Subgroups — #41011369

Order 27008.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	#27212897	C₂	1	1	yes	maximal, characteristic
3	#346×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₄₄	C₄	1	1	yes	characteristic
5	C₄×Dic₄₂₂	—	2	1	no	—
6	#20269911	C₄	1	1	yes	characteristic
7	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
8	#20269902	—	2	1	no	—
9	C₂×C₁₆×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₂₂	—	2	1	no	—
12	Dic₈₄₄	—	2	1	no	—
13	Dic₈₄₄	C₈	1	2	yes	complemented
14	C₂×C₂₁₁⋊C₈	—	4	1	no	—
15	C₄⋊Dic₂₁₁	—	2	1	no	—
16	Dic₂₁₁⋊C₄	—	4	1	no	—
17	C₄×Dic₂₁₁	—	4	1	no	—
18	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
20	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
21	C₁₆×C₂₁₁	—	2	2	no	—
22	Dic₄₂₂	—	2	2	no	—
23	Dic₄₂₂	—	4	1	no	—
24	C₂×Dic₂₁₁	—	4	1	no	—
25	C₂×Dic₂₁₁	—	4	1	no	—
26	C₂₁₁⋊C₈	—	8	1	no	—
27	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
30	C₈×C₂₁₁	—	2	1	no	—
31	C₈×C₂₁₁	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₁₁	M₁₆	1	1	yes	characteristic
33	C₂²×C₂₁₁	#102	1	1	yes	characteristic
34	C₄×C₂₁₁	—	2	1	no	—
35	C₄×C₂₁₁	—	2	1	no	—
36	C₄×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₂₁₁	#104	1	1	yes	characteristic
38	Dic₂₁₁	—	4	2	no	—
39	Dic₂₁₁	—	8	1	no	—
40	C₂×C₂₁₁	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₂₁₁	#360	1	1	yes	characteristic
42	C₂×C₂₁₁	#362	1	1	yes	characteristic
43	C₂₁₁	#1334	1	1	yes	characteristic, complemented, Sylow
44	#1334	—	211	1	no	maximal, Sylow
45	C₄×Q₁₆	—	211	1	no	—
46	#346	D₂₁₁	1	1	yes	characteristic
47	#334	—	211	1	no	—
48	C₂×Q₁₆	—	211	1	no	—
49	C₄×Q₈	—	422	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	211	1	no	—
52	C₄⋊C₈	—	422	1	no	—
53	#103	—	422	1	no	—
54	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
55	C₂×Q₈	—	422	1	no	—
56	Q₁₆	—	422	1	no	—
57	Q₁₆	—	211	2	no	—
58	C₂×C₈	—	844	1	no	—
59	C₂×C₈	C₄×D₂₁₁	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
61	C₄⋊C₄	—	844	1	no	—
62	C₄⋊C₄	—	422	1	no	—
63	C₄²	—	844	1	no	—
64	C₄²	D₈₄₄	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	844	1	no	—
67	Q₈	—	422	2	no	—
68	C₂×C₄	—	844	1	no	—
69	C₂×C₄	—	844	1	no	—
70	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
71	C₂×C₄	D₁₆₈₈	1	1	yes	characteristic
72	C₂×C₄	#16809951	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	1688	1	no	—
75	C₈	C₈×D₂₁₁	1	1	yes	characteristic
76	C₈	#16809945	1	1	yes	characteristic
77	C₂²	#20269899	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	844	2	no	—
81	C₄	—	1688	1	no	—
82	C₄	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
83	C₄	#20269905	1	1	yes	characteristic
84	C₂	#27212875	1	1	yes	characteristic
85	C₂	#27212957	1	1	yes	characteristic
86	C₂	#27212963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269899
commutator	C₈×C₂₁₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#346×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#102
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#346	D₂₁₁
hypocenter	C₂₁₁	#1334
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. Dic₈₄₄	C₈	semidirect
43. C₂₁₁	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₈₄₄	2
#27212897	2
#346×C₂₁₁	2
#1334	211