Group #31891149 — subgroups

Subgroups — #31891149

Order 18176.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#22588512	C₂	1	1	yes	maximal, characteristic
2	#22586632	C₂	1	1	yes	maximal, characteristic
3	#22586198	C₂	1	4	yes	maximal
4	C₇₁×#1835	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	2	yes	—
6	C₂×#14556064	—	2	2	no	—
7	#17986023	C₂²	1	2	yes	—
8	#17985721	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	#382×C₇₁	C₂²	1	2	yes	—
11	#340×C₇₁	—	2	2	no	—
12	C₂×#11765849	—	2	2	no	—
13	#14556064	—	4	4	no	—
14	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
17	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₇₁	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
20	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₇₁	Q₈	1	2	yes	—
23	#95×C₇₁	—	2	4	no	—
24	#11765849	—	4	4	no	—
25	C₂×C₇₁⋊C₈	—	2	4	no	—
26	C₂×C₇₁⋊C₈	—	4	2	no	—
27	C₂²×C₄×C₇₁	—	4	2	no	—
28	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	#41	1	2	yes	—
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
33	C₄⋊C₄×C₇₁	—	4	2	no	—
34	C₄⋊C₄×C₇₁	—	4	2	no	—
35	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₇₁	#41	1	2	yes	—
38	C₂³×C₇₁	#124	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	—	8	4	no	—
41	C₂×C₄×C₇₁	—	8	2	no	—
42	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#117	1	2	yes	—
44	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
45	C₂×C₄×C₇₁	#125	1	2	yes	—
46	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₇₁	#137	1	2	yes	—
48	C₈×C₇₁	—	2	4	no	—
49	C₇₁⋊C₈	—	4	4	no	—
50	C₇₁⋊C₈	—	8	2	no	—
51	C₂²×C₇₁	—	2	2	no	—
52	C₂²×C₇₁	#382	1	1	yes	characteristic
53	C₂²×C₇₁	#489	1	2	yes	—
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	—	16	2	no	—
57	C₄×C₇₁	#428	1	2	yes	—
58	C₄×C₇₁	#490	1	1	yes	characteristic
59	C₄×C₇₁	#491	1	1	yes	characteristic
60	#8794	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	#1833	1	2	yes	—
63	C₂×C₇₁	#1836	1	1	yes	characteristic
64	#1848	—	71	1	no	—
65	#1835	D₇₁	1	1	yes	characteristic
66	#1452	—	71	1	no	—
67	#1286	—	71	4	no	—
68	C₇₁	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	—	71	2	no	—
70	C₂×#108	—	142	2	no	—
71	#423	—	71	2	no	—
72	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	#382	C₂×D₇₁	1	2	yes	—
74	#340	—	2	2	no	—
75	#322	—	71	4	no	—
76	C₂×M₁₆	—	142	2	no	—
77	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
80	#108	—	284	4	no	—
81	C₄⋊C₈	—	142	4	no	—
82	C₄⋊C₈	—	142	2	no	—
83	C₂²⋊C₈	—	142	2	no	—
84	C₄×C₈	—	71	4	no	—
85	C₄×C₈	Dic₁₄₂	1	2	yes	—
86	C₄×C₈	C₄×D₇₁	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
90	C₂²×C₄	#11765864	1	2	yes	—
91	M₁₆	—	284	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	284	2	no	—
95	C₂×C₈	—	142	4	no	—
96	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
101	C₄²	#11765866	1	2	yes	—
102	C₂³	#14556115	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#14556086	1	2	yes	—
108	C₂×C₄	#14556117	1	1	yes	characteristic
109	C₂×C₄	#14556119	1	2	yes	—
110	C₂×C₄	#14556153	1	2	yes	—
111	C₂×C₄	#14556158	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	284	4	no	—
114	C₈	—	568	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#17985864	1	1	yes	characteristic
117	C₂²	#17986358	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#17986042	1	2	yes	—
122	C₄	#17986365	1	1	yes	characteristic
123	C₄	#17986369	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#22588463	1	2	yes	—
126	C₂	#22588473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal, characteristic

#22586632

C₂

yes

maximal, characteristic

#22586198

C₂

yes

maximal

C₇₁×#1835

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556064

—

#17986023

C₂²

yes

—

#17985721

C₄

yes

complemented

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×C₇₁

C₂²

yes

—

#340×C₇₁

—

C₂×#11765849

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

#95×C₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#382

yes

characteristic

C₂²×C₇₁

#489

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#428

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

#1833

yes

—

C₂×C₇₁

yes

—

D₇₁

yes

characteristic

#1452

—

#1286

—

C₇₁

#8794

yes

characteristic, complemented, Sylow

#431

—

C₂×#108

—

142

—

#423

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

C₂×D₇₁

yes

—

#340

—

#322

—

C₂×M₁₆

—

142

—

C₂²×C₈

C₂²⋊D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#108

—

284

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

C₄×C₈

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₄×D₇₁

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

#11765864

yes

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×D₇₁

yes

characteristic

100

C₄²

C₂×Dic₁₄₂

yes

characteristic

101

C₄²

#11765866

yes

—

102

C₂³

#14556115

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

107

C₂×C₄

#14556086

yes

—

108

C₂×C₄

#14556117

yes

characteristic

109

C₂×C₄

#14556119

yes

—

110

C₂×C₄

#14556153

yes

—

111

C₂×C₄

#14556158

yes

—

112

C₈

—

113

C₈

—

284

—

114

C₈

—

568

—

115

C₂²

—

116

C₂²

#17985864

yes

characteristic

117

C₂²

#17986358

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#17986042

yes

—

122

C₄

#17986365

yes

characteristic

123

C₄

#17986369

yes

characteristic

124

C₂

—

125

C₂

#22588463

yes

—

126

C₂

#22588473

yes

characteristic

	subgroup	G/H
center	C₂²	#17985864
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1835	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1835	D₇₁
hypocenter	C₇₁	#8794
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#17985864

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1835

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #17985721	C₄	semidirect
68. C₇₁	#8794	semidirect

H ≅ G/N

type

8. #17985721

C₄

semidirect

68. C₇₁

#8794

semidirect

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

subgroup	index
#22588512	2
#22586632	2
#22586198	2
C₇₁×#1835	2
#8794	71

subgroup

index

C₇₁×#1835

Subgroups — #31891149

Conjugacy classes of subgroups

Complements

Maximal subgroups