Subgroups — #41011356

Order 27008.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212922	C₂	1	1	yes	maximal, characteristic, complemented
3	#436×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
5	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
6	C₂×M₁₆×C₂₁₁	—	2	1	no	—
7	C₄×D₄×C₂₁₁	C₄	1	1	yes	characteristic
8	C₄⋊C₈×C₂₁₁	C₄	1	1	yes	characteristic
9	C₂²⋊C₈×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂₁₁⋊C₁₆	—	2	2	no	—
12	C₂₁₁⋊C₁₆	—	4	1	no	—
13	C₂×D₄×C₂₁₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₂₁₁	—	2	1	no	—
15	M₁₆×C₂₁₁	—	4	1	no	—
16	M₁₆×C₂₁₁	—	4	1	no	—
17	C₂×C₈×C₂₁₁	—	2	1	no	—
18	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
20	C₄⋊C₄×C₂₁₁	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
22	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₂₁₁	—	2	1	no	—
24	D₄×C₂₁₁	—	4	1	no	—
25	C₂×C₄×C₂₁₁	—	2	1	no	—
26	C₂×C₄×C₂₁₁	—	4	1	no	—
27	C₂×C₄×C₂₁₁	—	4	1	no	—
28	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	1	no	—
32	C₈×C₂₁₁	—	4	1	no	—
33	C₈×C₂₁₁	D₈	1	1	yes	characteristic
34	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₁₁	—	4	1	no	—
36	C₂²×C₂₁₁	—	4	1	no	—
37	C₂²×C₂₁₁	—	4	1	no	—
38	C₂²×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	—	2	1	no	—
41	C₄×C₂₁₁	—	4	1	no	—
42	C₄×C₂₁₁	#102	1	1	yes	characteristic
43	C₄×C₂₁₁	#104	1	1	yes	characteristic
44	C₂×C₂₁₁	—	8	1	no	—
45	C₂×C₂₁₁	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₂₁₁	#349	1	1	yes	characteristic
47	C₂×C₂₁₁	#350	1	1	yes	characteristic
48	C₂₁₁	#1330	1	1	yes	characteristic, complemented, Sylow
49	#1330	—	211	1	no	maximal, Sylow
50	#436	D₂₁₁	1	1	yes	characteristic
51	C₄⋊C₁₆	—	211	1	no	—
52	#346	—	211	1	no	—
53	C₂×M₁₆	—	2	1	no	—
54	C₄×D₄	Dic₂₁₁	1	1	yes	characteristic
55	C₂×C₁₆	—	422	1	no	—
56	C₂×C₁₆	—	422	1	no	—
57	C₄⋊C₈	Dic₂₁₁	1	1	yes	characteristic
58	C₂²⋊C₈	—	2	1	no	—
59	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
60	C₂×D₄	C₂₁₁⋊C₈	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
67	C₄⋊C₄	C₂₁₁⋊C₈	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
70	C₁₆	—	844	1	no	—
71	C₁₆	—	422	2	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
79	C₂×C₄	#16809946	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	#16809947	1	1	yes	characteristic
83	C₈	#16809950	1	1	yes	characteristic
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#20269898	1	1	yes	characteristic
92	C₄	#20269904	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
95	C₂	#27212928	1	1	yes	characteristic
96	C₂	#27212930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#436×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂²⋊C₈
prime radical	C₂×D₄×C₂₁₁	C₈
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#436	D₂₁₁
hypocenter	C₂₁₁	#1330
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₂₁₁⋊C₁₆)	C₂	semidirect
2. #27212922	C₂	semidirect
48. C₂₁₁	#1330	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₁₁⋊C₁₆)	2
#27212922	2
#436×C₂₁₁	2
#1330	211