Group #16842048 — subgroups

Subgroups — C₃₁×#2115 #16842048

Order 3968.

220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃₁×#349	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₃₁	C₄	1	1	yes	characteristic
5	C₃₁×#131	—	2	2	no	—
6	C₃₁×#131	—	2	2	no	—
7	C₂×M₁₆×C₃₁	C₄	1	1	yes	characteristic
8	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₈×C₃₁	C₄	1	2	yes	—
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×M₃₂	—	2	2	no	—
12	C₃₁×M₃₂	C₂²	1	4	yes	—
13	C₂×C₁₆×C₃₁	—	2	2	no	—
14	#41×C₃₁	—	2	2	no	—
15	#41×C₃₁	—	2	2	no	—
16	C₂×Q₈×C₃₁	C₈	1	1	yes	characteristic
17	C₂×D₄×C₃₁	C₈	1	2	yes	—
18	C₂×D₄×C₃₁	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	M₁₆×C₃₁	—	2	2	no	—
22	M₁₆×C₃₁	—	2	4	no	—
23	C₂×C₈×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₂×C₈×C₃₁	D₄	1	2	yes	—
29	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₁₆×C₃₁	—	2	4	no	—
31	C₂³×C₃₁	C₂×C₈	1	2	yes	—
32	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₃₁	—	2	2	no	—
34	D₄×C₃₁	—	2	4	no	—
35	D₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₃₁	—	2	2	no	—
44	C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₈×C₃₁	C₂×D₄	1	2	yes	—
47	#2115	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₃₁	—	2	2	no	—
49	C₂²×C₃₁	—	4	1	no	—
50	C₂²×C₃₁	—	4	2	no	—
51	C₂²×C₃₁	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
53	C₄×C₃₁	—	2	2	no	—
54	C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
55	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂×#131	C₂×C₃₁	1	1	yes	characteristic
58	C₂×M₃₂	C₂×C₃₁	1	2	yes	complemented
59	#349	C₂×C₃₁	1	4	yes	complemented
60	C₂×C₃₁	—	2	1	no	—
61	C₂×C₃₁	—	2	2	no	—
62	C₂×C₃₁	—	4	2	no	—
63	C₂×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
64	C₂×#41	C₄×C₃₁	1	1	yes	characteristic
65	#131	—	2	2	no	—
66	#131	—	2	2	no	—
67	C₂×M₁₆	C₄×C₃₁	1	1	yes	characteristic
68	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
69	C₂²×C₈	C₄×C₃₁	1	2	yes	—
70	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
71	M₃₂	—	2	2	no	—
72	M₃₂	C₂²×C₃₁	1	4	yes	—
73	C₂×C₁₆	—	2	2	no	—
74	C₃₁	#2115	1	1	yes	characteristic, direct product factor, Sylow
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	C₈×C₃₁	1	1	yes	characteristic
78	C₂×D₄	C₈×C₃₁	1	2	yes	—
79	C₂×D₄	C₈×C₃₁	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄×C₃₁	1	2	yes	—
81	C₂²×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	C₂×C₄×C₃₁	1	2	yes	—
87	C₂×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
88	C₂×C₈	D₄×C₃₁	1	2	yes	—
89	C₂×C₈	D₄×C₃₁	1	2	yes	—
90	C₂×C₈	C₂³×C₃₁	1	1	yes	characteristic
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×C₈×C₃₁	1	2	yes	—
93	C₂³	C₂×C₈×C₃₁	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
100	C₂×C₄	C₂×C₈×C₃₁	1	2	yes	—
101	C₂×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
102	C₂×C₄	M₁₆×C₃₁	1	2	yes	—
103	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
106	C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
107	C₈	C₂×D₄×C₃₁	1	2	yes	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₃₁×C₂²⋊C₈	1	2	yes	—
112	C₂²	C₂²×C₈×C₃₁	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
115	C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
116	C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#131

C₂

yes

maximal, characteristic

C₂×C₃₁×M₃₂

C₂

yes

maximal, complemented

C₃₁×#349

C₂

yes

maximal, complemented

C₂×#41×C₃₁

C₄

yes

characteristic

C₃₁×#131

—

C₃₁×#131

—

C₂×M₁₆×C₃₁

C₄

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₄

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×M₃₂

—

C₃₁×M₃₂

C₂²

yes

—

C₂×C₁₆×C₃₁

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₈

yes

characteristic

C₂×D₄×C₃₁

C₈

yes

—

C₂×D₄×C₃₁

C₈

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₁₆×C₃₁

—

C₂³×C₃₁

C₂×C₈

yes

—

C₂³×C₃₁

C₂×C₈

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₈×C₃₁

C₂×D₄

yes

—

#2115

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂²⋊C₈

yes

—

C₂²×C₃₁

C₂²×C₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊C₈

yes

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₂×M₁₆

yes

characteristic

C₂×#131

C₂×C₃₁

yes

characteristic

C₂×M₃₂

C₂×C₃₁

yes

complemented

#349

C₂×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×#41

C₄×C₃₁

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₄×C₃₁

yes

characteristic

C₂×M₁₆

C₂²×C₃₁

yes

—

C₂²×C₈

C₄×C₃₁

yes

—

C₂²×C₈

C₂²×C₃₁

yes

characteristic

M₃₂

—

M₃₂

C₂²×C₃₁

yes

—

C₂×C₁₆

—

C₃₁

#2115

yes

characteristic, direct product factor, Sylow

#41

—

#41

—

C₂×Q₈

C₈×C₃₁

yes

characteristic

C₂×D₄

C₈×C₃₁

yes

—

C₂×D₄

C₈×C₃₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₃₁

yes

—

C₂²×C₄

C₂×C₄×C₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₃₁

yes

—

C₂×C₈

C₂×C₄×C₃₁

yes

characteristic

C₂×C₈

D₄×C₃₁

yes

—

C₂×C₈

D₄×C₃₁

yes

—

C₂×C₈

C₂³×C₃₁

yes

characteristic

C₁₆

—

C₂³

C₂×C₈×C₃₁

yes

—

C₂³

C₂×C₈×C₃₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₃₁

yes

—

100

C₂×C₄

C₂×C₈×C₃₁

yes

—

101

C₂×C₄

C₂×C₈×C₃₁

yes

characteristic

102

C₂×C₄

M₁₆×C₃₁

yes

—

103

C₂×C₄

C₂²×C₄×C₃₁

yes

characteristic

104

C₈

—

105

C₈

C₂²⋊C₄×C₃₁

yes

characteristic

106

C₈

C₂²⋊C₄×C₃₁

yes

characteristic

107

C₈

C₂×D₄×C₃₁

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₃₁×C₂²⋊C₈

yes

—

112

C₂²

C₂²×C₈×C₃₁

yes

characteristic

113

C₄

—

114

C₄

C₃₁×C₂²⋊C₈

yes

—

115

C₄

C₂×C₂²⋊C₄×C₃₁

yes

characteristic

116

C₄

C₂×M₁₆×C₃₁

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₃₁×C₂²⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈×C₃₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle	C₂×C₃₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₃₁	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₈×C₃₁

C₂²⋊C₄

commutator

C₂²

C₂²×C₈×C₃₁

Frattini, cosocle

C₂×C₈

C₂³×C₃₁

socle

C₂×C₃₁

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₃₁

C₄

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₂×C₃₁×M₃₂	C₂	semidirect
3. C₃₁×#349	C₂	semidirect
47. #2115	C₃₁	direct (N × H)
58. C₂×M₃₂	C₂×C₃₁	semidirect
59. #349	C₂×C₃₁	semidirect
74. C₃₁	#2115	direct (N × H)

H ≅ G/N

type

2. C₂×C₃₁×M₃₂

C₂

semidirect

3. C₃₁×#349

C₂

semidirect

47. #2115

C₃₁

direct (N × H)

58. C₂×M₃₂

C₂×C₃₁

semidirect

59. #349

C₂×C₃₁

semidirect

74. C₃₁

#2115

direct (N × H)

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

subgroup	index
C₂×C₃₁×#131	2
C₂×C₃₁×M₃₂	2
C₃₁×#349	2
#2115	31

subgroup

index

C₂×C₃₁×#131

C₂×C₃₁×M₃₂

C₃₁×#349

#2115

Subgroups — C₃₁×#2115 #16842048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#2115 #16842048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2115 #16842048