Group #16856214 — subgroups

Subgroups — C₂×#14521450 #16856214

Order 3968.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758467	C₂	1	2	yes	maximal
2	#14521450	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758406	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758396	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₃₁	—	2	2	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂×Dic₃₁⋊C₄	—	2	2	no	—
12	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	#11758467	—	2	4	no	—
15	#11758467	C₂²	1	8	yes	—
16	C₃₁×#107	C₂²	1	4	yes	—
17	#11758406	C₂²	1	4	yes	—
18	C₈×Dic₃₁	C₂²	1	4	yes	—
19	#11758396	C₂²	1	8	yes	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₁	—	2	2	no	—
23	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₁	Q₈	1	4	yes	—
26	C₂×C₈×C₃₁	C₂³	1	2	yes	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	—	2	8	no	—
31	C₄⋊Dic₃₁	C₂³	1	4	yes	—
32	Dic₃₁⋊C₄	—	2	8	no	—
33	C₄×Dic₃₁	D₄	1	2	yes	—
34	C₄×Dic₃₁	C₂³	1	2	yes	—
35	C₂×C₃₁⋊C₈	Q₈	1	4	yes	—
36	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
37	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	4	4	no	—
40	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₃₁	—	2	2	no	—
45	C₂×Dic₃₁	—	2	2	no	—
46	C₂×Dic₃₁	—	2	4	no	—
47	C₂×Dic₃₁	—	4	4	no	—
48	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
50	C₈×C₃₁	C₂×Q₈	1	4	yes	—
51	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
52	C₂×#499	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
54	C₂²×C₃₁	#135	1	2	yes	—
55	C₂²×C₃₁	#135	1	2	yes	—
56	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₃₁	—	4	4	no	—
58	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
59	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
60	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
61	Dic₃₁	—	2	2	no	—
62	Dic₃₁	—	2	2	no	—
63	Dic₃₁	—	4	4	no	—
64	C₂×#125	—	31	2	no	—
65	#499	—	31	8	no	—
66	C₂×#107	—	31	1	no	—
67	C₂×#107	D₃₁	1	1	yes	characteristic
68	C₂×#106	—	31	2	no	—
69	C₂×C₄×C₈	—	31	1	no	—
70	C₂×C₃₁	#499	1	4	yes	complemented
71	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
74	C₂²×C₈	—	31	1	no	—
75	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
76	#125	—	31	8	no	—
77	#125	—	62	4	no	—
78	C₂×C₄⋊C₄	—	31	2	no	—
79	C₂×C₄⋊C₄	—	62	2	no	—
80	C₂×C₄⋊C₄	—	62	2	no	—
81	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
82	C₂×C₄²	—	31	1	no	—
83	#107	—	31	4	no	—
84	#107	C₂×D₃₁	1	4	yes	—
85	#106	—	31	8	no	—
86	C₄×C₈	—	31	4	no	—
87	C₃₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	31	1	no	—
90	C₂²×C₄	—	31	1	no	—
91	C₂²×C₄	—	62	2	no	—
92	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
93	C₂×C₈	—	31	4	no	—
94	C₂×C₈	—	31	2	no	—
95	C₂×C₈	Dic₆₂	1	4	yes	—
96	C₂×C₈	C₂²×D₃₁	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	31	4	no	—
99	C₄⋊C₄	—	62	8	no	—
100	C₄⋊C₄	—	62	8	no	—
101	C₄⋊C₄	—	62	2	no	—
102	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
103	C₄²	—	31	2	no	—
104	C₄²	—	31	2	no	—
105	C₂³	D₄×D₃₁	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	31	2	no	—
109	C₂×C₄	—	31	2	no	—
110	C₂×C₄	—	62	2	no	—
111	C₂×C₄	—	62	2	no	—
112	C₂×C₄	—	62	4	no	—
113	C₂×C₄	—	124	4	no	—
114	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
115	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₃₁	1	2	yes	—
117	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
118	C₈	—	31	4	no	—
119	C₈	C₂×Dic₆₂	1	4	yes	—
120	C₂²	#11758465	1	2	yes	—
121	C₂²	#11758489	1	2	yes	—
122	C₂²	#11758496	1	2	yes	—
123	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	62	2	no	—
126	C₄	—	62	2	no	—
127	C₄	—	124	4	no	—
128	C₄	#11758465	1	2	yes	—
129	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
130	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
131	C₂	#14521450	1	4	yes	direct product factor
132	C₂	C₂×#11758465	1	1	yes	characteristic
133	C₂	C₂×#11758489	1	1	yes	characteristic
134	C₂	C₂×#11758496	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758467

C₂

yes

maximal

#14521450

C₂

yes

maximal, direct product factor

C₂×#11758406

C₂

yes

maximal, characteristic

C₂×#11758396

C₂

yes

maximal

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×#107

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758467

—

#11758467

C₂²

yes

—

C₃₁×#107

C₂²

yes

—

#11758406

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

#11758396

C₂²

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

characteristic

C₂²×Dic₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

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×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₂×#499

—

maximal, Sylow

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

#135

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

—

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₃₁

#499

yes

complemented

C₂×C₃₁

C₂×C₄⋊Q₈

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

#107

—

#107

C₂×D₃₁

yes

—

#106

—

C₄×C₈

—

C₃₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂²×D₃₁

yes

—

103

C₄²

—

104

C₄²

—

105

C₂³

D₄×D₃₁

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

124

—

114

C₂×C₄

C₂×Dic₆₂

yes

—

115

C₂×C₄

D₄×D₃₁

yes

characteristic

116

C₂×C₄

Q₈×D₃₁

yes

—

117

C₂×C₄

C₂³×D₃₁

yes

characteristic

118

C₈

—

119

C₈

C₂×Dic₆₂

yes

—

120

C₂²

#11758465

yes

—

121

C₂²

#11758489

yes

—

122

C₂²

#11758496

yes

—

123

C₂²

C₂×D₄×D₃₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

124

—

128

C₄

#11758465

yes

—

129

C₄

C₂²×Dic₆₂

yes

characteristic

130

C₄

C₂×Q₈×D₃₁

yes

characteristic

131

C₂

#14521450

yes

direct product factor

132

C₂

C₂×#11758465

yes

characteristic

133

C₂

C₂×#11758489

yes

characteristic

134

C₂

C₂×#11758496

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#107	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×#107	D₃₁
hypocenter	C₃₁	C₂×#499
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#107

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #14521450	C₂	direct (N × H)
70. C₂×C₃₁	#499	semidirect
87. C₃₁	C₂×#499	semidirect
131. C₂	#14521450	direct (N × H)

H ≅ G/N

type

2. #14521450

C₂

direct (N × H)

70. C₂×C₃₁

#499

semidirect

87. C₃₁

C₂×#499

semidirect

131. C₂

#14521450

direct (N × H)

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

subgroup	index
C₂×#11758467	2
#14521450	2
C₂×#11758406	2
C₂×#11758396	2
C₂×C₈×Dic₃₁	2
C₂×C₃₁×#107	2
C₂×#499	31

subgroup

index

C₂×C₈×Dic₃₁

C₂×C₃₁×#107

C₂×#499

Subgroups — C₂×#14521450 #16856214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521450 #16856214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521450 #16856214