Group #34097487 — subgroups

Subgroups — C₂×#487×C₁₅₇ #34097487

Order 20096.

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 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×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#487×C₁₅₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#103×C₁₅₇	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₅₇	—	2	1	no	—
7	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
11	C₄⋊Q₈×C₁₅₇	C₂²	1	4	yes	—
12	C₄⋊Q₈×C₁₅₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
18	#103×C₁₅₇	C₂²	1	16	yes	—
19	C₄×C₈×C₁₅₇	C₂²	1	4	yes	—
20	C₂×Q₈×C₁₅₇	—	2	2	no	—
21	C₂×Q₈×C₁₅₇	—	2	4	no	—
22	C₂×Q₈×C₁₅₇	—	2	8	no	—
23	C₂×Q₈×C₁₅₇	—	4	4	no	—
24	C₂×Q₈×C₁₅₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₅₇	—	2	2	no	—
26	C₂²×C₄×C₁₅₇	—	2	2	no	—
27	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₅₇	—	2	4	no	—
31	C₂×C₈×C₁₅₇	C₂³	1	4	yes	—
32	C₄⋊C₄×C₁₅₇	—	2	4	no	—
33	C₄⋊C₄×C₁₅₇	—	2	4	no	—
34	C₄⋊C₄×C₁₅₇	—	2	4	no	—
35	C₄⋊C₄×C₁₅₇	—	2	2	no	—
36	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
37	C₄²×C₁₅₇	D₄	1	2	yes	—
38	C₄²×C₁₅₇	C₂³	1	2	yes	—
39	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₅₇	—	2	8	no	—
41	Q₈×C₁₅₇	—	4	4	no	—
42	Q₈×C₁₅₇	—	4	8	no	—
43	C₂×C₄×C₁₅₇	—	2	4	no	—
44	C₂×C₄×C₁₅₇	—	2	4	no	—
45	C₂×C₄×C₁₅₇	—	4	4	no	—
46	C₂×C₄×C₁₅₇	—	4	4	no	—
47	C₂×C₄×C₁₅₇	SD₁₆	1	4	yes	—
48	C₂×C₄×C₁₅₇	Q₁₆	1	4	yes	—
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₅₇	#41	1	4	yes	—
53	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₅₇	—	2	4	no	—
55	C₂²×C₁₅₇	#124	1	2	yes	—
56	C₂²×C₁₅₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₅₇	—	4	4	no	—
60	C₄×C₁₅₇	—	4	4	no	—
61	C₄×C₁₅₇	#124	1	2	yes	—
62	C₄×C₁₅₇	C₂×SD₁₆	1	4	yes	—
63	C₄×C₁₅₇	C₂×Q₁₆	1	4	yes	—
64	C₄×C₁₅₇	C₂×#41	1	2	yes	—
65	C₂×C₁₅₇	#487	1	4	yes	direct product factor
66	C₂×C₁₅₇	C₂×#124	1	1	yes	characteristic
67	C₂×C₁₅₇	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₁₅₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₁₅₇	C₂×#487	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×#487	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₄⋊Q₈	C₂×C₁₅₇	1	1	yes	characteristic
72	C₂×C₄⋊Q₈	C₂×C₁₅₇	1	1	yes	characteristic
73	#487	C₂×C₁₅₇	1	8	yes	direct product factor
74	C₂×#103	C₂×C₁₅₇	1	4	yes	—
75	C₂×C₄×C₈	C₂×C₁₅₇	1	1	yes	characteristic
76	C₂²×Q₈	—	2	1	no	—
77	C₂²×Q₈	C₂²×C₁₅₇	1	2	yes	—
78	C₂²×C₈	C₂²×C₁₅₇	1	2	yes	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	C₂²×C₁₅₇	1	4	yes	—
82	C₄⋊Q₈	C₂²×C₁₅₇	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	C₂²×C₁₅₇	1	2	yes	—
87	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
88	#103	C₂²×C₁₅₇	1	16	yes	—
89	C₄×C₈	C₂²×C₁₅₇	1	4	yes	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	2	8	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂×Q₈	C₂³×C₁₅₇	1	4	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	D₄×C₁₅₇	1	1	yes	characteristic
98	C₂²×C₄	D₄×C₁₅₇	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂³×C₁₅₇	1	4	yes	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂³×C₁₅₇	1	4	yes	—
107	C₄²	D₄×C₁₅₇	1	2	yes	—
108	C₄²	C₂³×C₁₅₇	1	2	yes	—
109	C₂³	C₂×D₄×C₁₅₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	4	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	SD₁₆×C₁₅₇	1	4	yes	—
118	C₂×C₄	Q₁₆×C₁₅₇	1	4	yes	—
119	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
120	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
121	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
122	C₂×C₄	#41×C₁₅₇	1	4	yes	—
123	C₂×C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	#124×C₁₅₇	1	2	yes	—
126	C₂²	C₂×SD₁₆×C₁₅₇	1	2	yes	—
127	C₂²	C₂×Q₁₆×C₁₅₇	1	2	yes	—
128	C₂²	C₂²×D₄×C₁₅₇	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#124×C₁₅₇	1	2	yes	—
132	C₄	C₂×SD₁₆×C₁₅₇	1	4	yes	—
133	C₄	C₂×Q₁₆×C₁₅₇	1	4	yes	—
134	C₄	C₂×#41×C₁₅₇	1	2	yes	—
135	C₂	#487×C₁₅₇	1	4	yes	direct product factor
136	C₂	C₂×#124×C₁₅₇	1	1	yes	characteristic
137	C₂	C₂²×SD₁₆×C₁₅₇	1	1	yes	characteristic
138	C₂	C₂²×Q₁₆×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

#487×C₁₅₇

C₂

yes

maximal, direct product factor

C₂×#103×C₁₅₇

C₂

yes

maximal

C₂×C₄×C₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₅₇

—

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

#103×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

#124

yes

—

C₂²×C₁₅₇

C₂×SD₁₆

yes

—

C₂²×C₁₅₇

C₂×Q₁₆

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#124

yes

—

C₄×C₁₅₇

C₂×SD₁₆

yes

—

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

C₂×#41

yes

—

C₂×C₁₅₇

#487

yes

direct product factor

C₂×C₁₅₇

C₂×#124

yes

characteristic

C₂×C₁₅₇

C₂²×SD₁₆

yes

characteristic

C₂×C₁₅₇

C₂²×Q₁₆

yes

characteristic

C₁₅₇

C₂×#487

yes

characteristic, direct product factor, Sylow

C₂×#487

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₅₇

yes

characteristic

C₂×C₄⋊Q₈

C₂×C₁₅₇

yes

characteristic

#487

C₂×C₁₅₇

yes

direct product factor

C₂×#103

C₂×C₁₅₇

yes

—

C₂×C₄×C₈

C₂×C₁₅₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂²×C₁₅₇

yes

—

C₂²×C₈

C₂²×C₁₅₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₅₇

yes

—

C₄⋊Q₈

C₂²×C₁₅₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₅₇

yes

—

C₂×C₄²

C₂²×C₁₅₇

yes

characteristic

#103

C₂²×C₁₅₇

yes

—

C₄×C₈

C₂²×C₁₅₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₅₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₅₇

yes

characteristic

C₂²×C₄

D₄×C₁₅₇

yes

characteristic

C₂²×C₄

C₂³×C₁₅₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂³×C₁₅₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂³×C₁₅₇

yes

—

107

C₄²

D₄×C₁₅₇

yes

—

108

C₄²

C₂³×C₁₅₇

yes

—

109

C₂³

C₂×D₄×C₁₅₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

SD₁₆×C₁₅₇

yes

—

118

C₂×C₄

Q₁₆×C₁₅₇

yes

—

119

C₂×C₄

C₂×D₄×C₁₅₇

yes

characteristic

120

C₂×C₄

C₂×D₄×C₁₅₇

yes

—

121

C₂×C₄

C₂×D₄×C₁₅₇

yes

—

122

C₂×C₄

#41×C₁₅₇

yes

—

123

C₂×C₄

C₂⁴×C₁₅₇

yes

characteristic

124

C₈

—

125

C₂²

#124×C₁₅₇

yes

—

126

C₂²

C₂×SD₁₆×C₁₅₇

yes

—

127

C₂²

C₂×Q₁₆×C₁₅₇

yes

—

128

C₂²

C₂²×D₄×C₁₅₇

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

#124×C₁₅₇

yes

—

132

C₄

C₂×SD₁₆×C₁₅₇

yes

—

133

C₄

C₂×Q₁₆×C₁₅₇

yes

—

134

C₄

C₂×#41×C₁₅₇

yes

—

135

C₂

#487×C₁₅₇

yes

direct product factor

136

C₂

C₂×#124×C₁₅₇

yes

characteristic

137

C₂

C₂²×SD₁₆×C₁₅₇

yes

characteristic

138

C₂

C₂²×Q₁₆×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₅₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₅₇ ◁ C₂×C₄²×C₁₅₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₅₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₅₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₅₇ ◁ C₂×C₄²×C₁₅₇ ◁ G

N	H ≅ G/N	type
3. #487×C₁₅₇	C₂	direct (N × H)
65. C₂×C₁₅₇	#487	direct (N × H)
69. C₁₅₇	C₂×#487	direct (N × H)
70. C₂×#487	C₁₅₇	direct (N × H)
73. #487	C₂×C₁₅₇	direct (N × H)
135. C₂	#487×C₁₅₇	direct (N × H)

H ≅ G/N

type

3. #487×C₁₅₇

C₂

direct (N × H)

65. C₂×C₁₅₇

#487

direct (N × H)

69. C₁₅₇

C₂×#487

direct (N × H)

70. C₂×#487

C₁₅₇

direct (N × H)

73. #487

C₂×C₁₅₇

direct (N × H)

135. C₂

#487×C₁₅₇

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₁₅₇	2
#487×C₁₅₇	2
C₂×#103×C₁₅₇	2
C₂×C₄×C₈×C₁₅₇	2
C₂×#487	157

subgroup

index

C₂×C₄⋊Q₈×C₁₅₇

#487×C₁₅₇

C₂×#103×C₁₅₇

C₂×C₄×C₈×C₁₅₇

C₂×#487

157

Subgroups — C₂×#487×C₁₅₇ #34097487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#487×C157 #34097487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#487×C₁₅₇ #34097487