Group #23752736 — subgroups

Subgroups — C₂×#18012732 #23752736

Order 9664.

246 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 111 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	#18012732	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₁₅₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₅₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₅₁	C₂²	1	4	yes	complemented
14	Dic₁₅₁⋊C₄	C₂²	1	4	yes	complemented
15	Dic₁₅₁⋊C₄	C₂²	1	4	yes	complemented
16	C₂²⋊Dic₁₅₁	C₂²	1	4	yes	—
17	C₂²⋊Dic₁₅₁	C₂²	1	4	yes	—
18	C₄×Dic₁₅₁	C₂²	1	4	yes	complemented
19	C₂⁴×C₁₅₁	C₂²	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₁₅₁	C₂²	1	4	yes	—
23	C₂×Dic₁₅₁	—	2	2	no	—
24	C₂×Dic₁₅₁	—	2	2	no	—
25	C₂×Dic₁₅₁	—	2	2	no	—
26	C₂×Dic₁₅₁	—	2	2	no	—
27	C₂×Dic₁₅₁	C₂³	1	2	yes	—
28	C₂×Dic₁₅₁	C₂³	1	2	yes	—
29	C₂×Dic₁₅₁	C₂³	1	2	yes	—
30	C₂×Dic₁₅₁	C₂³	1	2	yes	—
31	C₂³×C₁₅₁	—	2	2	no	—
32	C₂³×C₁₅₁	—	2	2	no	—
33	C₂³×C₁₅₁	—	2	2	no	—
34	C₂³×C₁₅₁	C₂³	1	2	yes	—
35	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	—	2	2	no	—
37	C₂×C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
39	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
40	C₂²×C₁₅₁	—	2	2	no	—
41	C₂²×C₁₅₁	—	2	2	no	—
42	C₂²×C₁₅₁	—	2	2	no	—
43	C₂²×C₁₅₁	—	4	4	no	—
44	C₂²×C₁₅₁	#41	1	2	yes	—
45	C₂²×C₁₅₁	#41	1	2	yes	—
46	C₂²×C₁₅₁	#41	1	2	yes	—
47	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
48	C₄×C₁₅₁	—	2	2	no	—
49	C₄×C₁₅₁	—	2	2	no	—
50	Dic₁₅₁	—	2	2	no	—
51	Dic₁₅₁	—	2	2	no	—
52	Dic₁₅₁	—	2	2	no	—
53	Dic₁₅₁	—	2	2	no	—
54	C₂×C₁₅₁	—	4	2	no	—
55	C₂×C₁₅₁	#126	1	4	yes	complemented
56	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
57	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
58	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
59	C₁₅₁	C₂×#126	1	1	yes	characteristic, complemented, Sylow
60	C₂×#126	—	151	1	no	maximal, Sylow
61	#126	—	151	8	no	—
62	C₂×C₄⋊C₄	—	151	1	no	—
63	C₂×C₄⋊C₄	—	151	1	no	—
64	C₂×C₄⋊C₄	—	151	1	no	—
65	C₂×C₂²⋊C₄	—	151	1	no	—
66	C₂×C₂²⋊C₄	—	151	1	no	—
67	C₂×C₂²⋊C₄	D₁₅₁	1	1	yes	characteristic
68	C₂×C₄²	—	151	1	no	—
69	C₂⁴	C₂×D₁₅₁	1	1	yes	characteristic
70	C₂²×C₄	—	151	1	no	—
71	C₂²×C₄	—	151	1	no	—
72	C₂²×C₄	—	151	1	no	—
73	C₂²×C₄	—	151	1	no	—
74	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
76	C₄⋊C₄	—	151	4	no	—
77	C₄⋊C₄	—	151	4	no	—
78	C₄⋊C₄	—	151	4	no	—
79	C₂²⋊C₄	—	151	4	no	—
80	C₂²⋊C₄	—	151	4	no	—
81	C₂²⋊C₄	C₂×D₁₅₁	1	4	yes	—
82	C₄²	—	151	4	no	—
83	C₂³	—	2	2	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	C₂²×D₁₅₁	1	2	yes	—
87	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	302	2	no	—
91	C₂×C₄	—	302	2	no	—
92	C₂×C₄	—	302	2	no	—
93	C₂×C₄	—	302	2	no	—
94	C₂×C₄	—	151	2	no	—
95	C₂×C₄	—	151	2	no	—
96	C₂×C₄	—	151	2	no	—
97	C₂×C₄	—	151	2	no	—
98	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
99	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	4	4	no	—
104	C₂²	#14559003	1	2	yes	—
105	C₂²	#14559003	1	2	yes	—
106	C₂²	#14559004	1	2	yes	—
107	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	302	2	no	—
111	C₄	—	302	2	no	—
112	C₄	—	302	2	no	—
113	C₄	—	302	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#18012732	1	4	yes	direct product factor
116	C₂	C₂×#14559003	1	1	yes	characteristic
117	C₂	C₂×#14559003	1	1	yes	characteristic
118	C₂	C₂×#14559004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18012732

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₅₁⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₅₁⋊C₄

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₅₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₅₁

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

C₂²

yes

complemented

Dic₁₅₁⋊C₄

C₂²

yes

complemented

Dic₁₅₁⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₁₅₁

C₂²

yes

—

C₂²⋊Dic₁₅₁

C₂²

yes

—

C₄×Dic₁₅₁

C₂²

yes

complemented

C₂⁴×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₁

C₂²

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

—

C₂³×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#126

yes

complemented

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

C₁₅₁

C₂×#126

yes

characteristic, complemented, Sylow

C₂×#126

—

151

maximal, Sylow

#126

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂×C₂²⋊C₄

D₁₅₁

yes

characteristic

C₂×C₄²

—

151

—

C₂⁴

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

C₂×D₁₅₁

yes

—

C₄²

—

151

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₁₅₁

yes

—

C₂³

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#14559003

yes

—

105

C₂²

#14559003

yes

—

106

C₂²

#14559004

yes

—

107

C₂²

C₂³×D₁₅₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

302

—

111

C₄

—

302

—

112

C₄

—

302

—

113

C₄

—

302

—

114

C₂

—

115

C₂

#18012732

yes

direct product factor

116

C₂

C₂×#14559003

yes

characteristic

117

C₂

C₂×#14559003

yes

characteristic

118

C₂

C₂×#14559004

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₅₁
commutator, cosocle	C₂²×C₁₅₁	C₂⁴
Frattini	C₂²	C₂³×D₁₅₁
Fitting	C₂×C₂²⋊C₄×C₁₅₁	C₂
socle	C₂³×C₁₅₁	C₂³
prime radical	C₂⁴×C₁₅₁	C₂²
hypercenter	C₂×C₂²⋊C₄	D₁₅₁
hypocenter	C₁₅₁	C₂×#126
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂²×D₁₅₁

commutator, cosocle

C₂²×C₁₅₁

C₂⁴

Frattini

C₂²

C₂³×D₁₅₁

Fitting

C₂×C₂²⋊C₄×C₁₅₁

C₂

socle

C₂³×C₁₅₁

C₂³

prime radical

C₂⁴×C₁₅₁

C₂²

hypercenter

C₂×C₂²⋊C₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. #18012732	C₂	direct (N × H)
2. C₂×C₄⋊Dic₁₅₁	C₂	semidirect
3. C₂×Dic₁₅₁⋊C₄	C₂	semidirect
4. C₂×Dic₁₅₁⋊C₄	C₂	semidirect
7. C₂×C₄×Dic₁₅₁	C₂	semidirect
13. C₄⋊Dic₁₅₁	C₂²	semidirect
14. Dic₁₅₁⋊C₄	C₂²	semidirect
15. Dic₁₅₁⋊C₄	C₂²	semidirect
18. C₄×Dic₁₅₁	C₂²	semidirect
55. C₂×C₁₅₁	#126	semidirect
59. C₁₅₁	C₂×#126	semidirect
115. C₂	#18012732	direct (N × H)

H ≅ G/N

type

1. #18012732

C₂

direct (N × H)

2. C₂×C₄⋊Dic₁₅₁

C₂

semidirect

3. C₂×Dic₁₅₁⋊C₄

C₂

semidirect

4. C₂×Dic₁₅₁⋊C₄

C₂

semidirect

7. C₂×C₄×Dic₁₅₁

C₂

semidirect

13. C₄⋊Dic₁₅₁

C₂²

semidirect

14. Dic₁₅₁⋊C₄

C₂²

semidirect

15. Dic₁₅₁⋊C₄

C₂²

semidirect

18. C₄×Dic₁₅₁

C₂²

semidirect

55. C₂×C₁₅₁

#126

semidirect

59. C₁₅₁

C₂×#126

semidirect

115. C₂

#18012732

direct (N × H)

Subgroups — C₂×#18012732 #23752736

Conjugacy classes of subgroups

Complements

Subgroups — C2×#18012732 #23752736

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#18012732 #23752736