Group #31918247 — subgroups

Subgroups — #31918247

Order 18176.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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×#17985839	C₂	1	2	yes	maximal
2	#22590593	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#22586520	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	—	2	1	no	—
7	C₂×Dic₅₆₈	C₂²	1	2	yes	—
8	#17986379	—	2	1	no	—
9	#17985839	—	2	4	no	—
10	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₁	—	2	1	no	—
13	#366×C₇₁	C₂²	1	1	yes	characteristic
14	#358×C₇₁	—	2	1	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₈₄	—	2	2	no	—
17	C₂×Dic₂₈₄	—	4	1	no	—
18	#14556128	—	4	1	no	—
19	Dic₅₆₈	—	2	2	no	—
20	Dic₅₆₈	—	2	2	no	—
21	Dic₅₆₈	—	4	2	no	—
22	C₇₁⋊C₃₂	—	2	2	no	—
23	C₂×Q₁₆×C₇₁	—	2	1	no	—
24	C₄×Q₈×C₇₁	—	4	1	no	—
25	Q₃₂×C₇₁	—	2	1	no	—
26	Q₃₂×C₇₁	D₄	1	2	yes	—
27	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
29	#107×C₇₁	—	2	1	no	—
30	#103×C₇₁	—	4	1	no	—
31	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂×Dic₁₄₂	—	4	2	no	—
33	Dic₂₈₄	—	4	2	no	—
34	Dic₂₈₄	—	4	2	no	—
35	Dic₂₈₄	—	8	2	no	—
36	C₄⋊Dic₇₁	—	8	1	no	—
37	C₄⋊Dic₇₁	—	8	1	no	—
38	C₂×Q₈×C₇₁	—	4	1	no	—
39	Q₁₆×C₇₁	—	2	2	no	—
40	Q₁₆×C₇₁	—	4	1	no	—
41	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₇₁	—	4	1	no	—
44	C₄⋊C₄×C₇₁	—	8	1	no	—
45	C₄²×C₇₁	—	8	1	no	—
46	C₄²×C₇₁	D₈	1	1	yes	characteristic
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₁	#41	1	1	yes	characteristic
50	Q₈×C₇₁	—	4	2	no	—
51	Q₈×C₇₁	—	8	1	no	—
52	C₂×C₄×C₇₁	—	8	1	no	—
53	C₂×C₄×C₇₁	—	8	1	no	—
54	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₇₁	—	8	2	no	—
58	Dic₁₄₂	—	8	2	no	—
59	Dic₁₄₂	—	8	2	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	#121	1	1	yes	characteristic
62	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	8	2	no	—
66	C₄×C₇₁	—	16	1	no	—
67	C₄×C₇₁	Q₆₄	1	2	yes	complemented
68	C₄×C₇₁	#459	1	1	yes	characteristic
69	C₄×C₇₁	#509	1	1	yes	characteristic
70	Dic₇₁	—	16	2	no	—
71	#13688	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	#2206	1	1	yes	characteristic
73	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
74	C₂×C₇₁	#2264	1	1	yes	characteristic
75	C₂×Q₆₄	—	71	2	no	—
76	#2247	—	71	1	no	—
77	C₄×Q₃₂	D₇₁	1	1	yes	characteristic
78	C₄⋊C₃₂	—	71	1	no	—
79	#1416	—	71	2	no	—
80	C₇₁	#13688	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₃₂	—	71	2	no	—
82	C₂×Q₃₂	—	142	1	no	—
83	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
84	#494	—	142	1	no	—
85	C₄×Q₁₆	—	2	1	no	—
86	Q₆₄	—	142	4	no	—
87	C₂×C₃₂	—	71	2	no	—
88	#366	C₂×D₇₁	1	1	yes	characteristic
89	#358	—	2	1	no	—
90	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	284	1	no	—
93	C₂×Q₁₆	—	142	2	no	—
94	C₄⋊Q₈	—	284	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	Q₃₂	—	2	1	no	—
97	Q₃₂	—	284	2	no	—
98	Q₃₂	—	142	2	no	—
99	Q₃₂	—	142	2	no	—
100	Q₃₂	D₂₈₄	1	2	yes	—
101	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#103	—	4	1	no	—
105	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
106	C₃₂	—	142	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	284	2	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	4	1	no	—
111	Q₁₆	—	284	2	no	—
112	Q₁₆	—	284	2	no	—
113	Q₁₆	—	568	2	no	—
114	C₂×C₈	#11765850	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	568	1	no	—
119	C₄⋊C₄	—	568	1	no	—
120	C₄²	—	8	1	no	—
121	C₄²	#11765850	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
124	C₁₆	#11765865	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	Q₈	—	568	2	no	—
128	Q₈	—	568	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	568	2	no	—
132	C₂×C₄	#14556069	1	1	yes	characteristic
133	C₂×C₄	#14556069	1	1	yes	characteristic
134	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#14556102	1	1	yes	characteristic
137	C₈	#14556153	1	1	yes	characteristic
138	C₂²	C₂×#14556069	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	1136	2	no	—
143	C₄	#17985839	1	2	yes	complemented
144	C₄	#17986194	1	1	yes	characteristic
145	C₄	#17986434	1	1	yes	characteristic
146	C₂	#22590375	1	1	yes	characteristic
147	C₂	C₂×#17985839	1	1	yes	characteristic
148	C₂	#22590657	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985839

C₂

yes

maximal

#22590593

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic

#22586520

C₂

yes

maximal

C₄×Q₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

C₂²

yes

—

#17986379

—

#17985839

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₇₁

C₂²

yes

characteristic

C₄×Q₁₆×C₇₁

—

#366×C₇₁

C₂²

yes

characteristic

#358×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

C₇₁⋊C₃₂

—

C₂×Q₁₆×C₇₁

—

C₄×Q₈×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#103×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

Q₆₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₂×Q₆₄

yes

characteristic

C₂×C₇₁

#2264

yes

characteristic

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

D₇₁

yes

characteristic

C₄⋊C₃₂

—

#1416

—

C₇₁

#13688

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

C₂×Q₃₂

C₂×D₇₁

yes

characteristic

#494

—

142

—

C₄×Q₁₆

—

Q₆₄

—

142

—

C₂×C₃₂

—

#366

C₂×D₇₁

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

284

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

284

—

Q₃₂

—

142

—

Q₃₂

—

142

—

100

Q₃₂

D₂₈₄

yes

—

101

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

102

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

103

#107

—

104

#103

—

105

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

106

C₃₂

—

142

—

107

C₂×Q₈

—

108

C₂×Q₈

—

284

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

284

—

112

Q₁₆

—

284

—

113

Q₁₆

—

568

—

114

C₂×C₈

#11765850

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

568

—

119

C₄⋊C₄

—

568

—

120

C₄²

—

121

C₄²

#11765850

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂×D₂₈₄

yes

characteristic

124

C₁₆

#11765865

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

568

—

128

Q₈

—

568

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

568

—

132

C₂×C₄

#14556069

yes

characteristic

133

C₂×C₄

#14556069

yes

characteristic

134

C₂×C₄

C₂×#11765850

yes

characteristic

135

C₈

—

136

C₈

#14556102

yes

characteristic

137

C₈

#14556153

yes

characteristic

138

C₂²

C₂×#14556069

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

1136

—

143

C₄

#17985839

yes

complemented

144

C₄

#17986194

yes

characteristic

145

C₄

#17986434

yes

characteristic

146

C₂

#22590375

yes

characteristic

147

C₂

C₂×#17985839

yes

characteristic

148

C₂

#22590657

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556069
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₄×Q₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₇₁
hypocenter	C₇₁	#13688
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14556069

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₄×Q₃₂×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₇₁

hypocenter

C₇₁

#13688

LCS

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
67. C₄×C₇₁	Q₆₄	semidirect
80. C₇₁	#13688	semidirect
143. C₄	#17985839	semidirect

H ≅ G/N

type

67. C₄×C₇₁

Q₆₄

semidirect

80. C₇₁

#13688

semidirect

143. C₄

#17985839

semidirect

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

subgroup	index
C₂×#17985839	2
#22590593	2
C₄⋊(C₇₁⋊C₃₂)	2
#22586520	2
C₄×Q₃₂×C₇₁	2
#13688	71

subgroup

index

C₄⋊(C₇₁⋊C₃₂)

C₄×Q₃₂×C₇₁

Subgroups — #31918247

Conjugacy classes of subgroups

Complements

Maximal subgroups