Subgroups — #27213435

Order 13504.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269976	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269939	C₂	1	1	yes	maximal, characteristic
3	#20269908	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#20269899	C₂	1	1	yes	maximal, characteristic
6	#20269897	C₂	1	1	yes	maximal, characteristic
7	#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₂₂	—	2	1	no	—
9	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₁₁	—	2	1	no	—
11	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₂₁₁⋊C₄	—	2	1	no	—
14	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
15	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂²×D₂₁₁	—	2	1	no	—
20	Dic₄₂₂	—	4	2	no	—
21	D₈₄₄	—	2	1	no	—
22	D₈₄₄	Q₈	1	2	yes	complemented
23	C₂×Dic₂₁₁	—	2	1	no	—
24	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
25	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
26	C₄×D₂₁₁	—	4	2	no	—
27	C₂₁₁⋊C₈	—	2	1	no	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	1	no	—
32	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₁₁	—	2	2	no	—
34	C₂×D₂₁₁	—	4	1	no	—
35	C₂×D₂₁₁	—	4	1	no	—
36	C₄×C₂₁₁	—	4	1	no	—
37	C₄×C₂₁₁	—	4	1	no	—
38	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₂₁₁	#41	1	1	yes	characteristic
40	Dic₂₁₁	—	2	1	no	—
41	Dic₂₁₁	—	4	1	no	—
42	Dic₂₁₁	SD₁₆	1	2	yes	—
43	C₂×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
45	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
46	D₂₁₁	—	4	2	no	—
47	C₂₁₁	#476	1	1	yes	characteristic, complemented, Sylow
48	#476	—	211	1	no	maximal, Sylow
49	C₄⋊Q₈	—	211	1	no	—
50	C₄×D₄	—	211	1	no	—
51	#106	—	211	1	no	—
52	#106	D₂₁₁	1	1	yes	characteristic, complemented
53	C₄⋊C₈	—	211	1	no	—
54	#102	—	211	1	no	—
55	#102	—	211	1	no	—
56	C₂×Q₈	—	422	1	no	—
57	C₂×D₄	—	211	1	no	—
58	C₂²×C₄	—	422	1	no	—
59	C₂×C₈	—	211	1	no	—
60	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
61	C₄⋊C₄	—	422	1	no	—
62	C₄⋊C₄	—	211	1	no	—
63	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
64	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂²⋊C₄	—	422	1	no	—
66	C₄²	—	211	1	no	—
67	C₂³	—	422	1	no	—
68	Q₈	—	844	2	no	—
69	D₄	—	422	1	no	—
70	D₄	—	211	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	844	2	no	—
74	C₂×C₄	—	422	1	no	—
75	C₂×C₄	—	211	1	no	—
76	C₂×C₄	—	211	1	no	—
77	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	422	1	no	—
80	C₂²	—	844	1	no	—
81	C₂²	—	844	1	no	—
82	C₂²	—	422	2	no	—
83	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	844	1	no	—
87	C₄	—	422	1	no	—
88	C₄	—	211	2	no	—
89	C₄	Q₈×D₂₁₁	1	1	yes	characteristic
90	C₄	#16809962	1	1	yes	characteristic
91	C₂	—	844	2	no	—
92	C₂	#20269955	1	1	yes	characteristic
93	C₂	SD₁₆×D₂₁₁	1	1	yes	characteristic
94	C₂	#20270004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#106×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₈₄₄	C₂²
hypercenter	#106	D₂₁₁
hypocenter	C₂₁₁	#476
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. #20269976	C₂	semidirect
3. #20269908	C₂	semidirect
4. Dic₂₁₁⋊C₈	C₂	semidirect
7. #106×C₂₁₁	C₂	semidirect
22. D₈₄₄	Q₈	semidirect
47. C₂₁₁	#476	semidirect
52. #106	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269976	2
#20269939	2
#20269908	2
Dic₂₁₁⋊C₈	2
#20269899	2
#20269897	2
#106×C₂₁₁	2
#476	211