Group #34101443 — subgroups

Subgroups — #34101443

Order 20096.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#23758073	C₂	1	1	yes	maximal, characteristic
2	#23757301	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757267	C₂	1	2	yes	maximal, complemented
4	#23757246	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14578954	C₂²	1	2	yes	—
7	#19097804	—	2	2	no	—
8	#19097799	—	2	2	no	—
9	#19097793	—	2	2	no	—
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	#19097730	C₂²	1	1	yes	characteristic
12	#19097700	—	2	2	no	—
13	#19097694	—	2	2	no	—
14	#19097691	—	2	2	no	—
15	#19097685	—	2	4	no	—
16	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
17	#117×C₁₅₇	C₂²	1	2	yes	—
18	#14578954	—	2	2	no	—
19	#14578954	—	2	2	no	—
20	#14578954	—	2	2	no	—
21	#14578954	—	2	2	no	—
22	C₂×Dic₃₁₄	D₄	1	2	yes	—
23	C₂×D₆₂₈	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₅₇	—	2	2	no	—
25	C₂×C₄×D₁₅₇	—	2	2	no	—
26	Dic₆₂₈	—	2	2	no	—
27	#14578943	—	2	4	no	—
28	D₁₂₅₆	—	2	2	no	—
29	C₂×#11929434	—	2	2	no	—
30	C₂×#11929434	—	2	1	no	—
31	C₄⋊Dic₁₅₇	—	2	1	no	—
32	C₂²⋊Dic₁₅₇	D₄	1	2	yes	—
33	C₄×Dic₁₅₇	—	2	1	no	—
34	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₅₇	—	2	2	no	—
36	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₅₇	—	2	2	no	—
38	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
39	C₄²×C₁₅₇	—	2	2	no	—
40	C₂²×D₁₅₇	—	2	2	no	—
41	Dic₃₁₄	—	2	2	no	—
42	Dic₃₁₄	—	2	2	no	—
43	D₆₂₈	—	2	2	no	—
44	D₆₂₈	—	2	2	no	—
45	C₂²⋊D₁₅₇	—	4	2	no	—
46	C₂²⋊D₁₅₇	—	4	2	no	—
47	C₂×Dic₁₅₇	—	2	2	no	—
48	C₂×Dic₁₅₇	—	2	2	no	—
49	C₄×D₁₅₇	—	4	2	no	—
50	C₄×D₁₅₇	—	4	2	no	—
51	#11929434	—	4	1	no	—
52	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	—	4	2	no	—
54	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₇	#41	1	2	yes	—
57	C₂×C₄×C₁₅₇	#41	1	1	yes	characteristic
58	C₈×C₁₅₇	—	2	2	no	—
59	C₂²×C₁₅₇	—	4	1	no	—
60	C₂²×C₁₅₇	#121	1	2	yes	—
61	C₂²×C₁₅₇	#121	1	1	yes	characteristic
62	C₂×D₁₅₇	—	4	2	no	—
63	C₂×D₁₅₇	—	8	2	no	—
64	C₄×C₁₅₇	—	8	2	no	—
65	C₄×C₁₅₇	#120	1	1	yes	characteristic
66	C₄×C₁₅₇	#123	1	2	yes	—
67	C₄×C₁₅₇	#124	1	1	yes	characteristic
68	Dic₁₅₇	—	4	2	no	—
69	Dic₁₅₇	—	4	2	no	—
70	C₂×C₁₅₇	—	2	1	no	—
71	C₂×C₁₅₇	—	2	2	no	—
72	C₂×C₁₅₇	#394	1	1	yes	characteristic
73	D₁₅₇	—	8	2	no	—
74	C₁₅₇	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	157	1	no	maximal, Sylow
76	#575	—	157	1	no	—
77	#424	—	157	1	no	—
78	#416	—	157	2	no	—
79	#410	—	157	2	no	—
80	H(3,ℤ/4)	D₁₅₇	1	1	yes	characteristic, complemented
81	C₂×#41	—	157	2	no	—
82	#137	—	314	2	no	—
83	AGL(1,ℤ/8)	—	314	2	no	—
84	#135	—	314	2	no	—
85	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
86	C₂²×C₈	—	157	1	no	—
87	#117	—	157	1	no	—
88	#117	C₂×D₁₅₇	1	2	yes	—
89	C₄⋊C₈	—	314	2	no	—
90	#103	—	314	2	no	—
91	#102	—	314	2	no	—
92	#99	—	314	4	no	—
93	#41	—	314	2	no	—
94	#41	—	314	2	no	—
95	#41	—	314	2	no	—
96	#41	—	314	2	no	—
97	C₂×Q₈	—	157	2	no	—
98	C₂×D₄	—	314	2	no	—
99	C₂×D₄	—	157	2	no	—
100	C₂²×C₄	—	314	2	no	—
101	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
102	Q₁₆	—	314	2	no	—
103	SD₁₆	—	314	4	no	—
104	D₈	—	314	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	314	2	no	—
107	C₂×C₈	—	314	1	no	—
108	C₂×C₈	D₆₂₈	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	314	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	157	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	314	1	no	—
115	C₂³	—	314	2	no	—
116	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
117	Q₈	—	314	2	no	—
118	Q₈	—	314	2	no	—
119	D₄	—	314	2	no	—
120	D₄	—	314	2	no	—
121	D₄	—	628	2	no	—
122	D₄	—	628	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	314	2	no	—
125	C₂×C₄	—	314	2	no	—
126	C₂×C₄	—	628	2	no	—
127	C₂×C₄	—	628	2	no	—
128	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
129	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
130	C₂×C₄	#14578954	1	2	yes	—
131	C₂×C₄	#14578955	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	628	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	628	2	no	—
136	C₂²	—	1256	2	no	—
137	C₂²	#19097744	1	2	yes	—
138	C₂²	#19097746	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	628	2	no	—
141	C₄	—	628	2	no	—
142	C₄	#19097741	1	1	yes	characteristic
143	C₄	#19097756	1	2	yes	—
144	C₄	#19097759	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	1256	2	no	—
148	C₂	#23757191	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23758073

C₂

yes

maximal, characteristic

#23757301

C₂

yes

maximal, characteristic, complemented

#23757267

C₂

yes

maximal, complemented

#23757246

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578954

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#19097730

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₅₇

C₂²

yes

characteristic

#117×C₁₅₇

C₂²

yes

—

—

—

—

—

C₂×Dic₃₁₄

D₄

yes

—

C₂×D₆₂₈

D₄

yes

—

C₂×C₂²⋊D₁₅₇

—

C₂×C₄×D₁₅₇

—

Dic₆₂₈

—

#14578943

—

D₁₂₅₆

—

C₂×#11929434

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₂²×D₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#121

yes

—

C₂²×C₁₅₇

#121

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

#123

yes

—

C₄×C₁₅₇

#124

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#394

yes

characteristic

D₁₅₇

—

C₁₅₇

#2053

yes

characteristic, complemented, Sylow

#2053

—

157

maximal, Sylow

#575

—

157

—

#424

—

157

—

#416

—

157

—

#410

—

157

—

H(3,ℤ/4)

D₁₅₇

yes

characteristic, complemented

C₂×#41

—

157

—

#137

—

314

—

AGL(1,ℤ/8)

—

314

—

#135

—

314

—

C₂×M₁₆

C₂×D₁₅₇

yes

characteristic

C₂²×C₈

—

157

—

#117

—

157

—

#117

C₂×D₁₅₇

yes

—

C₄⋊C₈

—

314

—

#103

—

314

—

#102

—

314

—

#99

—

314

—

#41

—

314

—

#41

—

314

—

#41

—

314

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

100

C₂²×C₄

—

314

—

101

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

102

Q₁₆

—

314

—

103

SD₁₆

—

314

—

104

D₈

—

314

—

105

M₁₆

—

106

C₂×C₈

—

314

—

107

C₂×C₈

—

314

—

108

C₂×C₈

D₆₂₈

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

314

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

157

—

113

C₄²

—

114

C₄²

—

314

—

115

C₂³

—

314

—

116

C₂³

D₄×D₁₅₇

yes

characteristic

117

Q₈

—

314

—

118

Q₈

—

314

—

119

D₄

—

314

—

120

D₄

—

314

—

121

D₄

—

628

—

122

D₄

—

628

—

123

C₂×C₄

—

124

C₂×C₄

—

314

—

125

C₂×C₄

—

314

—

126

C₂×C₄

—

628

—

127

C₂×C₄

—

628

—

128

C₂×C₄

D₄×D₁₅₇

yes

—

129

C₂×C₄

C₂×D₆₂₈

yes

characteristic

130

C₂×C₄

#14578954

yes

—

131

C₂×C₄

#14578955

yes

characteristic

132

C₈

—

133

C₈

—

628

—

134

C₂²

—

135

C₂²

—

628

—

136

C₂²

—

1256

—

137

C₂²

#19097744

yes

—

138

C₂²

#19097746

yes

characteristic

139

C₄

—

140

C₄

—

628

—

141

C₄

—

628

—

142

C₄

#19097741

yes

characteristic

143

C₄

#19097756

yes

—

144

C₄

#19097759

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1256

—

148

C₂

#23757191

yes

characteristic

	subgroup	G/H
center	C₄	#19097759
commutator, cosocle	C₂²×C₄×C₁₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	H(3,ℤ/4)×C₁₅₇	C₂
socle	C₂×C₁₅₇	#394
prime radical	#23758073	C₂
hypercenter	H(3,ℤ/4)	D₁₅₇
hypocenter	C₁₅₇	#2053
LCS	G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#19097759

commutator, cosocle

C₂²×C₄×C₁₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

H(3,ℤ/4)×C₁₅₇

C₂

C₂×C₁₅₇

C₂

D₁₅₇

C₁₅₇

G ▷ C₂²×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #23757301	C₂	semidirect
3. #23757267	C₂	semidirect
4. #23757246	C₂	semidirect
5. H(3,ℤ/4)×C₁₅₇	C₂	semidirect
74. C₁₅₇	#2053	semidirect
80. H(3,ℤ/4)	D₁₅₇	semidirect

H ≅ G/N

type

2. #23757301

C₂

semidirect

3. #23757267

C₂

semidirect

4. #23757246

C₂

semidirect

5. H(3,ℤ/4)×C₁₅₇

C₂

semidirect

74. C₁₅₇

#2053

semidirect

80. H(3,ℤ/4)

D₁₅₇

semidirect

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

subgroup	index
#23758073	2
#23757301	2
#23757267	2
#23757246	2
H(3,ℤ/4)×C₁₅₇	2
#2053	157

subgroup

index

H(3,ℤ/4)×C₁₅₇

157

Subgroups — #34101443

Conjugacy classes of subgroups

Complements

Maximal subgroups