Group #16856234 — subgroups

Subgroups — #16856234

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	#14521602	C₂	1	2	yes	maximal
2	#14521457	C₂	1	4	yes	maximal, complemented
3	#14521456	C₂	1	4	yes	maximal, complemented
4	#14521087	C₂	1	1	yes	maximal, characteristic
5	#14521085	C₂	1	2	yes	maximal
6	C₂×#11758403	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#117	C₂²	1	2	yes	—
10	#11758503	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	#11758467	C₂²	1	4	yes	—
14	#11758465	C₂²	1	4	yes	—
15	#11758455	—	2	4	no	—
16	C₄×Dic₆₂	—	2	4	no	—
17	C₃₁×#107	C₂²	1	2	yes	—
18	C₃₁×#106	C₂²	1	2	yes	—
19	#11758406	C₂²	1	2	yes	—
20	#11758405	C₂²	1	2	yes	—
21	#11758403	C₂²	1	2	yes	—
22	#11758403	C₂²	1	2	yes	—
23	#11758396	C₂²	1	4	yes	—
24	#11758395	C₂²	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	D₄	1	2	yes	—
27	C₂×Dic₆₂	—	2	4	no	—
28	C₂×C₈×C₃₁	Q₈	1	4	yes	complemented
29	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₄²×C₃₁	—	2	2	no	—
34	C₂²⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	—	2	4	no	—
36	C₄⋊Dic₃₁	C₂³	1	4	yes	—
37	Dic₃₁⋊C₄	—	2	4	no	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	C₄×Dic₃₁	—	2	2	no	—
40	C₄×Dic₃₁	D₄	1	2	yes	—
41	C₄×Dic₃₁	C₂³	1	2	yes	—
42	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
43	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
44	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	4	no	—
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
53	Dic₆₂	—	4	8	no	—
54	C₈×C₃₁	C₂×Q₈	1	2	yes	—
55	C₈×C₃₁	C₂×Q₈	1	2	yes	—
56	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
57	#3162	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	—	2	2	no	—
59	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₃₁	—	4	4	no	—
62	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
63	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
65	Dic₃₁	—	2	4	no	—
66	Dic₃₁	—	4	4	no	—
67	#533	—	31	2	no	—
68	#501	—	31	4	no	—
69	#501	—	31	4	no	—
70	#427	—	31	1	no	—
71	#427	—	31	2	no	—
72	#427	D₃₁	1	1	yes	characteristic
73	C₂×#97	—	31	1	no	—
74	C₂×C₃₁	—	2	2	no	—
75	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₃₁	#575	1	2	yes	—
77	C₂²×C₈	—	31	1	no	—
78	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
79	C₄⋊Q₈	—	31	4	no	—
80	#125	—	31	4	no	—
81	C₂²⋊Q₈	—	62	4	no	—
82	C₄×Q₈	—	62	4	no	—
83	#117	—	31	2	no	—
84	#117	C₂×D₃₁	1	2	yes	—
85	C₂×C₄²	—	31	1	no	—
86	#107	—	31	2	no	—
87	#107	—	31	4	no	—
88	#107	C₂×D₃₁	1	2	yes	—
89	#106	—	31	2	no	—
90	#106	—	31	4	no	—
91	#106	C₂×D₃₁	1	2	yes	—
92	#97	—	31	2	no	—
93	#97	—	31	2	no	—
94	C₃₁	#3162	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	62	4	no	—
96	C₂²×C₄	—	31	2	no	—
97	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	C₂×C₈	—	31	4	no	—
99	C₂×C₈	—	31	2	no	—
100	C₂×C₈	Dic₆₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
103	C₄⋊C₄	—	31	4	no	—
104	C₄⋊C₄	—	62	4	no	—
105	C₄⋊C₄	—	62	4	no	—
106	C₄⋊C₄	—	62	4	no	—
107	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	62	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	31	2	no	—
112	C₄²	—	31	2	no	—
113	C₄²	—	62	2	no	—
114	C₂³	D₄×D₃₁	1	1	yes	characteristic
115	Q₈	—	124	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	31	4	no	—
118	C₂×C₄	—	62	4	no	—
119	C₂×C₄	—	124	2	no	—
120	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
121	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₃₁	1	2	yes	—
123	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
124	C₈	—	31	4	no	—
125	C₈	C₂×Dic₆₂	1	2	yes	—
126	C₈	C₂×Dic₆₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#11758465	1	2	yes	—
129	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	62	4	no	—
132	C₄	—	124	4	no	—
133	C₄	#11758465	1	2	yes	—
134	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
135	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#11758465	1	1	yes	characteristic
138	C₂	#14521846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521602

C₂

yes

maximal

#14521457

C₂

yes

maximal, complemented

#14521456

C₂

yes

maximal, complemented

#14521087

C₂

yes

maximal, characteristic

#14521085

C₂

yes

maximal

C₂×#11758403

C₂

yes

maximal, characteristic

C₃₁×#427

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₂²

yes

—

#11758503

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758467

C₂²

yes

—

#11758465

C₂²

yes

—

#11758455

—

C₄×Dic₆₂

—

C₃₁×#107

C₂²

yes

—

C₃₁×#106

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

#11758403

C₂²

yes

—

#11758403

C₂²

yes

—

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

D₄

yes

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

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⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂×#97

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#575

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₃₁

yes

—

#106

—

#106

—

#106

C₂×D₃₁

yes

—

#97

—

#97

—

C₃₁

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₆₂

yes

complemented

101

C₂×C₈

C₂²×D₃₁

yes

characteristic

102

C₂×C₈

C₂²×D₃₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×D₃₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₃₁

yes

characteristic

115

Q₈

—

124

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

124

—

120

C₂×C₄

C₂×Dic₆₂

yes

—

121

C₂×C₄

D₄×D₃₁

yes

characteristic

122

C₂×C₄

Q₈×D₃₁

yes

—

123

C₂×C₄

C₂³×D₃₁

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₆₂

yes

—

126

C₈

C₂×Dic₆₂

yes

—

127

C₂²

—

128

C₂²

#11758465

yes

—

129

C₂²

C₂×D₄×D₃₁

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

124

—

133

C₄

#11758465

yes

—

134

C₄

C₂²×Dic₆₂

yes

characteristic

135

C₄

C₂×Q₈×D₃₁

yes

characteristic

136

C₂

—

137

C₂

C₂×#11758465

yes

characteristic

138

C₂

#14521846

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#427	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#427	D₃₁
hypocenter	C₃₁	#3162
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#427

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 ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #14521457	C₂	semidirect
3. #14521456	C₂	semidirect
28. C₂×C₈×C₃₁	Q₈	semidirect
42. C₂×C₃₁⋊C₈	Q₈	semidirect
94. C₃₁	#3162	semidirect
100. C₂×C₈	Dic₆₂	semidirect

H ≅ G/N

type

2. #14521457

C₂

semidirect

3. #14521456

C₂

semidirect

28. C₂×C₈×C₃₁

Q₈

semidirect

42. C₂×C₃₁⋊C₈

Q₈

semidirect

94. C₃₁

#3162

semidirect

100. C₂×C₈

Dic₆₂

semidirect

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

subgroup	index
#14521602	2
#14521457	2
#14521456	2
#14521087	2
#14521085	2
C₂×#11758403	2
C₃₁×#427	2
#3162	31

subgroup

index

C₃₁×#427

Subgroups — #16856234

Conjugacy classes of subgroups

Complements

Maximal subgroups