Group #31912188 — subgroups

Subgroups — #31912188

Order 18176.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22590615	C₂	1	1	yes	maximal, characteristic
2	#22590613	C₂	1	2	yes	maximal
3	#22587957	C₂	1	2	yes	maximal
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986402	—	2	2	no	—
8	#17986378	—	2	2	no	—
9	#17985825	—	2	2	no	—
10	#17985824	—	2	4	no	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₇₁	C₂²	1	2	yes	—
14	#366×C₇₁	—	2	2	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765855	—	4	4	no	—
17	#14556128	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
19	#14556062	—	2	2	no	—
20	#14556062	—	2	2	no	—
21	#14556061	—	4	4	no	—
22	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₇₁	—	2	2	no	—
25	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
26	#107×C₇₁	—	2	2	no	—
27	#107×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	4	2	no	—
30	#11765855	—	8	8	no	—
31	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
32	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₇₁⋊C₁₆	Q₁₆	1	4	yes	complemented
36	C₂×Q₈×C₇₁	—	4	2	no	—
37	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₁	—	4	2	no	—
40	C₄⋊C₄×C₇₁	—	4	2	no	—
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	Q₁₆	1	4	yes	complemented
43	Q₈×C₇₁	—	8	4	no	—
44	C₂×C₄×C₇₁	—	8	2	no	—
45	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₁	—	8	2	no	—
49	Dic₁₄₂	—	8	4	no	—
50	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
51	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
52	C₇₁⋊C₈	D₁₆	1	4	yes	—
53	C₇₁⋊C₈	Q₃₂	1	4	yes	—
54	C₂²×C₇₁	#493	1	1	yes	characteristic
55	C₄×C₇₁	—	16	2	no	—
56	C₄×C₇₁	#494	1	1	yes	characteristic
57	C₄×C₇₁	#494	1	1	yes	characteristic
58	C₄×C₇₁	C₂×D₁₆	1	2	yes	—
59	C₄×C₇₁	C₂×Q₃₂	1	2	yes	—
60	Dic₇₁	—	16	2	no	—
61	#12521	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	#1713	1	1	yes	characteristic
63	C₂×C₇₁	#2246	1	1	yes	characteristic
64	C₂×C₇₁	#2247	1	1	yes	characteristic
65	#2252	—	71	2	no	—
66	#2252	—	71	1	no	—
67	#2252	D₇₁	1	1	yes	characteristic
68	#1713	—	71	2	no	—
69	C₈×C₁₆	—	71	1	no	—
70	C₇₁	#12521	1	1	yes	characteristic, complemented, Sylow
71	#500	—	71	2	no	—
72	#500	—	142	2	no	—
73	#500	C₂×D₇₁	1	2	yes	—
74	#494	—	142	2	no	—
75	#366	—	2	2	no	—
76	#366	—	142	4	no	—
77	#366	—	142	2	no	—
78	C₄×C₁₆	—	71	1	no	—
79	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
80	C₈²	—	71	1	no	—
81	C₂×Q₁₆	—	284	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	142	2	no	—
84	C₂×C₁₆	—	71	2	no	—
85	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	284	4	no	—
89	#107	—	142	2	no	—
90	#107	—	142	2	no	—
91	C₄×C₈	—	71	1	no	—
92	C₄×C₈	—	71	1	no	—
93	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	284	2	no	—
96	Q₁₆	—	568	8	no	—
97	C₂×C₈	—	71	2	no	—
98	C₂×C₈	—	71	2	no	—
99	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	284	2	no	—
104	C₄⋊C₄	—	284	2	no	—
105	C₄²	D₄×D₇₁	1	1	yes	characteristic
106	C₁₆	—	71	4	no	—
107	C₁₆	#11765855	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	568	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	568	2	no	—
112	C₂×C₄	#14556124	1	1	yes	characteristic
113	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
114	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
115	C₈	—	71	4	no	—
116	C₈	—	71	4	no	—
117	C₈	Q₁₆×D₇₁	1	2	yes	—
118	C₈	C₂×#11765855	1	2	yes	—
119	C₂²	#17986374	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	1136	2	no	—
122	C₄	#17986377	1	1	yes	characteristic
123	C₄	#17986378	1	1	yes	characteristic
124	C₄	D₁₆×D₇₁	1	2	yes	—
125	C₄	Q₃₂×D₇₁	1	2	yes	—
126	C₂	#22587957	1	1	yes	characteristic
127	C₂	#22590588	1	1	yes	characteristic
128	C₂	#22590591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic

#22590613

C₂

yes

maximal

#22587957

C₂

yes

maximal

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#2252

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

—

—

—

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

#500×C₇₁

C₂²

yes

—

#366×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765855

—

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

#14556062

—

#14556061

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

#11765855

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

Q₁₆

yes

complemented

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₁₆

yes

—

C₈×C₇₁

C₂×Q₁₆

yes

—

C₇₁⋊C₈

D₁₆

yes

—

C₇₁⋊C₈

Q₃₂

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×D₁₆

yes

—

C₄×C₇₁

C₂×Q₃₂

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1713

—

C₈×C₁₆

—

C₇₁

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

142

—

#500

C₂×D₇₁

yes

—

#494

—

142

—

#366

—

#366

—

142

—

#366

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₂×Q₁₆

—

284

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#107

—

#107

—

#107

—

284

—

#107

—

142

—

#107

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

Q₁₆

—

568

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

284

—

104

C₄⋊C₄

—

284

—

105

C₄²

D₄×D₇₁

yes

characteristic

106

C₁₆

—

107

C₁₆

#11765855

yes

complemented

108

Q₈

—

109

Q₈

—

568

—

110

C₂×C₄

—

111

C₂×C₄

—

568

—

112

C₂×C₄

#14556124

yes

characteristic

113

C₂×C₄

D₈×D₇₁

yes

characteristic

114

C₂×C₄

D₈×D₇₁

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₇₁

yes

—

118

C₈

C₂×#11765855

yes

—

119

C₂²

#17986374

yes

characteristic

120

C₄

—

121

C₄

—

1136

—

122

C₄

#17986377

yes

characteristic

123

C₄

#17986378

yes

characteristic

124

C₄

D₁₆×D₇₁

yes

—

125

C₄

Q₃₂×D₇₁

yes

—

126

C₂

#22587957

yes

characteristic

127

C₂

#22590588

yes

characteristic

128

C₂

#22590591

yes

characteristic

	subgroup	G/H
center	C₂²	#17986374
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2252	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	#2252	D₇₁
hypocenter	C₇₁	#12521
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#17986374

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2252

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
35. C₇₁⋊C₁₆	Q₁₆	semidirect
42. C₁₆×C₇₁	Q₁₆	semidirect
70. C₇₁	#12521	semidirect
107. C₁₆	#11765855	semidirect

H ≅ G/N

type

35. C₇₁⋊C₁₆

Q₁₆

semidirect

42. C₁₆×C₇₁

Q₁₆

semidirect

70. C₇₁

#12521

semidirect

107. C₁₆

#11765855

semidirect

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

subgroup	index
#22590615	2
#22590613	2
#22587957	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#2252	2
#12521	71

subgroup

index

#22590615

#22590613

#22587957

C₁₆×C₇₁⋊C₈

C₇₁×#2252

#12521

Subgroups — #31912188

Conjugacy classes of subgroups

Complements

Maximal subgroups