Group #22553387 — subgroups

Subgroups — #22553387

Order 8384.

122 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523496	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₁₃₁	C₂	1	1	yes	maximal, characteristic
4	Dic₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₃₁)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₃₁	C₂²	1	1	yes	characteristic
10	#14523497	—	2	2	no	—
11	#14523496	—	2	2	no	—
12	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	Dic₁₃₁⋊C₄	C₄	1	1	yes	characteristic
15	C₂²⋊Dic₁₃₁	C₄	1	1	yes	characteristic
16	(C₂×D₁₃₁)⋊C₄	C₄	1	1	yes	characteristic
17	C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊D₁₃₁	—	2	2	no	—
23	C₂×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₃₁	—	2	1	no	—
27	C₄×D₁₃₁	—	2	1	no	—
28	C₁₃₁⋊C₈	—	2	1	no	—
29	C₁₃₁⋊C₈	D₄	1	2	yes	complemented
30	C₂³×C₁₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₁	—	2	1	no	—
32	C₂×C₄×C₁₃₁	—	2	1	no	—
33	C₂×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₈×C₁₃₁	—	2	1	no	—
36	C₈×C₁₃₁	—	2	1	no	—
37	C₂²×C₁₃₁	—	2	1	no	—
38	C₂²×C₁₃₁	—	2	1	no	—
39	C₂²×C₁₃₁	—	2	1	no	—
40	C₂²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₃₁	—	2	1	no	—
42	C₂×D₁₃₁	—	2	1	no	—
43	C₂×D₁₃₁	—	2	1	no	—
44	C₄×C₁₃₁	—	2	1	no	—
45	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₃₁	#41	1	1	yes	characteristic
47	Dic₁₃₁	—	2	1	no	—
48	Dic₁₃₁	—	2	1	no	—
49	Dic₁₃₁	M₁₆	1	2	yes	complemented
50	C₂×C₁₃₁	—	4	1	no	—
51	C₂×C₁₃₁	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₃₁	#131	1	1	yes	characteristic
54	D₁₃₁	—	4	1	no	—
55	C₁₃₁	#436	1	1	yes	characteristic, complemented, Sylow
56	#436	—	131	1	no	maximal, Sylow
57	C₂×M₁₆	—	131	1	no	—
58	C₂×M₁₆	—	131	1	no	—
59	C₄×D₄	—	131	1	no	—
60	C₄⋊C₈	—	131	1	no	—
61	C₂²⋊C₈	—	131	1	no	—
62	C₂²⋊C₈	D₁₃₁	1	1	yes	characteristic, complemented
63	C₄×C₈	—	131	1	no	—
64	C₂×D₄	—	131	1	no	—
65	C₂²×C₄	—	131	1	no	—
66	C₂²×C₄	C₂×D₁₃₁	1	1	yes	characteristic
67	M₁₆	—	262	2	no	—
68	M₁₆	—	262	2	no	—
69	C₂×C₈	—	131	1	no	—
70	C₂×C₈	—	131	1	no	—
71	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
73	C₄⋊C₄	—	131	1	no	—
74	C₂²⋊C₄	—	131	1	no	—
75	C₂²⋊C₄	—	131	1	no	—
76	C₄²	—	131	1	no	—
77	C₂³	—	131	1	no	—
78	C₂³	C₄×D₁₃₁	1	1	yes	characteristic
79	D₄	—	262	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	262	1	no	—
83	C₂×C₄	—	262	1	no	—
84	C₂×C₄	—	131	1	no	—
85	C₂×C₄	—	131	1	no	—
86	C₂×C₄	—	131	1	no	—
87	C₂×C₄	C₄×D₁₃₁	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	262	1	no	—
92	C₈	—	131	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	262	1	no	—
97	C₂²	—	262	1	no	—
98	C₂²	—	262	1	no	—
99	C₂²	C₂×C₄×D₁₃₁	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	262	1	no	—
102	C₄	—	262	1	no	—
103	C₄	—	131	2	no	—
104	C₄	D₄×D₁₃₁	1	1	yes	characteristic
105	C₄	#14523512	1	1	yes	characteristic
106	C₂	—	4	1	no	—
107	C₂	—	524	1	no	—
108	C₂	(C₂²⋊D₁₃₁)⋊C₄	1	1	yes	characteristic
109	C₂	M₁₆×D₁₃₁	1	1	yes	characteristic
110	C₂	#16862252	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523497

C₂

yes

maximal, characteristic, complemented

C₂×#14523496

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₁

C₂

yes

maximal, characteristic

Dic₁₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₃₁)⋊C₈

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₃₁

C₄

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

characteristic

#14523497

—

#14523496

—

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂²

yes

characteristic

Dic₁₃₁⋊C₄

C₄

yes

characteristic

C₂²⋊Dic₁₃₁

C₄

yes

characteristic

(C₂×D₁₃₁)⋊C₄

C₄

yes

characteristic

C₄×Dic₁₃₁

C₂²

yes

characteristic

C₂²×C₄×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₈×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

C₂×C₄

yes

characteristic

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

C₂×C₄

yes

characteristic

C₂×Dic₁₃₁

C₂×C₄

yes

characteristic

C₂×Dic₁₃₁

C₂×C₄

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

D₄

yes

complemented

C₂³×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₄×C₁₃₁

#41

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

M₁₆

yes

complemented

C₂×C₁₃₁

—

C₂×C₁₃₁

C₄×D₄

yes

characteristic

C₂×C₁₃₁

C₂×M₁₆

yes

characteristic

C₂×C₁₃₁

#131

yes

characteristic

D₁₃₁

—

C₁₃₁

#436

yes

characteristic, complemented, Sylow

#436

—

131

maximal, Sylow

C₂×M₁₆

—

131

—

C₂×M₁₆

—

131

—

C₄×D₄

—

131

—

C₄⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

D₁₃₁

yes

characteristic, complemented

C₄×C₈

—

131

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₂×D₁₃₁

yes

characteristic

M₁₆

—

262

—

M₁₆

—

262

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₂×C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

131

—

C₂²⋊C₄

—

131

—

C₂²⋊C₄

—

131

—

C₄²

—

131

—

C₂³

—

131

—

C₂³

C₄×D₁₃₁

yes

characteristic

D₄

—

262

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₄×D₁₃₁

yes

characteristic

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₈

—

C₈

—

C₈

—

262

—

C₈

—

131

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

C₂×C₄×D₁₃₁

yes

characteristic

100

C₄

—

101

C₄

—

262

—

102

C₄

—

262

—

103

C₄

—

131

—

104

C₄

D₄×D₁₃₁

yes

characteristic

105

C₄

#14523512

yes

characteristic

106

C₂

—

107

C₂

—

524

—

108

C₂

(C₂²⋊D₁₃₁)⋊C₄

yes

characteristic

109

C₂

M₁₆×D₁₃₁

yes

characteristic

110

C₂

#16862252

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₃₁	C₂
hypercenter	C₂²⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#436
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₃₁

C₂

hypercenter

C₂²⋊C₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14523497	C₂	semidirect
2. C₂×#14523496	C₂	semidirect
4. Dic₁₃₁⋊C₈	C₂	semidirect
5. (C₂×D₁₃₁)⋊C₈	C₂	semidirect
6. C₈×Dic₁₃₁	C₂	semidirect
7. C₂²⋊C₈×C₁₃₁	C₂	semidirect
29. C₁₃₁⋊C₈	D₄	semidirect
49. Dic₁₃₁	M₁₆	semidirect
55. C₁₃₁	#436	semidirect
62. C₂²⋊C₈	D₁₃₁	semidirect

H ≅ G/N

type

1. C₂×#14523497

C₂

semidirect

2. C₂×#14523496

C₂

semidirect

4. Dic₁₃₁⋊C₈

C₂

semidirect

5. (C₂×D₁₃₁)⋊C₈

C₂

semidirect

6. C₈×Dic₁₃₁

C₂

semidirect

7. C₂²⋊C₈×C₁₃₁

C₂

semidirect

29. C₁₃₁⋊C₈

D₄

semidirect

49. Dic₁₃₁

M₁₆

semidirect

55. C₁₃₁

#436

semidirect

62. C₂²⋊C₈

D₁₃₁

semidirect

Subgroups — #22553387

Conjugacy classes of subgroups

Complements