Group #16856220 — subgroups

Subgroups — #16856220

Order 3968.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#14521604	C₂	1	2	yes	maximal
2	#14521452	C₂	1	2	yes	maximal, complemented
3	#14521449	C₂	1	4	yes	maximal, complemented
4	#14521447	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521084	C₂	1	4	yes	maximal
6	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	#11758523	—	2	2	no	—
10	Q₈×Dic₃₁	—	2	2	no	—
11	C₂×C₈×D₃₁	C₂²	1	2	yes	—
12	#11758474	—	2	2	no	—
13	#11758469	C₂²	1	4	yes	—
14	#11758467	C₂²	1	4	yes	—
15	#11758464	—	2	2	no	—
16	C₄²×D₃₁	C₂²	1	1	yes	characteristic
17	#11758446	C₂²	1	2	yes	—
18	C₃₁×#106	C₂²	1	4	yes	—
19	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	#11758405	C₂²	1	4	yes	—
21	C₈×Dic₃₁	C₂²	1	2	yes	—
22	#11758395	C₂²	1	8	yes	—
23	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₁	—	2	2	no	—
25	C₂×C₄×D₃₁	D₄	1	2	yes	—
26	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₆₂	—	2	2	no	—
28	C₂×C₈×C₃₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
31	C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	(C₂×D₃₁)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₃₁	—	2	2	no	—
34	C₄⋊Dic₃₁	—	2	4	no	—
35	C₄⋊Dic₃₁	C₂³	1	4	yes	—
36	Dic₃₁⋊C₄	—	2	4	no	—
37	C₄×Dic₃₁	—	2	4	no	—
38	C₄×Dic₃₁	D₄	1	2	yes	—
39	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
41	C₈×D₃₁	Q₈	1	8	yes	complemented
42	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₃₁	—	4	4	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₃₁	—	2	4	no	—
48	C₂×Dic₃₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₃₁	—	2	4	no	—
51	C₄×D₃₁	C₂×Q₈	1	4	yes	—
52	Dic₆₂	—	4	4	no	—
53	C₈×C₃₁	C₂×Q₈	1	4	yes	—
54	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
55	#3160	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₃₁	—	2	2	no	—
58	C₂×D₃₁	C₄⋊Q₈	1	2	yes	—
59	C₄×C₃₁	—	4	4	no	—
60	C₄×C₃₁	#135	1	4	yes	—
61	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
62	Dic₃₁	—	2	2	no	—
63	Dic₃₁	—	4	4	no	—
64	Dic₃₁	C₄⋊Q₈	1	2	yes	—
65	#533	—	31	2	no	—
66	#500	—	31	2	no	—
67	#499	—	31	4	no	—
68	#498	—	31	1	no	—
69	#498	D₃₁	1	1	yes	characteristic, complemented
70	#427	—	31	4	no	—
71	C₂×C₄×C₈	—	31	1	no	—
72	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#135	1	2	yes	—
74	D₃₁	—	2	2	no	—
75	C₂²×C₈	—	31	2	no	—
76	C₄⋊Q₈	—	31	2	no	—
77	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
78	#125	—	31	4	no	—
79	C₂²⋊Q₈	—	62	2	no	—
80	C₂²⋊Q₈	—	62	2	no	—
81	C₄×Q₈	—	62	2	no	—
82	C₄×Q₈	—	62	2	no	—
83	#117	—	31	4	no	—
84	C₂×C₄²	—	31	1	no	—
85	#107	—	31	8	no	—
86	#106	—	31	4	no	—
87	#106	C₂×D₃₁	1	4	yes	—
88	C₄×C₈	—	31	2	no	—
89	C₄×C₈	—	31	1	no	—
90	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
91	C₃₁	#3160	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	62	2	no	—
94	C₂²×C₄	—	31	1	no	—
95	C₂²×C₄	—	31	2	no	—
96	C₂×C₈	—	31	2	no	—
97	C₂×C₈	—	31	8	no	—
98	C₂×C₈	C₂²×D₃₁	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	31	4	no	—
101	C₄⋊C₄	—	62	4	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄⋊C₄	—	62	2	no	—
104	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
105	C₂²⋊C₄	—	62	4	no	—
106	C₄²	—	31	2	no	—
107	C₄²	—	31	1	no	—
108	C₄²	—	62	4	no	—
109	C₄²	C₂²×D₃₁	1	1	yes	characteristic
110	C₂³	—	31	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	124	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	31	1	no	—
115	C₂×C₄	—	31	4	no	—
116	C₂×C₄	—	31	2	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	62	4	no	—
119	C₂×C₄	D₄×D₃₁	1	2	yes	—
120	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
121	C₈	—	31	4	no	—
122	C₈	Q₈×D₃₁	1	4	yes	—
123	C₂²	—	31	2	no	—
124	C₂²	—	62	2	no	—
125	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	31	2	no	—
128	C₄	—	62	2	no	—
129	C₄	—	124	4	no	—
130	C₄	#11758493	1	4	yes	—
131	C₄	C₂×Q₈×D₃₁	1	2	yes	—
132	C₂	—	62	2	no	—
133	C₂	C₄⋊Q₈×D₃₁	1	1	yes	characteristic
134	C₂	C₂×#11758493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521604

C₂

yes

maximal

#14521452

C₂

yes

maximal, complemented

#14521449

C₂

yes

maximal, complemented

#14521447

C₂

yes

maximal, characteristic, complemented

#14521084

C₂

yes

maximal

C₄×C₈×D₃₁

C₂

yes

maximal, characteristic

C₃₁×#498

C₂

yes

maximal, characteristic, complemented

C₃₁×C₄⋊Q₈

C₂²

yes

—

#11758523

—

Q₈×Dic₃₁

—

C₂×C₈×D₃₁

C₂²

yes

—

#11758474

—

#11758469

C₂²

yes

—

#11758467

C₂²

yes

—

#11758464

—

C₄²×D₃₁

C₂²

yes

characteristic

#11758446

C₂²

yes

—

C₃₁×#106

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758405

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

#11758395

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×C₄×D₃₁

D₄

yes

—

C₂×C₄×D₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

(C₂×D₃₁)⋊C₄

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

—

C₈×D₃₁

Q₈

yes

complemented

C₂²×D₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

C₂×D₄

yes

characteristic

C₄×D₃₁

—

C₄×D₃₁

C₂×Q₈

yes

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₄⋊Q₈

yes

—

—

—

—

—

D₃₁

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

D₃₁

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₃₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂²×D₃₁

yes

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂²×D₃₁

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

124

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₃₁

yes

—

120

C₂×C₄

C₂³×D₃₁

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₃₁

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×D₄×D₃₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

124

—

130

C₄

#11758493

yes

—

131

C₄

C₂×Q₈×D₃₁

yes

—

132

C₂

—

133

C₂

C₄⋊Q₈×D₃₁

yes

characteristic

134

C₂

C₂×#11758493

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#498	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×D₃₁	C₂×D₄
hypercenter	#498	D₃₁
hypocenter	C₃₁	#3160
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#498

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂²×D₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #14521452	C₂	semidirect
3. #14521449	C₂	semidirect
4. #14521447	C₂	semidirect
7. C₃₁×#498	C₂	semidirect
41. C₈×D₃₁	Q₈	semidirect
69. #498	D₃₁	semidirect
91. C₃₁	#3160	semidirect

H ≅ G/N

type

2. #14521452

C₂

semidirect

3. #14521449

C₂

semidirect

4. #14521447

C₂

semidirect

7. C₃₁×#498

C₂

semidirect

41. C₈×D₃₁

Q₈

semidirect

69. #498

D₃₁

semidirect

91. C₃₁

#3160

semidirect

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

subgroup	index
#14521604	2
#14521452	2
#14521449	2
#14521447	2
#14521084	2
C₄×C₈×D₃₁	2
C₃₁×#498	2
#3160	31

subgroup

index

C₄×C₈×D₃₁

C₃₁×#498

#3160

Subgroups — #16856220

Conjugacy classes of subgroups

Complements

Maximal subgroups