Subgroups — #41011366

Order 27008.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₆₈₈	C₂	1	1	yes	maximal, characteristic
2	#27212897	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₈₈	C₄	1	1	yes	characteristic
5	C₄×D₈₄₄	—	2	1	no	—
6	#20269911	C₄	1	1	yes	characteristic
7	C₄⋊(C₂₁₁⋊C₈)	—	2	1	no	—
8	#20269899	—	2	1	no	—
9	C₂×C₁₆×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×D₈₄₄	—	2	1	no	—
12	C₂×C₄×D₂₁₁	—	4	1	no	—
13	D₁₆₈₈	—	2	1	no	—
14	D₁₆₈₈	C₈	1	2	yes	complemented
15	C₂×C₂₁₁⋊C₈	—	4	1	no	—
16	C₄⋊Dic₂₁₁	—	2	1	no	—
17	(C₂×D₂₁₁)⋊C₄	—	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	C₂²×D₂₁₁	—	4	1	no	—
23	D₈₄₄	—	2	2	no	—
24	D₈₄₄	—	4	1	no	—
25	C₂×Dic₂₁₁	—	4	1	no	—
26	C₄×D₂₁₁	—	8	2	no	—
27	C₂₁₁⋊C₈	—	8	1	no	—
28	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	1	no	—
32	C₈×C₂₁₁	C₂×C₈	1	1	yes	characteristic
33	C₈×C₂₁₁	M₁₆	1	1	yes	characteristic
34	C₂²×C₂₁₁	#102	1	1	yes	characteristic
35	C₂×D₂₁₁	—	4	2	no	—
36	C₂×D₂₁₁	—	8	1	no	—
37	C₂×D₂₁₁	—	8	1	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₂₁₁	#104	1	1	yes	characteristic
42	Dic₂₁₁	—	8	1	no	—
43	C₂×C₂₁₁	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₂₁₁	#360	1	1	yes	characteristic
45	C₂×C₂₁₁	#361	1	1	yes	characteristic
46	D₂₁₁	—	8	2	no	—
47	C₂₁₁	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	211	1	no	maximal, Sylow
49	C₄×D₈	—	211	1	no	—
50	#346	D₂₁₁	1	1	yes	characteristic, complemented
51	#334	—	211	1	no	—
52	C₂×D₈	—	211	1	no	—
53	C₄×D₄	—	422	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	211	1	no	—
56	C₄⋊C₈	—	422	1	no	—
57	#102	—	422	1	no	—
58	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂×D₄	—	422	1	no	—
60	C₂²×C₄	—	844	1	no	—
61	D₈	—	422	1	no	—
62	D₈	—	211	2	no	—
63	C₂×C₈	—	844	1	no	—
64	C₂×C₈	C₄×D₂₁₁	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
66	C₄⋊C₄	—	422	1	no	—
67	C₂²⋊C₄	—	844	1	no	—
68	C₄²	D₈₄₄	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	844	1	no	—
71	D₄	—	844	1	no	—
72	D₄	—	422	2	no	—
73	C₂×C₄	—	844	1	no	—
74	C₂×C₄	—	1688	2	no	—
75	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₁₆₈₈	1	1	yes	characteristic
77	C₂×C₄	#16809951	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	1688	1	no	—
80	C₈	C₈×D₂₁₁	1	1	yes	characteristic
81	C₈	#16809945	1	1	yes	characteristic
82	C₂²	—	844	2	no	—
83	C₂²	—	1688	1	no	—
84	C₂²	—	1688	1	no	—
85	C₂²	#20269899	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	1688	1	no	—
89	C₄	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
90	C₄	#20269905	1	1	yes	characteristic
91	C₂	—	1688	2	no	—
92	C₂	#27212875	1	1	yes	characteristic
93	C₂	#27212957	1	1	yes	characteristic
94	C₂	#27212960	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	C₂²×C₂₁₁	#102
prime radical	C₂×D₁₆₈₈	C₄
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#346	D₂₁₁
hypocenter	C₂₁₁	#1333
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
2. #27212897	C₂	semidirect
3. #346×C₂₁₁	C₂	semidirect
14. D₁₆₈₈	C₈	semidirect
47. C₂₁₁	#1333	semidirect
50. #346	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₆₈₈	2
#27212897	2
#346×C₂₁₁	2
#1333	211