Group #34100421 — subgroups

Subgroups — #34100421

Order 20096.

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

normal subgroups only

This group has 260 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	#23758040	C₂	1	1	yes	maximal, characteristic
2	#23757299	C₂	1	2	yes	maximal, complemented
3	#23757247	C₂	1	1	yes	maximal, characteristic
4	#23757045	C₂	1	2	yes	maximal, complemented
5	#408×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	#19097778	—	2	4	no	—
7	C₂×#14578938	—	2	2	no	—
8	C₂×#14578938	C₂²	1	2	yes	—
9	#19097730	C₂²	1	2	yes	—
10	#19097700	—	2	4	no	—
11	#19097686	C₄	1	4	yes	—
12	C₂×#41×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₁₅₇	—	2	2	no	—
16	#14578938	—	2	4	no	—
17	#14578938	—	2	2	no	—
18	#14578938	—	2	2	no	—
19	C₂×#11929434	—	2	4	no	—
20	C₂×#11929434	D₄	1	4	yes	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₂²⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₅₇	—	2	2	no	—
24	#41×C₁₅₇	—	2	2	no	—
25	#41×C₁₅₇	—	2	2	no	—
26	C₂×Q₈×C₁₅₇	Q₈	1	1	yes	characteristic
27	C₂×D₄×C₁₅₇	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₅₇	Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₅₇	—	2	2	no	—
32	C₂×C₈×C₁₅₇	—	2	1	no	—
33	C₂×C₈×C₁₅₇	—	2	1	no	—
34	C₂×C₈×C₁₅₇	—	2	2	no	—
35	C₂×Dic₁₅₇	—	2	4	no	—
36	#11929434	—	2	4	no	—
37	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
38	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₁₅₇	—	2	2	no	—
40	D₄×C₁₅₇	—	2	4	no	—
41	D₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	—	2	2	no	—
44	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₅₇	#41	1	2	yes	—
49	C₈×C₁₅₇	—	4	1	no	—
50	C₈×C₁₅₇	—	4	1	no	—
51	C₂²×C₁₅₇	—	2	2	no	—
52	C₂²×C₁₅₇	—	4	1	no	—
53	C₂²×C₁₅₇	—	4	2	no	—
54	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
56	C₄×C₁₅₇	—	2	2	no	—
57	C₄×C₁₅₇	#120	1	1	yes	characteristic
58	C₄×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₅₇	#123	1	1	yes	characteristic
60	Dic₁₅₇	—	4	4	no	—
61	C₂×C₁₅₇	—	2	2	no	—
62	C₂×C₁₅₇	—	2	1	no	—
63	C₂×C₁₅₇	—	4	2	no	—
64	C₂×C₁₅₇	#385	1	1	yes	characteristic
65	C₁₅₇	#1868	1	1	yes	characteristic, complemented, Sylow
66	#1868	—	157	1	no	maximal, Sylow
67	#568	—	157	1	no	—
68	#424	—	157	2	no	—
69	#410	—	157	1	no	—
70	#408	D₁₅₇	1	1	yes	characteristic
71	#344	—	157	2	no	—
72	C₂×#41	C₂×D₁₅₇	1	1	yes	characteristic
73	#131	—	314	4	no	—
74	C₂×M₁₆	—	314	2	no	—
75	C₂×M₁₆	—	157	2	no	—
76	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
78	#117	—	157	2	no	—
79	C₄⋊C₈	—	314	4	no	—
80	#99	—	157	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₃₁₄	1	1	yes	characteristic
85	C₂×D₄	C₄×D₁₅₇	1	2	yes	—
86	C₂×D₄	Dic₃₁₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	314	4	no	—
91	M₁₆	—	314	2	no	—
92	M₁₆	—	314	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	314	4	no	—
97	C₂×C₈	—	157	4	no	—
98	C₄⋊C₄	—	314	2	no	—
99	C₂²⋊C₄	—	157	4	no	—
100	C₄²	—	314	2	no	—
101	C₂³	#14578933	1	2	yes	—
102	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	314	4	no	—
109	C₂×C₄	#14578933	1	2	yes	—
110	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
113	C₂×C₄	#14578953	1	2	yes	—
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	314	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#14578933	1	1	yes	characteristic
121	C₂²	#19097731	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	628	4	no	—
124	C₄	#19097739	1	1	yes	characteristic
125	C₄	#19097748	1	2	yes	—
126	C₄	#19097752	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#23757145	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758040

C₂

yes

maximal, characteristic

#23757299

C₂

yes

maximal, complemented

#23757247

C₂

yes

maximal, characteristic

#23757045

C₂

yes

maximal, complemented

#408×C₁₅₇

C₂

yes

maximal, characteristic

#19097778

—

C₂×#14578938

—

C₂×#14578938

C₂²

yes

—

#19097730

C₂²

yes

—

#19097700

—

#19097686

C₄

yes

—

C₂×#41×C₁₅₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂²⋊C₈×C₁₅₇

—

—

—

—

—

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

C₂×C₄

yes

—

C₄×Dic₁₅₇

—

#41×C₁₅₇

—

#41×C₁₅₇

—

C₂×Q₈×C₁₅₇

Q₈

yes

characteristic

C₂×D₄×C₁₅₇

C₂×C₄

yes

—

C₂×D₄×C₁₅₇

Q₈

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₄⋊C₄

yes

—

C₂³×C₁₅₇

C₂²×C₄

yes

characteristic

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

#123

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#385

yes

characteristic

C₁₅₇

#1868

yes

characteristic, complemented, Sylow

#1868

—

157

maximal, Sylow

#568

—

157

—

#424

—

157

—

#410

—

157

—

#408

D₁₅₇

yes

characteristic

#344

—

157

—

C₂×#41

C₂×D₁₅₇

yes

characteristic

#131

—

314

—

C₂×M₁₆

—

314

—

C₂×M₁₆

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#117

—

157

—

C₄⋊C₈

—

314

—

#99

—

157

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₃₁₄

yes

characteristic

C₂×D₄

C₄×D₁₅₇

yes

—

C₂×D₄

Dic₃₁₄

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

M₁₆

—

314

—

M₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

314

—

C₂²⋊C₄

—

157

—

100

C₄²

—

314

—

101

C₂³

#14578933

yes

—

102

C₂³

C₂×C₄×D₁₅₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

314

—

109

C₂×C₄

#14578933

yes

—

110

C₂×C₄

D₄×D₁₅₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

112

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

113

C₂×C₄

#14578953

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

314

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14578933

yes

characteristic

121

C₂²

#19097731

yes

—

122

C₄

—

123

C₄

—

628

—

124

C₄

#19097739

yes

characteristic

125

C₄

#19097748

yes

—

126

C₄

#19097752

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23757145

yes

characteristic

	subgroup	G/H
center	C₄	#19097739
commutator	C₂³×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#408×C₁₅₇	C₂
socle	C₂×C₁₅₇	#385
prime radical	C₂×#41×C₁₅₇	C₂²
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#408	D₁₅₇
hypocenter	C₁₅₇	#1868
LCS	G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#19097739

commutator

C₂³×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#408×C₁₅₇

C₂

socle

C₂×C₁₅₇

#385

prime radical

C₂×#41×C₁₅₇

C₂²

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #23757299	C₂	semidirect
4. #23757045	C₂	semidirect
65. C₁₅₇	#1868	semidirect

H ≅ G/N

type

2. #23757299

C₂

semidirect

4. #23757045

C₂

semidirect

65. C₁₅₇

#1868

semidirect

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

subgroup	index
#23758040	2
#23757299	2
#23757247	2
#23757045	2
#408×C₁₅₇	2
#1868	157

subgroup

index

#408×C₁₅₇

157

Subgroups — #34100421

Conjugacy classes of subgroups

Complements

Maximal subgroups