Group #16846693 — subgroups

Subgroups — #16846693

Order 3968.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	#14521073	C₂	1	1	yes	maximal, characteristic
2	#14520895	C₂	1	1	yes	maximal, characteristic
3	#14520808	C₂	1	4	yes	maximal
4	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
5	C₃₁×C₄⋊Q₈	C₄	1	2	yes	complemented
6	C₃₁×#125	C₄	1	2	yes	complemented
7	C₃₁×C₂²⋊Q₈	—	2	2	no	—
8	C₄×Q₈×C₃₁	—	2	2	no	—
9	C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
12	C₂×#92474	C₂²	1	2	yes	—
13	C₃₁⋊#95	—	2	2	no	—
14	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
15	C₂×Q₈×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₄⋊C₄×C₃₁	—	2	4	no	—
20	C₄⋊C₄×C₃₁	—	2	2	no	—
21	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₃₁	—	2	2	no	—
23	C₄²×C₃₁	—	2	2	no	—
24	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄×Dic₃₁	—	2	2	no	—
28	C₄×Dic₃₁	Q₈	1	4	yes	complemented
29	#92474	—	2	4	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₃₁	—	4	4	no	—
33	C₂×C₄×C₃₁	—	2	4	no	—
34	C₂×C₄×C₃₁	—	4	2	no	—
35	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₂×Dic₃₁	—	2	4	no	—
42	C₂×Dic₃₁	—	4	4	no	—
43	C₃₁⋊C₈	—	4	2	no	—
44	#1995	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	2	no	—
46	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₃₁	—	2	2	no	—
49	C₄×C₃₁	—	2	2	no	—
50	C₄×C₃₁	—	4	4	no	—
51	C₄×C₃₁	C₄×Q₈	1	2	yes	—
52	C₄×C₃₁	#124	1	1	yes	characteristic
53	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₃₁	—	4	4	no	—
55	#533	D₃₁	1	1	yes	characteristic
56	#423	—	31	1	no	—
57	#376	—	31	1	no	—
58	#339	—	31	4	no	—
59	C₂×C₃₁	—	2	2	no	—
60	C₂×C₃₁	#391	1	1	yes	characteristic
61	C₂×C₃₁	#421	1	2	yes	—
62	C₂×M₁₆	—	31	2	no	—
63	C₄⋊Q₈	Dic₃₁	1	2	yes	complemented
64	#125	Dic₃₁	1	2	yes	complemented
65	C₂²⋊Q₈	—	2	2	no	—
66	C₄×Q₈	—	2	2	no	—
67	#117	C₂×D₃₁	1	2	yes	—
68	C₂×C₄²	—	31	2	no	—
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	C₄⋊C₈	—	62	2	no	—
71	#95	—	62	2	no	—
72	C₃₁	#1995	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	62	2	no	—
75	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
77	M₁₆	—	62	4	no	—
78	C₂×C₈	—	62	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	31	4	no	—
85	C₄²	—	62	2	no	—
86	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
87	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₃₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	62	4	no	—
94	C₂×C₄	—	124	4	no	—
95	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
97	C₂×C₄	Q₈×D₃₁	1	2	yes	—
98	C₂×C₄	#92499	1	2	yes	—
99	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
101	C₈	—	124	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	#11758523	1	2	yes	—
104	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	124	4	no	—
109	C₄	#11758521	1	1	yes	characteristic
110	C₄	Q₈×Dic₃₁	1	2	yes	—
111	C₄	#11758524	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#14520956	1	1	yes	characteristic
114	C₂	#14521066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521073

C₂

yes

maximal, characteristic

#14520895

C₂

yes

maximal, characteristic

#14520808

C₂

yes

maximal

C₃₁×#533

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₄

yes

complemented

C₃₁×#125

C₄

yes

complemented

C₃₁×C₂²⋊Q₈

—

C₄×Q₈×C₃₁

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₃₁⋊#95

—

C₄⋊(C₃₁⋊C₈)

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

Q₈

yes

complemented

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

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×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1995

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

#533

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

#421

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₃₁

yes

complemented

#125

Dic₃₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#95

—

C₃₁

#1995

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

Q₈×D₃₁

yes

—

C₂×C₄

#92499

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

101

C₈

—

124

—

102

C₂²

—

103

C₂²

#11758523

yes

—

104

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

124

—

109

C₄

#11758521

yes

characteristic

110

C₄

Q₈×Dic₃₁

yes

—

111

C₄

#11758524

yes

characteristic

112

C₂

—

113

C₂

#14520956

yes

characteristic

114

C₂

#14521066

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#533	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#533	D₃₁
hypocenter	C₃₁	#1995
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#533

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
5. C₃₁×C₄⋊Q₈	C₄	semidirect
6. C₃₁×#125	C₄	semidirect
28. C₄×Dic₃₁	Q₈	semidirect
63. C₄⋊Q₈	Dic₃₁	semidirect
64. #125	Dic₃₁	semidirect
72. C₃₁	#1995	semidirect

H ≅ G/N

type

5. C₃₁×C₄⋊Q₈

C₄

semidirect

6. C₃₁×#125

C₄

semidirect

28. C₄×Dic₃₁

Q₈

semidirect

63. C₄⋊Q₈

Dic₃₁

semidirect

64. #125

Dic₃₁

semidirect

72. C₃₁

#1995

semidirect

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

subgroup	index
#14521073	2
#14520895	2
#14520808	2
C₃₁×#533	2
#1995	31

subgroup

index

C₃₁×#533

Subgroups — #16846693

Conjugacy classes of subgroups

Complements

Maximal subgroups