Group #31885016 — subgroups

Subgroups — #31885016

Order 18176.

140 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 47 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	C₂²⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	#22586434	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2216	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	1	no	—
5	C₂×C₇₁⋊C₃₂	—	2	1	no	—
6	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	#467×C₇₁	—	2	1	no	—
9	#366×C₇₁	C₄	1	1	yes	characteristic
10	#358×C₇₁	—	2	1	no	—
11	C₂×#11765849	—	4	1	no	—
12	#14556064	—	2	2	no	—
13	#14556064	—	4	1	no	—
14	C₇₁⋊C₃₂	—	4	1	no	—
15	C₂×Q₁₆×C₇₁	—	2	1	no	—
16	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈×C₇₁	—	4	1	no	—
18	Q₃₂×C₇₁	—	4	1	no	—
19	C₂×C₁₆×C₇₁	—	4	1	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
22	#107×C₇₁	—	2	1	no	—
23	#103×C₇₁	—	4	1	no	—
24	#11765849	—	4	2	no	—
25	#11765849	—	8	1	no	—
26	C₂×C₇₁⋊C₈	—	8	1	no	—
27	C₂×Q₈×C₇₁	—	4	1	no	—
28	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
29	Q₁₆×C₇₁	—	4	1	no	—
30	Q₁₆×C₇₁	—	4	1	no	—
31	C₂×C₈×C₇₁	—	2	1	no	—
32	C₂×C₈×C₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	—	4	1	no	—
36	C₄⋊C₄×C₇₁	—	8	1	no	—
37	C₂²⋊C₄×C₇₁	—	8	1	no	—
38	C₁₆×C₇₁	—	2	1	no	—
39	C₁₆×C₇₁	—	2	1	no	—
40	C₂³×C₇₁	SD₃₂	1	1	yes	characteristic
41	Q₈×C₇₁	—	8	1	no	—
42	Q₈×C₇₁	—	8	1	no	—
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	8	1	no	—
46	C₂×C₄×C₇₁	—	8	1	no	—
47	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
48	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
49	C₈×C₇₁	—	2	1	no	—
50	C₈×C₇₁	#99	1	1	yes	characteristic
51	C₈×C₇₁	#104	1	1	yes	characteristic
52	C₇₁⋊C₈	—	8	2	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	#357	1	1	yes	characteristic
57	C₄×C₇₁	—	2	1	no	—
58	C₄×C₇₁	—	16	1	no	—
59	C₄×C₇₁	—	16	1	no	—
60	C₄×C₇₁	#327	1	1	yes	characteristic
61	C₄×C₇₁	#360	1	1	yes	characteristic
62	#7421	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	4	1	no	—
64	C₂×C₇₁	#1347	1	1	yes	characteristic
65	C₂×C₇₁	#1417	1	1	yes	characteristic
66	C₂×C₇₁	#1420	1	1	yes	characteristic
67	#2216	D₇₁	1	1	yes	characteristic
68	C₂²⋊C₃₂	—	71	1	no	—
69	#1381	—	71	1	no	—
70	C₇₁	#7421	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic
72	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	#467	—	2	1	no	—
74	C₂×#108	—	142	1	no	—
75	C₂×C₃₂	—	142	1	no	—
76	#366	Dic₇₁	1	1	yes	characteristic
77	#358	—	2	1	no	—
78	C₂×Q₁₆	—	2	1	no	—
79	C₂×M₁₆	—	284	1	no	—
80	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
81	C₂²⋊Q₈	—	4	1	no	—
82	Q₃₂	—	4	1	no	—
83	C₂×C₁₆	—	4	1	no	—
84	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
85	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
86	#108	—	284	1	no	—
87	#108	—	142	2	no	—
88	#107	—	2	1	no	—
89	#103	—	4	1	no	—
90	C₃₂	—	284	1	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂²×C₄	#11765850	1	1	yes	characteristic
93	Q₁₆	—	4	1	no	—
94	Q₁₆	—	4	1	no	—
95	M₁₆	—	284	2	no	—
96	M₁₆	—	568	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	568	1	no	—
100	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
101	C₂×C₈	#11765853	1	1	yes	characteristic
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	8	1	no	—
104	C₂²⋊C₄	—	8	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	2	1	no	—
107	C₂³	#14556072	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	8	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	#14556049	1	1	yes	characteristic
115	C₂×C₄	#14556069	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	568	2	no	—
118	C₈	#14556043	1	1	yes	characteristic
119	C₈	#14556055	1	1	yes	characteristic
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#17985800	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	16	1	no	—
126	C₄	—	16	1	no	—
127	C₄	#17985733	1	1	yes	characteristic
128	C₄	#17985809	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	#22586349	1	1	yes	characteristic
131	C₂	#22586524	1	1	yes	characteristic
132	C₂	#22586533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic

#22586434

C₂

yes

maximal, characteristic

C₇₁×#2216

C₂

yes

maximal, characteristic

C₂×#14556064

—

C₂×C₇₁⋊C₃₂

—

C₂×Q₃₂×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

#467×C₇₁

—

#366×C₇₁

C₄

yes

characteristic

#358×C₇₁

—

C₂×#11765849

—

#14556064

—

#14556064

—

C₇₁⋊C₃₂

—

C₂×Q₁₆×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²⋊Q₈×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#103×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

SD₃₂

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#357

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

characteristic

C₂²⋊C₃₂

—

#1381

—

C₇₁

#7421

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₁

yes

characteristic

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#467

—

C₂×#108

—

142

—

C₂×C₃₂

—

142

—

#366

Dic₇₁

yes

characteristic

#358

—

C₂×Q₁₆

—

C₂×M₁₆

—

284

—

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

#108

—

284

—

#108

—

142

—

#107

—

#103

—

C₃₂

—

284

—

C₂×Q₈

—

C₂²×C₄

#11765850

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

284

—

M₁₆

—

568

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

568

—

100

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

101

C₂×C₈

#11765853

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₁₆

—

106

C₁₆

—

107

C₂³

#14556072

yes

characteristic

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#14556049

yes

characteristic

115

C₂×C₄

#14556069

yes

characteristic

116

C₈

—

117

C₈

—

568

—

118

C₈

#14556043

yes

characteristic

119

C₈

#14556055

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17985800

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#17985733

yes

characteristic

128

C₄

#17985809

yes

characteristic

129

C₂

—

130

C₂

#22586349

yes

characteristic

131

C₂

#22586524

yes

characteristic

132

C₂

#22586533

yes

characteristic

	subgroup	G/H
center	C₂²	#17985800
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₂²×C₁₆	C₂×D₇₁
Fitting	C₇₁×#2216	C₂
socle	C₂²×C₇₁	#357
prime radical	C₂³×C₇₁	SD₃₂
cosocle	C₂²×C₁₆×C₇₁	C₂²
hypercenter	#2216	D₇₁
hypocenter	C₇₁	#7421
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2216

subgroup

G/H

center

C₂²

#17985800

commutator

C₂×C₁₆×C₇₁

C₂×C₄

Frattini

C₂²×C₁₆

C₂×D₇₁

Fitting

C₇₁×#2216

C₂

socle

C₂²×C₇₁

#357

prime radical

C₂³×C₇₁

SD₃₂

cosocle

C₂²×C₁₆×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2216

N	H ≅ G/N	type
70. C₇₁	#7421	semidirect

H ≅ G/N

type

70. C₇₁

#7421

semidirect

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

subgroup	index
C₂²⋊(C₇₁⋊C₃₂)	2
#22586434	2
C₇₁×#2216	2
#7421	71

subgroup

index

C₂²⋊(C₇₁⋊C₃₂)

#22586434

C₇₁×#2216

#7421

Subgroups — #31885016

Conjugacy classes of subgroups

Complements

Maximal subgroups