Group #34069276 — subgroups

Subgroups — #34069276

Order 19328.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#18012672	C₂	1	2	yes	maximal
2	C₂×#18012670	C₂	1	2	yes	maximal
3	C₂×#18012650	C₂	1	1	yes	maximal, characteristic
4	#23752078	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012672	—	2	4	no	—
10	#18012670	—	2	4	no	—
11	#18012650	C₄	1	4	yes	complemented
12	C₁₅₁⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₅₁	—	2	1	no	—
17	C₂²×Dic₁₅₁	—	2	1	no	—
18	C₂×C₁₅₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₁₅₁⋊C₈	D₄	1	4	yes	—
20	C₂×C₁₅₁⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₄⋊Dic₁₅₁	—	2	2	no	—
23	C₂²×C₄×C₁₅₁	—	2	2	no	—
24	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	—	2	2	no	—
28	C₄⋊C₄×C₁₅₁	—	4	4	no	—
29	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₅₁	Q₈	1	2	yes	—
31	C₂×Dic₁₅₁	—	4	1	no	—
32	C₂×Dic₁₅₁	—	4	1	no	—
33	C₂×Dic₁₅₁	—	4	2	no	—
34	C₂×Dic₁₅₁	—	4	2	no	—
35	C₁₅₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
36	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	—	4	2	no	—
40	C₂×C₄×C₁₅₁	—	4	4	no	—
41	C₂×C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₁	#127	1	1	yes	characteristic
48	C₂²×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₅₁	#137	1	2	yes	—
51	C₄×C₁₅₁	—	2	4	no	—
52	C₄×C₁₅₁	—	8	2	no	—
53	C₄×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₅₁	C₄⋊Q₈	1	2	yes	—
55	Dic₁₅₁	—	8	1	no	—
56	Dic₁₅₁	—	8	1	no	—
57	C₂×C₁₅₁	#384	1	1	yes	characteristic
58	C₂×C₁₅₁	#428	1	2	yes	—
59	C₂×C₁₅₁	#496	1	1	yes	characteristic
60	C₂×C₁₅₁	#497	1	1	yes	characteristic
61	C₂×C₁₅₁	#501	1	2	yes	—
62	C₁₅₁	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	151	1	no	maximal, Sylow
64	C₂×#107	—	151	2	no	—
65	C₂×#106	—	151	2	no	—
66	C₂×#97	—	151	1	no	—
67	#384	D₁₅₁	1	1	yes	characteristic
68	#382	—	151	1	no	—
69	C₂²×C₈	—	151	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	151	1	no	—
72	C₂×C₄⋊C₄	—	151	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
74	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
75	#107	—	302	4	no	—
76	#106	—	302	4	no	—
77	#97	—	151	4	no	—
78	#95	—	302	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	302	1	no	—
81	C₂²×C₄	—	302	1	no	—
82	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
84	C₂×C₈	—	151	4	no	—
85	C₂×C₈	—	151	4	no	—
86	C₂×C₈	—	151	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	302	2	no	—
91	C₄⋊C₄	—	302	2	no	—
92	C₄²	C₄×D₁₅₁	1	2	yes	—
93	C₄²	Dic₃₀₂	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	604	1	no	—
100	C₂×C₄	—	604	1	no	—
101	C₂×C₄	—	604	2	no	—
102	C₂×C₄	—	604	2	no	—
103	C₂×C₄	Dic₁₅₁⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
106	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
108	C₈	—	151	8	no	—
109	C₂²	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
110	C₂²	#18012735	1	1	yes	characteristic
111	C₂²	#18012737	1	1	yes	characteristic
112	C₂²	#18012765	1	2	yes	—
113	C₂²	#18012770	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	1208	1	no	—
117	C₄	—	1208	1	no	—
118	C₄	C₄⋊C₄×D₁₅₁	1	2	yes	—
119	C₄	#18012738	1	2	yes	—
120	C₂	#23752081	1	1	yes	characteristic
121	C₂	#23752252	1	2	yes	—
122	C₂	#23752599	1	1	yes	characteristic
123	C₂	#23752604	1	1	yes	characteristic
124	C₂	#23752618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012672

C₂

yes

maximal

C₂×#18012670

C₂

yes

maximal

C₂×#18012650

C₂

yes

maximal, characteristic

#23752078

C₂

yes

maximal, characteristic

#384×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

—

C₄

yes

complemented

C₁₅₁⋊#95

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

Q₈

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

C₄⋊C₄

yes

complemented

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

#127

yes

characteristic

C₂²×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₁

#137

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₅₁

C₄⋊Q₈

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#384

yes

characteristic

C₂×C₁₅₁

#428

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₁₅₁

#1847

yes

characteristic, complemented, Sylow

#1847

—

151

maximal, Sylow

C₂×#107

—

151

—

C₂×#106

—

151

—

C₂×#97

—

151

—

#384

D₁₅₁

yes

characteristic

#382

—

151

—

C₂²×C₈

—

151

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#107

—

302

—

#106

—

302

—

#97

—

151

—

#95

—

302

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

Dic₃₀₂

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

604

—

100

C₂×C₄

—

604

—

101

C₂×C₄

—

604

—

102

C₂×C₄

—

604

—

103

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

105

C₂×C₄

D₄×D₁₅₁

yes

—

106

C₂×C₄

Q₈×D₁₅₁

yes

—

107

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

108

C₈

—

151

—

109

C₂²

C₂×Dic₁₅₁⋊C₄

yes

characteristic

110

C₂²

#18012735

yes

characteristic

111

C₂²

#18012737

yes

characteristic

112

C₂²

#18012765

yes

—

113

C₂²

#18012770

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

1208

—

117

C₄

—

1208

—

118

C₄

C₄⋊C₄×D₁₅₁

yes

—

119

C₄

#18012738

yes

—

120

C₂

#23752081

yes

characteristic

121

C₂

#23752252

yes

—

122

C₂

#23752599

yes

characteristic

123

C₂

#23752604

yes

characteristic

124

C₂

#23752618

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#384×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#384	D₁₅₁
hypocenter	C₁₅₁	#1847
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#384×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #18012650	C₄	semidirect
35. C₁₅₁⋊C₈	C₄⋊C₄	semidirect
62. C₁₅₁	#1847	semidirect

H ≅ G/N

type

11. #18012650

C₄

semidirect

35. C₁₅₁⋊C₈

C₄⋊C₄

semidirect

62. C₁₅₁

#1847

semidirect

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

subgroup	index
C₂×#18012672	2
C₂×#18012670	2
C₂×#18012650	2
#23752078	2
#384×C₁₅₁	2
#1847	151

subgroup

index

#384×C₁₅₁

151

Subgroups — #34069276

Conjugacy classes of subgroups

Complements

Maximal subgroups