Group #31887893 — subgroups

Subgroups — #31887893

Order 18176.

354 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 195 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	#22586623	C₂	1	2	yes	maximal
2	#22586171	C₂	1	2	yes	maximal
3	#22586169	C₂	1	2	yes	maximal
4	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985720	C₄	1	4	yes	complemented
7	C₈×C₇₁⋊C₈	C₄	1	4	yes	complemented
8	C₂×#97×C₇₁	C₄	1	2	yes	—
9	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#376×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
14	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
15	C₄×C₇₁⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
21	#97×C₇₁	—	2	4	no	—
22	#97×C₇₁	—	2	2	no	—
23	C₄×C₈×C₇₁	C₈	1	8	yes	complemented
24	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
26	#95×C₇₁	—	2	2	no	—
27	C₂×C₇₁⋊C₈	—	2	8	no	—
28	C₂²×C₄×C₇₁	—	2	2	no	—
29	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	#41	1	2	yes	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₄²	1	2	yes	—
35	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
39	C₄²×C₇₁	—	2	2	no	—
40	C₄²×C₇₁	C₂×C₈	1	4	yes	—
41	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
42	C₄²×C₇₁	#41	1	2	yes	—
43	C₂³×C₇₁	#117	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	4	no	—
45	C₂×C₄×C₇₁	—	4	2	no	—
46	C₂×C₄×C₇₁	—	4	4	no	—
47	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#117	1	4	yes	—
49	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
51	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
52	C₈×C₇₁	—	2	4	no	—
53	C₈×C₇₁	C₄×C₈	1	2	yes	—
54	C₈×C₇₁	C₄×C₈	1	2	yes	—
55	C₈×C₇₁	#97	1	4	yes	—
56	C₇₁⋊C₈	—	4	8	no	—
57	C₂²×C₇₁	—	2	2	no	—
58	C₂²×C₇₁	#376	1	1	yes	characteristic
59	C₂²×C₇₁	#431	1	2	yes	—
60	C₄×C₇₁	—	2	4	no	—
61	C₄×C₇₁	—	4	4	no	—
62	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
64	C₄×C₇₁	#431	1	2	yes	—
65	#8137	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	—	2	2	no	—
67	C₂×C₇₁	#1744	1	1	yes	characteristic
68	C₂×C₇₁	#1747	1	2	yes	—
69	#1726	D₇₁	1	1	yes	characteristic
70	#1450	—	71	2	no	—
71	#1275	—	71	2	no	—
72	#1274	—	71	2	no	—
73	C₇₁	#8137	1	1	yes	characteristic, complemented, Sylow
74	#431	—	71	4	no	—
75	C₂×#97	Dic₇₁	1	2	yes	—
76	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
77	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
79	#376	C₂×D₇₁	1	1	yes	characteristic
80	#322	—	71	4	no	—
81	C₈²	—	71	4	no	—
82	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
83	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
85	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
86	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
87	C₄⋊C₈	—	142	4	no	—
88	C₂²⋊C₈	—	142	4	no	—
89	#97	—	2	4	no	—
90	#97	—	2	2	no	—
91	C₄×C₈	—	71	8	no	—
92	C₄×C₈	C₇₁⋊C₈	1	8	yes	complemented
93	C₄×C₈	C₄×D₇₁	1	2	yes	—
94	C₄×C₈	C₄×D₇₁	1	2	yes	—
95	#95	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
98	C₂²×C₄	#11765865	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	142	8	no	—
102	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
104	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
105	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
106	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
107	C₂×C₈	#11765849	1	4	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
110	C₄²	C₂×C₄×D₇₁	1	2	yes	—
111	C₄²	#11765865	1	2	yes	—
112	C₂³	#14556087	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
117	C₂×C₄	#14556085	1	4	yes	—
118	C₂×C₄	#14556087	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
120	C₂×C₄	C₂×#11765849	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	284	8	no	—
123	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
124	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
125	C₈	#14556039	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#17985845	1	1	yes	characteristic
128	C₂²	#17986052	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
132	C₄	C₂×#14556039	1	1	yes	characteristic
133	C₄	#17986052	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	#22588086	1	1	yes	characteristic
136	C₂	#22588095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22586623

C₂

yes

maximal

#22586171

C₂

yes

maximal

#22586169

C₂

yes

maximal

C₇₁×#1726

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

#17985720

C₄

yes

complemented

C₈×C₇₁⋊C₈

C₄

yes

complemented

C₂×#97×C₇₁

C₄

yes

—

C₂×#97×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#376×C₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#97×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

C₈

yes

complemented

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

#95×C₇₁

—

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

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

—

C₈×C₇₁

C₄×C₈

yes

—

C₈×C₇₁

#97

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#376

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄×C₈

yes

characteristic

C₄×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

#431

yes

—

#8137

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1744

yes

characteristic

C₂×C₇₁

#1747

yes

—

#1726

D₇₁

yes

—

—

—

C₇₁

#8137

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

Dic₇₁

yes

—

C₂×#97

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#376

C₂×D₇₁

yes

characteristic

#322

—

C₈²

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₇₁⋊C₈

yes

complemented

C₄×C₈

C₄×D₇₁

yes

—

C₄×C₈

C₄×D₇₁

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂²×C₄

#11765865

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

142

—

102

C₂×C₈

C₄×Dic₇₁

yes

characteristic

103

C₂×C₈

C₄×Dic₇₁

yes

—

104

C₂×C₈

C₄×Dic₇₁

yes

characteristic

105

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

106

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

107

C₂×C₈

#11765849

yes

—

108

C₄²

—

109

C₄²

C₂×C₇₁⋊C₈

yes

—

110

C₄²

C₂×C₄×D₇₁

yes

—

111

C₄²

#11765865

yes

—

112

C₂³

#14556087

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

117

C₂×C₄

#14556085

yes

—

118

C₂×C₄

#14556087

yes

characteristic

119

C₂×C₄

C₂²×C₇₁⋊C₈

yes

—

120

C₂×C₄

C₂×#11765849

yes

—

121

C₈

—

122

C₈

—

284

—

123

C₈

C₄×C₇₁⋊C₈

yes

—

124

C₈

C₄×C₇₁⋊C₈

yes

—

125

C₈

#14556039

yes

—

126

C₂²

—

127

C₂²

#17985845

yes

characteristic

128

C₂²

#17986052

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×C₇₁⋊C₈

yes

characteristic

132

C₄

C₂×#14556039

yes

characteristic

133

C₄

#17986052

yes

—

134

C₂

—

135

C₂

#22588086

yes

characteristic

136

C₂

#22588095

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1726	C₂
socle	C₂²×C₇₁	#376
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#8137
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₄×Dic₇₁

commutator

C₄×C₇₁

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1726

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ #1726

N	H ≅ G/N	type
6. #17985720	C₄	semidirect
7. C₈×C₇₁⋊C₈	C₄	semidirect
23. C₄×C₈×C₇₁	C₈	semidirect
73. C₇₁	#8137	semidirect
92. C₄×C₈	C₇₁⋊C₈	semidirect

H ≅ G/N

type

6. #17985720

C₄

semidirect

7. C₈×C₇₁⋊C₈

C₄

semidirect

23. C₄×C₈×C₇₁

C₈

semidirect

73. C₇₁

#8137

semidirect

92. C₄×C₈

C₇₁⋊C₈

semidirect

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

subgroup	index
#22586623	2
#22586171	2
#22586169	2
C₇₁×#1726	2
#8137	71

subgroup

index

C₇₁×#1726

Subgroups — #31887893

Conjugacy classes of subgroups

Complements

Maximal subgroups