Group #31887887 — subgroups

Subgroups — #31887887

Order 18176.

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

normal subgroups only

This group has 382 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	#22588086	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	C₇₁⋊#1273	C₂	1	4	yes	maximal
4	C₇₁×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
7	#17985754	C₄	1	4	yes	—
8	#17985754	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
10	C₄³×C₇₁	C₂²	1	1	yes	characteristic
11	#336×C₇₁	C₄	1	4	yes	—
12	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
13	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
14	C₄×C₇₁⋊C₈	—	2	4	no	—
15	C₄×C₇₁⋊C₈	—	2	4	no	—
16	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₇₁	—	2	4	no	—
22	C₂×C₇₁⋊C₈	—	2	4	no	—
23	C₂×C₇₁⋊C₈	—	2	4	no	—
24	C₂×C₇₁⋊C₈	—	2	8	no	—
25	C₂×C₇₁⋊C₈	—	2	8	no	—
26	C₂²×C₄×C₇₁	C₄²	1	4	yes	—
27	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	—	2	4	no	—
31	C₂×C₈×C₇₁	—	2	8	no	—
32	C₄²×C₇₁	—	2	4	no	—
33	C₄²×C₇₁	—	2	4	no	—
34	C₄²×C₇₁	—	2	4	no	—
35	C₄²×C₇₁	#41	1	4	yes	—
36	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇₁	—	2	4	no	—
38	C₂×C₄×C₇₁	—	2	4	no	—
39	C₂×C₄×C₇₁	—	2	4	no	—
40	C₂×C₄×C₇₁	#117	1	2	yes	—
41	C₂×C₄×C₇₁	#117	1	2	yes	—
42	C₂×C₄×C₇₁	#117	1	2	yes	—
43	C₂×C₄×C₇₁	#131	1	4	yes	—
44	C₂×C₄×C₇₁	#131	1	4	yes	—
45	C₂×C₄×C₇₁	#131	1	4	yes	—
46	C₈×C₇₁	—	4	4	no	—
47	C₇₁⋊C₈	—	4	4	no	—
48	C₇₁⋊C₈	—	4	4	no	—
49	C₂²×C₇₁	#376	1	1	yes	characteristic
50	C₂²×C₇₁	#405	1	2	yes	—
51	C₂²×C₇₁	#405	1	2	yes	—
52	C₂²×C₇₁	#405	1	2	yes	—
53	C₄×C₇₁	—	4	4	no	—
54	C₄×C₇₁	#433	1	4	yes	—
55	C₄×C₇₁	#433	1	4	yes	—
56	C₄×C₇₁	#433	1	4	yes	—
57	#8135	—	71	1	no	maximal, Sylow
58	C₂×C₇₁	#1454	1	4	yes	—
59	C₂×C₇₁	#1746	1	1	yes	characteristic
60	C₂×C₇₁	#1746	1	1	yes	characteristic
61	C₂×C₇₁	#1746	1	1	yes	characteristic
62	#1744	—	71	1	no	—
63	#1744	—	71	1	no	—
64	#1744	D₇₁	1	1	yes	characteristic
65	#1273	—	71	4	no	—
66	C₇₁	#8135	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄×C₈	—	71	2	no	—
68	C₂×C₄×C₈	—	71	2	no	—
69	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
70	C₄³	C₂×D₇₁	1	1	yes	characteristic
71	#336	—	71	4	no	—
72	#336	—	71	4	no	—
73	#336	Dic₇₁	1	4	yes	—
74	C₂²×C₈	—	71	4	no	—
75	C₂²×C₈	—	71	4	no	—
76	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
77	C₂×C₄²	C₄×D₇₁	1	2	yes	—
78	C₂×C₄²	C₄×D₇₁	1	2	yes	—
79	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
80	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	142	4	no	—
83	C₄×C₈	—	142	4	no	—
84	C₂²×C₄	C₄×Dic₇₁	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	142	4	no	—
91	C₂×C₈	—	142	4	no	—
92	C₂×C₈	—	142	8	no	—
93	C₂×C₈	—	142	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11765865	1	4	yes	—
98	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#14556085	1	2	yes	—
103	C₂×C₄	#14556085	1	2	yes	—
104	C₂×C₄	#14556087	1	2	yes	—
105	C₂×C₄	#14556134	1	4	yes	—
106	C₂×C₄	#14556135	1	4	yes	—
107	C₂×C₄	#14556136	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	284	4	no	—
110	C₈	—	284	4	no	—
111	C₂²	#17985845	1	1	yes	characteristic
112	C₂²	#17985954	1	2	yes	—
113	C₂²	#17985955	1	2	yes	—
114	C₂²	#17985957	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#17986058	1	4	yes	—
117	C₄	#17986060	1	4	yes	—
118	C₄	#17986061	1	4	yes	—
119	C₂	#22586640	1	4	yes	—
120	C₂	#22588091	1	1	yes	characteristic
121	C₂	#22588092	1	1	yes	characteristic
122	C₂	#22588093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588086

C₂

yes

maximal, characteristic

#22588086

C₂

yes

maximal, characteristic

C₇₁⋊#1273

C₂

yes

maximal

C₇₁×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

#17985754

C₄

yes

—

#17985754

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₂²

yes

characteristic

#336×C₇₁

C₄

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₄²

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#376

yes

characteristic

C₂²×C₇₁

#405

yes

—

C₂²×C₇₁

#405

yes

—

C₂²×C₇₁

#405

yes

—

C₄×C₇₁

—

C₄×C₇₁

#433

yes

—

C₄×C₇₁

#433

yes

—

C₄×C₇₁

#433

yes

—

#8135

—

maximal, Sylow

C₂×C₇₁

#1454

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1273

—

C₇₁

#8135

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₄³

C₂×D₇₁

yes

—

—

Dic₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂×C₄²

C₄×D₇₁

yes

—

C₂×C₄²

C₄×D₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

142

—

C₄×C₈

—

142

—

C₂²×C₄

C₄×Dic₇₁

yes

—

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11765865

yes

—

C₂³

C₂×C₄×Dic₇₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#14556085

yes

—

103

C₂×C₄

#14556085

yes

—

104

C₂×C₄

#14556087

yes

—

105

C₂×C₄

#14556134

yes

—

106

C₂×C₄

#14556135

yes

—

107

C₂×C₄

#14556136

yes

—

108

C₈

—

109

C₈

—

284

—

110

C₈

—

284

—

111

C₂²

#17985845

yes

characteristic

112

C₂²

#17985954

yes

—

113

C₂²

#17985955

yes

—

114

C₂²

#17985957

yes

—

115

C₄

—

116

C₄

#17986058

yes

—

117

C₄

#17986060

yes

—

118

C₄

#17986061

yes

—

119

C₂

#22586640

yes

—

120

C₂

#22588091

yes

characteristic

121

C₂

#22588092

yes

characteristic

122

C₂

#22588093

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂×C₄²
Fitting	C₇₁×#1744	C₂
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1744	D₇₁
hypocenter	C₇₁	#8135
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₇₁

commutator, socle, prime radical

C₂³×C₇₁

C₂×C₄²

Fitting

C₇₁×#1744

C₂

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
66. C₇₁	#8135	semidirect

H ≅ G/N

type

66. C₇₁

#8135

semidirect

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

subgroup	index
#22588086	2
C₇₁⋊#1273	2
C₇₁×#1744	2
#8135	71

subgroup

index

#22588086

C₇₁⋊#1273

C₇₁×#1744

#8135

Subgroups — #31887887

Conjugacy classes of subgroups

Complements

Maximal subgroups