Group #31888622 — subgroups

Subgroups — #31888622

Order 18176.

366 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 179 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	#22587088	C₂	1	2	yes	maximal
2	#22586632	C₂	1	2	yes	maximal
3	#22586173	C₂	1	2	yes	maximal
4	C₄×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985752	C₄	1	4	yes	complemented
7	#17985750	C₄	1	4	yes	complemented
8	#17985721	—	2	4	no	—
9	C₂×#97×C₇₁	C₄	1	2	yes	—
10	C₂×#97×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄³×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	8	yes	—
15	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₄×C₇₁⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	—	2	2	no	—
21	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
23	#97×C₇₁	—	2	4	no	—
24	#97×C₇₁	—	2	4	no	—
25	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	Q₈	1	2	yes	—
29	C₂×C₇₁⋊C₈	—	2	8	no	—
30	C₂²×C₄×C₇₁	—	2	2	no	—
31	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	#41	1	2	yes	—
33	C₂×C₈×C₇₁	—	2	2	no	—
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₄²	1	2	yes	—
39	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
41	C₄²×C₇₁	—	2	2	no	—
42	C₄²×C₇₁	—	4	4	no	—
43	C₄²×C₇₁	M₁₆	1	4	yes	—
44	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
45	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₂³×C₇₁	#117	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	4	no	—
49	C₂×C₄×C₇₁	—	2	4	no	—
50	C₂×C₄×C₇₁	—	4	4	no	—
51	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	—	2	1	no	—
59	C₈×C₇₁	—	2	4	no	—
60	C₈×C₇₁	#97	1	4	yes	—
61	C₇₁⋊C₈	—	4	8	no	—
62	C₂²×C₇₁	—	2	2	no	—
63	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₇₁	#432	1	2	yes	—
65	C₄×C₇₁	—	2	2	no	—
66	C₄×C₇₁	—	4	4	no	—
67	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
68	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
69	C₄×C₇₁	#423	1	2	yes	—
70	C₄×C₇₁	#444	1	4	yes	—
71	#8269	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	#1767	1	1	yes	characteristic
74	C₂×C₇₁	#1773	1	2	yes	—
75	C₄×#97	D₇₁	1	1	yes	characteristic
76	#1564	—	71	2	no	—
77	#1452	—	71	2	no	—
78	#1276	—	71	2	no	—
79	C₇₁	#8269	1	1	yes	characteristic, complemented, Sylow
80	#431	—	71	4	no	—
81	C₂×#97	Dic₇₁	1	2	yes	—
82	C₂×#97	Dic₇₁	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
85	C₄³	C₂×D₇₁	1	1	yes	characteristic
86	#335	—	71	4	no	—
87	#334	—	71	4	no	—
88	#322	—	142	4	no	—
89	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
91	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
94	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
95	C₄⋊C₈	—	71	8	no	—
96	C₂²⋊C₈	—	142	4	no	—
97	#97	—	2	4	no	—
98	#97	—	2	4	no	—
99	C₄×C₈	—	142	4	no	—
100	C₄×C₈	Dic₁₄₂	1	2	yes	—
101	C₄×C₈	C₄×D₇₁	1	2	yes	—
102	C₄×C₈	C₄×D₇₁	1	2	yes	—
103	C₄×C₈	D₂₈₄	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
106	C₂²×C₄	#11765865	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	142	8	no	—
111	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
114	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
115	C₂×C₈	#11765849	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11765849	1	4	yes	—
119	C₄²	C₂×C₄×D₇₁	1	2	yes	—
120	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
121	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
122	C₂³	#14556087	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
128	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
129	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
130	C₂×C₄	C₂×#11765849	1	2	yes	—
131	C₂×C₄	C₂×#11765849	1	2	yes	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	4	no	—
135	C₈	—	284	8	no	—
136	C₈	#14556039	1	4	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
139	C₂²	#17986056	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂×#14556039	1	1	yes	characteristic
143	C₄	C₄×#11765849	1	1	yes	characteristic
144	C₄	#17986023	1	2	yes	—
145	C₄	#17986122	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#22588193	1	1	yes	characteristic
148	C₂	#22588224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22587088

C₂

yes

maximal

#22586632

C₂

yes

maximal

#22586173

C₂

yes

maximal

C₄×#97×C₇₁

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

#17985752

C₄

yes

complemented

#17985750

C₄

yes

complemented

#17985721

—

C₂×#97×C₇₁

C₄

yes

—

C₂×#97×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄³×C₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#97×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#97

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#432

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

#423

yes

—

C₄×C₇₁

#444

yes

—

#8269

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1767

yes

characteristic

C₂×C₇₁

#1773

yes

—

C₄×#97

D₇₁

yes

—

—

—

C₇₁

#8269

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

Dic₇₁

yes

—

C₂×#97

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄³

C₂×D₇₁

yes

—

—

—

142

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

C₄×C₈

—

142

—

100

C₄×C₈

Dic₁₄₂

yes

—

101

C₄×C₈

C₄×D₇₁

yes

—

102

C₄×C₈

C₄×D₇₁

yes

—

103

C₄×C₈

D₂₈₄

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

106

C₂²×C₄

#11765865

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

142

—

111

C₂×C₈

C₄×Dic₇₁

yes

characteristic

112

C₂×C₈

C₄×Dic₇₁

yes

characteristic

113

C₂×C₈

C₄×Dic₇₁

yes

—

114

C₂×C₈

C₄⋊Dic₇₁

yes

—

115

C₂×C₈

#11765849

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11765849

yes

—

119

C₄²

C₂×C₄×D₇₁

yes

—

120

C₄²

C₂×D₂₈₄

yes

characteristic

121

C₄²

C₂×Dic₁₄₂

yes

characteristic

122

C₂³

#14556087

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

128

C₂×C₄

C₄×D₂₈₄

yes

—

129

C₂×C₄

C₄×Dic₁₄₂

yes

—

130

C₂×C₄

C₂×#11765849

yes

—

131

C₂×C₄

C₂×#11765849

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

284

—

136

C₈

#14556039

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₇₁

yes

characteristic

139

C₂²

#17986056

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#14556039

yes

characteristic

143

C₄

C₄×#11765849

yes

characteristic

144

C₄

#17986023

yes

—

145

C₄

#17986122

yes

—

146

C₂

—

147

C₂

#22588193

yes

characteristic

148

C₂

#22588224

yes

—

	subgroup	G/H
center	C₄²	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#97×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#97	D₇₁
hypocenter	C₇₁	#8269
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₄²

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₄×#97×C₇₁

C₂

socle

C₂²×C₇₁

C₄×C₄⋊C₄

prime radical

C₂³×C₇₁

#117

cosocle

C₂×C₄²×C₇₁

C₂³

hypercenter

C₄×#97

D₇₁

hypocenter

C₇₁

#8269

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
6. #17985752	C₄	semidirect
7. #17985750	C₄	semidirect
79. C₇₁	#8269	semidirect

H ≅ G/N

type

6. #17985752

C₄

semidirect

7. #17985750

C₄

semidirect

79. C₇₁

#8269

semidirect

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

subgroup	index
#22587088	2
#22586632	2
#22586173	2
C₄×#97×C₇₁	2
#8269	71

subgroup

index

C₄×#97×C₇₁

Subgroups — #31888622

Conjugacy classes of subgroups

Complements

Maximal subgroups