Group #16847221 — subgroups

Subgroups — #16847221

Order 3968.

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

normal subgroups only

This group has 176 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	#14521092	C₂	1	2	yes	maximal
2	C₂×#11758405	C₂	1	1	yes	maximal, characteristic
3	#14520817	C₂	1	2	yes	maximal
4	#14520814	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#424	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×#92474	C₂²	1	2	yes	—
11	C₃₁×C₄⋊C₈	—	2	2	no	—
12	#11758405	—	2	2	no	—
13	#11758396	—	2	2	no	—
14	#11758395	—	2	2	no	—
15	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₁	—	2	2	no	—
17	M₁₆×C₃₁	—	2	2	no	—
18	C₂×C₈×C₃₁	—	2	2	no	—
19	C₂×C₈×C₃₁	—	2	1	no	—
20	C₂×C₈×C₃₁	D₄	1	2	yes	—
21	C₄⋊C₄×C₃₁	D₄	1	2	yes	—
22	C₂²⋊C₄×C₃₁	—	2	1	no	—
23	C₄²×C₃₁	—	2	1	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	C₄⋊Dic₃₁	—	2	2	no	—
26	#92474	—	2	2	no	—
27	#92474	Q₈	1	4	yes	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂³×C₃₁	#41	1	1	yes	characteristic
30	C₂×C₄×C₃₁	—	2	2	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×Q₈	1	2	yes	—
34	C₂×C₄×C₃₁	#41	1	2	yes	—
35	C₂×Dic₃₁	—	4	2	no	—
36	C₂×Dic₃₁	—	8	2	no	—
37	C₈×C₃₁	—	2	2	no	—
38	C₈×C₃₁	—	4	1	no	—
39	C₃₁⋊C₈	—	2	4	no	—
40	#2090	—	31	1	no	maximal, Sylow
41	C₂²×C₃₁	—	2	2	no	—
42	C₂²×C₃₁	#121	1	1	yes	characteristic
43	C₂²×C₃₁	#125	1	2	yes	—
44	C₄×C₃₁	—	4	2	no	—
45	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
46	C₄×C₃₁	#123	1	1	yes	characteristic
47	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₃₁	—	8	2	no	—
49	#428	—	31	2	no	—
50	C₂×#106	—	31	1	no	—
51	#424	D₃₁	1	1	yes	characteristic
52	#343	—	31	2	no	—
53	#341	—	31	1	no	—
54	C₂×C₃₁	—	2	2	no	—
55	C₂×C₃₁	#398	1	1	yes	characteristic
56	C₂×C₃₁	#471	1	2	yes	—
57	C₂×M₁₆	—	31	2	no	—
58	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
60	#117	C₂×D₃₁	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	31	2	no	—
62	#107	—	62	2	no	—
63	#106	—	62	2	no	—
64	#106	—	62	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₃₁	#2090	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	62	2	no	—
68	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	31	4	no	—
71	M₁₆	—	62	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	62	2	no	—
75	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
76	C₄⋊C₄	—	62	2	no	—
77	C₄⋊C₄	—	62	2	no	—
78	C₄⋊C₄	D₁₂₄	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#92495	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	124	2	no	—
84	C₂×C₄	—	248	2	no	—
85	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
86	C₂×C₄	#92497	1	2	yes	—
87	C₂×C₄	Q₈×D₃₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	62	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#11758466	1	2	yes	—
94	C₂²	#11758507	1	1	yes	characteristic
95	C₄	—	4	2	no	—
96	C₄	—	248	2	no	—
97	C₄	#11758474	1	2	yes	—
98	C₄	#11758512	1	1	yes	characteristic
99	C₄	#11758521	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#14520979	1	1	yes	characteristic
102	C₂	#14521312	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521092

C₂

yes

maximal

C₂×#11758405

C₂

yes

maximal, characteristic

#14520817

C₂

yes

maximal

#14520814

C₂

yes

maximal, characteristic

C₃₁×#424

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₃₁×C₄⋊C₈

—

#11758405

—

#11758396

—

#11758395

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

D₄

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

#92474

—

#92474

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#2090

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#121

yes

characteristic

C₂²×C₃₁

#125

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#123

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

#428

—

C₂×#106

—

#424

D₃₁

yes

characteristic

#343

—

#341

—

C₂×C₃₁

—

C₂×C₃₁

#398

yes

characteristic

C₂×C₃₁

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

#117

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₃₁

#2090

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₂₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#92495

yes

characteristic

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

248

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

#92497

yes

—

C₂×C₄

Q₈×D₃₁

yes

—

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#11758466

yes

—

C₂²

#11758507

yes

characteristic

C₄

—

C₄

—

248

—

C₄

#11758474

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#14520979

yes

characteristic

102

C₂

#14521312

yes

—

	subgroup	G/H
center	C₂²	#11758507
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#424	C₂
socle	C₂²×C₃₁	#121
prime radical	C₂³×C₃₁	#41
hypercenter	#424	D₃₁
hypocenter	C₃₁	#2090
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#11758507

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#424

C₂

C₂²×C₃₁

C₂³×C₃₁

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
66. C₃₁	#2090	semidirect

H ≅ G/N

type

66. C₃₁

#2090

semidirect

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

subgroup	index
#14521092	2
C₂×#11758405	2
#14520817	2
#14520814	2
C₃₁×#424	2
#2090	31

subgroup

index

C₃₁×#424

Subgroups — #16847221

Conjugacy classes of subgroups

Complements

Maximal subgroups