Group #34100302 — subgroups

Subgroups — #34100302

Order 20096.

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

normal subgroups only

This group has 256 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	C₂×#19097704	C₂	1	2	yes	maximal
2	C₂×#19097702	C₂	1	2	yes	maximal
3	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
4	#23757139	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
9	#19097704	—	2	4	no	—
10	#19097702	—	2	4	no	—
11	#19097682	C₄	1	4	yes	complemented
12	#19097678	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₅₇	—	2	1	no	—
17	C₂²×Dic₁₅₇	—	2	1	no	—
18	C₂×#11929434	C₂×C₄	1	4	yes	—
19	C₂×#11929434	D₄	1	4	yes	—
20	C₂×#11929434	Q₈	1	4	yes	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₄⋊Dic₁₅₇	—	2	2	no	—
23	C₂²×C₄×C₁₅₇	—	2	2	no	—
24	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄⋊C₄×C₁₅₇	—	2	2	no	—
28	C₄⋊C₄×C₁₅₇	—	4	4	no	—
29	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₅₇	Q₈	1	2	yes	—
31	C₂×Dic₁₅₇	—	4	1	no	—
32	C₂×Dic₁₅₇	—	4	1	no	—
33	C₂×Dic₁₅₇	—	4	2	no	—
34	C₂×Dic₁₅₇	—	4	2	no	—
35	#11929434	C₄⋊C₄	1	8	yes	complemented
36	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	—	2	2	no	—
38	C₂×C₄×C₁₅₇	—	2	2	no	—
39	C₂×C₄×C₁₅₇	—	4	2	no	—
40	C₂×C₄×C₁₅₇	—	4	4	no	—
41	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	#127	1	1	yes	characteristic
48	C₂²×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₅₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₅₇	#137	1	2	yes	—
51	C₄×C₁₅₇	—	2	4	no	—
52	C₄×C₁₅₇	—	8	2	no	—
53	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
55	Dic₁₅₇	—	8	1	no	—
56	Dic₁₅₇	—	8	1	no	—
57	C₂×C₁₅₇	#384	1	1	yes	characteristic
58	C₂×C₁₅₇	#428	1	2	yes	—
59	C₂×C₁₅₇	#496	1	1	yes	characteristic
60	C₂×C₁₅₇	#497	1	1	yes	characteristic
61	C₂×C₁₅₇	#501	1	2	yes	—
62	C₁₅₇	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	157	1	no	maximal, Sylow
64	C₂×#107	—	157	2	no	—
65	C₂×#106	—	157	2	no	—
66	C₂×#97	—	157	1	no	—
67	#384	D₁₅₇	1	1	yes	characteristic
68	#382	—	157	1	no	—
69	C₂²×C₈	—	157	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	157	1	no	—
72	C₂×C₄⋊C₄	—	157	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
74	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
75	#107	—	314	4	no	—
76	#106	—	314	4	no	—
77	#97	—	157	4	no	—
78	#95	—	314	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	314	1	no	—
81	C₂²×C₄	—	314	1	no	—
82	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
84	C₂×C₈	—	157	4	no	—
85	C₂×C₈	—	157	4	no	—
86	C₂×C₈	—	157	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	314	2	no	—
91	C₄⋊C₄	—	314	2	no	—
92	C₄²	C₄×D₁₅₇	1	2	yes	—
93	C₄²	Dic₃₁₄	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	628	1	no	—
100	C₂×C₄	—	628	1	no	—
101	C₂×C₄	—	628	2	no	—
102	C₂×C₄	—	628	2	no	—
103	C₂×C₄	#14578933	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
106	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
108	C₈	—	157	8	no	—
109	C₂²	C₂×#14578933	1	1	yes	characteristic
110	C₂²	#19097767	1	1	yes	characteristic
111	C₂²	#19097769	1	1	yes	characteristic
112	C₂²	#19097797	1	2	yes	—
113	C₂²	#19097802	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	1256	1	no	—
117	C₄	—	1256	1	no	—
118	C₄	C₄⋊C₄×D₁₅₇	1	2	yes	—
119	C₄	#19097770	1	2	yes	—
120	C₂	#23757142	1	1	yes	characteristic
121	C₂	#23757313	1	2	yes	—
122	C₂	#23757660	1	1	yes	characteristic
123	C₂	#23757665	1	1	yes	characteristic
124	C₂	#23757679	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097704

C₂

yes

maximal

C₂×#19097702

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

#23757139

C₂

yes

maximal, characteristic

#384×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

—

C₄

yes

complemented

#19097678

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

—

C₂×#11929434

C₂×C₄

yes

—

C₂×#11929434

D₄

yes

—

C₂×#11929434

Q₈

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

C₄⋊C₄

yes

complemented

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

#127

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₅₇

#137

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#384

yes

characteristic

C₂×C₁₅₇

#428

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₁₅₇

#1847

yes

characteristic, complemented, Sylow

#1847

—

157

maximal, Sylow

C₂×#107

—

157

—

C₂×#106

—

157

—

C₂×#97

—

157

—

#384

D₁₅₇

yes

characteristic

#382

—

157

—

C₂²×C₈

—

157

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#107

—

314

—

#106

—

314

—

#97

—

157

—

#95

—

314

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

100

C₂×C₄

—

628

—

101

C₂×C₄

—

628

—

102

C₂×C₄

—

628

—

103

C₂×C₄

#14578933

yes

—

104

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

105

C₂×C₄

D₄×D₁₅₇

yes

—

106

C₂×C₄

Q₈×D₁₅₇

yes

—

107

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

108

C₈

—

157

—

109

C₂²

C₂×#14578933

yes

characteristic

110

C₂²

#19097767

yes

characteristic

111

C₂²

#19097769

yes

characteristic

112

C₂²

#19097797

yes

—

113

C₂²

#19097802

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

1256

—

117

C₄

—

1256

—

118

C₄

C₄⋊C₄×D₁₅₇

yes

—

119

C₄

#19097770

yes

—

120

C₂

#23757142

yes

characteristic

121

C₂

#23757313

yes

—

122

C₂

#23757660

yes

characteristic

123

C₂

#23757665

yes

characteristic

124

C₂

#23757679

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#384×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#384	D₁₅₇
hypocenter	C₁₅₇	#1847
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#384×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #19097682	C₄	semidirect
35. #11929434	C₄⋊C₄	semidirect
62. C₁₅₇	#1847	semidirect

H ≅ G/N

type

11. #19097682

C₄

semidirect

35. #11929434

C₄⋊C₄

semidirect

62. C₁₅₇

#1847

semidirect

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

subgroup	index
C₂×#19097704	2
C₂×#19097702	2
C₂×#19097682	2
#23757139	2
#384×C₁₅₇	2
#1847	157

subgroup

index

#384×C₁₅₇

157

Subgroups — #34100302

Conjugacy classes of subgroups

Complements

Maximal subgroups