—
7	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄²×C₂₁₁	C₄	1	1	yes	characteristic
9	C₄²×C₂₁₁	C₄	1	1	yes	characteristic
10	C₄²×C₂₁₁	C₂²	1	2	yes	—
11	C₂₁₁⋊C₈	—	2	4	no	—
12	C₂³×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₁₁	—	2	4	no	—
14	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
18	C₂×C₄×C₂₁₁	Q₈	1	2	yes	—
19	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂²×C₂₁₁	—	2	2	no	—
21	C₂²×C₂₁₁	C₄⋊C₄	1	2	yes	—
22	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₂₁₁	—	2	2	no	—
24	C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
25	C₄×C₂₁₁	M₁₆	1	4	yes	complemented
26	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
27	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₂₁₁	—	2	2	no	—
29	C₂×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₂₁₁	C₂×M₁₆	1	2	yes	—
31	C₂₁₁	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	211	1	no	maximal, Sylow
33	C₂×M₁₆	—	211	2	no	—
34	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
35	C₄⋊C₈	—	211	4	no	—
36	C₂²×C₄	Dic₂₁₁	1	2	yes	—
37	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
38	M₁₆	—	422	4	no	—
39	C₂×C₈	—	211	4	no	—
40	C₄²	Dic₂₁₁	1	1	yes	characteristic
41	C₄²	Dic₂₁₁	1	1	yes	characteristic
42	C₄²	C₂×D₂₁₁	1	2	yes	—
43	C₂³	C₂×Dic₂₁₁	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
46	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
47	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
48	C₂×C₄	D₈₄₄	1	2	yes	—
49	C₂×C₄	Dic₄₂₂	1	2	yes	—
50	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
51	C₈	—	422	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₂₁₁	1	2	yes	—
54	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₄⋊Dic₂₁₁	1	2	yes	—
57	C₄	#16809946	1	4	yes	complemented
58	C₄	C₂×D₈₄₄	1	1	yes	characteristic
59	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
62	C₂	C₂×#16809946	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁₁
commutator, socle	C₂²×C₂₁₁	C₂²×C₄
Fitting	C₂×C₄²×C₂₁₁	C₂
prime radical	C₂³×C₂₁₁	C₂×C₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	#423
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
25. C₄×C₂₁₁	M₁₆	semidirect
31. C₂₁₁	#423	semidirect
57. 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
C₂×C₄²×C₂₁₁	2
#423	211