Subgroups — #27213347

Order 13504.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809951	C₂	1	2	yes	maximal, complemented
2	#20269974	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269899	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂×D₈₄₄	—	2	1	no	—
8	C₂×D₈₄₄	—	2	1	no	—
9	C₂×D₈₄₄	C₂²	1	2	yes	—
10	#16809951	—	2	4	no	—
11	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
12	Dic₂₁₁⋊C₄	—	2	1	no	—
13	C₄×Dic₂₁₁	—	2	1	no	—
14	C₂×C₈×C₂₁₁	C₂²	1	2	yes	—
15	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²×D₂₁₁	—	2	2	no	—
17	Dic₄₂₂	—	2	1	no	—
18	Dic₄₂₂	D₄	1	2	yes	complemented
19	D₈₄₄	—	2	2	no	—
20	D₈₄₄	—	2	2	no	—
21	D₈₄₄	—	4	2	no	—
22	D₈₄₄	—	4	2	no	—
23	C₂×Dic₂₁₁	—	2	1	no	—
24	C₂×Dic₂₁₁	—	2	1	no	—
25	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₈×C₂₁₁	—	2	2	no	—
29	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₂₁₁	—	4	2	no	—
31	C₂×D₂₁₁	—	4	2	no	—
32	C₂×D₂₁₁	—	4	2	no	—
33	C₄×C₂₁₁	—	2	1	no	—
34	C₄×C₂₁₁	SD₁₆	1	2	yes	complemented
35	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₁₁	#41	1	1	yes	characteristic
37	Dic₂₁₁	—	2	2	no	—
38	Dic₂₁₁	—	4	1	no	—
39	C₂×C₂₁₁	#121	1	1	yes	characteristic
40	C₂×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₂₁₁	—	8	2	no	—
43	C₂₁₁	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	211	1	no	maximal, Sylow
45	C₂×SD₁₆	—	211	2	no	—
46	#127	—	211	1	no	—
47	C₄×Q₈	—	211	1	no	—
48	C₄⋊C₈	D₂₁₁	1	1	yes	characteristic, complemented
49	#102	—	211	2	no	—
50	C₂×Q₈	—	211	1	no	—
51	C₂×D₄	—	422	1	no	—
52	C₂×D₄	—	422	1	no	—
53	C₂×D₄	—	211	2	no	—
54	SD₁₆	—	422	4	no	—
55	C₂×C₈	C₂×D₂₁₁	1	2	yes	—
56	C₄⋊C₄	—	422	1	no	—
57	C₄⋊C₄	—	211	1	no	—
58	C₄²	—	422	1	no	—
59	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
60	C₂³	—	422	2	no	—
61	Q₈	—	422	1	no	—
62	Q₈	—	211	2	no	—
63	D₄	—	844	2	no	—
64	D₄	—	844	2	no	—
65	D₄	—	422	2	no	—
66	D₄	—	422	2	no	—
67	C₂×C₄	—	422	1	no	—
68	C₂×C₄	—	422	1	no	—
69	C₂×C₄	D₈₄₄	1	1	yes	characteristic
70	C₂×C₄	D₈₄₄	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	844	2	no	—
74	C₂²	—	844	2	no	—
75	C₂²	—	844	2	no	—
76	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	844	1	no	—
79	C₄	—	422	2	no	—
80	C₄	#16809951	1	2	yes	complemented
81	C₄	D₄×D₂₁₁	1	1	yes	characteristic
82	C₄	#16809963	1	1	yes	characteristic
83	C₂	—	1688	2	no	—
84	C₂	#20269952	1	1	yes	characteristic
85	C₂	C₂×#16809951	1	1	yes	characteristic
86	C₂	#20270005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₄₄
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄⋊C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	#20269974	C₂
hypercenter	C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#460
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16809951	C₂	semidirect
3. C₄×Dic₄₂₂	C₂	semidirect
4. #20269899	C₂	semidirect
5. C₄⋊C₈×C₂₁₁	C₂	semidirect
18. Dic₄₂₂	D₄	semidirect
34. C₄×C₂₁₁	SD₁₆	semidirect
43. C₂₁₁	#460	semidirect
48. C₄⋊C₈	D₂₁₁	semidirect
80. C₄	#16809951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809951	2
#20269974	2
C₄×Dic₄₂₂	2
#20269899	2
C₄⋊C₈×C₂₁₁	2
#460	211