Group #31884278 — subgroups

Subgroups — #31884278

Order 18176.

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

normal subgroups only

This group has 202 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	#22586422	C₂	1	2	yes	maximal
2	C₇₁×#1991	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556068	—	2	2	no	—
4	C₂×#125×C₇₁	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
6	#382×C₇₁	C₄	1	1	yes	characteristic
7	#340×C₇₁	—	2	2	no	—
8	#14556068	—	2	4	no	—
9	#14556068	—	4	2	no	—
10	C₂×C₇₁⋊C₁₆	—	4	2	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×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₇₁	—	4	1	no	—
20	#95×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	C₄²×C₇₁	—	2	1	no	—
36	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
38	C₂³×C₇₁	#100	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	2	1	no	—
40	C₂×C₄×C₇₁	—	2	1	no	—
41	C₂×C₄×C₇₁	—	4	2	no	—
42	C₂×C₄×C₇₁	—	4	2	no	—
43	C₂×C₄×C₇₁	—	8	2	no	—
44	C₂×C₄×C₇₁	—	8	2	no	—
45	C₂×C₄×C₇₁	—	8	1	no	—
46	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#103	1	2	yes	—
49	C₂×C₄×C₇₁	#104	1	2	yes	—
50	C₈×C₇₁	—	2	2	no	—
51	C₈×C₇₁	—	2	2	no	—
52	C₂²×C₇₁	—	2	2	no	—
53	C₂²×C₇₁	#323	1	1	yes	characteristic
54	C₂²×C₇₁	#329	1	1	yes	characteristic
55	C₂²×C₇₁	#333	1	1	yes	characteristic
56	C₄×C₇₁	—	4	1	no	—
57	C₄×C₇₁	—	4	1	no	—
58	C₄×C₇₁	—	8	2	no	—
59	C₄×C₇₁	—	16	1	no	—
60	C₄×C₇₁	Q₈⋊C₈	1	2	yes	—
61	C₄×C₇₁	#328	1	2	yes	—
62	#7135	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	2	1	no	—
64	C₂×C₇₁	—	2	1	no	—
65	C₂×C₇₁	#1272	1	1	yes	characteristic
66	C₂×C₇₁	#1366	1	2	yes	—
67	#1991	D₇₁	1	1	yes	characteristic
68	#1376	—	71	2	no	—
69	C₇₁	#7135	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	Dic₇₁	1	1	yes	characteristic
71	C₂×M₃₂	—	142	2	no	—
72	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	#382	Dic₇₁	1	1	yes	characteristic
74	#340	—	2	2	no	—
75	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
76	#125	—	2	1	no	—
77	#125	—	4	1	no	—
78	#125	C₇₁⋊C₈	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
83	M₃₂	—	284	2	no	—
84	M₃₂	—	142	4	no	—
85	C₂×C₁₆	—	284	2	no	—
86	C₄×C₈	—	4	1	no	—
87	#95	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	#11765852	1	2	yes	—
95	C₂×C₈	#11765855	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
103	C₄²	#11765849	1	1	yes	characteristic
104	C₁₆	—	284	4	no	—
105	C₂³	#14556045	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
114	C₂×C₄	#14556043	1	1	yes	characteristic
115	C₂×C₄	#14556052	1	2	yes	—
116	C₂×C₄	#14556055	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#17985723	1	1	yes	characteristic
121	C₂²	#17985739	1	1	yes	characteristic
122	C₂²	#17985748	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	16	1	no	—
127	C₄	Q₈⋊(C₇₁⋊C₈)	1	2	yes	—
128	C₄	#17985736	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#22586166	1	1	yes	characteristic
132	C₂	#22586401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22586422

C₂

yes

maximal

C₇₁×#1991

C₂

yes

maximal, characteristic

C₂×#14556068

—

C₂×#125×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×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×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

—

#95×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

—

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

C₂²⋊C₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#104

yes

—

C₈×C₇₁

—

C₈×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

—

#7135

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1272

yes

characteristic

C₂×C₇₁

#1366

yes

—

#1991

D₇₁

yes

characteristic

#1376

—

C₇₁

#7135

yes

characteristic, complemented, Sylow

C₂×#125

Dic₇₁

yes

characteristic

C₂×M₃₂

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

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×C₄²

C₂×Dic₇₁

yes

characteristic

M₃₂

—

284

—

M₃₂

—

142

—

C₂×C₁₆

—

284

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11765852

yes

—

C₂×C₈

#11765855

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

103

C₄²

#11765849

yes

characteristic

104

C₁₆

—

284

—

105

C₂³

#14556045

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

114

C₂×C₄

#14556043

yes

characteristic

115

C₂×C₄

#14556052

yes

—

116

C₂×C₄

#14556055

yes

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

#17985723

yes

characteristic

121

C₂²

#17985739

yes

characteristic

122

C₂²

#17985748

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈⋊(C₇₁⋊C₈)

yes

—

128

C₄

#17985736

yes

—

129

C₂

—

130

C₂

—

131

C₂

#22586166

yes

characteristic

132

C₂

#22586401

yes

—

	subgroup	G/H
center	C₂²	#17985748
commutator	C₄²×C₇₁	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1991	C₂
socle	C₂²×C₇₁	#333
prime radical	C₂³×C₇₁	#100
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1991	D₇₁
hypocenter	C₇₁	#7135
LCS	G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

subgroup

G/H

center

C₂²

#17985748

commutator

C₄²×C₇₁

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₇₁×#1991

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

N	H ≅ G/N	type
69. C₇₁	#7135	semidirect

H ≅ G/N

type

69. C₇₁

#7135

semidirect

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

subgroup	index
#22586422	2
C₇₁×#1991	2
#7135	71

subgroup

index

#22586422

C₇₁×#1991

#7135

Subgroups — #31884278

Conjugacy classes of subgroups

Complements

Maximal subgroups