Group #31884268 — subgroups

Subgroups — #31884268

Order 18176.

206 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22586420	C₂	1	2	yes	maximal
2	C₇₁×#1992	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556068	—	2	2	no	—
4	C₂×#125×C₇₁	C₄	1	1	yes	characteristic
5	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
6	#384×C₇₁	C₄	1	1	yes	characteristic
7	#340×C₇₁	—	2	2	no	—
8	#14556068	—	4	2	no	—
9	#14556068	—	4	2	no	—
10	C₂×C₇₁⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₇₁	D₄	1	2	yes	—
12	#125×C₇₁	—	2	1	no	—
13	#125×C₇₁	—	4	1	no	—
14	#125×C₇₁	C₈	1	2	yes	—
15	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
16	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
17	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
18	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
20	#97×C₇₁	—	4	1	no	—
21	C₇₁⋊C₁₆	—	4	4	no	—
22	C₂²×C₄×C₇₁	—	2	1	no	—
23	C₂²×C₄×C₇₁	—	4	1	no	—
24	C₂²×C₄×C₇₁	—	4	1	no	—
25	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	—	2	2	no	—
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
29	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₄⋊C₄×C₇₁	—	4	2	no	—
32	C₄⋊C₄×C₇₁	—	4	2	no	—
33	C₄⋊C₄×C₇₁	—	4	1	no	—
34	C₄⋊C₄×C₇₁	—	4	2	no	—
35	C₄⋊C₄×C₇₁	—	8	2	no	—
36	C₄²×C₇₁	—	2	1	no	—
37	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
39	C₂³×C₇₁	#100	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	1	no	—
41	C₂×C₄×C₇₁	—	2	1	no	—
42	C₂×C₄×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	—	4	2	no	—
45	C₂×C₄×C₇₁	—	8	1	no	—
46	C₂×C₄×C₇₁	—	8	1	no	—
47	C₂×C₄×C₇₁	—	8	2	no	—
48	C₂×C₄×C₇₁	—	8	1	no	—
49	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#103	1	2	yes	—
52	C₂×C₄×C₇₁	#104	1	2	yes	—
53	C₈×C₇₁	—	2	2	no	—
54	C₈×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	#323	1	1	yes	characteristic
58	C₂²×C₇₁	#329	1	1	yes	characteristic
59	C₂²×C₇₁	#333	1	1	yes	characteristic
60	C₄×C₇₁	—	4	1	no	—
61	C₄×C₇₁	—	4	1	no	—
62	C₄×C₇₁	—	8	2	no	—
63	C₄×C₇₁	—	16	1	no	—
64	C₄×C₇₁	Q₈⋊C₈	1	2	yes	—
65	C₄×C₇₁	#328	1	2	yes	—
66	#7130	—	71	1	no	maximal, Sylow
67	C₂×C₇₁	—	2	1	no	—
68	C₂×C₇₁	—	2	1	no	—
69	C₂×C₇₁	#1272	1	1	yes	characteristic
70	C₂×C₇₁	#1355	1	1	yes	characteristic
71	C₂×C₇₁	#1356	1	1	yes	characteristic
72	#1992	D₇₁	1	1	yes	characteristic
73	#1375	—	71	2	no	—
74	C₇₁	#7130	1	1	yes	characteristic, complemented, Sylow
75	C₂×#125	Dic₇₁	1	1	yes	characteristic
76	C₂×M₃₂	—	142	2	no	—
77	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
78	#384	Dic₇₁	1	1	yes	characteristic
79	#340	—	2	2	no	—
80	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
81	#125	—	2	1	no	—
82	#125	—	4	1	no	—
83	#125	C₇₁⋊C₈	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
89	M₃₂	—	284	2	no	—
90	M₃₂	—	284	2	no	—
91	C₂×C₁₆	—	142	4	no	—
92	#97	—	4	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	#11765852	1	2	yes	—
100	C₂×C₈	#11765855	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
109	C₄²	#11765849	1	1	yes	characteristic
110	C₁₆	—	284	4	no	—
111	C₂³	#14556045	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	#14556043	1	1	yes	characteristic
123	C₂×C₄	#14556052	1	2	yes	—
124	C₂×C₄	#14556055	1	2	yes	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#17985723	1	1	yes	characteristic
130	C₂²	#17985739	1	1	yes	characteristic
131	C₂²	#17985748	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	16	1	no	—
136	C₄	Q₈⋊(C₇₁⋊C₈)	1	2	yes	—
137	C₄	#17985736	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#22586166	1	1	yes	characteristic
141	C₂	#22586373	1	1	yes	characteristic
142	C₂	#22586375	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22586420

C₂

yes

maximal

C₇₁×#1992

C₂

yes

maximal, characteristic

C₂×#14556068

—

C₂×#125×C₇₁

C₄

yes

characteristic

C₂×#97×C₇₁

C₂²

yes

characteristic

#384×C₇₁

C₄

yes

characteristic

#340×C₇₁

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

D₄

yes

—

#125×C₇₁

—

#125×C₇₁

—

#125×C₇₁

C₈

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

#97×C₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

Q₁₆

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

M₁₆

yes

characteristic

C₂³×C₇₁

#100

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#104

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

Q₈⋊C₈

yes

—

C₄×C₇₁

#328

yes

—

#7130

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

#1375

—

C₇₁

#7130

yes

characteristic, complemented, Sylow

C₂×#125

Dic₇₁

yes

characteristic

C₂×M₃₂

—

142

—

C₂×#97

C₂×D₇₁

yes

characteristic

#384

Dic₇₁

yes

characteristic

#340

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

#125

—

#125

—

#125

C₇₁⋊C₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

M₃₂

—

284

—

M₃₂

—

284

—

C₂×C₁₆

—

142

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11765852

yes

—

100

C₂×C₈

#11765855

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

109

C₄²

#11765849

yes

characteristic

110

C₁₆

—

284

—

111

C₂³

#14556045

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

122

C₂×C₄

#14556043

yes

characteristic

123

C₂×C₄

#14556052

yes

—

124

C₂×C₄

#14556055

yes

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#17985723

yes

characteristic

130

C₂²

#17985739

yes

characteristic

131

C₂²

#17985748

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₈⋊(C₇₁⋊C₈)

yes

—

137

C₄

#17985736

yes

—

138

C₂

—

139

C₂

—

140

C₂

#22586166

yes

characteristic

141

C₂

#22586373

yes

characteristic

142

C₂

#22586375

yes

characteristic

	subgroup	G/H
center	C₂²	#17985739
commutator	C₄²×C₇₁	C₂×C₈
Frattini	C₂×#97	C₂×D₇₁
Fitting	C₇₁×#1992	C₂
socle	C₂²×C₇₁	#329
prime radical	C₂³×C₇₁	#100
cosocle	C₂×#97×C₇₁	C₂²
hypercenter	#1992	D₇₁
hypocenter	C₇₁	#7130
LCS	G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1992

subgroup

G/H

center

C₂²

#17985739

commutator

C₄²×C₇₁

C₂×C₈

Frattini

C₂×#97

C₂×D₇₁

Fitting

C₇₁×#1992

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×#97×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1992

N	H ≅ G/N	type
74. C₇₁	#7130	semidirect

H ≅ G/N

type

74. C₇₁

#7130

semidirect

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

subgroup	index
#22586420	2
C₇₁×#1992	2
#7130	71

subgroup

index

#22586420

C₇₁×#1992

#7130

Subgroups — #31884268

Conjugacy classes of subgroups

Complements

Maximal subgroups