Group #34070103 — subgroups

Subgroups — #34070103

Order 19328.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#18012731	C₂	1	1	yes	maximal, characteristic
2	#23752236	C₂	1	1	yes	maximal, characteristic
3	#23751980	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14558988	C₂²	1	2	yes	—
6	#18012731	—	2	1	no	—
7	#18012731	—	2	1	no	—
8	#18012731	—	2	2	no	—
9	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
10	C₂×Dic₁₅₁⋊C₄	C₄	1	4	yes	—
11	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
12	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
14	#97×C₁₅₁	—	2	1	no	—
15	C₄×C₈×C₁₅₁	—	2	1	no	—
16	C₂²×Dic₁₅₁	C₂×C₄	1	4	yes	—
17	#14558988	—	2	4	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₅₁	—	2	2	no	—
20	C₄⋊Dic₁₅₁	—	2	2	no	—
21	Dic₁₅₁⋊C₄	—	2	4	no	—
22	Dic₁₅₁⋊C₄	—	2	4	no	—
23	C₂²×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₅₁	—	2	2	no	—
26	M₁₆×C₁₅₁	Q₈	1	4	yes	—
27	C₂×C₈×C₁₅₁	—	2	2	no	—
28	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
29	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
30	C₄²×C₁₅₁	D₄	1	2	yes	—
31	C₂×Dic₁₅₁	—	2	4	no	—
32	C₂×Dic₁₅₁	—	4	4	no	—
33	C₁₅₁⋊C₈	—	4	2	no	—
34	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₁	—	4	2	no	—
36	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₅₁	#41	1	2	yes	—
42	C₈×C₁₅₁	—	2	4	no	—
43	C₂²×C₁₅₁	—	2	2	no	—
44	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₅₁	C₄×Q₈	1	2	yes	—
46	C₄×C₁₅₁	—	2	2	no	—
47	C₄×C₁₅₁	—	2	2	no	—
48	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₅₁	#124	1	1	yes	characteristic
50	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₅₁	—	4	4	no	—
52	C₂×C₁₅₁	—	2	2	no	—
53	C₂×C₁₅₁	#391	1	1	yes	characteristic
54	C₂×C₁₅₁	#413	1	2	yes	—
55	C₁₅₁	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	151	1	no	maximal, Sylow
57	C₂×#125	—	151	1	no	—
58	#423	—	151	1	no	—
59	C₄×M₁₆	D₁₅₁	1	1	yes	characteristic
60	#343	—	151	4	no	—
61	C₂×M₁₆	—	151	2	no	—
62	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
63	#125	—	302	1	no	—
64	#125	—	302	1	no	—
65	#125	—	302	2	no	—
66	C₂×C₄⋊C₄	—	151	4	no	—
67	C₂×C₄⋊C₄	—	151	2	no	—
68	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
69	C₄⋊C₈	—	302	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	151	4	no	—
73	C₂²×C₄	C₄×D₁₅₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	302	4	no	—
77	M₁₆	Dic₃₀₂	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	302	2	no	—
80	C₄⋊C₄	—	302	4	no	—
81	C₄⋊C₄	—	302	4	no	—
82	C₄⋊C₄	—	302	2	no	—
83	C₄⋊C₄	—	302	2	no	—
84	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
86	C₄²	D₆₀₄	1	2	yes	—
87	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	302	4	no	—
90	C₂×C₄	—	604	4	no	—
91	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
94	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₃₀₂	1	2	yes	—
96	C₂×C₄	#14559004	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	604	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
101	C₂²	C₄×Dic₃₀₂	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	604	4	no	—
105	C₄	#18012718	1	2	yes	—
106	C₄	#18012727	1	1	yes	characteristic
107	C₄	#18012739	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#23752118	1	1	yes	characteristic
110	C₂	#23752194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012731

C₂

yes

maximal, characteristic

#23752236

C₂

yes

maximal, characteristic

#23751980

C₂

yes

maximal

C₄×M₁₆×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

—

#18012731

—

#18012731

—

#18012731

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×Dic₁₅₁⋊C₄

C₄

yes

—

C₄⋊(C₁₅₁⋊C₈)

—

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

#97×C₁₅₁

—

C₄×C₈×C₁₅₁

—

C₂²×Dic₁₅₁

C₂×C₄

yes

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

Dic₁₅₁⋊C₄

—

Dic₁₅₁⋊C₄

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

Q₈

yes

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₁

C₄×Q₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₄×C₁₅₁

#124

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#391

yes

characteristic

C₂×C₁₅₁

#413

yes

—

C₁₅₁

#1998

yes

characteristic, complemented, Sylow

#1998

—

151

maximal, Sylow

C₂×#125

—

151

—

#423

—

151

—

C₄×M₁₆

D₁₅₁

yes

characteristic

#343

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

#125

—

302

—

#125

—

302

—

#125

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

302

—

#97

—

C₄×C₈

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

—

302

—

M₁₆

Dic₃₀₂

yes

—

C₂×C₈

—

C₂×C₈

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄²

C₂²⋊D₁₅₁

yes

characteristic

C₄²

C₂²⋊D₁₅₁

yes

characteristic

C₄²

D₆₀₄

yes

—

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

C₂×D₆₀₄

yes

characteristic

C₂×C₄

C₂×Dic₃₀₂

yes

—

C₂×C₄

#14559004

yes

—

C₈

—

C₈

—

604

—

C₂²

—

100

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

101

C₂²

C₄×Dic₃₀₂

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

604

—

105

C₄

#18012718

yes

—

106

C₄

#18012727

yes

characteristic

107

C₄

#18012739

yes

characteristic

108

C₂

—

109

C₂

#23752118

yes

characteristic

110

C₂

#23752194

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₅₁)⋊C₄
commutator, prime radical	C₂³×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₄×M₁₆×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₄×M₁₆	D₁₅₁
hypocenter	C₁₅₁	#1998
LCS	G ▷ C₂³×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

commutator, prime radical

C₂³×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₄×M₁₆×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₄×M₁₆

D₁₅₁

hypocenter

C₁₅₁

#1998

LCS

G ▷ C₂³×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₅₁	#1998	semidirect

H ≅ G/N

type

55. C₁₅₁

#1998

semidirect

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

subgroup	index
C₂×#18012731	2
#23752236	2
#23751980	2
C₄×M₁₆×C₁₅₁	2
#1998	151

subgroup

index

C₄×M₁₆×C₁₅₁

151

Subgroups — #34070103

Conjugacy classes of subgroups

Complements

Maximal subgroups