Group #31918420 — subgroups

Subgroups — #31918420

Order 18176.

202 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 75 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, complemented
2	#22586518	C₂	1	4	yes	maximal, complemented
3	C₄×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#2246	C₂	1	1	yes	maximal, characteristic
5	#17986404	—	2	1	no	—
6	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
7	#17985825	—	2	1	no	—
8	#17985825	C₂²	1	2	yes	—
9	C₂×D₁₆×C₇₁	—	2	1	no	—
10	C₂×D₁₆×C₇₁	C₂²	1	2	yes	—
11	#493×C₇₁	—	2	1	no	—
12	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	#14556128	—	4	1	no	—
14	#14556062	—	2	2	no	—
15	#14556062	—	4	1	no	—
16	C₇₁⋊C₃₂	—	2	2	no	—
17	C₂×D₈×C₇₁	—	2	2	no	—
18	C₂×D₈×C₇₁	—	4	1	no	—
19	#127×C₇₁	—	4	1	no	—
20	D₁₆×C₇₁	—	2	4	no	—
21	D₁₆×C₇₁	—	4	2	no	—
22	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₄₂	—	8	1	no	—
26	C₄⋊Dic₇₁	—	4	2	no	—
27	C₄⋊Dic₇₁	—	8	1	no	—
28	C₂×D₄×C₇₁	—	4	2	no	—
29	C₂×D₄×C₇₁	—	8	1	no	—
30	C₂×D₄×C₇₁	—	8	1	no	—
31	D₈×C₇₁	—	4	2	no	—
32	D₈×C₇₁	—	4	2	no	—
33	D₈×C₇₁	—	8	2	no	—
34	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₇₁	D₈	1	1	yes	characteristic
37	C₁₆×C₇₁	SD₁₆	1	2	yes	—
38	C₁₆×C₇₁	#41	1	2	yes	—
39	C₂³×C₇₁	—	8	2	no	—
40	D₄×C₇₁	—	8	2	no	—
41	D₄×C₇₁	—	8	2	no	—
42	D₄×C₇₁	—	16	2	no	—
43	D₄×C₇₁	—	16	2	no	—
44	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₇₁	—	8	2	no	—
48	Dic₁₄₂	—	16	2	no	—
49	C₈×C₇₁	SD₃₂	1	2	yes	—
50	C₈×C₇₁	#124	1	1	yes	characteristic
51	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₂²×C₇₁	—	16	2	no	—
53	C₂²×C₇₁	—	16	2	no	—
54	C₂²×C₇₁	—	16	2	no	—
55	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₁	D₃₂	1	2	yes	—
57	C₄×C₇₁	SD₆₄	1	2	yes	—
58	C₄×C₇₁	#486	1	1	yes	characteristic
59	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
60	Dic₇₁	—	16	2	no	—
61	#13719	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	32	2	no	—
63	C₂×C₇₁	#2240	1	1	yes	characteristic
64	C₂×C₇₁	C₂×D₃₂	1	1	yes	characteristic
65	C₂×C₇₁	C₂×SD₆₄	1	1	yes	characteristic
66	#2252	—	71	1	no	—
67	#2246	D₇₁	1	1	yes	characteristic
68	#1415	—	71	4	no	—
69	C₄×C₃₂	—	71	1	no	—
70	C₇₁	#13719	1	1	yes	characteristic, complemented, Sylow
71	C₂×D₁₆	—	2	1	no	—
72	C₂×D₁₆	C₂×D₇₁	1	2	yes	—
73	#500	—	142	1	no	—
74	#493	—	2	1	no	—
75	C₂×C₃₂	—	71	2	no	—
76	#366	—	71	2	no	—
77	#366	—	142	1	no	—
78	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
79	C₂×D₈	—	2	2	no	—
80	C₂×D₈	—	4	1	no	—
81	C₄⋊Q₈	—	284	1	no	—
82	#127	—	4	1	no	—
83	D₁₆	—	2	4	no	—
84	D₁₆	—	4	2	no	—
85	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
87	#107	—	284	1	no	—
88	#107	—	142	2	no	—
89	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
90	C₃₂	—	142	2	no	—
91	C₂×Q₈	—	568	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	8	1	no	—
94	C₂×D₄	—	8	1	no	—
95	D₈	—	4	2	no	—
96	D₈	—	4	2	no	—
97	D₈	—	8	2	no	—
98	C₂×C₈	#11765850	1	1	yes	characteristic
99	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
100	C₄⋊C₄	—	284	2	no	—
101	C₄⋊C₄	—	568	1	no	—
102	C₄²	#11765850	1	1	yes	characteristic
103	C₁₆	#11765853	1	2	yes	—
104	C₁₆	#11765864	1	2	yes	—
105	C₂³	—	8	2	no	—
106	Q₈	—	1136	2	no	—
107	D₄	—	8	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	16	2	no	—
110	D₄	—	16	2	no	—
111	C₂×C₄	—	568	2	no	—
112	C₂×C₄	#14556069	1	1	yes	characteristic
113	C₂×C₄	#14556069	1	1	yes	characteristic
114	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
115	C₈	#14556072	1	2	yes	—
116	C₈	#14556115	1	1	yes	characteristic
117	C₈	C₂×#11765853	1	1	yes	characteristic
118	C₂²	—	16	2	no	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	C₂×#14556069	1	1	yes	characteristic
122	C₄	—	1136	2	no	—
123	C₄	#17985834	1	2	yes	—
124	C₄	#17985837	1	2	yes	—
125	C₄	#17986345	1	1	yes	characteristic
126	C₄	C₂×#14556072	1	1	yes	characteristic
127	C₂	—	32	2	no	—
128	C₂	#22590563	1	1	yes	characteristic
129	C₂	C₂×#17985834	1	1	yes	characteristic
130	C₂	C₂×#17985837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590615

C₂

yes

maximal, characteristic, complemented

#22586518

C₂

yes

maximal, complemented

C₄×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₇₁×#2246

C₂

yes

maximal, characteristic

#17986404

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

#17985825

—

#17985825

C₂²

yes

—

C₂×D₁₆×C₇₁

—

C₂×D₁₆×C₇₁

C₂²

yes

—

#493×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

—

—

—

C₇₁⋊C₃₂

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

—

D₁₆×C₇₁

—

D₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

#41

yes

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

D₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

SD₃₂

yes

—

C₈×C₇₁

#124

yes

characteristic

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

D₃₂

yes

—

C₄×C₇₁

SD₆₄

yes

—

C₄×C₇₁

#486

yes

characteristic

C₄×C₇₁

C₂×SD₃₂

yes

characteristic

Dic₇₁

—

#13719

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#2240

yes

characteristic

C₂×C₇₁

C₂×D₃₂

yes

characteristic

C₂×C₇₁

C₂×SD₆₄

yes

characteristic

#2252

—

#2246

D₇₁

yes

characteristic

#1415

—

C₄×C₃₂

—

C₇₁

#13719

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

C₂×D₇₁

yes

—

#500

—

142

—

#493

—

C₂×C₃₂

—

#366

—

#366

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

284

—

#127

—

D₁₆

—

D₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

284

—

#107

—

142

—

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

C₃₂

—

142

—

C₂×Q₈

—

568

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

#11765850

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

100

C₄⋊C₄

—

284

—

101

C₄⋊C₄

—

568

—

102

C₄²

#11765850

yes

characteristic

103

C₁₆

#11765853

yes

—

104

C₁₆

#11765864

yes

—

105

C₂³

—

106

Q₈

—

1136

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

568

—

112

C₂×C₄

#14556069

yes

characteristic

113

C₂×C₄

#14556069

yes

characteristic

114

C₂×C₄

C₂×#11765850

yes

characteristic

115

C₈

#14556072

yes

—

116

C₈

#14556115

yes

characteristic

117

C₈

C₂×#11765853

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#14556069

yes

characteristic

122

C₄

—

1136

—

123

C₄

#17985834

yes

—

124

C₄

#17985837

yes

—

125

C₄

#17986345

yes

characteristic

126

C₄

C₂×#14556072

yes

characteristic

127

C₂

—

128

C₂

#22590563

yes

characteristic

129

C₂

C₂×#17985834

yes

characteristic

130

C₂

C₂×#17985837

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556069
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting, prime radical	C₇₁×#2246	C₂
socle	C₂²×C₇₁	C₂×D₁₆
hypercenter	#2246	D₇₁
hypocenter	C₇₁	#13719
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

subgroup

G/H

center

C₂²

C₂×#14556069

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting, prime radical

C₇₁×#2246

C₂

socle

C₂²×C₇₁

C₂×D₁₆

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2246

N	H ≅ G/N	type
1. #22590615	C₂	semidirect
2. #22586518	C₂	semidirect
3. C₄×C₇₁⋊C₃₂	C₂	semidirect
70. C₇₁	#13719	semidirect

H ≅ G/N

type

1. #22590615

C₂

semidirect

2. #22586518

C₂

semidirect

3. C₄×C₇₁⋊C₃₂

C₂

semidirect

70. C₇₁

#13719

semidirect

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

subgroup	index
#22590615	2
#22586518	2
C₄×C₇₁⋊C₃₂	2
C₇₁×#2246	2
#13719	71

subgroup

index

#22590615

#22586518

C₄×C₇₁⋊C₃₂

C₇₁×#2246

#13719

Subgroups — #31918420

Conjugacy classes of subgroups

Complements

Maximal subgroups