Group #31885029 — subgroups

Subgroups — #31885029

Order 18176.

136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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	#22590448	C₂	1	1	yes	maximal, characteristic
2	#22586444	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1400	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₆₈	C₄	1	1	yes	characteristic
5	#17986240	—	2	1	no	—
6	C₂×#14556064	—	2	1	no	—
7	#17985827	C₄	1	1	yes	characteristic
8	#17985804	—	2	1	no	—
9	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	M₆₄×C₇₁	—	2	1	no	—
11	C₂×Dic₂₈₄	—	2	1	no	—
12	C₂×#11765849	—	4	1	no	—
13	#14556107	—	4	1	no	—
14	Dic₅₆₈	—	4	1	no	—
15	#14556064	—	2	2	no	—
16	#14556064	—	4	1	no	—
17	#14556062	—	2	1	no	—
18	#14556053	—	4	1	no	—
19	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
20	M₃₂×C₇₁	—	4	1	no	—
21	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
23	C₃₂×C₇₁	—	4	1	no	—
24	C₂×Dic₁₄₂	—	4	1	no	—
25	Dic₂₈₄	—	4	1	no	—
26	Dic₂₈₄	—	4	1	no	—
27	#11765849	—	4	2	no	—
28	#11765849	—	8	1	no	—
29	C₂×C₇₁⋊C₈	—	8	1	no	—
30	C₄⋊Dic₇₁	—	4	1	no	—
31	Dic₇₁⋊C₄	—	8	1	no	—
32	C₂²⋊Dic₇₁	—	8	1	no	—
33	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	1	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
38	C₁₆×C₇₁	—	2	1	no	—
39	C₁₆×C₇₁	—	2	1	no	—
40	C₂³×C₇₁	SD₃₂	1	1	yes	characteristic
41	C₂×C₄×C₇₁	—	2	1	no	—
42	C₂×C₄×C₇₁	—	2	1	no	—
43	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
44	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
45	C₂×Dic₇₁	—	8	1	no	—
46	C₂×Dic₇₁	—	8	1	no	—
47	Dic₁₄₂	—	8	1	no	—
48	Dic₁₄₂	—	8	1	no	—
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	—	4	1	no	—
55	C₂²×C₇₁	#357	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	#327	1	1	yes	characteristic
58	C₄×C₇₁	#360	1	1	yes	characteristic
59	Dic₇₁	—	16	1	no	—
60	Dic₇₁	—	16	1	no	—
61	#7425	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	—	4	1	no	—
64	C₂×C₇₁	#1347	1	1	yes	characteristic
65	#2219	—	71	1	no	—
66	#1400	D₇₁	1	1	yes	characteristic
67	#1385	—	71	1	no	—
68	C₇₁	#7425	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	71	1	no	—
70	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
71	#467	—	142	1	no	—
72	C₂×#108	—	142	1	no	—
73	M₆₄	—	2	1	no	—
74	#367	—	71	1	no	—
75	#358	—	142	1	no	—
76	C₂×Q₁₆	—	142	1	no	—
77	C₂×M₁₆	—	284	1	no	—
78	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
79	C₂²⋊Q₈	—	284	1	no	—
80	Q₃₂	—	284	1	no	—
81	M₃₂	—	4	1	no	—
82	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
84	#108	—	284	1	no	—
85	#108	—	142	2	no	—
86	#107	—	142	1	no	—
87	#103	—	284	1	no	—
88	C₃₂	—	4	1	no	—
89	C₂×Q₈	—	284	1	no	—
90	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
91	Q₁₆	—	284	1	no	—
92	Q₁₆	—	284	1	no	—
93	M₁₆	—	284	2	no	—
94	M₁₆	—	568	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	568	1	no	—
98	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
99	C₂×C₈	#11765854	1	1	yes	characteristic
100	C₄⋊C₄	—	284	1	no	—
101	C₄⋊C₄	—	568	1	no	—
102	C₂²⋊C₄	—	568	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	2	1	no	—
105	C₂³	#14556073	1	1	yes	characteristic
106	Q₈	—	568	1	no	—
107	Q₈	—	568	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	568	1	no	—
111	C₂×C₄	—	568	1	no	—
112	C₂×C₄	#14556050	1	1	yes	characteristic
113	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	568	2	no	—
116	C₈	#14556042	1	1	yes	characteristic
117	C₈	#14556054	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	#17985801	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	1136	1	no	—
123	C₄	—	1136	1	no	—
124	C₄	#17985734	1	1	yes	characteristic
125	C₄	#17985810	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#22586350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590448

C₂

yes

maximal, characteristic

#22586444

C₂

yes

maximal, characteristic

C₇₁×#1400

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₄

yes

—

—

C₄

yes

characteristic

#17985804

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

—

C₂×Dic₂₈₄

—

C₂×#11765849

—

#14556107

—

Dic₅₆₈

—

—

—

—

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²×C₄×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

SD₁₆

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

SD₃₂

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#102

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#357

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#7425

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1385

—

C₇₁

#7425

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

characteristic

#467

—

142

—

C₂×#108

—

142

—

M₆₄

—

#367

—

#358

—

142

—

C₂×Q₁₆

—

142

—

C₂×M₁₆

—

284

—

C₂²×C₈

D₂₈₄

yes

characteristic

C₂²⋊Q₈

—

284

—

Q₃₂

—

284

—

M₃₂

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

#108

—

284

—

#108

—

142

—

#107

—

142

—

#103

—

284

—

C₃₂

—

C₂×Q₈

—

284

—

C₂²×C₄

D₅₆₈

yes

characteristic

Q₁₆

—

284

—

Q₁₆

—

284

—

M₁₆

—

284

—

M₁₆

—

568

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

568

—

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₈

#11765854

yes

characteristic

100

C₄⋊C₄

—

284

—

101

C₄⋊C₄

—

568

—

102

C₂²⋊C₄

—

568

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

#14556073

yes

characteristic

106

Q₈

—

568

—

107

Q₈

—

568

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

568

—

111

C₂×C₄

—

568

—

112

C₂×C₄

#14556050

yes

characteristic

113

C₂×C₄

D₁₁₃₆

yes

characteristic

114

C₈

—

115

C₈

—

568

—

116

C₈

#14556042

yes

characteristic

117

C₈

#14556054

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#17985801

yes

characteristic

121

C₄

—

122

C₄

—

1136

—

123

C₄

—

1136

—

124

C₄

#17985734

yes

characteristic

125

C₄

#17985810

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#22586350

yes

characteristic

	subgroup	G/H
center	C₂	#22586350
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₂×M₃₂	C₂×D₇₁
Fitting	C₇₁×#1400	C₂
socle	C₂×C₇₁	#1347
prime radical	C₂³×C₇₁	SD₃₂
cosocle	C₂×M₃₂×C₇₁	C₂²
hypercenter	#1400	D₇₁
hypocenter	C₇₁	#7425
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #1400

subgroup

G/H

center

C₂

#22586350

commutator

C₂×C₁₆×C₇₁

C₂×C₄

Frattini

C₂×M₃₂

C₂×D₇₁

Fitting

C₇₁×#1400

C₂

socle

C₂×C₇₁

#1347

prime radical

C₂³×C₇₁

SD₃₂

cosocle

C₂×M₃₂×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #1400

N	H ≅ G/N	type
68. C₇₁	#7425	semidirect

H ≅ G/N

type

68. C₇₁

#7425

semidirect

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

subgroup	index
#22590448	2
#22586444	2
C₇₁×#1400	2
#7425	71

subgroup

index

#22590448

#22586444

C₇₁×#1400

#7425

Subgroups — #31885029

Conjugacy classes of subgroups

Complements

Maximal subgroups