Group #16858822 — subgroups

Subgroups — C₂×Q₃₂×D₃₁ #16858822

Order 3968.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q₁₆×D₃₁	C₂	1	2	yes	maximal
2	C₂×Dic₂₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11758417	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₃₁	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₃₁	—	2	2	no	—
8	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×#92482	—	2	2	no	—
10	Q₁₆×D₃₁	—	2	4	no	—
11	Q₁₆×D₃₁	C₂²	1	8	yes	—
12	C₂×Dic₁₂₄	C₂²	1	2	yes	—
13	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×Q₃₂	C₂²	1	4	yes	complemented
15	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	#11758417	C₂²	1	8	yes	complemented
17	C₂×C₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
18	Dic₂₄₈	C₂²	1	4	yes	complemented
19	C₁₆×D₃₁	C₂²	1	4	yes	—
20	C₂×Q₈×C₃₁	—	2	2	no	—
21	Q₈×D₃₁	—	2	8	no	—
22	Q₈×D₃₁	—	4	4	no	—
23	C₂×C₄×D₃₁	—	4	2	no	—
24	C₂×C₄×D₃₁	D₄	1	1	yes	characteristic
25	C₂×Dic₆₂	—	2	2	no	—
26	C₂×Dic₆₂	—	4	2	no	—
27	Q₁₆×C₃₁	—	2	2	no	—
28	Q₁₆×C₃₁	C₂³	1	4	yes	—
29	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	#92482	—	2	8	no	—
31	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
32	Dic₁₂₄	—	2	2	no	—
33	Dic₁₂₄	C₂³	1	4	yes	—
34	C₈×D₃₁	D₄	1	2	yes	—
35	C₈×D₃₁	C₂³	1	2	yes	—
36	C₁₆×C₃₁	C₂³	1	2	yes	—
37	C₃₁⋊C₁₆	C₂³	1	2	yes	—
38	C₂²×D₃₁	D₈	1	1	yes	characteristic
39	Q₈×C₃₁	—	2	4	no	—
40	Q₈×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	—	4	2	no	—
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	D₈	1	1	yes	characteristic
45	C₄×D₃₁	—	4	8	no	—
46	C₄×D₃₁	D₈	1	2	yes	—
47	C₄×D₃₁	C₂×D₄	1	2	yes	—
48	Dic₆₂	—	2	4	no	—
49	Dic₆₂	—	4	2	no	—
50	Dic₆₂	—	4	8	no	—
51	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₈×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₃₁⋊C₈	C₂×D₄	1	2	yes	—
54	C₂²×Q₃₂	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
56	C₂×D₃₁	Q₃₂	1	4	yes	direct product factor
57	C₂×D₃₁	C₂×D₈	1	2	yes	—
58	C₄×C₃₁	—	4	4	no	—
59	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
60	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
61	Dic₃₁	—	4	4	no	—
62	Dic₃₁	C₂×D₈	1	2	yes	—
63	C₂²×Q₁₆	—	31	2	no	—
64	C₂×Q₃₂	—	31	2	no	—
65	C₂×Q₃₂	—	31	8	no	—
66	C₂×Q₃₂	—	31	1	no	—
67	C₂×Q₃₂	D₃₁	1	1	yes	characteristic, direct product factor
68	C₂²×C₁₆	—	31	1	no	—
69	C₂×C₃₁	C₂×Q₃₂	1	2	yes	complemented
70	C₂×C₃₁	C₂²×D₈	1	1	yes	characteristic
71	D₃₁	C₂×Q₃₂	1	4	yes	direct product factor
72	C₂²×Q₈	—	62	2	no	—
73	C₂×Q₁₆	—	31	8	no	—
74	C₂×Q₁₆	—	31	2	no	—
75	C₂×Q₁₆	—	62	4	no	—
76	C₂×Q₁₆	—	62	2	no	—
77	C₂×Q₁₆	C₂×D₃₁	1	2	yes	—
78	C₂²×C₈	—	31	1	no	—
79	Q₃₂	—	31	8	no	—
80	Q₃₂	—	31	4	no	—
81	Q₃₂	C₂×D₃₁	1	4	yes	direct product factor
82	C₂×C₁₆	—	31	1	no	—
83	C₂×C₁₆	—	31	4	no	—
84	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
85	C₃₁	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	62	8	no	—
88	C₂×Q₈	—	62	2	no	—
89	C₂×Q₈	—	124	2	no	—
90	C₂×Q₈	—	124	4	no	—
91	C₂²×C₄	—	31	1	no	—
92	C₂²×C₄	—	124	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	31	4	no	—
95	Q₁₆	—	62	8	no	—
96	Q₁₆	—	62	2	no	—
97	Q₁₆	C₂²×D₃₁	1	4	yes	—
98	C₂×C₈	—	31	1	no	—
99	C₂×C₈	—	31	2	no	—
100	C₂×C₈	—	31	2	no	—
101	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
102	C₁₆	—	31	2	no	—
103	C₁₆	C₂²×D₃₁	1	2	yes	—
104	C₂³	—	31	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	62	4	no	—
108	Q₈	—	124	8	no	—
109	Q₈	—	124	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	31	1	no	—
112	C₂×C₄	—	31	2	no	—
113	C₂×C₄	—	31	2	no	—
114	C₂×C₄	—	124	8	no	—
115	C₂×C₄	—	124	2	no	—
116	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
117	C₈	—	31	2	no	—
118	C₈	D₄×D₃₁	1	1	yes	characteristic
119	C₈	C₂³×D₃₁	1	1	yes	characteristic
120	C₂²	—	31	4	no	—
121	C₂²	—	31	2	no	—
122	C₂²	D₈×D₃₁	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	31	2	no	—
125	C₄	—	124	4	no	—
126	C₄	D₈×D₃₁	1	1	yes	characteristic
127	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
128	C₂	—	31	4	no	—
129	C₂	Q₃₂×D₃₁	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₃₁

C₂

yes

maximal

C₂×Dic₂₄₈

C₂

yes

maximal, characteristic, complemented

C₂×#11758417

C₂

yes

maximal, complemented

Q₃₂×D₃₁

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×Q₃₂

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₃₁

—

C₂×Q₁₆×C₃₁

C₂²

yes

—

C₂×#92482

—

Q₁₆×D₃₁

—

Q₁₆×D₃₁

C₂²

yes

—

C₂×Dic₁₂₄

C₂²

yes

—

C₂×C₈×D₃₁

C₂²

yes

characteristic

C₃₁×Q₃₂

C₂²

yes

complemented

C₂×C₁₆×C₃₁

C₂²

yes

characteristic

#11758417

C₂²

yes

complemented

C₂×C₃₁⋊C₁₆

C₂²

yes

characteristic

Dic₂₄₈

C₂²

yes

complemented

C₁₆×D₃₁

C₂²

yes

—

C₂×Q₈×C₃₁

—

Q₈×D₃₁

—

Q₈×D₃₁

—

C₂×C₄×D₃₁

—

C₂×C₄×D₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

C₂³

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

#92482

—

C₂×C₃₁⋊C₈

D₄

yes

characteristic

Dic₁₂₄

—

Dic₁₂₄

C₂³

yes

—

C₈×D₃₁

D₄

yes

—

C₈×D₃₁

C₂³

yes

—

C₁₆×C₃₁

C₂³

yes

—

C₃₁⋊C₁₆

C₂³

yes

—

C₂²×D₃₁

D₈

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

D₈

yes

characteristic

C₄×D₃₁

—

C₄×D₃₁

D₈

yes

—

C₄×D₃₁

C₂×D₄

yes

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

C₂×D₄

yes

characteristic

C₈×C₃₁

C₂⁴

yes

characteristic

C₃₁⋊C₈

C₂×D₄

yes

—

C₂²×Q₃₂

—

maximal, Sylow

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₂×D₃₁

Q₃₂

yes

direct product factor

C₂×D₃₁

C₂×D₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

C₂²×D₄

yes

characteristic

Dic₃₁

—

Dic₃₁

C₂×D₈

yes

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₃₁

yes

characteristic, direct product factor

C₂²×C₁₆

—

C₂×C₃₁

C₂×Q₃₂

yes

complemented

C₂×C₃₁

C₂²×D₈

yes

characteristic

D₃₁

C₂×Q₃₂

yes

direct product factor

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

C₂×D₃₁

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₁

yes

characteristic

C₃₁

C₂²×Q₃₂

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

124

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂²×D₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₃₁

yes

characteristic

102

C₁₆

—

103

C₁₆

C₂²×D₃₁

yes

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

124

—

109

Q₈

—

124

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

124

—

115

C₂×C₄

—

124

—

116

C₂×C₄

D₄×D₃₁

yes

characteristic

117

C₈

—

118

C₈

D₄×D₃₁

yes

characteristic

119

C₈

C₂³×D₃₁

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

D₈×D₃₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

124

—

126

C₄

D₈×D₃₁

yes

characteristic

127

C₄

C₂×D₄×D₃₁

yes

characteristic

128

C₂

—

129

C₂

Q₃₂×D₃₁

yes

direct product factor

130

C₂

C₂×D₈×D₃₁

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₁
commutator, cosocle	C₈×C₃₁	C₂⁴
Frattini	C₈	C₂³×D₃₁
Fitting	C₂×C₃₁×Q₃₂	C₂
socle	C₂²×C₃₁	C₂×D₈
prime radical	C₂²×D₃₁	D₈
hypercenter	C₂×Q₃₂	D₃₁
hypocenter	C₃₁	C₂²×Q₃₂
LCS	G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₃₁

commutator, cosocle

C₈×C₃₁

C₂⁴

Frattini

C₈

C₂³×D₃₁

Fitting

C₂×C₃₁×Q₃₂

C₂

socle

C₂²×C₃₁

C₂×D₈

prime radical

C₂²×D₃₁

D₈

hypercenter

C₂×Q₃₂

D₃₁

hypocenter

C₃₁

C₂²×Q₃₂

LCS

G ▷ C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×Dic₂₄₈	C₂	semidirect
3. C₂×#11758417	C₂	semidirect
4. Q₃₂×D₃₁	C₂	direct (N × H)
6. C₂×C₃₁×Q₃₂	C₂	semidirect
14. C₃₁×Q₃₂	C₂²	semidirect
16. #11758417	C₂²	semidirect
18. Dic₂₄₈	C₂²	semidirect
56. C₂×D₃₁	Q₃₂	direct (N × H)
67. C₂×Q₃₂	D₃₁	direct (N × H)
69. C₂×C₃₁	C₂×Q₃₂	semidirect
71. D₃₁	C₂×Q₃₂	direct (N × H)
81. Q₃₂	C₂×D₃₁	direct (N × H)
85. C₃₁	C₂²×Q₃₂	semidirect
129. C₂	Q₃₂×D₃₁	direct (N × H)

H ≅ G/N

type

2. C₂×Dic₂₄₈

C₂

semidirect

3. C₂×#11758417

C₂

semidirect

4. Q₃₂×D₃₁

C₂

direct (N × H)

6. C₂×C₃₁×Q₃₂

C₂

semidirect

14. C₃₁×Q₃₂

C₂²

semidirect

16. #11758417

C₂²

semidirect

18. Dic₂₄₈

C₂²

semidirect

56. C₂×D₃₁

Q₃₂

direct (N × H)

67. C₂×Q₃₂

D₃₁

direct (N × H)

69. C₂×C₃₁

C₂×Q₃₂

semidirect

71. D₃₁

C₂×Q₃₂

direct (N × H)

81. Q₃₂

C₂×D₃₁

direct (N × H)

85. C₃₁

C₂²×Q₃₂

semidirect

129. C₂

Q₃₂×D₃₁

direct (N × H)

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

subgroup	index
C₂×Q₁₆×D₃₁	2
C₂×Dic₂₄₈	2
C₂×#11758417	2
Q₃₂×D₃₁	2
C₂×C₁₆×D₃₁	2
C₂×C₃₁×Q₃₂	2
C₂²×Q₃₂	31

subgroup

index

C₂×Q₁₆×D₃₁

C₂×Dic₂₄₈

C₂×#11758417

Q₃₂×D₃₁

C₂×C₁₆×D₃₁

C₂×C₃₁×Q₃₂

C₂²×Q₃₂

Subgroups — C₂×Q₃₂×D₃₁ #16858822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×Q32×D31 #16858822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×D₃₁ #16858822