Subgroups — #41011357

Order 27008.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213225	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16809945	—	2	1	no	—
5	C₄×D₈₄₄	C₄	1	1	yes	characteristic
6	C₂×C₂₁₁⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₂₁₁⋊C₈)	C₄	1	1	yes	characteristic
8	(C₂×D₂₁₁)⋊C₈	—	2	1	no	—
9	C₂×C₁₆×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×D₈₄₄	C₈	1	1	yes	characteristic
12	C₂×C₄×D₂₁₁	—	2	1	no	—
13	#16809945	—	4	1	no	—
14	#16809945	—	4	1	no	—
15	C₂×C₂₁₁⋊C₈	—	2	1	no	—
16	C₄⋊Dic₂₁₁	C₈	1	1	yes	characteristic
17	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
18	C₂₁₁⋊C₁₆	—	4	1	no	—
19	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
21	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₁₆×C₂₁₁	—	2	2	no	—
23	C₂²×D₂₁₁	—	2	1	no	—
24	D₈₄₄	—	4	1	no	—
25	C₂×Dic₂₁₁	—	2	1	no	—
26	C₄×D₂₁₁	—	4	1	no	—
27	C₄×D₂₁₁	—	4	1	no	—
28	C₂₁₁⋊C₈	—	4	1	no	—
29	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
32	C₈×C₂₁₁	—	2	1	no	—
33	C₈×C₂₁₁	D₈	1	1	yes	characteristic
34	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×D₂₁₁	—	4	1	no	—
37	C₂×D₂₁₁	—	4	1	no	—
38	C₂×D₂₁₁	—	4	1	no	—
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	—	2	1	no	—
41	C₄×C₂₁₁	#102	1	1	yes	characteristic
42	C₄×C₂₁₁	#104	1	1	yes	characteristic
43	Dic₂₁₁	—	4	1	no	—
44	C₂×C₂₁₁	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₂₁₁	#349	1	1	yes	characteristic
46	C₂×C₂₁₁	#350	1	1	yes	characteristic
47	D₂₁₁	—	8	1	no	—
48	C₂₁₁	#1330	1	1	yes	characteristic, complemented, Sylow
49	#1330	—	211	1	no	maximal, Sylow
50	#436	—	211	1	no	—
51	C₄⋊C₁₆	—	211	1	no	—
52	#346	D₂₁₁	1	1	yes	characteristic, complemented
53	C₂×M₁₆	—	422	1	no	—
54	C₄×D₄	—	211	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	422	1	no	—
57	C₄⋊C₈	—	211	1	no	—
58	C₂²⋊C₈	—	422	1	no	—
59	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
60	C₂×D₄	—	211	1	no	—
61	C₂²×C₄	—	422	1	no	—
62	M₁₆	—	844	1	no	—
63	M₁₆	—	844	1	no	—
64	C₂×C₈	—	422	1	no	—
65	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
66	C₂×C₈	D₈₄₄	1	1	yes	characteristic
67	C₄⋊C₄	—	211	1	no	—
68	C₂²⋊C₄	—	422	1	no	—
69	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	C₁₆	—	844	1	no	—
72	C₂³	—	422	1	no	—
73	D₄	—	844	1	no	—
74	C₂×C₄	—	844	1	no	—
75	C₂×C₄	—	844	1	no	—
76	C₂×C₄	—	422	1	no	—
77	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
78	C₂×C₄	C₈×D₂₁₁	1	1	yes	characteristic
79	C₂×C₄	#16809945	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	844	1	no	—
82	C₈	D₁₆₈₈	1	1	yes	characteristic
83	C₈	#16809951	1	1	yes	characteristic
84	C₂²	—	844	1	no	—
85	C₂²	—	844	1	no	—
86	C₂²	—	844	1	no	—
87	C₂²	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	844	1	no	—
91	C₄	#20269899	1	1	yes	characteristic
92	C₄	#20269905	1	1	yes	characteristic
93	C₂	—	1688	1	no	—
94	C₂	#27212875	1	1	yes	characteristic
95	C₂	#27212927	1	1	yes	characteristic
96	C₂	#27212931	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₁₁)⋊C₄
commutator	C₂×C₄×C₂₁₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#346×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂²⋊C₈
prime radical	C₂×D₈₄₄	C₈
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#346	D₂₁₁
hypocenter	C₂₁₁	#1330
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₂₁₁⋊C₁₆)	C₂	semidirect
3. #346×C₂₁₁	C₂	semidirect
48. C₂₁₁	#1330	semidirect
52. #346	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213225	2
C₄⋊(C₂₁₁⋊C₁₆)	2
#346×C₂₁₁	2
#1330	211