Group #16856233 — subgroups

Subgroups — #16856233

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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521459	C₂	1	4	yes	maximal, complemented
3	#14521455	C₂	1	4	yes	maximal, complemented
4	#14521085	C₂	1	4	yes	maximal
5	C₂×#11758391	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	#11758503	C₂²	1	2	yes	—
14	#11758501	—	2	2	no	—
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	#11758447	C₂²	1	2	yes	—
17	#11758446	C₂²	1	2	yes	—
18	C₄×Dic₆₂	—	2	2	no	—
19	#11758396	C₂²	1	8	yes	—
20	#11758395	C₂²	1	8	yes	—
21	#11758391	C₂²	1	2	yes	—
22	#11758391	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	C₂×Dic₆₂	—	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	C₂²⋊Dic₃₁	—	2	2	no	—
36	C₄⋊Dic₃₁	—	2	2	no	—
37	C₄⋊Dic₃₁	C₂³	1	4	yes	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	C₄×Dic₃₁	—	2	2	no	—
40	C₂×C₃₁⋊C₈	Q₈	1	8	yes	complemented
41	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₃₁	—	4	4	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₃₁	—	2	4	no	—
52	Dic₆₂	—	4	4	no	—
53	C₃₁⋊C₈	C₂×Q₈	1	8	yes	—
54	#3162	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	—	4	4	no	—
61	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
63	Dic₃₁	—	4	4	no	—
64	#533	—	31	1	no	—
65	#533	D₃₁	1	1	yes	characteristic
66	#501	—	31	4	no	—
67	#501	—	31	4	no	—
68	#427	—	31	4	no	—
69	C₂×#97	—	31	1	no	—
70	C₂×C₃₁	—	2	2	no	—
71	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₃₁	#575	1	2	yes	—
73	C₂²×C₈	—	31	2	no	—
74	C₄⋊Q₈	—	31	2	no	—
75	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
76	#125	—	31	2	no	—
77	#125	C₂×D₃₁	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₂²⋊Q₈	—	62	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	62	2	no	—
82	#117	—	31	2	no	—
83	#117	C₂×D₃₁	1	2	yes	—
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	#107	—	31	8	no	—
86	#106	—	31	8	no	—
87	#97	—	31	2	no	—
88	#97	—	31	2	no	—
89	C₃₁	#3162	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	62	2	no	—
92	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
94	C₂×C₈	—	31	8	no	—
95	C₂×C₈	—	31	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	31	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	C₂²⋊C₄	—	62	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	62	2	no	—
106	C₄²	C₂²⋊D₃₁	1	2	yes	—
107	C₄²	C₂²×D₃₁	1	1	yes	characteristic
108	C₄²	C₂²×D₃₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	124	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	Q₈×D₃₁	1	4	yes	—
116	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
120	C₈	—	31	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#11758521	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	124	4	no	—
128	C₄	#11758521	1	2	yes	—
129	C₄	C₂×Q₈×D₃₁	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#11758521	1	1	yes	characteristic
132	C₂	#14521847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521459

C₂

yes

maximal, complemented

#14521455

C₂

yes

maximal, complemented

#14521085

C₂

yes

maximal

C₂×#11758391

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

#11758503

C₂²

yes

—

#11758501

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758447

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

#11758391

C₂²

yes

—

#11758391

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

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

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

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

—

Dic₆₂

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#3162

—

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

—

#533

D₃₁

yes

—

—

—

C₂×#97

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#575

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

#125

C₂×D₃₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₃₁

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²⋊D₃₁

yes

—

107

C₄²

C₂²×D₃₁

yes

characteristic

108

C₄²

C₂²×D₃₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

110

Q₈

—

111

Q₈

—

124

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

Q₈×D₃₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

118

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

119

C₂×C₄

C₂³×D₃₁

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

#11758521

yes

—

123

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

124

—

128

C₄

#11758521

yes

—

129

C₄

C₂×Q₈×D₃₁

yes

—

130

C₂

—

131

C₂

C₂×#11758521

yes

characteristic

132

C₂

#14521847

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×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	#3162
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×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 ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14521459	C₂	semidirect
3. #14521455	C₂	semidirect
40. C₂×C₃₁⋊C₈	Q₈	semidirect
89. C₃₁	#3162	semidirect

H ≅ G/N

type

2. #14521459

C₂

semidirect

3. #14521455

C₂

semidirect

40. C₂×C₃₁⋊C₈

Q₈

semidirect

89. C₃₁

#3162

semidirect

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

subgroup	index
#14521606	2
#14521459	2
#14521455	2
#14521085	2
C₂×#11758391	2
C₃₁×#533	2
#3162	31

subgroup

index

C₃₁×#533

Subgroups — #16856233

Conjugacy classes of subgroups

Complements

Maximal subgroups