Subgroups — #27213481

Order 13504.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269956	C₂	1	1	yes	maximal, characteristic
2	#20269910	C₂	1	2	yes	maximal, complemented
3	#20269900	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
9	Dic₂₁₁⋊C₄	—	2	1	no	—
10	C₂²⋊Dic₂₁₁	—	2	1	no	—
11	C₂²×C₄×C₂₁₁	—	2	1	no	—
12	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₁₁	—	2	1	no	—
14	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	Dic₄₂₂	—	2	2	no	—
16	C₂×Dic₂₁₁	—	2	1	no	—
17	C₂×Dic₂₁₁	—	2	1	no	—
18	C₂₁₁⋊C₈	—	2	2	no	—
19	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₁₁	—	2	2	no	—
21	C₂×C₄×C₂₁₁	—	2	2	no	—
22	C₂×C₄×C₂₁₁	—	4	2	no	—
23	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₁₁	—	2	1	no	—
26	C₂²×C₂₁₁	—	2	1	no	—
27	C₂²×C₂₁₁	Q₁₆	1	2	yes	—
28	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₁₁	—	2	1	no	—
30	C₄×C₂₁₁	—	4	2	no	—
31	C₄×C₂₁₁	#41	1	2	yes	—
32	Dic₂₁₁	—	4	1	no	—
33	Dic₂₁₁	—	4	1	no	—
34	C₂×C₂₁₁	—	2	2	no	—
35	C₂×C₂₁₁	#123	1	1	yes	characteristic
36	C₂×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₂₁₁	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	211	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	211	1	no	—
41	C₂×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
42	#107	—	211	2	no	—
43	#103	—	211	2	no	—
44	C₂²⋊C₈	—	211	1	no	—
45	C₂×Q₈	—	211	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
48	C₂×C₈	—	211	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	422	1	no	—
51	C₄⋊C₄	—	211	1	no	—
52	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
53	C₂²⋊C₄	—	422	1	no	—
54	C₂³	C₂²⋊D₂₁₁	1	1	yes	characteristic
55	Q₈	—	422	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	422	1	no	—
60	C₂×C₄	—	422	1	no	—
61	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₈	—	422	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#16809952	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	844	1	no	—
71	C₄	—	844	1	no	—
72	C₄	#16809962	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20269960	1	1	yes	characteristic
75	C₂	C₂×#16809952	1	1	yes	characteristic
76	C₂	#20270003	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊C₄×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂³×C₂₁₁	D₄
hypercenter	C₂×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#484
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20269910	C₂	semidirect
3. #20269900	C₂	semidirect
38. C₂₁₁	#484	semidirect

Maximal subgroups

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

subgroup	index
#20269956	2
#20269910	2
#20269900	2
C₂²⋊(C₂₁₁⋊C₈)	2
C₂×C₄⋊C₄×C₂₁₁	2
#484	211