Group #16854648 — subgroups

Subgroups — #16854648

Order 3968.

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

normal subgroups only

This group has 348 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	8	yes	maximal, complemented
3	#14521077	C₂	1	2	yes	maximal
4	#14521007	C₂	1	2	yes	maximal
5	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#423	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	2	yes	—
10	#11758501	C₄	1	4	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	#11758447	C₄	1	2	yes	—
14	#11758446	C₄	1	2	yes	—
15	C₄×Dic₆₂	C₂²	1	4	yes	—
16	C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	#11758403	C₂²	1	4	yes	—
18	Dic₃₁⋊C₈	C₂²	1	8	yes	—
19	C₈×Dic₃₁	C₂²	1	4	yes	—
20	#11758391	C₂²	1	4	yes	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
24	M₁₆×C₃₁	—	2	4	no	—
25	C₂×C₈×C₃₁	C₂³	1	4	yes	—
26	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₃₁	C₂³	1	2	yes	—
29	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
32	Dic₃₁⋊C₄	C₂×C₄	1	8	yes	—
33	C₄×Dic₃₁	C₂³	1	4	yes	—
34	C₂×C₃₁⋊C₈	—	2	2	no	—
35	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
36	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
37	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
38	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	4	2	no	—
40	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₃₁	#41	1	2	yes	—
45	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
47	Dic₆₂	—	4	4	no	—
48	C₈×C₃₁	—	2	4	no	—
49	C₃₁⋊C₈	—	2	2	no	—
50	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
51	#2961	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	2	no	—
53	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
54	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	C₄×Q₈	1	2	yes	—
58	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
60	Dic₃₁	—	2	8	no	—
61	#533	—	31	1	no	—
62	#446	—	31	8	no	—
63	#424	—	31	2	no	—
64	#423	D₃₁	1	1	yes	characteristic
65	#405	—	31	2	no	—
66	C₂×#97	—	31	1	no	—
67	C₂×C₃₁	—	2	2	no	—
68	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₃₁	#568	1	2	yes	—
70	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
71	C₂²×C₈	—	31	1	no	—
72	C₂²×C₈	—	31	1	no	—
73	C₄⋊Q₈	—	31	2	no	—
74	#125	—	31	2	no	—
75	C₂²⋊Q₈	—	31	4	no	—
76	C₄×Q₈	—	31	4	no	—
77	#117	—	31	2	no	—
78	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
79	C₄⋊C₈	—	31	8	no	—
80	C₄⋊C₈	C₂×D₃₁	1	4	yes	—
81	#97	—	31	4	no	—
82	#97	—	31	4	no	—
83	C₄×C₈	—	31	4	no	—
84	C₃₁	#2961	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	31	4	no	—
86	C₂²×C₄	C₄×D₃₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	31	4	no	—
90	C₂×C₈	—	31	2	no	—
91	C₂×C₈	—	31	2	no	—
92	C₂×C₈	—	62	2	no	—
93	C₂×C₈	C₂²×D₃₁	1	4	yes	—
94	C₄⋊C₄	—	31	8	no	—
95	C₄⋊C₄	—	31	4	no	—
96	C₄⋊C₄	—	31	4	no	—
97	C₂²⋊C₄	—	31	4	no	—
98	C₄²	—	31	4	no	—
99	C₄²	C₄×D₃₁	1	1	yes	characteristic
100	C₄²	C₄×D₃₁	1	1	yes	characteristic
101	C₄²	C₂²×D₃₁	1	2	yes	—
102	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
103	Q₈	—	124	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	31	8	no	—
106	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
109	C₂×C₄	#92497	1	2	yes	—
110	C₂×C₄	Q₈×D₃₁	1	2	yes	—
111	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	31	4	no	—
114	C₈	—	62	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#11758464	1	2	yes	—
117	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	62	8	no	—
121	C₄	#11758464	1	2	yes	—
122	C₄	C₂×#92497	1	1	yes	characteristic
123	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×#11758464	1	1	yes	characteristic
126	C₂	#14521815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521178

C₂

yes

maximal, complemented

#14521077

C₂

yes

maximal

#14521007

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic

C₃₁×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

C₄

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758447

C₄

yes

—

#11758446

C₄

yes

—

C₄×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

—

#11758403

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

#11758391

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

C₂×C₄

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

C₂²×C₄

yes

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#2961

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂×#41

yes

characteristic

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic

#405

—

C₂×#97

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₁

#568

yes

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

#97

—

#97

—

C₄×C₈

—

C₃₁

#2961

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

C₄×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₃₁

yes

characteristic

100

C₄²

C₄×D₃₁

yes

characteristic

101

C₄²

C₂²×D₃₁

yes

—

102

C₂³

C₂×C₄×D₃₁

yes

characteristic

103

Q₈

—

124

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₃₁

yes

—

107

C₂×C₄

C₂×C₄×D₃₁

yes

—

108

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

109

C₂×C₄

#92497

yes

—

110

C₂×C₄

Q₈×D₃₁

yes

—

111

C₂×C₄

C₂³×D₃₁

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

#11758464

yes

—

117

C₂²

C₂²×C₄×D₃₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#11758464

yes

—

122

C₄

C₂×#92497

yes

characteristic

123

C₄

C₂×Q₈×D₃₁

yes

characteristic

124

C₂

—

125

C₂

C₂×#11758464

yes

characteristic

126

C₂

#14521815

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#423	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#423	D₃₁
hypocenter	C₃₁	#2961
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#423

C₂

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #14521178	C₂	semidirect
35. C₂×C₃₁⋊C₈	Q₈	semidirect
84. C₃₁	#2961	semidirect

H ≅ G/N

type

2. #14521178

C₂

semidirect

35. C₂×C₃₁⋊C₈

Q₈

semidirect

84. C₃₁

#2961

semidirect

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

subgroup	index
#14521606	2
#14521178	2
#14521077	2
#14521007	2
C₂×#11758391	2
C₃₁×#423	2
#2961	31

subgroup

index

C₃₁×#423

Subgroups — #16854648

Conjugacy classes of subgroups

Complements

Maximal subgroups