Group #31912191 — subgroups

Subgroups — #31912191

Order 18176.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590620	C₂	1	2	yes	maximal
2	#22590615	C₂	1	1	yes	maximal, characteristic
3	#22587949	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	#17986351	—	2	2	no	—
8	#17986349	—	2	2	no	—
9	#17985826	—	2	4	no	—
10	#17985825	—	2	2	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	#14556128	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
18	#14556062	—	2	2	no	—
19	#14556062	—	2	2	no	—
20	#14556052	—	4	4	no	—
21	#14556051	—	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	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
31	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
32	C₄⋊Dic₇₁	—	4	2	no	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₇₁⋊C₁₆	Q₁₆	1	4	yes	—
35	C₂×Q₈×C₇₁	—	4	2	no	—
36	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	—	4	2	no	—
39	C₄⋊C₄×C₇₁	—	4	2	no	—
40	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₁	Q₁₆	1	4	yes	—
42	Q₈×C₇₁	—	8	4	no	—
43	C₂×C₄×C₇₁	—	8	2	no	—
44	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₁	—	8	2	no	—
48	Dic₁₄₂	—	8	4	no	—
49	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
50	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
51	C₇₁⋊C₈	—	2	2	no	—
52	C₇₁⋊C₈	—	2	2	no	—
53	C₂²×C₇₁	#493	1	1	yes	characteristic
54	C₄×C₇₁	—	16	2	no	—
55	C₄×C₇₁	#494	1	1	yes	characteristic
56	C₄×C₇₁	#494	1	1	yes	characteristic
57	C₄×C₇₁	#508	1	2	yes	—
58	C₄×C₇₁	#508	1	2	yes	—
59	Dic₇₁	—	16	2	no	—
60	#12522	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	#1713	1	1	yes	characteristic
62	C₂×C₇₁	#2249	1	1	yes	characteristic
63	C₂×C₇₁	#2249	1	1	yes	characteristic
64	#2254	—	71	2	no	—
65	#2252	—	71	1	no	—
66	#2252	D₇₁	1	1	yes	characteristic
67	#1710	—	71	2	no	—
68	C₈×C₁₆	—	71	1	no	—
69	C₇₁	#12522	1	1	yes	characteristic, complemented, Sylow
70	#500	—	71	2	no	—
71	#500	C₂×D₇₁	1	2	yes	—
72	#487	—	142	2	no	—
73	#487	—	142	2	no	—
74	#367	—	142	4	no	—
75	#366	—	2	2	no	—
76	#366	—	142	2	no	—
77	C₄×C₁₆	—	71	1	no	—
78	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
79	C₈²	—	71	1	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	142	2	no	—
82	C₂×C₁₆	—	71	2	no	—
83	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	142	2	no	—
87	#107	—	142	2	no	—
88	#103	—	284	4	no	—
89	#103	—	284	4	no	—
90	C₄×C₈	—	71	1	no	—
91	C₄×C₈	—	71	1	no	—
92	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	284	2	no	—
95	C₂×C₈	—	71	2	no	—
96	C₂×C₈	—	71	2	no	—
97	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
98	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	284	2	no	—
102	C₄⋊C₄	—	284	2	no	—
103	C₄²	D₄×D₇₁	1	1	yes	characteristic
104	C₁₆	—	71	4	no	—
105	C₁₆	#11765855	1	4	yes	—
106	Q₈	—	8	4	no	—
107	Q₈	—	568	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	568	2	no	—
110	C₂×C₄	#14556124	1	1	yes	characteristic
111	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
112	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
113	C₈	—	142	2	no	—
114	C₈	—	142	2	no	—
115	C₈	Q₁₆×D₇₁	1	2	yes	—
116	C₈	C₂×#11765855	1	2	yes	—
117	C₂²	#17986374	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1136	2	no	—
120	C₄	#17986377	1	1	yes	characteristic
121	C₄	#17986378	1	1	yes	characteristic
122	C₄	#17986428	1	2	yes	—
123	C₄	#17986432	1	2	yes	—
124	C₂	#22587957	1	1	yes	characteristic
125	C₂	#22590598	1	1	yes	characteristic
126	C₂	#22590602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590620

C₂

yes

maximal

#22590615

C₂

yes

maximal, characteristic

#22587949

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

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

—

—

—

—

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

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

Q₁₆

yes

—

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

—

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

—

C₇₁⋊C₈

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₄×C₇₁

#508

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1710

—

C₈×C₁₆

—

C₇₁

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₁

yes

—

#487

—

142

—

#487

—

142

—

#367

—

142

—

#366

—

#366

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#107

—

#107

—

#107

—

142

—

#107

—

142

—

#103

—

284

—

#103

—

284

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

284

—

102

C₄⋊C₄

—

284

—

103

C₄²

D₄×D₇₁

yes

characteristic

104

C₁₆

—

105

C₁₆

#11765855

yes

—

106

Q₈

—

107

Q₈

—

568

—

108

C₂×C₄

—

109

C₂×C₄

—

568

—

110

C₂×C₄

#14556124

yes

characteristic

111

C₂×C₄

D₈×D₇₁

yes

characteristic

112

C₂×C₄

D₈×D₇₁

yes

characteristic

113

C₈

—

142

—

114

C₈

—

142

—

115

C₈

Q₁₆×D₇₁

yes

—

116

C₈

C₂×#11765855

yes

—

117

C₂²

#17986374

yes

characteristic

118

C₄

—

119

C₄

—

1136

—

120

C₄

#17986377

yes

characteristic

121

C₄

#17986378

yes

characteristic

122

C₄

#17986428

yes

—

123

C₄

#17986432

yes

—

124

C₂

#22587957

yes

characteristic

125

C₂

#22590598

yes

characteristic

126

C₂

#22590602

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	#12522
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
69. C₇₁	#12522	semidirect

H ≅ G/N

type

69. C₇₁

#12522

semidirect

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

subgroup	index
#22590620	2
#22590615	2
#22587949	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#2252	2
#12522	71

subgroup

index

#22590620

#22590615

#22587949

C₁₆×C₇₁⋊C₈

C₇₁×#2252

#12522

Subgroups — #31912191

Conjugacy classes of subgroups

Complements

Maximal subgroups