Group #16856212 — subgroups

Subgroups — C₂×#14521448 #16856212

Order 3968.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#11758447	C₂	1	1	yes	maximal, characteristic
2	#14521448	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758395	C₂	1	2	yes	maximal
4	C₂×#11758396	C₂	1	2	yes	maximal
5	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#125	—	2	2	no	—
8	C₃₁×#125	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂×Dic₃₁⋊C₄	—	2	2	no	—
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	#11758447	—	2	2	no	—
17	#11758447	C₂²	1	4	yes	—
18	#11758396	C₂²	1	8	yes	—
19	#11758395	C₂²	1	8	yes	—
20	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
21	C₂²×C₄×C₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₄⋊C₄×C₃₁	—	2	8	no	—
26	C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
28	C₄²×C₃₁	D₄	1	2	yes	—
29	C₄²×C₃₁	C₂³	1	2	yes	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	C₂³	1	4	yes	—
32	Dic₃₁⋊C₄	—	2	8	no	—
33	C₂×C₃₁⋊C₈	Q₈	1	4	yes	—
34	C₂×C₃₁⋊C₈	Q₈	1	4	yes	—
35	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
36	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
37	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	—	4	4	no	—
41	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₃₁	—	2	4	no	—
47	C₂×Dic₃₁	—	4	4	no	—
48	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
49	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
50	C₂×#499	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
52	C₂²×C₃₁	#135	1	4	yes	—
53	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₃₁	—	2	4	no	—
55	C₄×C₃₁	—	4	4	no	—
56	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
57	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
59	Dic₃₁	—	4	4	no	—
60	C₂×#125	—	31	1	no	—
61	C₂×#125	D₃₁	1	1	yes	characteristic
62	#499	—	31	8	no	—
63	C₂×#107	—	31	2	no	—
64	C₂×#106	—	31	2	no	—
65	C₂×C₄×C₈	—	31	1	no	—
66	C₂×C₃₁	#499	1	4	yes	complemented
67	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₃₁	C₂×#135	1	2	yes	—
69	C₂²×C₈	—	31	1	no	—
70	C₂²×C₈	—	31	1	no	—
71	#125	—	2	2	no	—
72	#125	—	31	4	no	—
73	#125	—	62	2	no	—
74	#125	C₂×D₃₁	1	4	yes	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	—	62	2	no	—
78	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
79	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
80	#107	—	31	8	no	—
81	#106	—	31	8	no	—
82	C₄×C₈	—	31	4	no	—
83	C₃₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	62	2	no	—
86	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	C₂×C₈	—	31	4	no	—
89	C₂×C₈	—	31	4	no	—
90	C₂×C₈	—	31	2	no	—
91	C₂×C₈	—	31	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	31	4	no	—
95	C₄⋊C₄	—	62	8	no	—
96	C₄⋊C₄	—	62	2	no	—
97	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
98	C₄²	C₂²⋊D₃₁	1	2	yes	—
99	C₄²	C₂²×D₃₁	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	62	4	no	—
105	C₂×C₄	—	124	4	no	—
106	C₂×C₄	Q₈×D₃₁	1	2	yes	—
107	C₂×C₄	Q₈×D₃₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
110	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
111	C₈	—	31	4	no	—
112	C₈	—	31	4	no	—
113	C₂²	#11758521	1	2	yes	—
114	C₂²	#11758527	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	124	4	no	—
119	C₄	#11758521	1	2	yes	—
120	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
122	C₂	#14521448	1	4	yes	direct product factor
123	C₂	C₂×#11758521	1	1	yes	characteristic
124	C₂	C₂×#11758527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic

#14521448

C₂

yes

maximal, direct product factor

C₂×#11758395

C₂

yes

maximal

C₂×#11758396

C₂

yes

maximal

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₃₁×#125

—

C₃₁×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758447

—

#11758447

C₂²

yes

—

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₂×#499

—

maximal, Sylow

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

C₂×#125

—

C₂×#125

D₃₁

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₃₁

#499

yes

complemented

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₃₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

—

100

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

124

—

106

C₂×C₄

Q₈×D₃₁

yes

—

107

C₂×C₄

Q₈×D₃₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

110

C₂×C₄

C₂³×D₃₁

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

#11758521

yes

—

114

C₂²

#11758527

yes

—

115

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

124

—

119

C₄

#11758521

yes

—

120

C₄

C₂×Q₈×D₃₁

yes

characteristic

121

C₄

C₂×Q₈×D₃₁

yes

characteristic

122

C₂

#14521448

yes

direct product factor

123

C₂

C₂×#11758521

yes

characteristic

124

C₂

C₂×#11758527

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	C₂×#499
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #14521448	C₂	direct (N × H)
66. C₂×C₃₁	#499	semidirect
83. C₃₁	C₂×#499	semidirect
122. C₂	#14521448	direct (N × H)

H ≅ G/N

type

2. #14521448

C₂

direct (N × H)

66. C₂×C₃₁

#499

semidirect

83. C₃₁

C₂×#499

semidirect

122. C₂

#14521448

direct (N × H)

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

subgroup	index
C₂×#11758447	2
#14521448	2
C₂×#11758395	2
C₂×#11758396	2
C₂×C₄×C₃₁⋊C₈	2
C₂×C₃₁×#125	2
C₂×#499	31

subgroup

index

C₂×C₄×C₃₁⋊C₈

C₂×C₃₁×#125

C₂×#499

Subgroups — C₂×#14521448 #16856212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521448 #16856212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521448 #16856212