Group #28338512 — subgroups

Subgroups — #28338512

Order 13696.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₈₅₆	C₂	1	2	yes	maximal
2	#20293007	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₄₂₈	C₂	1	1	yes	maximal, characteristic
4	#20292432	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₈	—	2	1	no	—
7	C₂×Dic₄₂₈	C₂²	1	2	yes	—
8	C₂×Dic₄₂₈	C₂²	1	1	yes	characteristic
9	#16812109	—	2	1	no	—
10	C₄×Dic₂₁₄	—	2	1	no	—
11	Dic₈₅₆	—	2	4	no	—
12	#16812043	C₂²	1	1	yes	characteristic
13	#16812034	—	2	1	no	—
14	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₄	—	2	1	no	—
17	C₂×Dic₂₁₄	—	2	2	no	—
18	Dic₄₂₈	—	2	2	no	—
19	Dic₄₂₈	—	2	1	no	—
20	Dic₄₂₈	—	2	2	no	—
21	Dic₄₂₈	—	4	2	no	—
22	Dic₄₂₈	D₄	1	2	yes	—
23	C₄⋊Dic₁₀₇	—	2	1	no	—
24	C₄⋊Dic₁₀₇	—	4	1	no	—
25	C₄⋊Dic₁₀₇	—	4	1	no	—
26	Dic₁₀₇⋊C₄	—	4	1	no	—
27	C₄×Dic₁₀₇	—	4	1	no	—
28	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₇	—	2	2	no	—
32	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₇	—	4	1	no	—
36	C₂×Dic₁₀₇	—	4	1	no	—
37	C₂×Dic₁₀₇	—	4	2	no	—
38	Dic₂₁₄	—	2	2	no	—
39	Dic₂₁₄	—	4	1	no	—
40	Dic₂₁₄	—	4	2	no	—
41	Dic₂₁₄	—	4	2	no	—
42	C₈×C₁₀₇	—	2	1	no	—
43	C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₀₇	#41	1	1	yes	characteristic
45	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₀₇	—	2	1	no	—
47	C₄×C₁₀₇	Q₃₂	1	2	yes	complemented
48	C₄×C₁₀₇	#121	1	1	yes	characteristic
49	C₄×C₁₀₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₀₇	—	4	2	no	—
51	Dic₁₀₇	—	8	2	no	—
52	Dic₁₀₇	—	8	1	no	—
53	C₂×C₁₀₇	#459	1	1	yes	characteristic
54	C₂×C₁₀₇	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₀₇	#510	1	1	yes	characteristic
56	#2209	—	107	1	no	maximal, Sylow
57	C₁₀₇	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	107	2	no	—
59	#494	—	107	1	no	—
60	C₄×Q₁₆	—	107	1	no	—
61	C₄⋊C₁₆	D₁₀₇	1	1	yes	characteristic
62	#358	—	107	2	no	—
63	C₂×Q₁₆	—	107	1	no	—
64	C₂×Q₁₆	—	107	2	no	—
65	C₂×Q₁₆	—	214	1	no	—
66	C₄⋊Q₈	—	214	1	no	—
67	C₄×Q₈	—	214	1	no	—
68	Q₃₂	—	214	4	no	—
69	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
70	#107	—	107	1	no	—
71	#103	—	214	1	no	—
72	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
73	C₂×Q₈	—	214	2	no	—
74	C₂×Q₈	—	214	1	no	—
75	Q₁₆	—	107	2	no	—
76	Q₁₆	—	428	2	no	—
77	Q₁₆	—	214	2	no	—
78	Q₁₆	—	214	2	no	—
79	Q₁₆	—	214	1	no	—
80	C₂×C₈	D₄₂₈	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
82	C₄⋊C₄	—	428	1	no	—
83	C₄⋊C₄	—	428	1	no	—
84	C₄⋊C₄	—	428	1	no	—
85	C₄⋊C₄	—	214	1	no	—
86	C₄²	—	428	1	no	—
87	C₄²	D₄₂₈	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	428	2	no	—
90	Q₈	—	428	2	no	—
91	Q₈	—	428	1	no	—
92	Q₈	—	214	2	no	—
93	C₂×C₄	—	428	1	no	—
94	C₂×C₄	—	428	2	no	—
95	C₂×C₄	—	428	1	no	—
96	C₂×C₄	D₈₅₆	1	1	yes	characteristic
97	C₂×C₄	D₈₅₆	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₀₇	1	1	yes	characteristic
101	C₈	#13133819	1	1	yes	characteristic
102	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	856	1	no	—
105	C₄	—	856	2	no	—
106	C₄	—	428	2	no	—
107	C₄	Dic₈₅₆	1	2	yes	complemented
108	C₄	#16812084	1	1	yes	characteristic
109	C₄	#16812137	1	1	yes	characteristic
110	C₂	#20292823	1	1	yes	characteristic
111	C₂	C₂×Dic₈₅₆	1	1	yes	characteristic
112	C₂	#20293070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₅₆

C₂

yes

maximal

#20293007

C₂

yes

maximal, characteristic

C₄×Dic₄₂₈

C₂

yes

maximal, characteristic

#20292432

C₂

yes

maximal

C₄⋊C₁₆×C₁₀₇

C₂

yes

maximal, characteristic

C₂×Dic₄₂₈

—

C₂×Dic₄₂₈

C₂²

yes

—

C₂×Dic₄₂₈

C₂²

yes

characteristic

#16812109

—

C₄×Dic₂₁₄

—

Dic₈₅₆

—

#16812043

C₂²

yes

characteristic

#16812034

—

C₂×C₁₆×C₁₀₇

C₂²

yes

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

Dic₄₂₈

—

Dic₄₂₈

—

Dic₄₂₈

—

Dic₄₂₈

—

Dic₄₂₈

D₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

Dic₁₀₇⋊C₄

—

C₄×Dic₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂³

yes

characteristic

C₄²×C₁₀₇

D₄

yes

characteristic

C₁₆×C₁₀₇

—

C₂×C₄×C₁₀₇

D₈

yes

characteristic

C₂×C₄×C₁₀₇

D₈

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₂×D₄

yes

characteristic

C₈×C₁₀₇

#41

yes

characteristic

C₂²×C₁₀₇

C₂×D₈

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

Q₃₂

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

#459

yes

characteristic

C₂×C₁₀₇

C₂×Q₃₂

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

107

—

#494

—

107

—

C₄×Q₁₆

—

107

—

C₄⋊C₁₆

D₁₀₇

yes

characteristic

#358

—

107

—

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

107

—

C₂×Q₁₆

—

214

—

C₄⋊Q₈

—

214

—

C₄×Q₈

—

214

—

Q₃₂

—

214

—

C₂×C₁₆

C₂×D₁₀₇

yes

—

#107

—

107

—

#103

—

214

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

Q₁₆

—

107

—

Q₁₆

—

428

—

Q₁₆

—

214

—

Q₁₆

—

214

—

Q₁₆

—

214

—

C₂×C₈

D₄₂₈

yes

characteristic

C₂×C₈

C₂²×D₁₀₇

yes

characteristic

C₄⋊C₄

—

428

—

C₄⋊C₄

—

428

—

C₄⋊C₄

—

428

—

C₄⋊C₄

—

214

—

C₄²

—

428

—

C₄²

D₄₂₈

yes

characteristic

C₁₆

—

Q₈

—

428

—

Q₈

—

428

—

Q₈

—

428

—

Q₈

—

214

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

—

428

—

C₂×C₄

D₈₅₆

yes

characteristic

C₂×C₄

D₈₅₆

yes

characteristic

C₂×C₄

C₂×D₄₂₈

yes

characteristic

C₈

—

100

C₈

D₄×D₁₀₇

yes

characteristic

101

C₈

#13133819

yes

characteristic

102

C₂²

C₂×D₈₅₆

yes

characteristic

103

C₄

—

104

C₄

—

856

—

105

C₄

—

856

—

106

C₄

—

428

—

107

C₄

Dic₈₅₆

yes

complemented

108

C₄

#16812084

yes

characteristic

109

C₄

#16812137

yes

characteristic

110

C₂

#20292823

yes

characteristic

111

C₂

C₂×Dic₈₅₆

yes

characteristic

112

C₂

#20293070

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₄⋊C₁₆×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₀₇
hypocenter	C₁₀₇	#2209
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₈₅₆

commutator, cosocle

C₂×C₈×C₁₀₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₇

Fitting

C₄⋊C₁₆×C₁₀₇

C₂

socle, prime radical

C₂²×C₁₀₇

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₀₇

hypocenter

C₁₀₇

#2209

LCS

G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₀₇	Q₃₂	semidirect
57. C₁₀₇	#2209	semidirect
107. C₄	Dic₈₅₆	semidirect

H ≅ G/N

type

47. C₄×C₁₀₇

Q₃₂

semidirect

57. C₁₀₇

#2209

semidirect

107. C₄

Dic₈₅₆

semidirect

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

subgroup	index
C₂×Dic₈₅₆	2
#20293007	2
C₄×Dic₄₂₈	2
#20292432	2
C₄⋊C₁₆×C₁₀₇	2
#2209	107

subgroup

index

C₂×Dic₈₅₆

#20293007

C₄×Dic₄₂₈

#20292432

C₄⋊C₁₆×C₁₀₇

#2209

107

Subgroups — #28338512

Conjugacy classes of subgroups

Complements

Maximal subgroups