Group #16845560 — subgroups

Subgroups — C₄×C₄⋊(C₃₁⋊C₈) #16845560

Order 3968.

316 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 213 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×C₄⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₃₁⋊C₈	C₂	1	2	yes	maximal
3	#14520804	C₂	1	2	yes	maximal
4	C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₃₁	C₄	1	2	yes	—
6	C₂×C₄²×C₃₁	C₄	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
11	C₄⋊(C₃₁⋊C₈)	C₄	1	8	yes	direct product factor
12	C₄×C₃₁⋊C₈	—	2	4	no	—
13	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₄²×C₃₁	—	2	4	no	—
19	C₄²×C₃₁	—	2	2	no	—
20	C₄²×C₃₁	C₈	1	8	yes	complemented
21	C₄²×C₃₁	C₂×C₄	1	4	yes	—
22	C₄²×C₃₁	D₄	1	2	yes	—
23	C₄²×C₃₁	Q₈	1	2	yes	—
24	C₂×C₃₁⋊C₈	—	2	4	no	—
25	C₂×C₃₁⋊C₈	—	2	4	no	—
26	C₂×C₃₁⋊C₈	C₂×C₄	1	8	yes	—
27	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	—	2	2	no	—
29	C₂×C₄×C₃₁	—	2	2	no	—
30	C₂×C₄×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	C₄²	1	2	yes	—
32	C₂×C₄×C₃₁	C₄²	1	2	yes	—
33	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
37	C₂×C₄×C₃₁	M₁₆	1	4	yes	—
38	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	2	yes	—
42	C₃₁⋊C₈	—	2	8	no	—
43	C₄×C₄⋊C₈	—	31	1	no	maximal, Sylow
44	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
45	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₃₁	#117	1	1	yes	characteristic
47	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
48	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
49	C₄×C₃₁	—	2	2	no	—
50	C₄×C₃₁	—	2	4	no	—
51	C₄×C₃₁	C₄×C₈	1	4	yes	complemented
52	C₄×C₃₁	C₄⋊C₈	1	8	yes	complemented
53	C₄×C₃₁	C₄×D₄	1	2	yes	—
54	C₄×C₃₁	C₄×Q₈	1	2	yes	—
55	C₂×C₄⋊C₈	—	31	2	no	—
56	C₂×C₄×C₈	—	31	2	no	—
57	C₄³	D₃₁	1	1	yes	characteristic
58	#336	—	31	2	no	—
59	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₃₁	C₄×M₁₆	1	1	yes	characteristic
62	C₂×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
63	C₂×C₃₁	#431	1	2	yes	—
64	C₂²×C₈	—	31	4	no	—
65	C₂×C₄²	Dic₃₁	1	2	yes	—
66	C₂×C₄²	Dic₃₁	1	2	yes	—
67	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	C₄⋊C₈	—	31	8	no	—
71	C₄×C₈	—	62	4	no	—
72	C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
74	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
78	C₂×C₈	—	31	8	no	—
79	C₂×C₈	—	62	4	no	—
80	C₂×C₈	—	62	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	2	no	—
83	C₄²	C₃₁⋊C₈	1	8	yes	complemented
84	C₄²	Dic₆₂	1	2	yes	—
85	C₄²	C₄×D₃₁	1	4	yes	—
86	C₄²	D₁₂₄	1	2	yes	—
87	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
92	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
93	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
94	C₂×C₄	#92474	1	4	yes	—
95	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
96	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
97	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
98	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
100	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
101	C₂×C₄	#92495	1	2	yes	—
102	C₈	—	62	8	no	—
103	C₂²	C₂²×C₃₁⋊C₈	1	2	yes	—
104	C₂²	C₂×#92474	1	2	yes	—
105	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
106	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
107	C₂²	#11758503	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	2	2	no	—
110	C₄	C₄×C₃₁⋊C₈	1	4	yes	complemented
111	C₄	C₄⋊(C₃₁⋊C₈)	1	8	yes	direct product factor
112	C₄	C₄×Dic₆₂	1	2	yes	—
113	C₄	C₄×D₁₂₄	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₃₁	1	1	yes	characteristic
115	C₂	C₂×C₄×C₃₁⋊C₈	1	1	yes	characteristic
116	C₂	C₄×#92474	1	1	yes	characteristic
117	C₂	C₂×C₄⋊(C₃₁⋊C₈)	1	2	yes	—
118	C₂	#14521102	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal

#14520804

C₂

yes

maximal

C₄³×C₃₁

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₄

yes

direct product factor

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₈

yes

complemented

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

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

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

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

—

C₃₁⋊C₈

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₂²×C₃₁

C₂²×C₈

yes

—

C₂²×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×C₈

yes

complemented

C₄×C₃₁

C₄⋊C₈

yes

complemented

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₄×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄³

D₃₁

yes

characteristic

#336

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₃₁

C₄×M₁₆

yes

characteristic

C₂×C₃₁

C₂×C₄⋊C₈

yes

—

C₂×C₃₁

#431

yes

—

C₂²×C₈

—

C₂×C₄²

Dic₃₁

yes

—

C₂×C₄²

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₃₁

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

C₃₁⋊C₈

yes

complemented

C₄²

Dic₆₂

yes

—

C₄²

C₄×D₃₁

yes

—

C₄²

D₁₂₄

yes

—

C₂³

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

C₂×C₄

#92474

yes

—

C₂×C₄

C₄×Dic₃₁

yes

—

C₂×C₄

C₄×Dic₃₁

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×C₄×D₃₁

yes

—

100

C₂×C₄

C₂×D₁₂₄

yes

characteristic

101

C₂×C₄

#92495

yes

—

102

C₈

—

103

C₂²

C₂²×C₃₁⋊C₈

yes

—

104

C₂²

C₂×#92474

yes

—

105

C₂²

C₂×C₄×Dic₃₁

yes

characteristic

106

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

107

C₂²

#11758503

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

C₄×C₃₁⋊C₈

yes

complemented

111

C₄

C₄⋊(C₃₁⋊C₈)

yes

direct product factor

112

C₄

C₄×Dic₆₂

yes

—

113

C₄

C₄×D₁₂₄

yes

—

114

C₂

C₄×C₄⋊Dic₃₁

yes

characteristic

115

C₂

C₂×C₄×C₃₁⋊C₈

yes

characteristic

116

C₂

C₄×#92474

yes

characteristic

117

C₂

C₂×C₄⋊(C₃₁⋊C₈)

yes

—

118

C₂

#14521102

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₃₁
commutator	C₂×C₃₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₄³×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₄³	D₃₁
hypocenter	C₃₁	C₄×C₄⋊C₈
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄²

C₂×D₃₁

commutator

C₂×C₃₁

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₄³×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₄³

D₃₁

hypocenter

C₃₁

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
11. C₄⋊(C₃₁⋊C₈)	C₄	direct (N × H)
20. C₄²×C₃₁	C₈	semidirect
51. C₄×C₃₁	C₄×C₈	semidirect
52. C₄×C₃₁	C₄⋊C₈	semidirect
72. C₃₁	C₄×C₄⋊C₈	semidirect
83. C₄²	C₃₁⋊C₈	semidirect
110. C₄	C₄×C₃₁⋊C₈	semidirect
111. C₄	C₄⋊(C₃₁⋊C₈)	direct (N × H)

H ≅ G/N

type

11. C₄⋊(C₃₁⋊C₈)

C₄

direct (N × H)

20. C₄²×C₃₁

C₈

semidirect

51. C₄×C₃₁

C₄×C₈

semidirect

52. C₄×C₃₁

C₄⋊C₈

semidirect

72. C₃₁

C₄×C₄⋊C₈

semidirect

83. C₄²

C₃₁⋊C₈

semidirect

110. C₄

C₄×C₃₁⋊C₈

semidirect

111. C₄

C₄⋊(C₃₁⋊C₈)

direct (N × H)

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

subgroup	index
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×C₄×C₃₁⋊C₈	2
#14520804	2
C₄³×C₃₁	2
C₄×C₄⋊C₈	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×C₄×C₃₁⋊C₈

#14520804

C₄³×C₃₁

C₄×C₄⋊C₈

Subgroups — C₄×C₄⋊(C₃₁⋊C₈) #16845560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×C4⋊(C31⋊C8) #16845560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄⋊(C₃₁⋊C₈) #16845560