Group #31912454 — subgroups

Subgroups — #31912454

Order 18176.

390 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 227 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×#17985775	C₂	1	2	yes	maximal
2	#22586418	C₂	1	4	yes	maximal
3	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
5	#17985775	C₄	1	8	yes	complemented
6	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₄³×C₇₁	C₄	1	1	yes	characteristic
11	C₂×C₇₁⋊C₁₆	—	2	8	no	—
12	C₂×C₇₁⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₁	C₈	1	4	yes	—
17	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₇₁	—	2	4	no	—
20	C₄×C₈×C₇₁	—	2	4	no	—
21	C₄×C₈×C₇₁	C₈	1	4	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
23	C₇₁⋊C₁₆	—	2	8	no	—
24	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
25	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
26	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	C₄²	1	2	yes	—
30	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
32	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₇₁	#41	1	4	yes	—
34	C₄²×C₇₁	—	2	8	no	—
35	C₄²×C₇₁	—	2	2	no	—
36	C₄²×C₇₁	—	2	2	no	—
37	C₄²×C₇₁	C₄²	1	2	yes	—
38	C₄²×C₇₁	C₂×C₈	1	2	yes	—
39	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	4	no	—
41	C₂×C₄×C₇₁	—	2	4	no	—
42	C₂×C₄×C₇₁	—	2	4	no	—
43	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₇₁	M₃₂	1	8	yes	—
45	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#117	1	2	yes	—
47	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
49	C₈×C₇₁	—	2	4	no	—
50	C₈×C₇₁	#97	1	4	yes	—
51	C₈×C₇₁	#117	1	4	yes	—
52	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂²×C₇₁	#431	1	2	yes	—
54	C₂²×C₇₁	C₂×M₃₂	1	4	yes	—
55	C₄×C₇₁	—	2	4	no	—
56	C₄×C₇₁	—	2	8	no	—
57	C₄×C₇₁	#376	1	1	yes	characteristic
58	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
59	C₄×C₇₁	#431	1	2	yes	—
60	#12591	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	#1744	1	1	yes	characteristic
62	C₂×C₇₁	C₄×M₃₂	1	2	yes	—
63	C₂×C₇₁	#2163	1	4	yes	—
64	C₂×#346	—	71	2	no	—
65	C₄²×C₈	D₇₁	1	1	yes	characteristic
66	#1374	—	71	4	no	—
67	C₇₁	#12591	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	—	71	4	no	—
69	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
70	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	C₄³	Dic₇₁	1	1	yes	characteristic
74	#346	—	71	8	no	—
75	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
76	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
78	C₂×C₄²	C₇₁⋊C₈	1	4	yes	—
79	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
80	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
81	C₂×C₁₆	—	71	8	no	—
82	C₂×C₁₆	—	142	8	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₇₁⋊C₈	1	4	yes	—
86	C₄×C₈	C₄×D₇₁	1	4	yes	—
87	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
88	C₂²×C₄	C₂×C₇₁⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
93	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
94	C₂×C₈	#11765849	1	4	yes	—
95	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
96	C₂×C₈	#11765865	1	4	yes	—
97	C₄²	—	2	8	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₄×Dic₇₁	1	2	yes	—
101	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
102	C₁₆	—	142	8	no	—
103	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄×C₇₁⋊C₈	1	4	yes	—
108	C₂×C₄	#14556068	1	8	yes	—
109	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
110	C₂×C₄	#14556087	1	2	yes	—
111	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
112	C₂×C₄	C₂×#11765849	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	#14556039	1	4	yes	—
115	C₈	#14556085	1	4	yes	—
116	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
117	C₂²	#17986052	1	2	yes	—
118	C₂²	C₂×#14556068	1	4	yes	—
119	C₄	—	2	8	no	—
120	C₄	—	2	4	no	—
121	C₄	#17985845	1	1	yes	characteristic
122	C₄	C₂×#14556039	1	1	yes	characteristic
123	C₄	#17986052	1	2	yes	—
124	C₂	#22588086	1	1	yes	characteristic
125	C₂	C₄×#14556068	1	2	yes	—
126	C₂	#22590143	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985775

C₂

yes

maximal

#22586418

C₂

yes

maximal

C₄²×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

#17985775

C₄

yes

complemented

C₂×C₄×C₈×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄³×C₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₈

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₄²

yes

—

C₄²×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×C₈

yes

—

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

#97

yes

—

C₈×C₇₁

#117

yes

—

C₂²×C₇₁

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₂²×C₇₁

C₂×M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#376

yes

characteristic

C₄×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

#431

yes

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₄×M₃₂

yes

—

C₂×C₇₁

#2163

yes

—

C₂×#346

—

C₄²×C₈

D₇₁

yes

characteristic

#1374

—

C₇₁

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₇₁

yes

—

C₂×C₄×C₈

Dic₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄³

Dic₇₁

yes

characteristic

#346

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

C₇₁⋊C₈

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₇₁⋊C₈

yes

—

C₄×C₈

C₄×D₇₁

yes

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₁

yes

—

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

#11765849

yes

—

C₂×C₈

C₂×C₄×D₇₁

yes

—

C₂×C₈

#11765865

yes

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₄×Dic₇₁

yes

—

101

C₄²

C₂×C₇₁⋊C₈

yes

—

102

C₁₆

—

142

—

103

C₂³

C₂²×C₇₁⋊C₈

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₇₁⋊C₈

yes

—

108

C₂×C₄

#14556068

yes

—

109

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

110

C₂×C₄

#14556087

yes

—

111

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

112

C₂×C₄

C₂×#11765849

yes

—

113

C₈

—

114

C₈

#14556039

yes

—

115

C₈

#14556085

yes

—

116

C₂²

C₂×C₄×C₇₁⋊C₈

yes

characteristic

117

C₂²

#17986052

yes

—

118

C₂²

C₂×#14556068

yes

—

119

C₄

—

120

C₄

—

121

C₄

#17985845

yes

characteristic

122

C₄

C₂×#14556039

yes

characteristic

123

C₄

#17986052

yes

—

124

C₂

#22588086

yes

characteristic

125

C₂

C₄×#14556068

yes

—

126

C₂

#22590143

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#12591
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄×C₈

Fitting

C₄²×C₈×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂²×C₈

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₄²×C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
5. #17985775	C₄	semidirect
67. C₇₁	#12591	semidirect

H ≅ G/N

type

5. #17985775

C₄

semidirect

67. C₇₁

#12591

semidirect

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

subgroup	index
C₂×#17985775	2
#22586418	2
C₄²×C₈×C₇₁	2
#12591	71

subgroup

index

C₂×#17985775

#22586418

C₄²×C₈×C₇₁

#12591

Subgroups — #31912454

Conjugacy classes of subgroups

Complements

Maximal subgroups