Group #31918455 — subgroups

Subgroups — #31918455

Order 18176.

238 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₂×Dic₁₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17985837	C₂	1	2	yes	maximal, complemented
3	#22590598	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₆₈	C₂²	1	2	yes	—
7	C₂×#14556072	—	2	2	no	—
8	#17986382	—	2	2	no	—
9	Dic₁₁₃₆	—	2	2	no	—
10	#17985837	—	2	4	no	—
11	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₇₁	C₂²	1	2	yes	—
14	D₃₂×C₇₁	—	2	2	no	—
15	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₈₄	—	2	2	no	—
17	C₂×#11765853	—	4	2	no	—
18	#14556115	—	4	2	no	—
19	Dic₅₆₈	—	2	2	no	—
20	Dic₅₆₈	—	2	2	no	—
21	#14556072	—	4	4	no	—
22	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
23	C₈×Dic₇₁	D₄	1	1	yes	characteristic
24	C₇₁⋊C₃₂	D₄	1	2	yes	—
25	C₂×D₈×C₇₁	—	2	2	no	—
26	D₁₆×C₇₁	—	2	2	no	—
27	D₁₆×C₇₁	—	2	2	no	—
28	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
29	C₃₂×C₇₁	D₄	1	2	yes	—
30	C₂×Dic₁₄₂	—	4	2	no	—
31	Dic₂₈₄	—	4	2	no	—
32	Dic₂₈₄	—	4	2	no	—
33	#11765853	—	8	4	no	—
34	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
35	C₂²⋊Dic₇₁	—	8	4	no	—
36	C₄×Dic₇₁	D₈	1	1	yes	characteristic
37	C₇₁⋊C₁₆	D₈	1	2	yes	—
38	C₂×D₄×C₇₁	—	4	2	no	—
39	D₈×C₇₁	—	4	2	no	—
40	D₈×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₇₁	—	8	2	no	—
45	D₄×C₇₁	—	8	2	no	—
46	D₄×C₇₁	—	8	2	no	—
47	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₁	—	8	2	no	—
49	C₂×Dic₇₁	D₁₆	1	2	yes	—
50	Dic₁₄₂	—	8	2	no	—
51	Dic₁₄₂	—	8	2	no	—
52	C₈×C₇₁	#127	1	1	yes	characteristic
53	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
54	C₇₁⋊C₈	D₁₆	1	2	yes	—
55	C₂²×C₇₁	—	16	2	no	—
56	C₂²×C₇₁	—	16	4	no	—
57	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
58	C₄×C₇₁	#493	1	1	yes	characteristic
59	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
60	Dic₇₁	—	2	2	no	—
61	Dic₇₁	—	16	2	no	—
62	#13728	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	32	2	no	—
64	C₂×C₇₁	#2246	1	1	yes	characteristic
65	C₂×C₇₁	#2262	1	2	yes	—
66	C₂×Q₆₄	—	71	1	no	—
67	C₂×SD₆₄	—	71	2	no	—
68	C₂×D₃₂	D₇₁	1	1	yes	characteristic
69	#2249	—	71	2	no	—
70	C₄×C₃₂	—	71	1	no	—
71	C₇₁	#13728	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	71	2	no	—
73	C₂×SD₃₂	—	142	2	no	—
74	C₂×D₁₆	C₂×D₇₁	1	2	yes	—
75	#495	—	142	2	no	—
76	Q₆₄	—	142	2	no	—
77	SD₆₄	—	142	4	no	—
78	D₃₂	—	2	2	no	—
79	C₂×C₃₂	—	71	1	no	—
80	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
81	C₄×C₁₆	—	71	1	no	—
82	C₂×Q₁₆	—	142	2	no	—
83	C₂×SD₁₆	—	284	2	no	—
84	C₂×D₈	—	2	2	no	—
85	#124	—	284	2	no	—
86	Q₃₂	—	142	2	no	—
87	Q₃₂	—	142	2	no	—
88	SD₃₂	—	284	4	no	—
89	D₁₆	—	2	2	no	—
90	D₁₆	—	2	2	no	—
91	C₂×C₁₆	—	71	1	no	—
92	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
93	C₄×C₈	—	71	1	no	—
94	C₃₂	—	71	2	no	—
95	C₃₂	C₂²⋊D₇₁	1	2	yes	—
96	C₂×Q₈	—	284	2	no	—
97	C₂×D₄	—	4	2	no	—
98	Q₁₆	—	284	2	no	—
99	Q₁₆	—	284	2	no	—
100	SD₁₆	—	568	4	no	—
101	D₈	—	4	2	no	—
102	D₈	—	4	2	no	—
103	C₂×C₈	—	71	1	no	—
104	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
105	C₂²⋊C₄	—	568	4	no	—
106	C₄²	—	71	1	no	—
107	C₁₆	—	71	2	no	—
108	C₁₆	D₄×D₇₁	1	1	yes	characteristic
109	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂³	—	8	2	no	—
111	Q₈	—	568	2	no	—
112	Q₈	—	568	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	568	2	no	—
116	C₂×C₄	—	71	2	no	—
117	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
118	C₈	—	71	2	no	—
119	C₈	#14556124	1	1	yes	characteristic
120	C₈	D₈×D₇₁	1	1	yes	characteristic
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
124	C₄	—	1136	2	no	—
125	C₄	—	142	2	no	—
126	C₄	#17986374	1	1	yes	characteristic
127	C₄	D₁₆×D₇₁	1	1	yes	characteristic
128	C₂	—	32	2	no	—
129	C₂	#22590588	1	1	yes	characteristic
130	C₂	#22590646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₃₆

C₂

yes

maximal, characteristic, complemented

C₂×#17985837

C₂

yes

maximal, complemented

#22590598

C₂

yes

maximal, complemented

C₃₂×Dic₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₅₆₈

C₂²

yes

—

C₂×#14556072

—

#17986382

—

Dic₁₁₃₆

—

#17985837

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×D₁₆×C₇₁

C₂²

yes

—

D₃₂×C₇₁

—

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×#11765853

—

#14556115

—

Dic₅₆₈

—

Dic₅₆₈

—

#14556072

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

D₄

yes

—

C₂×D₈×C₇₁

—

D₁₆×C₇₁

—

D₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765853

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

D₈

yes

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

#127

yes

characteristic

C₈×C₇₁

C₂×D₈

yes

characteristic

C₇₁⋊C₈

D₁₆

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

#493

yes

characteristic

C₄×C₇₁

C₂×D₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

#13728

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#2246

yes

characteristic

C₂×C₇₁

#2262

yes

—

C₂×Q₆₄

—

C₂×SD₆₄

—

C₂×D₃₂

D₇₁

yes

characteristic

#2249

—

C₄×C₃₂

—

C₇₁

#13728

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

142

—

C₂×D₁₆

C₂×D₇₁

yes

—

#495

—

142

—

Q₆₄

—

142

—

SD₆₄

—

142

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

142

—

C₂×SD₁₆

—

284

—

C₂×D₈

—

#124

—

284

—

Q₃₂

—

142

—

Q₃₂

—

142

—

SD₃₂

—

284

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₇₁

yes

—

C₂×Q₈

—

284

—

C₂×D₄

—

Q₁₆

—

284

—

Q₁₆

—

284

—

100

SD₁₆

—

568

—

101

D₈

—

102

D₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₇₁

yes

characteristic

105

C₂²⋊C₄

—

568

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

D₄×D₇₁

yes

characteristic

109

C₁₆

C₂×C₂²⋊D₇₁

yes

characteristic

110

C₂³

—

111

Q₈

—

568

—

112

Q₈

—

568

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

568

—

116

C₂×C₄

—

117

C₂×C₄

D₈×D₇₁

yes

characteristic

118

C₈

—

119

C₈

#14556124

yes

characteristic

120

C₈

D₈×D₇₁

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

D₁₆×D₇₁

yes

characteristic

124

C₄

—

1136

—

125

C₄

—

142

—

126

C₄

#17986374

yes

characteristic

127

C₄

D₁₆×D₇₁

yes

characteristic

128

C₂

—

129

C₂

#22590588

yes

characteristic

130

C₂

#22590646

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting, prime radical	C₂×D₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₂×D₃₂	D₇₁
hypocenter	C₇₁	#13728
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₃₂

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting, prime radical

C₂×D₃₂×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₁₆

hypercenter

C₂×D₃₂

D₇₁

hypocenter

C₇₁

#13728

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₃₂

N	H ≅ G/N	type
1. C₂×Dic₁₁₃₆	C₂	semidirect
2. C₂×#17985837	C₂	semidirect
3. #22590598	C₂	semidirect
4. C₃₂×Dic₇₁	C₂	semidirect
71. C₇₁	#13728	semidirect

H ≅ G/N

type

1. C₂×Dic₁₁₃₆

C₂

semidirect

2. C₂×#17985837

C₂

semidirect

3. #22590598

C₂

semidirect

4. C₃₂×Dic₇₁

C₂

semidirect

71. C₇₁

#13728

semidirect

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

subgroup	index
C₂×Dic₁₁₃₆	2
C₂×#17985837	2
#22590598	2
C₃₂×Dic₇₁	2
C₂×D₃₂×C₇₁	2
#13728	71

subgroup

index

C₂×Dic₁₁₃₆

C₂×#17985837

#22590598

C₃₂×Dic₇₁

C₂×D₃₂×C₇₁

#13728

Subgroups — #31918455

Conjugacy classes of subgroups

Complements

Maximal subgroups