Group #16845563 — subgroups

Subgroups — #16845563

Order 3968.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×Dic₃₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×#11758403	C₂	1	2	yes	maximal
3	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	#11758403	—	2	4	no	—
14	Dic₃₁⋊C₈	C₄	1	8	yes	complemented
15	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
19	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₃₁	—	2	2	no	—
21	C₂×C₈×C₃₁	—	2	2	no	—
22	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₄²×C₃₁	—	2	2	no	—
24	C₄×Dic₃₁	—	2	4	no	—
25	C₄×Dic₃₁	—	2	4	no	—
26	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
27	C₄×Dic₃₁	D₄	1	2	yes	—
28	C₄×Dic₃₁	Q₈	1	2	yes	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	M₁₆	1	4	yes	—
36	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#41	1	2	yes	—
40	C₂×Dic₃₁	—	2	2	no	—
41	C₂×Dic₃₁	—	2	2	no	—
42	C₂×Dic₃₁	—	2	2	no	—
43	C₂×Dic₃₁	C₄²	1	2	yes	—
44	C₂×Dic₃₁	C₄²	1	2	yes	—
45	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₃₁	M₁₆	1	4	yes	—
47	C₈×C₃₁	—	2	4	no	—
48	C₃₁⋊C₈	—	2	4	no	—
49	#1767	—	31	1	no	maximal, Sylow
50	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₃₁	#117	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
54	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
55	C₄×C₃₁	—	2	4	no	—
56	C₄×C₃₁	C₄×D₄	1	2	yes	—
57	C₄×C₃₁	C₄×Q₈	1	2	yes	—
58	Dic₃₁	—	2	2	no	—
59	Dic₃₁	—	2	4	no	—
60	Dic₃₁	#97	1	4	yes	complemented
61	C₂×C₄⋊C₈	—	31	2	no	—
62	C₂×#97	—	31	2	no	—
63	C₄³	—	31	1	no	—
64	#336	—	31	1	no	—
65	#336	D₃₁	1	1	yes	characteristic
66	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃₁	C₂×#97	1	1	yes	characteristic
68	C₂×C₃₁	C₄×M₁₆	1	1	yes	characteristic
69	C₂×C₃₁	#423	1	2	yes	—
70	C₂×C₃₁	#432	1	2	yes	—
71	C₂²×C₈	—	31	2	no	—
72	C₂²×C₈	C₂×D₃₁	1	2	yes	—
73	C₂×C₄²	—	31	1	no	—
74	C₂×C₄²	—	31	1	no	—
75	C₂×C₄²	—	31	2	no	—
76	C₂×C₄²	—	31	2	no	—
77	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
78	C₄⋊C₈	—	31	8	no	—
79	#97	—	62	4	no	—
80	C₃₁	#1767	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₄	—	31	1	no	—
82	C₂²×C₄	—	31	1	no	—
83	C₂²×C₄	—	31	2	no	—
84	C₂²×C₄	C₄×D₃₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	31	4	no	—
89	C₂×C₈	—	62	2	no	—
90	C₂×C₈	—	62	2	no	—
91	C₂×C₈	C₄×D₃₁	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	31	4	no	—
94	C₄²	—	31	2	no	—
95	C₄²	—	31	2	no	—
96	C₄²	—	62	4	no	—
97	C₄²	—	62	4	no	—
98	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	31	2	no	—
102	C₂×C₄	—	31	4	no	—
103	C₂×C₄	—	31	2	no	—
104	C₂×C₄	—	31	4	no	—
105	C₂×C₄	—	62	2	no	—
106	C₂×C₄	—	62	2	no	—
107	C₂×C₄	—	62	2	no	—
108	C₂×C₄	#92469	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
110	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
111	C₂×C₄	#92497	1	2	yes	—
112	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₈	—	62	4	no	—
115	C₂²	C₄²×D₃₁	1	1	yes	characteristic
116	C₂²	#11758454	1	1	yes	characteristic
117	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
118	C₂²	C₂×#92469	1	2	yes	—
119	C₂²	M₁₆×D₃₁	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	31	4	no	—
122	C₄	—	62	2	no	—
123	C₄	—	62	4	no	—
124	C₄	(C₂²⋊D₃₁)⋊C₄	1	2	yes	—
125	C₄	#11758464	1	2	yes	—
126	C₂	#14520900	1	1	yes	characteristic
127	C₂	#97×D₃₁	1	1	yes	characteristic
128	C₂	C₄×#92469	1	1	yes	characteristic
129	C₂	#14521071	1	2	yes	—
130	C₂	#14521103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₃₁⋊C₈

C₂

yes

maximal

C₂×#11758403

C₂

yes

maximal

C₄²×Dic₃₁

C₂

yes

maximal, characteristic

#14520804

C₂

yes

maximal, characteristic

C₃₁×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758403

—

Dic₃₁⋊C₈

C₄

yes

complemented

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄²

yes

—

C₂×Dic₃₁

C₄²

yes

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₂×Dic₃₁

M₁₆

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1767

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₂²×C₃₁

C₂×M₁₆

yes

—

C₂²×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₄×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#97

yes

complemented

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

—

#336

—

#336

D₃₁

yes

characteristic

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

C₂×#97

yes

characteristic

C₂×C₃₁

C₄×M₁₆

yes

characteristic

C₂×C₃₁

#423

yes

—

C₂×C₃₁

#432

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#97

—

C₃₁

#1767

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#92469

yes

—

109

C₂×C₄

C₂×C₄×D₃₁

yes

—

110

C₂×C₄

D₄×D₃₁

yes

characteristic

111

C₂×C₄

#92497

yes

—

112

C₂×C₄

Q₈×D₃₁

yes

characteristic

113

C₈

—

114

C₈

—

115

C₂²

C₄²×D₃₁

yes

characteristic

116

C₂²

#11758454

yes

characteristic

117

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

118

C₂²

C₂×#92469

yes

—

119

C₂²

M₁₆×D₃₁

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

(C₂²⋊D₃₁)⋊C₄

yes

—

125

C₄

#11758464

yes

—

126

C₂

#14520900

yes

characteristic

127

C₂

#97×D₃₁

yes

characteristic

128

C₂

C₄×#92469

yes

characteristic

129

C₂

#14521071

yes

—

130

C₂

#14521103

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂×C₄²
Fitting	C₃₁×#336	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#336	D₃₁
hypocenter	C₃₁	#1767
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂×C₄²

Fitting

C₃₁×#336

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
14. Dic₃₁⋊C₈	C₄	semidirect
60. Dic₃₁	#97	semidirect
80. C₃₁	#1767	semidirect

H ≅ G/N

type

14. Dic₃₁⋊C₈

C₄

semidirect

60. Dic₃₁

#97

semidirect

80. C₃₁

#1767

semidirect

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

subgroup	index
C₂×Dic₃₁⋊C₈	2
C₂×#11758403	2
C₄²×Dic₃₁	2
#14520804	2
C₃₁×#336	2
#1767	31

subgroup

index

C₂×Dic₃₁⋊C₈

C₂×#11758403

C₄²×Dic₃₁

#14520804

C₃₁×#336

#1767

Subgroups — #16845563

Conjugacy classes of subgroups

Complements

Maximal subgroups