Group #22553792 — subgroups

Subgroups — #22553792

Order 8384.

294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14523513	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₆₂	C₂	1	2	yes	maximal, complemented
3	C₄×D₅₂₄	C₂	1	2	yes	maximal, complemented
4	(C₂²⋊D₁₃₁)⋊C₄	C₂	1	4	yes	maximal, complemented
5	#16862204	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₁₃₁	C₂	1	2	yes	maximal, complemented
8	#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14523513	C₄	1	8	yes	complemented
10	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
11	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₃₁	C₂²	1	2	yes	—
13	C₂²×Dic₁₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₁₃₁	C₂²	1	4	yes	—
15	C₂×C₄×D₁₃₁	C₂²	1	2	yes	—
16	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
17	C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
18	Dic₁₃₁⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₁₃₁)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₁₃₁	C₂²	1	4	yes	—
21	C₂²×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
25	C₄²×C₁₃₁	C₂²	1	2	yes	—
26	C₂²×D₁₃₁	C₂³	1	2	yes	—
27	Dic₂₆₂	C₂×C₄	1	4	yes	complemented
28	D₅₂₄	C₂×C₄	1	4	yes	complemented
29	C₂²⋊D₁₃₁	C₂×C₄	1	8	yes	complemented
30	C₂×Dic₁₃₁	—	4	2	no	—
31	C₂×Dic₁₃₁	C₂³	1	4	yes	—
32	C₂×Dic₁₃₁	C₂³	1	2	yes	—
33	C₄×D₁₃₁	—	4	4	no	—
34	C₄×D₁₃₁	C₂×C₄	1	8	yes	complemented
35	C₂³×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	C₂×C₄	1	4	yes	complemented
37	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₁₃₁	C₂³	1	2	yes	—
39	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₁₃₁	—	2	1	no	—
42	C₂²×C₁₃₁	—	2	1	no	—
43	C₂²×C₁₃₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₃₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₃₁	—	2	2	no	—
46	C₂×D₁₃₁	—	2	2	no	—
47	C₂×D₁₃₁	C₂²×C₄	1	4	yes	—
48	C₄×C₁₃₁	—	2	2	no	—
49	C₄×C₁₃₁	—	2	2	no	—
50	C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
51	C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
52	Dic₁₃₁	—	2	4	no	—
53	Dic₁₃₁	C₂²×C₄	1	4	yes	—
54	C₂×C₁₃₁	—	2	2	no	—
55	C₂×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₃₁	—	2	4	no	—
59	C₁₃₁	#520	1	1	yes	characteristic, complemented, Sylow
60	#520	—	131	1	no	maximal, Sylow
61	C₂×#41	—	131	1	no	—
62	C₄×Q₈	—	131	2	no	—
63	C₄×D₄	—	131	4	no	—
64	C₄×D₄	—	131	2	no	—
65	#117	—	131	2	no	—
66	#117	D₁₃₁	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄	—	131	1	no	—
68	C₂×C₄⋊C₄	—	131	2	no	—
69	#41	—	131	8	no	—
70	C₂×Q₈	—	131	1	no	—
71	C₂×D₄	—	131	2	no	—
72	C₂×D₄	—	131	1	no	—
73	C₂²×C₄	—	131	2	no	—
74	C₂²×C₄	—	131	4	no	—
75	C₂²×C₄	—	131	2	no	—
76	C₂²×C₄	C₂×D₁₃₁	1	1	yes	characteristic
77	C₄⋊C₄	—	131	4	no	—
78	C₄⋊C₄	—	131	2	no	—
79	C₄⋊C₄	—	131	2	no	—
80	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
81	C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
82	C₂²⋊C₄	—	131	4	no	—
83	C₂²⋊C₄	C₂×D₁₃₁	1	2	yes	—
84	C₄²	—	131	4	no	—
85	C₄²	C₂×D₁₃₁	1	2	yes	—
86	C₂³	—	131	2	no	—
87	C₂³	C₂²×D₁₃₁	1	1	yes	characteristic
88	Q₈	—	131	4	no	—
89	D₄	—	131	8	no	—
90	D₄	—	131	4	no	—
91	C₂×C₄	—	524	4	no	—
92	C₂×C₄	—	524	2	no	—
93	C₂×C₄	—	131	4	no	—
94	C₂×C₄	—	131	2	no	—
95	C₂×C₄	—	131	8	no	—
96	C₂×C₄	C₄×D₁₃₁	1	4	yes	complemented
97	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
98	C₂×C₄	C₂²×D₁₃₁	1	2	yes	—
99	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	262	2	no	—
104	C₂²	—	262	2	no	—
105	C₂²	—	131	4	no	—
106	C₂²	C₂×C₄×D₁₃₁	1	2	yes	—
107	C₂²	C₂³×D₁₃₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	262	4	no	—
111	C₄	—	131	4	no	—
112	C₄	C₂×C₄×D₁₃₁	1	2	yes	—
113	C₄	C₂×C₄×D₁₃₁	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	—	262	4	no	—
116	C₂	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
117	C₂	#16862292	1	1	yes	characteristic
118	C₂	#16862293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523513

C₂

yes

maximal, characteristic

C₄×Dic₂₆₂

C₂

yes

maximal, complemented

C₄×D₅₂₄

C₂

yes

maximal, complemented

(C₂²⋊D₁₃₁)⋊C₄

C₂

yes

maximal, complemented

#16862204

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₃₁

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₃₁

C₂

yes

maximal, complemented

#117×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#14523513

C₄

yes

complemented

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₁

C₂²

yes

—

C₂²×Dic₁₃₁

C₂²

yes

—

C₂×C₄×D₁₃₁

C₂²

yes

—

C₂×C₄×D₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

—

C₂²×C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₁

C₂²

yes

—

C₄²×C₁₃₁

C₂²

yes

—

C₂²×D₁₃₁

C₂³

yes

—

Dic₂₆₂

C₂×C₄

yes

complemented

D₅₂₄

C₂×C₄

yes

complemented

C₂²⋊D₁₃₁

C₂×C₄

yes

complemented

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

—

C₄×D₁₃₁

—

C₄×D₁₃₁

C₂×C₄

yes

complemented

C₂³×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂×C₄

yes

complemented

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂²×C₄

yes

—

C₂²×C₁₃₁

C₂⁴

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

C₂²×C₄

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²×C₄

yes

—

C₄×C₁₃₁

C₂²×C₄

yes

—

Dic₁₃₁

—

Dic₁₃₁

C₂²×C₄

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂³×C₄

yes

characteristic

C₂×C₁₃₁

H(5,𝔽₂)

yes

characteristic

C₂×C₁₃₁

2₋¹⁺⁴

yes

characteristic

D₁₃₁

—

C₁₃₁

#520

yes

characteristic, complemented, Sylow

#520

—

131

maximal, Sylow

C₂×#41

—

131

—

C₄×Q₈

—

131

—

C₄×D₄

—

131

—

C₄×D₄

—

131

—

#117

—

131

—

#117

D₁₃₁

yes

characteristic, complemented

C₂×C₄⋊C₄

—

131

—

C₂×C₄⋊C₄

—

131

—

#41

—

131

—

C₂×Q₈

—

131

—

C₂×D₄

—

131

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₄⋊C₄

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₄

—

131

—

C₂²⋊C₄

C₂×D₁₃₁

yes

—

C₄²

—

131

—

C₄²

C₂×D₁₃₁

yes

—

C₂³

—

131

—

C₂³

C₂²×D₁₃₁

yes

characteristic

Q₈

—

131

—

D₄

—

131

—

D₄

—

131

—

C₂×C₄

—

524

—

C₂×C₄

—

524

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₄×D₁₃₁

yes

complemented

C₂×C₄

C₂²×D₁₃₁

yes

—

C₂×C₄

C₂²×D₁₃₁

yes

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

100

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

262

—

104

C₂²

—

262

—

105

C₂²

—

131

—

106

C₂²

C₂×C₄×D₁₃₁

yes

—

107

C₂²

C₂³×D₁₃₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

262

—

111

C₄

—

131

—

112

C₄

C₂×C₄×D₁₃₁

yes

—

113

C₄

C₂×C₄×D₁₃₁

yes

—

114

C₂

—

115

C₂

—

262

—

116

C₂

C₂²×C₄×D₁₃₁

yes

characteristic

117

C₂

#16862292

yes

characteristic

118

C₂

#16862293

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₁
commutator	C₂×C₁₃₁	C₂³×C₄
Fitting	#117×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂⁴
hypercenter	#117	D₁₃₁
hypocenter	C₁₃₁	#520
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₁

commutator

C₂×C₁₃₁

C₂³×C₄

Fitting

#117×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂⁴

D₁₃₁

C₁₃₁

G ▷ C₂×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₄×Dic₂₆₂	C₂	semidirect
3. C₄×D₅₂₄	C₂	semidirect
4. (C₂²⋊D₁₃₁)⋊C₄	C₂	semidirect
5. #16862204	C₂	semidirect
6. C₂×C₄⋊Dic₁₃₁	C₂	semidirect
7. C₄⋊C₄×D₁₃₁	C₂	semidirect
8. #117×C₁₃₁	C₂	semidirect
9. #14523513	C₄	semidirect
27. Dic₂₆₂	C₂×C₄	semidirect
28. D₅₂₄	C₂×C₄	semidirect
29. C₂²⋊D₁₃₁	C₂×C₄	semidirect
34. C₄×D₁₃₁	C₂×C₄	semidirect
36. C₂×C₄×C₁₃₁	C₂×C₄	semidirect
59. C₁₃₁	#520	semidirect
66. #117	D₁₃₁	semidirect
96. C₂×C₄	C₄×D₁₃₁	semidirect

H ≅ G/N

type

2. C₄×Dic₂₆₂

C₂

semidirect

3. C₄×D₅₂₄

C₂

semidirect

4. (C₂²⋊D₁₃₁)⋊C₄

C₂

semidirect

5. #16862204

C₂

semidirect

6. C₂×C₄⋊Dic₁₃₁

C₂

semidirect

7. C₄⋊C₄×D₁₃₁

C₂

semidirect

8. #117×C₁₃₁

C₂

semidirect

9. #14523513

C₄

semidirect

27. Dic₂₆₂

C₂×C₄

semidirect

28. D₅₂₄

C₂×C₄

semidirect

29. C₂²⋊D₁₃₁

C₂×C₄

semidirect

34. C₄×D₁₃₁

C₂×C₄

semidirect

36. C₂×C₄×C₁₃₁

C₂×C₄

semidirect

59. C₁₃₁

#520

semidirect

66. #117

D₁₃₁

semidirect

96. C₂×C₄

C₄×D₁₃₁

semidirect

Subgroups — #22553792

Conjugacy classes of subgroups

Complements