Group #16845300 — subgroups

Subgroups — #16845300

Order 3968.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14521454	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521421	C₂	1	2	yes	maximal, complemented
3	#14521395	C₂	1	2	yes	maximal, complemented
4	C₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×D₈×C₃₁	—	2	2	no	—
7	C₃₁×#127	C₂²	1	2	yes	—
8	C₂×#92480	—	2	4	no	—
9	#11758446	C₂²	1	2	yes	—
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	#11758420	—	2	4	no	—
12	#11758406	—	2	2	no	—
13	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₃₁	—	2	2	no	—
16	C₂×D₄×C₃₁	—	2	2	no	—
17	C₂×D₄×C₃₁	—	2	2	no	—
18	C₂×Dic₆₂	—	2	2	no	—
19	D₈×C₃₁	—	4	4	no	—
20	C₂×C₈×C₃₁	D₄	1	2	yes	—
21	C₄²×C₃₁	C₂³	1	1	yes	characteristic
22	#92480	—	4	8	no	—
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
26	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
27	C₂³×C₃₁	—	4	2	no	—
28	D₄×C₃₁	—	4	4	no	—
29	D₄×C₃₁	—	4	4	no	—
30	D₄×C₃₁	—	4	4	no	—
31	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₃₁	—	4	2	no	—
35	Dic₆₂	—	4	4	no	—
36	C₈×C₃₁	SD₁₆	1	4	yes	complemented
37	C₃₁⋊C₈	—	2	2	no	—
38	C₃₁⋊C₈	D₈	1	4	yes	complemented
39	#1706	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	—	8	2	no	—
41	C₂²×C₃₁	—	8	2	no	—
42	C₂²×C₃₁	—	8	2	no	—
43	C₂²×C₃₁	#127	1	1	yes	characteristic
44	C₄×C₃₁	C₂×D₈	1	2	yes	—
45	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
46	C₄×C₃₁	#135	1	2	yes	—
47	Dic₃₁	—	8	2	no	—
48	#500	—	31	1	no	—
49	#493	D₃₁	1	1	yes	characteristic
50	#492	—	31	2	no	—
51	#486	—	31	2	no	—
52	C₈²	—	31	1	no	—
53	C₂×C₃₁	—	16	2	no	—
54	C₂×C₃₁	#492	1	1	yes	characteristic
55	C₂×C₃₁	#493	1	1	yes	characteristic
56	C₂×C₃₁	#495	1	1	yes	characteristic
57	C₂×SD₁₆	—	62	4	no	—
58	C₂×D₈	—	2	2	no	—
59	C₄⋊Q₈	—	31	2	no	—
60	#127	C₂×D₃₁	1	2	yes	—
61	#107	—	62	2	no	—
62	#102	—	62	4	no	—
63	C₄×C₈	—	31	1	no	—
64	C₄×C₈	—	31	1	no	—
65	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
66	C₃₁	#1706	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	62	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	124	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	31	2	no	—
74	C₂×C₈	—	31	2	no	—
75	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
76	C₄⋊C₄	—	62	2	no	—
77	C₄⋊C₄	—	62	2	no	—
78	C₄²	C₂²×D₃₁	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	124	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	124	2	no	—
85	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
86	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
88	C₈	—	31	4	no	—
89	C₈	—	62	2	no	—
90	C₈	#92480	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#11758517	1	1	yes	characteristic
95	C₄	—	248	2	no	—
96	C₄	D₈×D₃₁	1	2	yes	—
97	C₄	#11758489	1	2	yes	—
98	C₄	C₂×#92480	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#14521421	1	1	yes	characteristic
101	C₂	#14521424	1	1	yes	characteristic
102	C₂	#14521432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521454

C₂

yes

maximal, characteristic, complemented

#14521421

C₂

yes

maximal, complemented

#14521395

C₂

yes

maximal, complemented

C₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₃₁×#493

C₂

yes

maximal, characteristic

C₂×D₈×C₃₁

—

C₃₁×#127

C₂²

yes

—

C₂×#92480

—

#11758446

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758420

—

#11758406

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×Dic₆₂

—

D₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

#92480

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

SD₁₆

yes

complemented

C₃₁⋊C₈

—

C₃₁⋊C₈

D₈

yes

complemented

#1706

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#127

yes

characteristic

C₄×C₃₁

C₂×D₈

yes

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

#135

yes

—

Dic₃₁

—

#500

—

#493

D₃₁

yes

characteristic

#492

—

#486

—

C₈²

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

C₂×D₃₁

yes

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1706

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

124

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

Q₈

—

124

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

124

—

C₂×C₄

D₄×D₃₁

yes

characteristic

C₂×C₄

D₄×D₃₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

—

C₈

#92480

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#11758517

yes

characteristic

C₄

—

248

—

C₄

D₈×D₃₁

yes

—

C₄

#11758489

yes

—

C₄

C₂×#92480

yes

—

C₂

—

100

C₂

#14521421

yes

characteristic

101

C₂

#14521424

yes

characteristic

102

C₂

#14521432

yes

characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting, prime radical	C₃₁×#493	C₂
socle	C₂²×C₃₁	#127
hypercenter	#493	D₃₁
hypocenter	C₃₁	#1706
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting, prime radical

C₃₁×#493

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #14521454	C₂	semidirect
2. #14521421	C₂	semidirect
3. #14521395	C₂	semidirect
4. C₈×C₃₁⋊C₈	C₂	semidirect
36. C₈×C₃₁	SD₁₆	semidirect
38. C₃₁⋊C₈	D₈	semidirect
66. C₃₁	#1706	semidirect
90. C₈	#92480	semidirect

H ≅ G/N

type

1. #14521454

C₂

semidirect

2. #14521421

C₂

semidirect

3. #14521395

C₂

semidirect

4. C₈×C₃₁⋊C₈

C₂

semidirect

36. C₈×C₃₁

SD₁₆

semidirect

38. C₃₁⋊C₈

D₈

semidirect

66. C₃₁

#1706

semidirect

90. C₈

#92480

semidirect

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

subgroup	index
#14521454	2
#14521421	2
#14521395	2
C₈×C₃₁⋊C₈	2
C₃₁×#493	2
#1706	31

subgroup

index

#14521454

#14521421

#14521395

C₈×C₃₁⋊C₈

C₃₁×#493

#1706

Subgroups — #16845300

Conjugacy classes of subgroups

Complements

Maximal subgroups