Group #23752705 — subgroups

Subgroups — C₂×#18012714 #23752705

Order 9664.

426 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂²×D₆₀₄	C₂	1	2	yes	maximal, complemented
2	C₂²×D₆₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#18012714	C₂	1	8	yes	maximal, direct product factor
5	C₂×(C₂×D₁₅₁)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₅₁	C₂²	1	2	yes	—
8	C₂³×D₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×D₆₀₄	—	2	8	no	—
10	C₂×D₆₀₄	—	2	4	no	—
11	C₂×D₆₀₄	C₂²	1	8	yes	complemented
12	C₂×D₆₀₄	C₂²	1	4	yes	complemented
13	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅₁	—	2	4	no	—
15	C₂×C₄×D₁₅₁	C₂²	1	4	yes	complemented
16	(C₂×D₁₅₁)⋊C₄	C₂²	1	8	yes	complemented
17	C₂²×C₄×C₁₅₁	C₂²	1	2	yes	—
18	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₅₁	C₂²	1	4	yes	complemented
20	C₂²×D₁₅₁	—	2	2	no	—
21	C₂²×D₁₅₁	—	2	2	no	—
22	C₂²×D₁₅₁	—	2	4	no	—
23	C₂²×D₁₅₁	—	2	4	no	—
24	C₂²×D₁₅₁	—	2	4	no	—
25	C₂²×D₁₅₁	D₄	1	4	yes	complemented
26	C₂²×D₁₅₁	C₂³	1	4	yes	—
27	C₂²×D₁₅₁	C₂³	1	2	yes	—
28	D₆₀₄	—	2	16	no	—
29	D₆₀₄	—	2	8	no	—
30	C₂×Dic₁₅₁	—	2	2	no	—
31	C₂×Dic₁₅₁	C₂³	1	2	yes	—
32	C₄×D₁₅₁	—	2	8	no	—
33	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₅₁	—	2	4	no	—
35	C₂×C₄×C₁₅₁	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₅₁	C₂³	1	4	yes	—
37	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
38	C₂²×C₁₅₁	C₂×D₄	1	2	yes	—
39	C₂²×C₁₅₁	C₂×D₄	1	2	yes	—
40	C₂²×C₁₅₁	#41	1	2	yes	—
41	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₅₁	—	2	8	no	—
43	C₂×D₁₅₁	—	2	2	no	—
44	C₂×D₁₅₁	—	2	2	no	—
45	C₂×D₁₅₁	—	2	4	no	—
46	C₂×D₁₅₁	—	2	4	no	—
47	C₂×D₁₅₁	—	2	4	no	—
48	C₂×D₁₅₁	—	4	8	no	—
49	C₂×D₁₅₁	C₂×D₄	1	4	yes	complemented
50	C₄×C₁₅₁	—	2	4	no	—
51	C₄×C₁₅₁	C₂×D₄	1	4	yes	complemented
52	Dic₁₅₁	—	2	2	no	—
53	C₂×C₁₅₁	#121	1	4	yes	complemented
54	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
57	D₁₅₁	—	2	4	no	—
58	D₁₅₁	—	4	4	no	—
59	C₁₅₁	C₂×#121	1	1	yes	characteristic, complemented, Sylow
60	C₂×#121	—	151	1	no	maximal, Sylow
61	C₂²×D₄	—	151	2	no	—
62	C₂²×D₄	—	151	1	no	—
63	C₂³×C₄	—	151	1	no	—
64	#121	—	151	8	no	—
65	C₂×C₄⋊C₄	D₁₅₁	1	1	yes	characteristic, complemented
66	C₂×C₂²⋊C₄	—	151	2	no	—
67	C₂⁴	—	151	2	no	—
68	C₂⁴	—	151	1	no	—
69	C₂×D₄	—	302	8	no	—
70	C₂×D₄	—	302	4	no	—
71	C₂×D₄	—	151	8	no	—
72	C₂×D₄	—	151	4	no	—
73	C₂²×C₄	—	302	4	no	—
74	C₂²×C₄	—	151	1	no	—
75	C₂²×C₄	—	151	4	no	—
76	C₂²×C₄	C₂×D₁₅₁	1	2	yes	—
77	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
78	C₄⋊C₄	C₂×D₁₅₁	1	4	yes	complemented
79	C₂²⋊C₄	—	151	8	no	—
80	C₂³	—	302	2	no	—
81	C₂³	—	302	2	no	—
82	C₂³	—	302	4	no	—
83	C₂³	—	302	4	no	—
84	C₂³	—	302	4	no	—
85	C₂³	—	151	4	no	—
86	C₂³	—	151	4	no	—
87	C₂³	—	151	2	no	—
88	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
89	D₄	—	302	16	no	—
90	D₄	—	302	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	302	2	no	—
93	C₂×C₄	—	302	8	no	—
94	C₂×C₄	—	151	2	no	—
95	C₂×C₄	D₆₀₄	1	4	yes	complemented
96	C₂×C₄	C₂²×D₁₅₁	1	4	yes	—
97	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
98	C₂²	—	302	8	no	—
99	C₂²	—	302	2	no	—
100	C₂²	—	302	2	no	—
101	C₂²	—	302	4	no	—
102	C₂²	—	302	4	no	—
103	C₂²	—	302	4	no	—
104	C₂²	—	604	8	no	—
105	C₂²	—	151	4	no	—
106	C₂²	D₄×D₁₅₁	1	2	yes	—
107	C₂²	C₂×D₆₀₄	1	2	yes	—
108	C₂²	#14559005	1	2	yes	—
109	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
110	C₄	—	2	4	no	—
111	C₄	—	302	2	no	—
112	C₄	C₂×D₆₀₄	1	4	yes	complemented
113	C₂	—	302	4	no	—
114	C₂	—	604	4	no	—
115	C₂	#18012714	1	4	yes	direct product factor
116	C₂	C₂×D₄×D₁₅₁	1	1	yes	characteristic
117	C₂	C₂²×D₆₀₄	1	1	yes	characteristic
118	C₂	C₂×#14559005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₀₄

C₂

yes

maximal, complemented

C₂²×D₆₀₄

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₅₁

C₂

yes

maximal, characteristic, complemented

#18012714

C₂

yes

maximal, direct product factor

C₂×(C₂×D₁₅₁)⋊C₄

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₁

C₂²

yes

—

C₂³×D₁₅₁

C₂²

yes

characteristic

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

C₂²

yes

complemented

C₂×D₆₀₄

C₂²

yes

complemented

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

C₂²

yes

complemented

(C₂×D₁₅₁)⋊C₄

C₂²

yes

complemented

C₂²×C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₁

C₂²

yes

complemented

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

D₄

yes

complemented

C₂²×D₁₅₁

C₂³

yes

—

C₂²×D₁₅₁

C₂³

yes

—

D₆₀₄

—

D₆₀₄

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

D₄

yes

complemented

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂²×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂×D₄

yes

complemented

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

complemented

Dic₁₅₁

—

C₂×C₁₅₁

#121

yes

complemented

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

D₁₅₁

—

D₁₅₁

—

C₁₅₁

C₂×#121

yes

characteristic, complemented, Sylow

C₂×#121

—

151

maximal, Sylow

C₂²×D₄

—

151

—

C₂²×D₄

—

151

—

C₂³×C₄

—

151

—

#121

—

151

—

C₂×C₄⋊C₄

D₁₅₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

151

—

C₂⁴

—

151

—

C₂⁴

—

151

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

151

—

C₂×D₄

—

151

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

C₂×D₁₅₁

yes

complemented

C₂²⋊C₄

—

151

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

151

—

C₂³

—

151

—

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

characteristic

D₄

—

302

—

D₄

—

302

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

D₆₀₄

yes

complemented

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂²

—

302

—

C₂²

—

302

—

100

C₂²

—

302

—

101

C₂²

—

302

—

102

C₂²

—

302

—

103

C₂²

—

302

—

104

C₂²

—

604

—

105

C₂²

—

151

—

106

C₂²

D₄×D₁₅₁

yes

—

107

C₂²

C₂×D₆₀₄

yes

—

108

C₂²

#14559005

yes

—

109

C₂²

C₂³×D₁₅₁

yes

characteristic

110

C₄

—

111

C₄

—

302

—

112

C₄

C₂×D₆₀₄

yes

complemented

113

C₂

—

302

—

114

C₂

—

604

—

115

C₂

#18012714

yes

direct product factor

116

C₂

C₂×D₄×D₁₅₁

yes

characteristic

117

C₂

C₂²×D₆₀₄

yes

characteristic

118

C₂

C₂×#14559005

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₅₁
commutator, cosocle	C₂²×C₁₅₁	C₂⁴
Frattini	C₂²	C₂³×D₁₅₁
Fitting	C₂×C₄⋊C₄×C₁₅₁	C₂
socle	C₂³×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₅₁
hypocenter	C₁₅₁	C₂×#121
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₅₁

commutator, cosocle

C₂²×C₁₅₁

C₂⁴

Frattini

C₂²

C₂³×D₁₅₁

Fitting

C₂×C₄⋊C₄×C₁₅₁

C₂

socle

C₂³×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂²×D₆₀₄	C₂	semidirect
2. C₂²×D₆₀₄	C₂	semidirect
3. C₂²×C₄×D₁₅₁	C₂	semidirect
4. #18012714	C₂	direct (N × H)
5. C₂×(C₂×D₁₅₁)⋊C₄	C₂	semidirect
6. C₂×C₄⋊C₄×C₁₅₁	C₂	semidirect
11. C₂×D₆₀₄	C₂²	semidirect
12. C₂×D₆₀₄	C₂²	semidirect
15. C₂×C₄×D₁₅₁	C₂²	semidirect
16. (C₂×D₁₅₁)⋊C₄	C₂²	semidirect
19. C₄⋊C₄×C₁₅₁	C₂²	semidirect
25. C₂²×D₁₅₁	D₄	semidirect
35. C₂×C₄×C₁₅₁	D₄	semidirect
49. C₂×D₁₅₁	C₂×D₄	semidirect
51. C₄×C₁₅₁	C₂×D₄	semidirect
53. C₂×C₁₅₁	#121	semidirect
59. C₁₅₁	C₂×#121	semidirect
65. C₂×C₄⋊C₄	D₁₅₁	semidirect
78. C₄⋊C₄	C₂×D₁₅₁	semidirect
95. C₂×C₄	D₆₀₄	semidirect
112. C₄	C₂×D₆₀₄	semidirect
115. C₂	#18012714	direct (N × H)

H ≅ G/N

type

1. C₂²×D₆₀₄

C₂

semidirect

2. C₂²×D₆₀₄

C₂

semidirect

3. C₂²×C₄×D₁₅₁

C₂

semidirect

4. #18012714

C₂

direct (N × H)

5. C₂×(C₂×D₁₅₁)⋊C₄

C₂

semidirect

6. C₂×C₄⋊C₄×C₁₅₁

C₂

semidirect

11. C₂×D₆₀₄

C₂²

semidirect

12. C₂×D₆₀₄

C₂²

semidirect

15. C₂×C₄×D₁₅₁

C₂²

semidirect

16. (C₂×D₁₅₁)⋊C₄

C₂²

semidirect

19. C₄⋊C₄×C₁₅₁

C₂²

semidirect

25. C₂²×D₁₅₁

D₄

semidirect

35. C₂×C₄×C₁₅₁

D₄

semidirect

49. C₂×D₁₅₁

C₂×D₄

semidirect

51. C₄×C₁₅₁

C₂×D₄

semidirect

53. C₂×C₁₅₁

#121

semidirect

59. C₁₅₁

C₂×#121

semidirect

65. C₂×C₄⋊C₄

D₁₅₁

semidirect

78. C₄⋊C₄

C₂×D₁₅₁

semidirect

95. C₂×C₄

D₆₀₄

semidirect

112. C₄

C₂×D₆₀₄

semidirect

115. C₂

#18012714

direct (N × H)

Subgroups — C₂×#18012714 #23752705

Conjugacy classes of subgroups

Complements

Subgroups — C2×#18012714 #23752705

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#18012714 #23752705