Group #16845612 — subgroups

Subgroups — #16845612

Order 3968.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×#11758410	C₂	1	2	yes	maximal
2	M₁₆×Dic₃₁	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520814	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
9	C₂×#92474	C₂²	1	2	yes	—
10	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	#11758410	C₄	1	8	yes	complemented
12	#11758403	—	2	2	no	—
13	C₈×Dic₃₁	—	2	2	no	—
14	C₈×Dic₃₁	C₄	1	4	yes	—
15	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₁	Q₈	1	2	yes	—
17	M₁₆×C₃₁	—	2	2	no	—
18	M₁₆×C₃₁	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	2	no	—
21	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
23	C₄×Dic₃₁	D₄	1	1	yes	characteristic
24	C₄×Dic₃₁	D₄	1	1	yes	characteristic
25	#92474	—	2	2	no	—
26	#92474	C₂×C₄	1	4	yes	—
27	C₂×C₃₁⋊C₈	—	2	2	no	—
28	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
30	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	#41	1	2	yes	—
35	C₂×Dic₃₁	—	2	4	no	—
36	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
38	C₈×C₃₁	—	2	2	no	—
39	C₈×C₃₁	—	2	4	no	—
40	C₃₁⋊C₈	—	2	4	no	—
41	C₃₁⋊C₈	C₄²	1	4	yes	—
42	C₄×#108	—	31	1	no	maximal, Sylow
43	C₂²×C₃₁	—	2	2	no	—
44	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
46	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₃₁	#117	1	1	yes	characteristic
48	C₄×C₃₁	C₄×D₄	1	2	yes	—
49	Dic₃₁	—	2	2	no	—
50	Dic₃₁	#108	1	4	yes	complemented
51	C₂×#108	—	31	2	no	—
52	C₄×M₁₆	—	31	2	no	—
53	C₂×C₄×C₈	—	31	1	no	—
54	#341	—	31	1	no	—
55	#341	D₃₁	1	1	yes	characteristic
56	C₂×C₃₁	—	2	2	no	—
57	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃₁	C₂×#108	1	2	yes	—
59	C₂×M₁₆	—	31	2	no	—
60	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
61	C₂²×C₈	—	31	1	no	—
62	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
63	C₂×C₄²	—	31	1	no	—
64	#108	—	31	8	no	—
65	#97	—	62	2	no	—
66	C₄×C₈	—	31	4	no	—
67	C₄×C₈	—	62	2	no	—
68	C₃₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	—	31	2	no	—
70	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	31	4	no	—
73	M₁₆	—	62	2	no	—
74	M₁₆	C₄×D₃₁	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	31	2	no	—
78	C₂×C₈	—	31	4	no	—
79	C₂×C₈	—	62	2	no	—
80	C₂×C₈	C₄×D₃₁	1	2	yes	—
81	C₄²	—	31	2	no	—
82	C₄²	—	31	1	no	—
83	C₄²	—	31	1	no	—
84	C₂³	Q₈×D₃₁	1	1	yes	characteristic
85	C₂×C₄	—	31	2	no	—
86	C₂×C₄	—	31	2	no	—
87	C₂×C₄	—	62	4	no	—
88	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
90	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
91	C₂×C₄	#92497	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	31	4	no	—
95	C₈	—	62	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	#11758464	1	2	yes	—
98	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
99	C₄	—	31	4	no	—
100	C₄	—	62	2	no	—
101	C₄	C₄²×D₃₁	1	1	yes	characteristic
102	C₄	#11758454	1	1	yes	characteristic
103	C₄	(C₂²⋊D₃₁)⋊C₄	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#14520900	1	1	yes	characteristic
106	C₂	#108×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758410

C₂

yes

maximal

M₁₆×Dic₃₁

C₂

yes

maximal

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520814

C₂

yes

maximal, characteristic

C₃₁×#341

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758410

C₄

yes

complemented

#11758403

—

C₈×Dic₃₁

—

C₈×Dic₃₁

C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

Q₈

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

D₄

yes

characteristic

C₄×Dic₃₁

D₄

yes

characteristic

#92474

—

#92474

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₄²

yes

—

C₄×#108

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

—

Dic₃₁

—

Dic₃₁

#108

yes

complemented

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#341

—

#341

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

C₂×#108

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₃₁

C₄×#108

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₄×D₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

C₂×C₄×D₃₁

yes

—

C₂×C₄

D₄×D₃₁

yes

characteristic

C₂×C₄

#92497

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

#11758464

yes

—

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

C₄

—

100

C₄

—

101

C₄

C₄²×D₃₁

yes

characteristic

102

C₄

#11758454

yes

characteristic

103

C₄

(C₂²⋊D₃₁)⋊C₄

yes

—

104

C₂

—

105

C₂

#14520900

yes

characteristic

106

C₂

#108×D₃₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#341	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#341	D₃₁
hypocenter	C₃₁	C₄×#108
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₃₁

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#341

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₃₁

C₂×Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
11. #11758410	C₄	semidirect
50. Dic₃₁	#108	semidirect
68. C₃₁	C₄×#108	semidirect

H ≅ G/N

type

11. #11758410

C₄

semidirect

50. Dic₃₁

#108

semidirect

68. C₃₁

C₄×#108

semidirect

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

subgroup	index
C₂×#11758410	2
M₁₆×Dic₃₁	2
C₂×C₈×Dic₃₁	2
#14520814	2
C₃₁×#341	2
C₄×#108	31

subgroup

index

C₂×#11758410

M₁₆×Dic₃₁

C₂×C₈×Dic₃₁

#14520814

C₃₁×#341

C₄×#108

Subgroups — #16845612

Conjugacy classes of subgroups

Complements

Maximal subgroups