Group #34068997 — subgroups

Subgroups — #34068997

Order 19328.

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

normal subgroups only

This group has 224 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	#23752250	C₂	1	1	yes	maximal, characteristic
2	#23752170	C₂	1	2	yes	maximal
3	C₂×#18012649	C₂	1	1	yes	maximal, characteristic
4	#23751973	C₂	1	2	yes	maximal
5	C₂×#108×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
7	#18012698	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012673	C₄	1	2	yes	complemented
10	#18012671	C₄	1	2	yes	complemented
11	#18012649	—	2	2	no	—
12	#18012649	C₄	1	4	yes	complemented
13	C₈×Dic₁₅₁	—	2	2	no	—
14	C₂×M₁₆×C₁₅₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
16	#108×C₁₅₁	C₄	1	4	yes	complemented
17	C₂²×Dic₁₅₁	—	2	1	no	—
18	C₂×C₁₅₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₁₅₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₁₅₁	C₂×C₄	1	4	yes	—
22	C₂²⋊Dic₁₅₁	—	2	2	no	—
23	C₄×Dic₁₅₁	—	2	2	no	—
24	C₄×Dic₁₅₁	—	2	1	no	—
25	C₄×Dic₁₅₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₁	—	2	2	no	—
28	M₁₆×C₁₅₁	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₁₅₁	—	2	2	no	—
30	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₂×Dic₁₅₁	—	2	4	no	—
36	C₂×Dic₁₅₁	—	4	2	no	—
37	C₁₅₁⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂³×C₁₅₁	#41	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
45	C₈×C₁₅₁	—	2	4	no	—
46	C₈×C₁₅₁	C₄²	1	2	yes	—
47	C₈×C₁₅₁	C₄²	1	2	yes	—
48	C₂²×C₁₅₁	—	2	2	no	—
49	C₂²×C₁₅₁	#117	1	1	yes	characteristic
50	C₂²×C₁₅₁	C₄×D₄	1	1	yes	characteristic
51	C₂²×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₁₅₁	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₅₁	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
56	Dic₁₅₁	—	4	2	no	—
57	Dic₁₅₁	—	4	4	no	—
58	C₂×C₁₅₁	—	2	1	no	—
59	C₂×C₁₅₁	—	2	1	no	—
60	C₂×C₁₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₅₁	#444	1	2	yes	—
62	C₁₅₁	#1773	1	1	yes	characteristic, complemented, Sylow
63	#1773	—	151	1	no	maximal, Sylow
64	C₂×#108	D₁₅₁	1	1	yes	characteristic
65	#427	—	151	1	no	—
66	#405	—	151	2	no	—
67	C₂×#97	—	151	1	no	—
68	#339	—	151	2	no	—
69	C₂×M₁₆	C₂×D₁₅₁	1	2	yes	—
70	C₂²×C₈	—	151	1	no	—
71	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
72	#117	—	151	2	no	—
73	C₂×C₄²	—	151	1	no	—
74	#108	Dic₁₅₁	1	4	yes	complemented
75	#107	—	151	2	no	—
76	#106	—	151	2	no	—
77	#97	—	302	2	no	—
78	#97	—	151	4	no	—
79	C₄×C₈	—	302	2	no	—
80	C₂²×C₄	—	302	1	no	—
81	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	C₂×Dic₁₅₁	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	151	2	no	—
86	C₂×C₈	—	151	2	no	—
87	C₂×C₈	—	151	2	no	—
88	C₂×C₈	C₄×D₁₅₁	1	2	yes	—
89	C₂×C₈	C₄×D₁₅₁	1	1	yes	characteristic
90	C₂×C₈	C₄×D₁₅₁	1	1	yes	characteristic
91	C₂×C₈	C₂×Dic₁₅₁	1	2	yes	—
92	C₄⋊C₄	—	151	4	no	—
93	C₂²⋊C₄	—	302	2	no	—
94	C₄²	—	302	2	no	—
95	C₄²	—	302	1	no	—
96	C₄²	—	151	2	no	—
97	C₂³	#14559005	1	1	yes	characteristic
98	C₂×C₄	—	302	2	no	—
99	C₂×C₄	—	302	4	no	—
100	C₂×C₄	—	604	2	no	—
101	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
103	C₂×C₄	C₂²×Dic₁₅₁	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
106	C₂×C₄	#14559003	1	1	yes	characteristic
107	C₈	—	2	4	no	—
108	C₈	—	151	4	no	—
109	C₈	C₄×Dic₁₅₁	1	2	yes	—
110	C₈	C₄×Dic₁₅₁	1	2	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	#18012697	1	1	yes	characteristic
113	C₂²	#18012701	1	1	yes	characteristic
114	C₂²	Q₈×Dic₁₅₁	1	1	yes	characteristic
115	C₄	—	604	2	no	—
116	C₄	—	604	4	no	—
117	C₄	C₂×C₄×Dic₁₅₁	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
119	C₄	D₄×Dic₁₅₁	1	1	yes	characteristic
120	C₄	#18012704	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	C₄⋊C₄×Dic₁₅₁	1	1	yes	characteristic
124	C₂	#23752333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752250

C₂

yes

maximal, characteristic

#23752170

C₂

yes

maximal

C₂×#18012649

C₂

yes

maximal, characteristic

#23751973

C₂

yes

maximal

C₂×#108×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012698

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012673

C₄

yes

complemented

#18012671

C₄

yes

complemented

#18012649

—

#18012649

C₄

yes

complemented

C₈×Dic₁₅₁

—

C₂×M₁₆×C₁₅₁

C₂²

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

#108×C₁₅₁

C₄

yes

complemented

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₅₁

C₂×C₄

yes

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

M₁₆×C₁₅₁

C₂×C₄

yes

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₅₁

#41

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

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

C₄²

yes

—

C₈×C₁₅₁

C₄²

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#117

yes

characteristic

C₂²×C₁₅₁

C₄×D₄

yes

characteristic

C₂²×C₁₅₁

C₄×Q₈

yes

characteristic

C₄×C₁₅₁

C₂×C₄²

yes

characteristic

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₁

C₄×D₄

yes

characteristic

C₄×C₁₅₁

C₄×Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₅₁

#444

yes

—

C₁₅₁

#1773

yes

characteristic, complemented, Sylow

#1773

—

151

maximal, Sylow

C₂×#108

D₁₅₁

yes

characteristic

#427

—

151

—

#405

—

151

—

C₂×#97

—

151

—

#339

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#117

—

151

—

C₂×C₄²

—

151

—

#108

Dic₁₅₁

yes

complemented

#107

—

151

—

#106

—

151

—

#97

—

302

—

#97

—

151

—

C₄×C₈

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₁₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₄×D₁₅₁

yes

—

C₂×C₈

C₄×D₁₅₁

yes

characteristic

C₂×C₈

C₄×D₁₅₁

yes

characteristic

C₂×C₈

C₂×Dic₁₅₁

yes

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

302

—

C₄²

—

302

—

C₄²

—

302

—

C₄²

—

151

—

C₂³

#14559005

yes

characteristic

C₂×C₄

—

302

—

C₂×C₄

—

302

—

100

C₂×C₄

—

604

—

101

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

102

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

103

C₂×C₄

C₂²×Dic₁₅₁

yes

characteristic

104

C₂×C₄

D₄×D₁₅₁

yes

characteristic

105

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

106

C₂×C₄

#14559003

yes

characteristic

107

C₈

—

108

C₈

—

151

—

109

C₈

C₄×Dic₁₅₁

yes

—

110

C₈

C₄×Dic₁₅₁

yes

—

111

C₂²

—

112

C₂²

#18012697

yes

characteristic

113

C₂²

#18012701

yes

characteristic

114

C₂²

Q₈×Dic₁₅₁

yes

characteristic

115

C₄

—

604

—

116

C₄

—

604

—

117

C₄

C₂×C₄×Dic₁₅₁

yes

characteristic

118

C₄

C₄⋊C₄×D₁₅₁

yes

characteristic

119

C₄

D₄×Dic₁₅₁

yes

characteristic

120

C₄

#18012704

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₄⋊C₄×Dic₁₅₁

yes

characteristic

124

C₂

#23752333

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅₁
commutator	C₄×C₁₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×#108×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₄×D₄
prime radical	C₂³×C₁₅₁	#41
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×#108	D₁₅₁
hypocenter	C₁₅₁	#1773
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₅₁

commutator

C₄×C₁₅₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×#108×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₄×D₄

prime radical

C₂³×C₁₅₁

#41

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #18012673	C₄	semidirect
10. #18012671	C₄	semidirect
12. #18012649	C₄	semidirect
16. #108×C₁₅₁	C₄	semidirect
62. C₁₅₁	#1773	semidirect
74. #108	Dic₁₅₁	semidirect

H ≅ G/N

type

9. #18012673

C₄

semidirect

10. #18012671

C₄

semidirect

12. #18012649

C₄

semidirect

16. #108×C₁₅₁

C₄

semidirect

62. C₁₅₁

#1773

semidirect

74. #108

Dic₁₅₁

semidirect

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

subgroup	index
#23752250	2
#23752170	2
C₂×#18012649	2
#23751973	2
C₂×#108×C₁₅₁	2
#1773	151

subgroup

index

C₂×#108×C₁₅₁

151

Subgroups — #34068997

Conjugacy classes of subgroups

Complements

Maximal subgroups