Group #16846691 — subgroups

Subgroups — #16846691

Order 3968.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14520895	C₂	1	1	yes	maximal, characteristic
3	#14520810	C₂	1	4	yes	maximal
4	C₃₁×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	#11758503	C₂²	1	2	yes	—
8	#11758501	—	2	2	no	—
9	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
10	#11758447	C₄	1	2	yes	complemented
11	#11758446	C₄	1	2	yes	complemented
12	C₄×Dic₆₂	—	2	2	no	—
13	C₃₁×C₄⋊C₈	—	2	2	no	—
14	C₃₁⋊#95	—	2	2	no	—
15	C₂²×C₄×C₃₁	D₄	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	—	2	2	no	—
18	C₂×Dic₆₂	—	2	2	no	—
19	M₁₆×C₃₁	—	2	4	no	—
20	C₂×C₈×C₃₁	—	2	2	no	—
21	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₃₁	D₄	1	2	yes	—
24	C₂²⋊Dic₃₁	—	2	2	no	—
25	C₄⋊Dic₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
27	Dic₃₁⋊C₄	—	2	4	no	—
28	C₄×Dic₃₁	—	2	2	no	—
29	C₄×Dic₃₁	—	2	2	no	—
30	C₄×Dic₃₁	Q₈	1	4	yes	complemented
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₃₁	#41	1	2	yes	—
39	C₂×Dic₃₁	—	2	4	no	—
40	C₂×Dic₃₁	—	2	4	no	—
41	C₂×Dic₃₁	—	4	4	no	—
42	Dic₆₂	—	4	4	no	—
43	C₈×C₃₁	—	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	C₄×Q₈	1	2	yes	—
51	C₄×C₃₁	#124	1	1	yes	characteristic
52	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₃₁	—	4	4	no	—
54	Dic₃₁	—	4	4	no	—
55	#533	—	31	1	no	—
56	#423	D₃₁	1	1	yes	characteristic
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	C₂×D₃₁	1	2	yes	—
63	C₄⋊Q₈	—	31	2	no	—
64	#125	—	31	2	no	—
65	C₂²⋊Q₈	—	62	2	no	—
66	C₄×Q₈	—	62	2	no	—
67	#117	—	31	2	no	—
68	C₂×C₄²	—	31	2	no	—
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	2	no	—
71	#95	—	62	2	no	—
72	C₃₁	#1995	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	62	2	no	—
74	C₂²×C₄	—	62	2	no	—
75	C₂²×C₄	D₁₂₄	1	2	yes	—
76	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	31	4	no	—
80	C₄⋊C₄	—	62	4	no	—
81	C₄⋊C₄	—	62	2	no	—
82	C₂²⋊C₄	—	62	2	no	—
83	C₄²	—	31	4	no	—
84	C₄²	—	62	2	no	—
85	C₄²	—	62	2	no	—
86	C₄²	C₄×D₃₁	1	1	yes	characteristic
87	C₄²	C₄×D₃₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₃₁	1	2	yes	—
89	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
90	Q₈	—	124	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	62	4	no	—
93	C₂×C₄	—	62	4	no	—
94	C₂×C₄	—	124	4	no	—
95	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
98	C₂×C₄	#92497	1	2	yes	—
99	C₂×C₄	Q₈×D₃₁	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	#11758474	1	2	yes	—
104	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	124	4	no	—
108	C₄	—	124	4	no	—
109	C₄	#11758464	1	2	yes	—
110	C₄	#11758513	1	1	yes	characteristic
111	C₄	#11758521	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#14520954	1	1	yes	characteristic
114	C₂	#14521065	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14520895

C₂

yes

maximal, characteristic

#14520810

C₂

yes

maximal

C₃₁×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758447

C₄

yes

complemented

#11758446

C₄

yes

complemented

C₄×Dic₆₂

—

C₃₁×C₄⋊C₈

—

C₃₁⋊#95

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

Q₈

yes

complemented

C₂³×C₃₁

C₂×D₄

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×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

—

#1995

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

#533

—

#423

D₃₁

yes

characteristic

#376

—

#339

—

C₂×C₃₁

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

#421

yes

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#95

—

C₃₁

#1995

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₃₁

yes

characteristic

C₄²

C₄×D₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

#92497

yes

—

C₂×C₄

Q₈×D₃₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

101

C₈

—

102

C₂²

—

103

C₂²

#11758474

yes

—

104

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

124

—

108

C₄

—

124

—

109

C₄

#11758464

yes

—

110

C₄

#11758513

yes

characteristic

111

C₄

#11758521

yes

characteristic

112

C₂

—

113

C₂

#14520954

yes

characteristic

114

C₂

#14521065

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#423	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#423	D₃₁
hypocenter	C₃₁	#1995
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#423

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 ◁ #423

N	H ≅ G/N	type
10. #11758447	C₄	semidirect
11. #11758446	C₄	semidirect
30. C₄×Dic₃₁	Q₈	semidirect
72. C₃₁	#1995	semidirect

H ≅ G/N

type

10. #11758447

C₄

semidirect

11. #11758446

C₄

semidirect

30. C₄×Dic₃₁

Q₈

semidirect

72. C₃₁

#1995

semidirect

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

subgroup	index
#14521606	2
#14520895	2
#14520810	2
C₃₁×#423	2
#1995	31

subgroup

index

C₃₁×#423

Subgroups — #16846691

Conjugacy classes of subgroups

Complements

Maximal subgroups