Group #34068255 — subgroups

Subgroups — #34068255

Order 19328.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23752591	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752502	C₂	1	2	yes	maximal, complemented
3	#23751961	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₅₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#555×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558994	—	2	2	no	—
7	#18012739	C₂²	1	1	yes	characteristic
8	#18012670	—	2	2	no	—
9	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
10	#18012660	—	2	2	no	—
11	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#124×C₁₅₁	—	2	2	no	—
14	#121×C₁₅₁	—	2	2	no	—
15	C₄×D₄×C₁₅₁	C₂²	1	2	yes	—
16	#117×C₁₅₁	—	2	2	no	—
17	C₂×Dic₃₀₂	—	2	1	no	—
18	#14558994	—	4	4	no	—
19	C₂×C₁₅₁⋊C₈	—	2	1	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
23	C₂×D₄×C₁₅₁	—	2	2	no	—
24	C₂×D₄×C₁₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
29	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄²×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
32	Dic₃₀₂	—	4	2	no	—
33	C₂×Dic₁₅₁	—	4	1	no	—
34	C₁₅₁⋊C₈	—	2	2	no	—
35	C₁₅₁⋊C₈	—	4	2	no	—
36	C₂³×C₁₅₁	—	2	2	no	—
37	Q₈×C₁₅₁	—	4	2	no	—
38	D₄×C₁₅₁	—	2	2	no	—
39	D₄×C₁₅₁	—	2	2	no	—
40	D₄×C₁₅₁	—	8	2	no	—
41	C₂×C₄×C₁₅₁	—	2	2	no	—
42	C₂×C₄×C₁₅₁	—	2	2	no	—
43	C₂×C₄×C₁₅₁	—	4	2	no	—
44	C₂×C₄×C₁₅₁	—	4	2	no	—
45	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
47	C₂²×C₁₅₁	—	4	2	no	—
48	C₂²×C₁₅₁	—	4	2	no	—
49	C₂²×C₁₅₁	—	4	2	no	—
50	C₂²×C₁₅₁	#120	1	1	yes	characteristic
51	C₄×C₁₅₁	—	2	1	no	—
52	C₄×C₁₅₁	—	4	2	no	—
53	C₄×C₁₅₁	—	4	2	no	—
54	C₄×C₁₅₁	#135	1	2	yes	—
55	C₄×C₁₅₁	#137	1	2	yes	—
56	Dic₁₅₁	—	8	1	no	—
57	C₂×C₁₅₁	—	8	2	no	—
58	C₂×C₁₅₁	#452	1	2	yes	—
59	C₂×C₁₅₁	#453	1	1	yes	characteristic
60	C₁₅₁	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	151	1	no	maximal, Sylow
62	#555	D₁₅₁	1	1	yes	characteristic
63	#494	—	151	1	no	—
64	#476	—	151	2	no	—
65	#333	—	151	1	no	—
66	D₄⋊C₈	—	151	2	no	—
67	C₂×Q₁₆	—	302	2	no	—
68	C₄⋊Q₈	—	151	1	no	—
69	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₅₁	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	302	2	no	—
75	C₄⋊C₈	—	151	2	no	—
76	#102	—	302	2	no	—
77	C₄×C₈	—	151	1	no	—
78	C₂×Q₈	—	302	1	no	—
79	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₅₁	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	604	4	no	—
84	C₂×C₈	—	302	1	no	—
85	C₂×C₈	—	302	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	302	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	604	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	604	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
106	C₈	—	302	2	no	—
107	C₈	—	604	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#18012708	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1208	1	no	—
116	C₄	#18012759	1	2	yes	—
117	C₄	#18012769	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#23752374	1	2	yes	—
120	C₂	#23752380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752591

C₂

yes

maximal, characteristic, complemented

#23752502

C₂

yes

maximal, complemented

#23751961

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, complemented

#555×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558994

—

#18012739

C₂²

yes

characteristic

#18012670

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012660

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#124×C₁₅₁

—

#121×C₁₅₁

—

C₄×D₄×C₁₅₁

C₂²

yes

—

#117×C₁₅₁

—

C₂×Dic₃₀₂

—

#14558994

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₂²⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#135

yes

—

C₄×C₁₅₁

#137

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#452

yes

—

C₂×C₁₅₁

#453

yes

characteristic

C₁₅₁

#1634

yes

characteristic, complemented, Sylow

#1634

—

151

maximal, Sylow

#555

D₁₅₁

yes

characteristic

#494

—

151

—

#476

—

151

—

#333

—

151

—

D₄⋊C₈

—

151

—

C₂×Q₁₆

—

302

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₅₁

yes

—

#117

—

#106

—

302

—

C₄⋊C₈

—

151

—

#102

—

302

—

C₄×C₈

—

151

—

C₂×Q₈

—

302

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

—

Q₁₆

—

604

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

604

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

604

—

104

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

106

C₈

—

302

—

107

C₈

—

604

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#18012708

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1208

—

116

C₄

#18012759

yes

—

117

C₄

#18012769

yes

—

118

C₂

—

119

C₂

#23752374

yes

—

120

C₂

#23752380

yes

characteristic

	subgroup	G/H
center	C₂²	#18012708
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting, prime radical	#555×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#120
hypercenter	#555	D₁₅₁
hypocenter	C₁₅₁	#1634
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#18012708

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting, prime radical

#555×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #23752591	C₂	semidirect
2. #23752502	C₂	semidirect
3. #23751961	C₂	semidirect
4. D₄⋊(C₁₅₁⋊C₈)	C₂	semidirect
60. C₁₅₁	#1634	semidirect

H ≅ G/N

type

1. #23752591

C₂

semidirect

2. #23752502

C₂

semidirect

3. #23751961

C₂

semidirect

4. D₄⋊(C₁₅₁⋊C₈)

C₂

semidirect

60. C₁₅₁

#1634

semidirect

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

subgroup	index
#23752591	2
#23752502	2
#23751961	2
D₄⋊(C₁₅₁⋊C₈)	2
#555×C₁₅₁	2
#1634	151

subgroup

index

#23752591

#23752502

#23751961

D₄⋊(C₁₅₁⋊C₈)

#555×C₁₅₁

#1634

151

Subgroups — #34068255

Conjugacy classes of subgroups

Complements

Maximal subgroups