Group #16856022 — subgroups

Subgroups — C₂×#14521400 #16856022

Order 3968.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#11758446	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758446	C₂	1	1	yes	maximal, characteristic
3	#14521400	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758404	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	—	2	1	no	—
7	C₂²×Dic₆₂	C₂²	1	2	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	—	2	1	no	—
11	C₂×C₄⋊Dic₃₁	—	2	1	no	—
12	C₂×C₄⋊Dic₃₁	—	2	1	no	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	#11758446	—	2	2	no	—
15	#11758446	—	2	2	no	—
16	#11758446	C₂²	1	4	yes	—
17	#11758446	C₂²	1	4	yes	—
18	C₄×C₈×C₃₁	C₂²	1	4	yes	—
19	#11758404	C₂²	1	16	yes	—
20	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₂×Dic₆₂	—	2	8	no	—
26	C₂×Dic₆₂	—	2	2	no	—
27	C₂×Dic₆₂	—	2	4	no	—
28	C₂×Dic₆₂	—	4	4	no	—
29	C₂×Dic₆₂	C₂³	1	4	yes	—
30	C₂×C₈×C₃₁	—	2	4	no	—
31	C₂×C₈×C₃₁	C₂³	1	4	yes	—
32	C₄²×C₃₁	D₄	1	2	yes	—
33	C₄²×C₃₁	C₂³	1	2	yes	—
34	C₄⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	—	2	4	no	—
36	C₄⋊Dic₃₁	—	2	4	no	—
37	C₄⋊Dic₃₁	—	2	4	no	—
38	C₄⋊Dic₃₁	C₂³	1	4	yes	—
39	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	SD₁₆	1	4	yes	—
41	C₂×C₄×C₃₁	Q₁₆	1	4	yes	—
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	#41	1	4	yes	—
46	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₃₁	—	2	4	no	—
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	4	4	no	—
50	C₂×Dic₃₁	—	4	4	no	—
51	Dic₆₂	—	2	8	no	—
52	Dic₆₂	—	4	4	no	—
53	Dic₆₂	—	4	8	no	—
54	C₈×C₃₁	—	2	4	no	—
55	C₂×#487	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	#124	1	2	yes	—
57	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
58	C₂²×C₃₁	C₂×Q₁₆	1	2	yes	—
59	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₃₁	#124	1	2	yes	—
61	C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
62	C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
63	C₄×C₃₁	C₂×#41	1	2	yes	—
64	Dic₃₁	—	4	4	no	—
65	Dic₃₁	—	4	4	no	—
66	C₂×C₄⋊Q₈	—	31	1	no	—
67	C₂×C₄⋊Q₈	—	31	1	no	—
68	#487	—	31	8	no	—
69	C₂×#103	—	31	4	no	—
70	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
71	C₂×C₃₁	#487	1	4	yes	complemented
72	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
73	C₂×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂²×Q₈	—	31	2	no	—
76	C₂²×Q₈	—	62	1	no	—
77	C₂²×C₈	C₂×D₃₁	1	2	yes	—
78	C₄⋊Q₈	—	31	4	no	—
79	C₄⋊Q₈	—	31	4	no	—
80	C₄⋊Q₈	—	62	2	no	—
81	C₄⋊Q₈	—	62	2	no	—
82	C₂×C₄⋊C₄	—	31	2	no	—
83	C₂×C₄⋊C₄	—	62	1	no	—
84	C₂×C₄⋊C₄	—	62	1	no	—
85	C₂×C₄⋊C₄	—	62	1	no	—
86	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
87	#103	—	31	16	no	—
88	C₄×C₈	C₂×D₃₁	1	4	yes	—
89	C₃₁	C₂×#487	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	31	4	no	—
91	C₂×Q₈	—	62	4	no	—
92	C₂×Q₈	—	62	2	no	—
93	C₂×Q₈	—	62	8	no	—
94	C₂×Q₈	—	124	4	no	—
95	C₂²×C₄	—	62	2	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
98	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂²×D₃₁	1	4	yes	—
102	C₄⋊C₄	—	31	4	no	—
103	C₄⋊C₄	—	62	2	no	—
104	C₄⋊C₄	—	62	4	no	—
105	C₄⋊C₄	—	62	4	no	—
106	C₄⋊C₄	—	62	4	no	—
107	C₄²	D₁₂₄	1	2	yes	—
108	C₄²	C₂²×D₃₁	1	2	yes	—
109	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
110	Q₈	—	62	8	no	—
111	Q₈	—	124	4	no	—
112	Q₈	—	124	8	no	—
113	C₂×C₄	—	62	4	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	—	124	4	no	—
116	C₂×C₄	—	124	4	no	—
117	C₂×C₄	#92470	1	4	yes	—
118	C₂×C₄	Dic₁₂₄	1	4	yes	—
119	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
120	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
121	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
122	C₂×C₄	#92495	1	4	yes	—
123	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	#11758452	1	2	yes	—
126	C₂²	C₂×#92470	1	2	yes	—
127	C₂²	C₂×Dic₁₂₄	1	2	yes	—
128	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
129	C₄	—	124	4	no	—
130	C₄	—	124	4	no	—
131	C₄	#11758452	1	2	yes	—
132	C₄	C₂×#92470	1	4	yes	—
133	C₄	C₂×Dic₁₂₄	1	4	yes	—
134	C₄	C₂×#92495	1	2	yes	—
135	C₂	#14521400	1	4	yes	direct product factor
136	C₂	C₂×#11758452	1	1	yes	characteristic
137	C₂	C₂²×#92470	1	1	yes	characteristic
138	C₂	C₂²×Dic₁₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

C₂×#11758446

C₂

yes

maximal, characteristic

#14521400

C₂

yes

maximal, direct product factor

C₂×#11758404

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂²×Dic₆₂

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758446

—

#11758446

—

#11758446

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758404

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂³

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₂×#487

—

maximal, Sylow

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

C₂×Q₁₆

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#487

—

C₂×#103

—

C₂×C₄×C₈

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

C₂²×SD₁₆

yes

characteristic

C₂×C₃₁

C₂²×Q₁₆

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

C₂×#487

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₃₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

D₁₂₄

yes

—

108

C₄²

C₂²×D₃₁

yes

—

109

C₂³

C₂×D₁₂₄

yes

characteristic

110

Q₈

—

111

Q₈

—

124

—

112

Q₈

—

124

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

124

—

116

C₂×C₄

—

124

—

117

C₂×C₄

#92470

yes

—

118

C₂×C₄

Dic₁₂₄

yes

—

119

C₂×C₄

C₂×D₁₂₄

yes

characteristic

120

C₂×C₄

C₂×D₁₂₄

yes

—

121

C₂×C₄

C₂×D₁₂₄

yes

—

122

C₂×C₄

#92495

yes

—

123

C₂×C₄

C₂³×D₃₁

yes

characteristic

124

C₈

—

125

C₂²

#11758452

yes

—

126

C₂²

C₂×#92470

yes

—

127

C₂²

C₂×Dic₁₂₄

yes

—

128

C₂²

C₂²×D₁₂₄

yes

characteristic

129

C₄

—

124

—

130

C₄

—

124

—

131

C₄

#11758452

yes

—

132

C₄

C₂×#92470

yes

—

133

C₄

C₂×Dic₁₂₄

yes

—

134

C₄

C₂×#92495

yes

—

135

C₂

#14521400

yes

direct product factor

136

C₂

C₂×#11758452

yes

characteristic

137

C₂

C₂²×#92470

yes

characteristic

138

C₂

C₂²×Dic₁₂₄

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	C₂×#487
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

C₂×#487

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #14521400	C₂	direct (N × H)
71. C₂×C₃₁	#487	semidirect
89. C₃₁	C₂×#487	semidirect
135. C₂	#14521400	direct (N × H)

H ≅ G/N

type

3. #14521400

C₂

direct (N × H)

71. C₂×C₃₁

#487

semidirect

89. C₃₁

C₂×#487

semidirect

135. C₂

#14521400

direct (N × H)

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

subgroup	index
C₂×#11758446	2
#14521400	2
C₂×#11758404	2
C₂×C₄×C₈×C₃₁	2
C₂×#487	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — C₂×#14521400 #16856022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521400 #16856022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521400 #16856022