Group #31884115 — subgroups

Subgroups — #31884115

Order 18176.

132 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 55 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	#22587238	C₂	1	1	yes	maximal, characteristic
2	#22586258	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1370	C₂	1	1	yes	maximal, characteristic
4	#17986129	—	2	1	no	—
5	C₄×Dic₂₈₄	C₄	1	1	yes	characteristic
6	C₄×C₇₁⋊C₁₆	—	2	1	no	—
7	#17985752	C₄	1	1	yes	characteristic
8	#17985731	—	2	1	no	—
9	C₄⋊C₁₆×C₇₁	—	2	1	no	—
10	C₈²×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₈₄	C₈	1	1	yes	characteristic
12	C₄×Dic₁₄₂	—	2	1	no	—
13	C₂×C₇₁⋊C₁₆	—	2	2	no	—
14	#14556062	C₈	1	1	yes	characteristic
15	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
16	Dic₇₁⋊C₈	—	4	1	no	—
17	#14556053	—	2	1	no	—
18	#14556038	—	4	1	no	—
19	C₂×C₁₆×C₇₁	—	4	1	no	—
20	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	Dic₂₈₄	—	4	1	no	—
25	C₂×C₇₁⋊C₈	—	4	1	no	—
26	C₄⋊Dic₇₁	—	2	1	no	—
27	Dic₇₁⋊C₄	—	4	1	no	—
28	C₄×Dic₇₁	—	4	1	no	—
29	C₇₁⋊C₁₆	—	4	2	no	—
30	C₂×C₈×C₇₁	—	2	1	no	—
31	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
35	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₆×C₇₁	—	8	1	no	—
37	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#104	1	1	yes	characteristic
40	C₂×Dic₇₁	—	4	1	no	—
41	C₂×Dic₇₁	—	4	1	no	—
42	Dic₁₄₂	—	4	1	no	—
43	Dic₁₄₂	—	4	1	no	—
44	C₈×C₇₁	—	2	2	no	—
45	C₈×C₇₁	—	2	1	no	—
46	C₈×C₇₁	—	2	1	no	—
47	C₈×C₇₁	—	2	1	no	—
48	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
49	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
50	C₇₁⋊C₈	—	8	1	no	—
51	C₂²×C₇₁	D₄⋊C₈	1	1	yes	characteristic
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	#349	1	1	yes	characteristic
54	C₄×C₇₁	#350	1	1	yes	characteristic
55	C₄×C₇₁	#358	1	1	yes	characteristic
56	C₄×C₇₁	#359	1	1	yes	characteristic
57	Dic₇₁	—	8	1	no	—
58	Dic₇₁	—	8	1	no	—
59	#7073	—	71	1	no	maximal, Sylow
60	C₂×C₇₁	#1330	1	1	yes	characteristic
61	C₂×C₇₁	#1332	1	1	yes	characteristic
62	C₂×C₇₁	#1336	1	1	yes	characteristic
63	#1591	—	71	1	no	—
64	#1370	D₇₁	1	1	yes	characteristic
65	#1312	—	71	1	no	—
66	C₇₁	#7073	1	1	yes	characteristic, complemented, Sylow
67	#446	—	142	1	no	—
68	C₄×Q₁₆	—	71	1	no	—
69	C₄⋊C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	142	1	no	—
71	#335	—	71	1	no	—
72	Q₈⋊C₈	—	142	1	no	—
73	C₈²	C₂×D₇₁	1	1	yes	characteristic
74	C₂×Q₁₆	—	71	1	no	—
75	C₄×Q₈	—	142	1	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	—	142	2	no	—
78	#107	—	71	1	no	—
79	C₄⋊C₈	—	284	1	no	—
80	C₄⋊C₈	—	142	1	no	—
81	#103	—	142	1	no	—
82	#97	—	284	1	no	—
83	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
84	C₄×C₈	D₂₈₄	1	1	yes	characteristic
85	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂×Q₈	—	142	1	no	—
87	Q₁₆	—	284	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	284	1	no	—
90	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
91	C₂×C₈	#11765848	1	1	yes	characteristic
92	C₂×C₈	#11765850	1	1	yes	characteristic
93	C₂×C₈	#11765852	1	1	yes	characteristic
94	C₄⋊C₄	—	284	1	no	—
95	C₄⋊C₄	—	142	1	no	—
96	C₄²	—	284	1	no	—
97	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
98	C₁₆	—	8	1	no	—
99	C₁₆	—	284	2	no	—
100	Q₈	—	284	1	no	—
101	Q₈	—	284	1	no	—
102	C₂×C₄	—	284	1	no	—
103	C₂×C₄	—	284	1	no	—
104	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
105	C₂×C₄	#14556048	1	1	yes	characteristic
106	C₂×C₄	#14556054	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	568	1	no	—
112	C₈	#14556072	1	1	yes	characteristic
113	C₈	#14556074	1	1	yes	characteristic
114	C₂²	#17985726	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	568	1	no	—
117	C₄	—	568	1	no	—
118	C₄	#17985780	1	1	yes	characteristic
119	C₄	#17985782	1	1	yes	characteristic
120	C₄	#17985802	1	1	yes	characteristic
121	C₄	#17985805	1	1	yes	characteristic
122	C₂	#22586297	1	1	yes	characteristic
123	C₂	#22586303	1	1	yes	characteristic
124	C₂	#22586315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22587238

C₂

yes

maximal, characteristic

#22586258

C₂

yes

maximal, characteristic

C₇₁×#1370

C₂

yes

maximal, characteristic

#17986129

—

C₄×Dic₂₈₄

C₄

yes

characteristic

C₄×C₇₁⋊C₁₆

—

#17985752

C₄

yes

characteristic

#17985731

—

C₄⋊C₁₆×C₇₁

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₈

yes

characteristic

C₄×Dic₁₄₂

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₈

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

#14556053

—

#14556038

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

characteristic

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₃₂

yes

characteristic

C₈×C₇₁

Q₃₂

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

D₄⋊C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1312

—

C₇₁

#7073

yes

characteristic, complemented, Sylow

#446

—

142

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

142

—

#335

—

Q₈⋊C₈

—

142

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

#107

—

C₄⋊C₈

—

284

—

C₄⋊C₈

—

142

—

#103

—

142

—

#97

—

284

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×Q₈

—

142

—

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₈×D₇₁

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄²

—

284

—

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

C₁₆

—

C₁₆

—

284

—

100

Q₈

—

284

—

101

Q₈

—

284

—

102

C₂×C₄

—

284

—

103

C₂×C₄

—

284

—

104

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

105

C₂×C₄

#14556048

yes

characteristic

106

C₂×C₄

#14556054

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

568

—

112

C₈

#14556072

yes

characteristic

113

C₈

#14556074

yes

characteristic

114

C₂²

#17985726

yes

characteristic

115

C₄

—

116

C₄

—

568

—

117

C₄

—

568

—

118

C₄

#17985780

yes

characteristic

119

C₄

#17985782

yes

characteristic

120

C₄

#17985802

yes

characteristic

121

C₄

#17985805

yes

characteristic

122

C₂

#22586297

yes

characteristic

123

C₂

#22586303

yes

characteristic

124

C₂

#22586315

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556048
commutator	C₂×C₈×C₇₁	C₂×C₈
Frattini	C₈²	C₂×D₇₁
Fitting	C₇₁×#1370	C₂
socle, prime radical	C₂²×C₇₁	D₄⋊C₈
cosocle	C₈²×C₇₁	C₂²
hypercenter	#1370	D₇₁
hypocenter	C₇₁	#7073
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂×C₄

#14556048

commutator

C₂×C₈×C₇₁

C₂×C₈

Frattini

C₈²

C₂×D₇₁

Fitting

C₇₁×#1370

C₂

socle, prime radical

C₂²×C₇₁

D₄⋊C₈

cosocle

C₈²×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1370

N	H ≅ G/N	type
66. C₇₁	#7073	semidirect

H ≅ G/N

type

66. C₇₁

#7073

semidirect

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

subgroup	index
#22587238	2
#22586258	2
C₇₁×#1370	2
#7073	71

subgroup

index

#22587238

#22586258

C₇₁×#1370

#7073

Subgroups — #31884115

Conjugacy classes of subgroups

Complements

Maximal subgroups