Group #16856264 — subgroups

Subgroups — #16856264

Order 3968.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758447	C₂	1	1	yes	maximal, characteristic
2	#14521455	C₂	1	4	yes	maximal, complemented
3	#14521448	C₂	1	4	yes	maximal, complemented
4	#14521092	C₂	1	4	yes	maximal
5	C₄×#92474	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
7	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₃₁×#125	C₂²	1	2	yes	—
9	C₃₁×C₂²⋊Q₈	—	2	2	no	—
10	C₄×Q₈×C₃₁	—	2	2	no	—
11	C₃₁×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₂×Dic₃₁⋊C₄	—	2	2	no	—
15	C₂×#92474	C₂²	1	2	yes	—
16	#11758447	—	2	2	no	—
17	#11758447	C₂²	1	2	yes	—
18	#11758447	C₂²	1	2	yes	—
19	#11758396	C₂²	1	8	yes	—
20	#11758395	C₂²	1	8	yes	—
21	#11758391	C₂²	1	2	yes	—
22	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
23	C₂×Q₈×C₃₁	—	2	2	no	—
24	C₂²×C₄×C₃₁	D₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
30	C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₄²×C₃₁	—	2	2	no	—
32	C₄²×C₃₁	D₄	1	2	yes	—
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₃₁	—	2	2	no	—
36	C₄⋊Dic₃₁	C₂³	1	4	yes	—
37	Dic₃₁⋊C₄	—	2	4	no	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	#92474	Q₈	1	8	yes	complemented
40	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₃₁	—	4	4	no	—
43	C₂×C₄×C₃₁	—	2	4	no	—
44	C₂×C₄×C₃₁	—	4	2	no	—
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	4	4	no	—
52	C₃₁⋊C₈	C₂×Q₈	1	8	yes	—
53	#3166	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	4	4	no	—
60	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
61	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
62	Dic₃₁	—	4	4	no	—
63	#533	D₃₁	1	1	yes	characteristic
64	C₂×#125	—	31	1	no	—
65	#501	—	31	4	no	—
66	#499	—	31	4	no	—
67	#428	—	31	4	no	—
68	C₄×M₁₆	—	31	1	no	—
69	C₂×C₃₁	—	2	2	no	—
70	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₃₁	#575	1	2	yes	—
72	C₂×M₁₆	—	31	2	no	—
73	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
74	#125	—	31	2	no	—
75	#125	—	31	2	no	—
76	#125	—	62	2	no	—
77	#125	C₂×D₃₁	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	#117	C₂×D₃₁	1	2	yes	—
81	C₂×C₄⋊C₄	—	31	2	no	—
82	C₂×C₄⋊C₄	—	62	2	no	—
83	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
84	#107	—	31	8	no	—
85	#106	—	31	8	no	—
86	#97	—	31	2	no	—
87	C₄×C₈	—	31	2	no	—
88	C₃₁	#3166	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	—	62	2	no	—
91	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
92	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
93	M₁₆	—	31	8	no	—
94	C₂×C₈	—	31	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	31	4	no	—
98	C₄⋊C₄	—	62	4	no	—
99	C₄⋊C₄	—	62	4	no	—
100	C₄⋊C₄	—	62	2	no	—
101	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₃₁	1	2	yes	—
105	C₄²	C₂²×D₃₁	1	1	yes	characteristic
106	C₄²	C₂²×D₃₁	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	62	4	no	—
112	C₂×C₄	—	124	4	no	—
113	C₂×C₄	Q₈×D₃₁	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
118	C₈	—	31	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#11758521	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	124	4	no	—
126	C₄	#11758521	1	2	yes	—
127	C₄	C₂×Q₈×D₃₁	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#11758521	1	1	yes	characteristic
130	C₂	#14521847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic

#14521455

C₂

yes

maximal, complemented

#14521448

C₂

yes

maximal, complemented

#14521092

C₂

yes

maximal

C₄×#92474

C₂

yes

maximal, characteristic

C₃₁×#533

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#125

C₂²

yes

—

C₃₁×C₂²⋊Q₈

—

C₄×Q₈×C₃₁

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂×#92474

C₂²

yes

—

#11758447

—

#11758447

C₂²

yes

—

#11758447

C₂²

yes

—

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

#92474

Q₈

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#3166

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

#533

D₃₁

yes

—

—

—

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#575

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₁

yes

—

—

—

—

C₂×D₃₁

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₄×C₈

—

C₃₁

#3166

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₃₁

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₃₁

yes

—

105

C₄²

C₂²×D₃₁

yes

characteristic

106

C₄²

C₂²×D₃₁

yes

characteristic

107

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

124

—

113

C₂×C₄

Q₈×D₃₁

yes

—

114

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

117

C₂×C₄

C₂³×D₃₁

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#11758521

yes

—

121

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

124

—

126

C₄

#11758521

yes

—

127

C₄

C₂×Q₈×D₃₁

yes

—

128

C₂

—

129

C₂

C₂×#11758521

yes

characteristic

130

C₂

#14521847

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#533	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#533	D₃₁
hypocenter	C₃₁	#3166
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#533

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14521455	C₂	semidirect
3. #14521448	C₂	semidirect
39. #92474	Q₈	semidirect
88. C₃₁	#3166	semidirect

H ≅ G/N

type

2. #14521455

C₂

semidirect

3. #14521448

C₂

semidirect

39. #92474

Q₈

semidirect

88. C₃₁

#3166

semidirect

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

subgroup	index
C₂×#11758447	2
#14521455	2
#14521448	2
#14521092	2
C₄×#92474	2
C₃₁×#533	2
#3166	31

subgroup

index

C₃₁×#533

Subgroups — #16856264

Conjugacy classes of subgroups

Complements

Maximal subgroups