Group #34066947 — subgroups

Subgroups — #34066947

Order 19328.

172 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 85 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	#23752265	C₂	1	1	yes	maximal, characteristic
2	C₄×#14558988	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14558988	C₄	1	2	yes	—
5	C₄⋊(C₁₅₁⋊C₈)	C₄	1	2	yes	—
6	C₂²⋊(C₁₅₁⋊C₈)	—	2	1	no	—
7	#18012650	—	2	1	no	—
8	C₄×C₁₅₁⋊C₈	—	2	1	no	—
9	C₄×C₁₅₁⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₁₅₁	—	2	1	no	—
11	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₅₁	—	2	1	no	—
13	C₄×C₈×C₁₅₁	C₄	1	2	yes	—
14	#14558988	—	2	2	no	—
15	#14558988	C₈	1	4	yes	complemented
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₁	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₅₁	—	2	4	no	—
22	C₂×C₈×C₁₅₁	—	2	2	no	—
23	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
26	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
27	C₁₅₁⋊C₈	—	2	4	no	—
28	C₁₅₁⋊C₈	—	4	2	no	—
29	C₂³×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₁	—	2	2	no	—
31	C₂×C₄×C₁₅₁	—	2	2	no	—
32	C₂×C₄×C₁₅₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₁₅₁	M₁₆	1	2	yes	—
40	C₈×C₁₅₁	—	2	4	no	—
41	C₂²×C₁₅₁	—	2	1	no	—
42	C₂²×C₁₅₁	—	2	1	no	—
43	C₂²×C₁₅₁	#95	1	1	yes	characteristic
44	C₂²×C₁₅₁	C₄⋊C₈	1	2	yes	—
45	C₄×C₁₅₁	—	2	1	no	—
46	C₄×C₁₅₁	—	2	1	no	—
47	C₄×C₁₅₁	C₄×C₈	1	1	yes	characteristic
48	C₄×C₁₅₁	#97	1	1	yes	characteristic
49	C₄×C₁₅₁	C₂²⋊C₈	1	2	yes	—
50	C₄×C₁₅₁	#104	1	2	yes	—
51	C₄×C₁₅₁	#108	1	2	yes	—
52	C₂×C₁₅₁	—	2	2	no	—
53	C₂×C₁₅₁	#336	1	1	yes	characteristic
54	C₂×C₁₅₁	#339	1	1	yes	characteristic
55	C₂×C₁₅₁	#341	1	1	yes	characteristic
56	C₁₅₁	#1278	1	1	yes	characteristic, complemented, Sylow
57	#1278	—	151	1	no	maximal, Sylow
58	#431	—	151	1	no	—
59	C₄×M₁₆	—	151	1	no	—
60	C₂×C₄×C₈	D₁₅₁	1	1	yes	characteristic
61	C₂×M₁₆	—	151	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
64	C₄⋊C₈	—	151	2	no	—
65	C₂²⋊C₈	—	302	1	no	—
66	#97	—	302	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	302	1	no	—
69	C₄×C₈	—	302	1	no	—
70	C₄×C₈	Dic₁₅₁	1	2	yes	—
71	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
73	C₂²×C₄	Dic₃₀₂	1	1	yes	characteristic
74	M₁₆	—	302	2	no	—
75	M₁₆	—	151	4	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	302	2	no	—
79	C₂×C₈	—	302	2	no	—
80	C₄²	C₄×D₁₅₁	1	1	yes	characteristic
81	C₄²	C₂×Dic₁₅₁	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
83	C₄²	D₆₀₄	1	1	yes	characteristic
84	C₂³	Dic₁₅₁⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₁₅₁	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₁₅₁	1	1	yes	characteristic
91	C₂×C₄	Dic₁₅₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₁₅₁	1	1	yes	characteristic
93	C₂×C₄	C₈×D₁₅₁	1	2	yes	—
94	C₂×C₄	#14558987	1	2	yes	—
95	C₈	—	2	4	no	—
96	C₈	—	302	4	no	—
97	C₈	—	604	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₁₅₁⋊#95	1	1	yes	characteristic
101	C₂²	Dic₁₅₁⋊C₈	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₅₁	1	1	yes	characteristic
105	C₄	#18012649	1	1	yes	characteristic
106	C₄	(C₂×D₁₅₁)⋊C₈	1	2	yes	—
107	C₄	#18012667	1	2	yes	—
108	C₄	#18012675	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#23751966	1	1	yes	characteristic
111	C₂	#23751972	1	1	yes	characteristic
112	C₂	#23751977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752265

C₂

yes

maximal, characteristic

C₄×#14558988

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₄

yes

—

C₄⋊(C₁₅₁⋊C₈)

C₄

yes

—

C₂²⋊(C₁₅₁⋊C₈)

—

#18012650

—

C₄×C₁₅₁⋊C₈

—

C₄×C₁₅₁⋊C₈

—

C₂²×C₈×C₁₅₁

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

—

C₄×C₈×C₁₅₁

C₄

yes

—

#14558988

—

#14558988

C₈

yes

complemented

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

characteristic

C₂²×C₄×C₁₅₁

Q₈

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

characteristic

C₄²×C₁₅₁

C₂×C₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄²

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×C₈

yes

—

C₂×C₄×C₁₅₁

M₁₆

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#95

yes

characteristic

C₂²×C₁₅₁

C₄⋊C₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄×C₈

yes

characteristic

C₄×C₁₅₁

#97

yes

characteristic

C₄×C₁₅₁

C₂²⋊C₈

yes

—

C₄×C₁₅₁

#104

yes

—

C₄×C₁₅₁

#108

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1278

—

151

maximal, Sylow

#431

—

151

—

C₄×M₁₆

—

151

—

C₂×C₄×C₈

D₁₅₁

yes

characteristic

C₂×M₁₆

—

151

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

151

—

C₂²⋊C₈

—

302

—

#97

—

302

—

C₄×C₈

—

C₄×C₈

—

302

—

C₄×C₈

—

302

—

C₄×C₈

Dic₁₅₁

yes

—

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₁

yes

characteristic

C₂²×C₄

Dic₃₀₂

yes

characteristic

M₁₆

—

302

—

M₁₆

—

151

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄²

C₄×D₁₅₁

yes

characteristic

C₄²

C₂×Dic₁₅₁

yes

characteristic

C₄²

C₂²⋊D₁₅₁

yes

characteristic

C₄²

D₆₀₄

yes

characteristic

C₂³

Dic₁₅₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₅₁

yes

characteristic

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₁₅₁

yes

characteristic

C₂×C₄

Dic₁₅₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₁₅₁

yes

characteristic

C₂×C₄

C₈×D₁₅₁

yes

—

C₂×C₄

#14558987

yes

—

C₈

—

C₈

—

302

—

C₈

—

604

—

C₂²

—

C₂²

—

100

C₂²

C₁₅₁⋊#95

yes

characteristic

101

C₂²

Dic₁₅₁⋊C₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₅₁

yes

characteristic

105

C₄

#18012649

yes

characteristic

106

C₄

(C₂×D₁₅₁)⋊C₈

yes

—

107

C₄

#18012667

yes

—

108

C₄

#18012675

yes

—

109

C₂

—

110

C₂

#23751966

yes

characteristic

111

C₂

#23751972

yes

characteristic

112

C₂

#23751977

yes

characteristic

	subgroup	G/H
center	C₄²	D₆₀₄
commutator	C₄×C₁₅₁	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₅₁
Fitting	C₂×C₄×C₈×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#95
prime radical	C₂³×C₁₅₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₅₁	C₂²
hypercenter	C₂×C₄×C₈	D₁₅₁
hypocenter	C₁₅₁	#1278
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

D₆₀₄

commutator

C₄×C₁₅₁

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₅₁

Fitting

C₂×C₄×C₈×C₁₅₁

C₂

socle

C₂²×C₁₅₁

#95

prime radical

C₂³×C₁₅₁

C₄⋊C₄

cosocle

C₂×C₄²×C₁₅₁

C₂²

hypercenter

C₂×C₄×C₈

D₁₅₁

hypocenter

C₁₅₁

#1278

LCS

G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
15. #14558988	C₈	semidirect
56. C₁₅₁	#1278	semidirect

H ≅ G/N

type

15. #14558988

C₈

semidirect

56. C₁₅₁

#1278

semidirect

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

subgroup	index
#23752265	2
C₄×#14558988	2
C₂×C₄×C₈×C₁₅₁	2
#1278	151

subgroup

index

#23752265

C₄×#14558988

C₂×C₄×C₈×C₁₅₁

#1278

151

Subgroups — #34066947

Conjugacy classes of subgroups

Complements

Maximal subgroups