Group #34070087 — subgroups

Subgroups — #34070087

Order 19328.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#23752769	C₂	1	1	yes	maximal, characteristic
2	#23752236	C₂	1	1	yes	maximal, characteristic
3	#23751972	C₂	1	4	yes	maximal
4	C₄³×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14558988	C₂²	1	2	yes	—
6	#18012739	C₄	1	2	yes	complemented
7	#18012731	C₄	1	2	yes	complemented
8	#18012718	—	2	2	no	—
9	C₄×Dic₃₀₂	—	2	2	no	—
10	#18012698	C₂²	1	2	yes	—
11	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
12	C₂×C₄²×C₁₅₁	—	2	2	no	—
13	C₂×C₄²×C₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀₂	—	2	2	no	—
16	#14558988	—	2	4	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₁₅₁	—	2	2	no	—
19	C₄⋊Dic₁₅₁	C₂×C₄	1	4	yes	—
20	Dic₁₅₁⋊C₄	—	2	4	no	—
21	C₂²⋊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	C₄²×C₁₅₁	—	2	8	no	—
27	C₄²×C₁₅₁	—	2	2	no	—
28	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₅₁	D₄	1	2	yes	—
31	C₄²×C₁₅₁	Q₈	1	4	yes	complemented
32	Dic₃₀₂	—	4	4	no	—
33	C₂×Dic₁₅₁	—	2	4	no	—
34	C₁₅₁⋊C₈	—	4	2	no	—
35	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	—	2	8	no	—
37	C₂×C₄×C₁₅₁	—	2	4	no	—
38	C₂×C₄×C₁₅₁	—	2	4	no	—
39	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₅₁	#41	1	2	yes	—
45	C₂²×C₁₅₁	—	2	2	no	—
46	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₅₁	—	2	8	no	—
49	C₄×C₁₅₁	—	2	2	no	—
50	C₄×C₁₅₁	#104	1	4	yes	—
51	C₄×C₁₅₁	C₄×Q₈	1	2	yes	—
52	C₄×C₁₅₁	#124	1	1	yes	characteristic
53	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₁₅₁	—	4	4	no	—
55	C₂×C₁₅₁	—	2	2	no	—
56	C₂×C₁₅₁	#391	1	1	yes	characteristic
57	C₂×C₁₅₁	C₂×#104	1	2	yes	—
58	C₁₅₁	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	151	1	no	maximal, Sylow
60	#533	—	151	1	no	—
61	#423	—	151	1	no	—
62	C₄³	D₁₅₁	1	1	yes	characteristic
63	#339	—	151	4	no	—
64	C₂×M₁₆	—	151	2	no	—
65	C₄⋊Q₈	—	151	2	no	—
66	#125	—	151	2	no	—
67	C₂²⋊Q₈	—	302	2	no	—
68	C₄×Q₈	—	302	2	no	—
69	#117	—	151	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₅₁	1	2	yes	—
72	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
73	C₄⋊C₈	—	302	2	no	—
74	C₂×Q₈	—	302	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
78	M₁₆	—	302	4	no	—
79	C₂×C₈	—	302	2	no	—
80	C₄⋊C₄	—	302	4	no	—
81	C₄⋊C₄	—	302	2	no	—
82	C₄⋊C₄	—	151	4	no	—
83	C₂²⋊C₄	—	302	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	302	2	no	—
87	C₄²	C₄×D₁₅₁	1	1	yes	characteristic
88	C₄²	C₄×D₁₅₁	1	1	yes	characteristic
89	C₄²	D₆₀₄	1	2	yes	—
90	C₄²	Dic₃₀₂	1	4	yes	complemented
91	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
92	Q₈	—	604	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	302	4	no	—
97	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
100	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₃₀₂	1	2	yes	—
102	C₂×C₄	#14559004	1	2	yes	—
103	C₈	—	604	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
106	C₂²	#18012718	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	604	4	no	—
110	C₄	#18012667	1	4	yes	—
111	C₄	C₄×Dic₃₀₂	1	2	yes	—
112	C₄	#18012727	1	1	yes	characteristic
113	C₄	#18012739	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#23752118	1	1	yes	characteristic
116	C₂	C₂×#18012667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752769

C₂

yes

maximal, characteristic

#23752236

C₂

yes

maximal, characteristic

#23751972

C₂

yes

maximal

C₄³×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

—

#18012739

C₄

yes

complemented

#18012731

C₄

yes

complemented

#18012718

—

C₄×Dic₃₀₂

—

#18012698

C₂²

yes

—

C₄⋊(C₁₅₁⋊C₈)

—

C₂×C₄²×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

Dic₁₅₁⋊C₄

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄²×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

characteristic

C₄²×C₁₅₁

C₂×C₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

Q₈

yes

complemented

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

C₂²⋊Q₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#104

yes

—

C₄×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

#124

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#391

yes

characteristic

C₂×C₁₅₁

C₂×#104

yes

—

C₁₅₁

#1994

yes

characteristic, complemented, Sylow

#1994

—

151

maximal, Sylow

#533

—

151

—

#423

—

151

—

C₄³

D₁₅₁

yes

characteristic

#339

—

151

—

C₂×M₁₆

—

151

—

C₄⋊Q₈

—

151

—

#125

—

151

—

C₂²⋊Q₈

—

302

—

C₄×Q₈

—

302

—

#117

—

151

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

302

—

C₂×Q₈

—

302

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

302

—

C₄²

—

C₄²

—

C₄²

—

302

—

C₄²

C₄×D₁₅₁

yes

characteristic

C₄²

C₄×D₁₅₁

yes

characteristic

C₄²

D₆₀₄

yes

—

C₄²

Dic₃₀₂

yes

complemented

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

Q₈

—

604

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

100

C₂×C₄

C₂×D₆₀₄

yes

characteristic

101

C₂×C₄

C₂×Dic₃₀₂

yes

—

102

C₂×C₄

#14559004

yes

—

103

C₈

—

604

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

106

C₂²

#18012718

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

604

—

110

C₄

#18012667

yes

—

111

C₄

C₄×Dic₃₀₂

yes

—

112

C₄

#18012727

yes

characteristic

113

C₄

#18012739

yes

characteristic

114

C₂

—

115

C₂

#23752118

yes

characteristic

116

C₂

C₂×#18012667

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀₄
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₄³×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₄³	D₁₅₁
hypocenter	C₁₅₁	#1994
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₆₀₄

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₄³×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₄³

D₁₅₁

hypocenter

C₁₅₁

#1994

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #18012739	C₄	semidirect
7. #18012731	C₄	semidirect
31. C₄²×C₁₅₁	Q₈	semidirect
58. C₁₅₁	#1994	semidirect
90. C₄²	Dic₃₀₂	semidirect

H ≅ G/N

type

6. #18012739

C₄

semidirect

7. #18012731

C₄

semidirect

31. C₄²×C₁₅₁

Q₈

semidirect

58. C₁₅₁

#1994

semidirect

90. C₄²

Dic₃₀₂

semidirect

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

subgroup	index
#23752769	2
#23752236	2
#23751972	2
C₄³×C₁₅₁	2
#1994	151

subgroup

index

C₄³×C₁₅₁

151

Subgroups — #34070087

Conjugacy classes of subgroups

Complements

Maximal subgroups