Group #16854498 — subgroups

Subgroups — C₂×#14521156 #16854498

Order 3968.

480 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₃₁	C₂	1	2	yes	maximal
2	#14521156	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758405	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758423	C₂	1	2	yes	maximal
5	C₂×#11758403	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₃₁	C₄	1	8	yes	complemented
9	C₂×Q₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	Q₈×Dic₃₁	—	2	4	no	—
12	Q₈×Dic₃₁	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₃₁	—	2	2	no	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₃₁	—	2	2	no	—
16	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	#11758423	C₂²	1	8	yes	—
19	#11758405	C₂²	1	4	yes	—
20	#11758403	C₂²	1	4	yes	—
21	C₂×Q₈×C₃₁	—	2	2	no	—
22	C₂×Q₈×C₃₁	—	2	4	no	—
23	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
24	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₁	—	2	2	no	—
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃₁	—	2	2	no	—
28	C₂²×Dic₃₁	D₄	1	2	yes	—
29	Q₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
30	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₃₁	C₂³	1	2	yes	—
32	C₄⋊Dic₃₁	—	2	2	no	—
33	C₄⋊Dic₃₁	—	2	8	no	—
34	C₄⋊Dic₃₁	C₂³	1	4	yes	—
35	C₄×Dic₃₁	—	2	8	no	—
36	C₄×Dic₃₁	D₄	1	2	yes	—
37	C₄×Dic₃₁	C₂³	1	2	yes	—
38	C₂×C₃₁⋊C₈	—	2	2	no	—
39	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
40	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₃₁	—	2	8	no	—
42	Q₈×C₃₁	—	2	4	no	—
43	Q₈×C₃₁	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₃₁	—	2	8	no	—
45	C₂×C₄×C₃₁	—	2	4	no	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#41	1	2	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	2	4	no	—
52	C₂×Dic₃₁	—	4	4	no	—
53	C₂×Dic₃₁	C₂×D₄	1	4	yes	—
54	C₈×C₃₁	C₂²×C₄	1	4	yes	—
55	C₃₁⋊C₈	—	2	2	no	—
56	C₂×#441	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	C₄×D₄	1	2	yes	—
58	C₂²×C₃₁	#137	1	4	yes	—
59	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₃₁	—	2	8	no	—
61	C₄×C₃₁	C₄×D₄	1	2	yes	—
62	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
63	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
64	Dic₃₁	—	2	4	no	—
65	Dic₃₁	—	4	4	no	—
66	C₂²×Q₁₆	D₃₁	1	1	yes	characteristic
67	C₂×C₄×Q₈	—	31	2	no	—
68	#441	—	31	8	no	—
69	C₂×#106	—	31	1	no	—
70	C₂×#103	—	31	2	no	—
71	C₂×#97	—	31	1	no	—
72	C₂×C₃₁	#441	1	4	yes	complemented
73	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
74	C₂×C₃₁	C₂×#137	1	2	yes	—
75	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
76	C₂×Q₁₆	Dic₃₁	1	8	yes	complemented
77	C₂×Q₁₆	C₂×D₃₁	1	4	yes	—
78	C₂²×C₈	—	31	1	no	—
79	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
80	C₄×Q₈	—	31	8	no	—
81	C₄×Q₈	—	62	4	no	—
82	C₂×C₄⋊C₄	—	31	2	no	—
83	C₂×C₄⋊C₄	—	62	2	no	—
84	C₂×C₄²	—	31	1	no	—
85	C₂×C₄²	—	62	2	no	—
86	#106	—	31	4	no	—
87	#103	—	31	8	no	—
88	#97	—	31	4	no	—
89	C₃₁	C₂×#441	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
93	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	31	2	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	Q₁₆	C₂×Dic₃₁	1	16	yes	complemented
99	C₂×C₈	—	31	2	no	—
100	C₂×C₈	—	62	2	no	—
101	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
102	C₂×C₈	C₂²×D₃₁	1	2	yes	—
103	C₄⋊C₄	—	31	4	no	—
104	C₄⋊C₄	—	62	8	no	—
105	C₄⋊C₄	—	62	2	no	—
106	C₄²	—	31	2	no	—
107	C₄²	—	31	2	no	—
108	C₄²	—	62	8	no	—
109	C₂³	D₄×D₃₁	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	2	4	no	—
112	Q₈	C₂²×Dic₃₁	1	8	yes	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	31	4	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	124	4	no	—
119	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
120	C₂×C₄	#92497	1	2	yes	—
121	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
122	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
123	C₈	—	62	2	no	—
124	C₈	C₂²×Dic₃₁	1	4	yes	—
125	C₂²	#11758495	1	4	yes	—
126	C₂²	D₄×Dic₃₁	1	2	yes	—
127	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
128	C₄	—	2	8	no	—
129	C₄	—	62	4	no	—
130	C₄	—	124	4	no	—
131	C₄	D₄×Dic₃₁	1	2	yes	—
132	C₄	C₂×#92497	1	1	yes	characteristic
133	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
134	C₂	#14521156	1	4	yes	direct product factor
135	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic
136	C₂	C₂×#11758495	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₃₁

C₂

yes

maximal

#14521156

C₂

yes

maximal, direct product factor

C₂×#11758405

C₂

yes

maximal, characteristic

C₂×#11758423

C₂

yes

maximal

C₂×#11758403

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂×Q₁₆×C₃₁

C₄

yes

complemented

C₂×Q₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

Q₈×Dic₃₁

—

Q₈×Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758423

C₂²

yes

—

#11758405

C₂²

yes

—

#11758403

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

—

Q₁₆×C₃₁

C₂×C₄

yes

complemented

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂²×C₄

yes

—

C₃₁⋊C₈

—

C₂×#441

—

maximal, Sylow

C₂²×C₃₁

C₄×D₄

yes

—

C₂²×C₃₁

#137

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂²×Q₁₆

D₃₁

yes

characteristic

C₂×C₄×Q₈

—

#441

—

C₂×#106

—

C₂×#103

—

C₂×#97

—

C₂×C₃₁

#441

yes

complemented

C₂×C₃₁

C₂×C₄×D₄

yes

characteristic

C₂×C₃₁

C₂×#137

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂×Q₁₆

Dic₃₁

yes

complemented

C₂×Q₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#106

—

#103

—

#97

—

C₃₁

C₂×#441

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

Q₁₆

C₂×Dic₃₁

yes

complemented

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×Dic₃₁

yes

—

102

C₂×C₈

C₂²×D₃₁

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

D₄×D₃₁

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

C₂²×Dic₃₁

yes

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

D₄×D₃₁

yes

characteristic

120

C₂×C₄

#92497

yes

—

121

C₂×C₄

C₂²×Dic₃₁

yes

—

122

C₂×C₄

C₂³×D₃₁

yes

characteristic

123

C₈

—

124

C₈

C₂²×Dic₃₁

yes

—

125

C₂²

#11758495

yes

—

126

C₂²

D₄×Dic₃₁

yes

—

127

C₂²

C₂×D₄×D₃₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

124

—

131

C₄

D₄×Dic₃₁

yes

—

132

C₄

C₂×#92497

yes

characteristic

133

C₄

C₂³×Dic₃₁

yes

characteristic

134

C₂

#14521156

yes

direct product factor

135

C₂

C₂×D₄×Dic₃₁

yes

characteristic

136

C₂

C₂×#11758495

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×Q₁₆×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×Q₁₆	D₃₁
hypocenter	C₃₁	C₂×#441
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₄×C₃₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂²×Q₁₆×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂×C₄×C₃₁

C₂⁴

hypercenter

C₂²×Q₁₆

D₃₁

hypocenter

C₃₁

C₂×#441

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #14521156	C₂	direct (N × H)
8. C₂×Q₁₆×C₃₁	C₄	semidirect
29. Q₁₆×C₃₁	C₂×C₄	semidirect
72. C₂×C₃₁	#441	semidirect
76. C₂×Q₁₆	Dic₃₁	semidirect
89. C₃₁	C₂×#441	semidirect
98. Q₁₆	C₂×Dic₃₁	semidirect
134. C₂	#14521156	direct (N × H)

H ≅ G/N

type

2. #14521156

C₂

direct (N × H)

8. C₂×Q₁₆×C₃₁

C₄

semidirect

29. Q₁₆×C₃₁

C₂×C₄

semidirect

72. C₂×C₃₁

#441

semidirect

76. C₂×Q₁₆

Dic₃₁

semidirect

89. C₃₁

C₂×#441

semidirect

98. Q₁₆

C₂×Dic₃₁

semidirect

134. C₂

#14521156

direct (N × H)

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

subgroup	index
C₂×Q₈×Dic₃₁	2
#14521156	2
C₂×#11758405	2
C₂×#11758423	2
C₂×#11758403	2
C₂²×Q₁₆×C₃₁	2
C₂×#441	31

subgroup

index

C₂×Q₈×Dic₃₁

C₂²×Q₁₆×C₃₁

Subgroups — C₂×#14521156 #16854498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521156 #16854498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521156 #16854498