Group #31884181 — subgroups

Subgroups — #31884181

Order 18176.

136 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 53 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	#22587937	C₂	1	1	yes	maximal, characteristic
2	#22586410	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1367	C₂	1	1	yes	maximal, characteristic
4	#17986404	C₄	1	1	yes	characteristic
5	#17986397	—	2	1	no	—
6	#17986379	C₄	1	1	yes	characteristic
7	#17986350	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊C₁₆×C₇₁	—	2	1	no	—
10	C₈²×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₈₄	—	4	1	no	—
12	#14556128	—	2	1	no	—
13	#14556128	—	2	1	no	—
14	C₂×C₇₁⋊C₁₆	—	4	1	no	—
15	#14556062	—	4	1	no	—
16	#14556060	—	4	1	no	—
17	#14556060	—	4	1	no	—
18	#14556053	—	4	1	no	—
19	#14556053	—	4	1	no	—
20	C₂×C₁₆×C₇₁	—	4	1	no	—
21	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₄₂	—	4	1	no	—
25	C₂×Dic₁₄₂	—	4	1	no	—
26	Dic₂₈₄	—	8	2	no	—
27	C₄⋊Dic₇₁	—	4	1	no	—
28	C₄⋊Dic₇₁	—	4	1	no	—
29	C₄⋊Dic₇₁	—	4	1	no	—
30	C₄⋊Dic₇₁	—	4	1	no	—
31	C₇₁⋊C₁₆	—	8	1	no	—
32	C₂×C₈×C₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
37	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₁₆×C₇₁	—	8	1	no	—
39	C₂×C₄×C₇₁	#100	1	1	yes	characteristic
40	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
42	C₂×Dic₇₁	—	8	1	no	—
43	C₂×Dic₇₁	—	8	1	no	—
44	Dic₁₄₂	—	8	2	no	—
45	Dic₁₄₂	—	8	1	no	—
46	Dic₁₄₂	—	8	1	no	—
47	C₈×C₇₁	—	2	2	no	—
48	C₈×C₇₁	—	2	1	no	—
49	C₈×C₇₁	—	2	1	no	—
50	C₈×C₇₁	—	2	1	no	—
51	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
52	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
53	C₂²×C₇₁	#331	1	1	yes	characteristic
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	#358	1	1	yes	characteristic
56	C₄×C₇₁	#359	1	1	yes	characteristic
57	C₄×C₇₁	#360	1	1	yes	characteristic
58	C₄×C₇₁	#362	1	1	yes	characteristic
59	Dic₇₁	—	16	1	no	—
60	Dic₇₁	—	16	1	no	—
61	#7095	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	#1359	1	1	yes	characteristic
63	C₂×C₇₁	#1360	1	1	yes	characteristic
64	C₂×C₇₁	#1362	1	1	yes	characteristic
65	#1707	—	71	1	no	—
66	#1370	—	71	1	no	—
67	#1367	D₇₁	1	1	yes	characteristic
68	C₇₁	#7095	1	1	yes	characteristic, complemented, Sylow
69	#500	—	71	1	no	—
70	#498	—	142	1	no	—
71	#494	—	71	1	no	—
72	#487	—	142	1	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	—	142	1	no	—
75	C₈²	C₂×D₇₁	1	1	yes	characteristic
76	C₂×Q₁₆	—	284	1	no	—
77	C₄⋊Q₈	—	142	1	no	—
78	C₄⋊Q₈	—	142	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	284	1	no	—
81	#107	—	284	1	no	—
82	#106	—	284	1	no	—
83	#106	—	284	1	no	—
84	#103	—	284	1	no	—
85	#103	—	284	1	no	—
86	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
87	C₄×C₈	D₂₈₄	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
89	C₂×Q₈	—	284	1	no	—
90	C₂×Q₈	—	284	1	no	—
91	Q₁₆	—	568	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	#11765850	1	1	yes	characteristic
94	C₂×C₈	D₅₆₈	1	1	yes	characteristic
95	C₂×C₈	#11765852	1	1	yes	characteristic
96	C₂×C₈	#11765854	1	1	yes	characteristic
97	C₄⋊C₄	—	284	1	no	—
98	C₄⋊C₄	—	284	1	no	—
99	C₄⋊C₄	—	284	1	no	—
100	C₄⋊C₄	—	284	1	no	—
101	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
102	C₁₆	—	8	1	no	—
103	C₁₆	—	568	1	no	—
104	Q₈	—	568	2	no	—
105	Q₈	—	568	1	no	—
106	Q₈	—	568	1	no	—
107	C₂×C₄	—	568	1	no	—
108	C₂×C₄	—	568	1	no	—
109	C₂×C₄	#14556044	1	1	yes	characteristic
110	C₂×C₄	#14556048	1	1	yes	characteristic
111	C₂×C₄	#14556050	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	#14556073	1	1	yes	characteristic
117	C₈	Dic₅₆₈	1	1	yes	characteristic
118	C₂²	#17985742	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	1136	1	no	—
121	C₄	—	1136	1	no	—
122	C₄	#17985804	1	1	yes	characteristic
123	C₄	#17985805	1	1	yes	characteristic
124	C₄	#17985808	1	1	yes	characteristic
125	C₄	#17985816	1	1	yes	characteristic
126	C₂	#22586380	1	1	yes	characteristic
127	C₂	#22586382	1	1	yes	characteristic
128	C₂	#22586390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22587937

C₂

yes

maximal, characteristic

#22586410

C₂

yes

maximal, characteristic

C₇₁×#1367

C₂

yes

maximal, characteristic

#17986404

C₄

yes

characteristic

#17986397

—

#17986379

C₄

yes

characteristic

#17986350

—

C₄⋊(C₇₁⋊C₁₆)

—

C₄⋊C₁₆×C₇₁

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

#14556128

—

#14556128

—

C₂×C₇₁⋊C₁₆

—

—

—

—

—

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₃₂

yes

characteristic

C₈×C₇₁

Q₃₂

yes

characteristic

C₂²×C₇₁

#331

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

C₇₁

#7095

yes

characteristic, complemented, Sylow

#500

—

#498

—

142

—

#494

—

#487

—

142

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

284

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

284

—

#107

—

284

—

#106

—

284

—

#106

—

284

—

#103

—

284

—

#103

—

284

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₄×C₈

D₂₈₄

yes

characteristic

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

Q₁₆

—

568

—

C₂×C₈

—

C₂×C₈

#11765850

yes

characteristic

C₂×C₈

D₅₆₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

100

C₄⋊C₄

—

284

—

101

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

102

C₁₆

—

103

C₁₆

—

568

—

104

Q₈

—

568

—

105

Q₈

—

568

—

106

Q₈

—

568

—

107

C₂×C₄

—

568

—

108

C₂×C₄

—

568

—

109

C₂×C₄

#14556044

yes

characteristic

110

C₂×C₄

#14556048

yes

characteristic

111

C₂×C₄

#14556050

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#14556073

yes

characteristic

117

C₈

Dic₅₆₈

yes

characteristic

118

C₂²

#17985742

yes

characteristic

119

C₄

—

120

C₄

—

1136

—

121

C₄

—

1136

—

122

C₄

#17985804

yes

characteristic

123

C₄

#17985805

yes

characteristic

124

C₄

#17985808

yes

characteristic

125

C₄

#17985816

yes

characteristic

126

C₂

#22586380

yes

characteristic

127

C₂

#22586382

yes

characteristic

128

C₂

#22586390

yes

characteristic

	subgroup	G/H
center	C₂²	#17985742
commutator	C₄×C₈×C₇₁	C₂×C₄
Frattini	C₈²	C₂×D₇₁
Fitting	C₇₁×#1367	C₂
socle, prime radical	C₂²×C₇₁	#331
cosocle	C₈²×C₇₁	C₂²
hypercenter	#1367	D₇₁
hypocenter	C₇₁	#7095
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#17985742

commutator

C₄×C₈×C₇₁

C₂×C₄

Frattini

C₈²

C₂×D₇₁

Fitting

C₇₁×#1367

C₂

socle, prime radical

C₂²×C₇₁

#331

cosocle

C₈²×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
68. C₇₁	#7095	semidirect

H ≅ G/N

type

68. C₇₁

#7095

semidirect

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

subgroup	index
#22587937	2
#22586410	2
C₇₁×#1367	2
#7095	71

subgroup

index

#22587937

#22586410

C₇₁×#1367

#7095

Subgroups — #31884181

Conjugacy classes of subgroups

Complements

Maximal subgroups