Group #28365083 — subgroups

Subgroups — #28365083

Order 13952.

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×Dic₈₇₂	C₂	1	2	yes	maximal
2	#20295406	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic
4	#20294831	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃₆	—	2	1	no	—
7	C₂×Dic₄₃₆	C₂²	1	2	yes	—
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	#16812699	—	2	1	no	—
10	C₄×Dic₂₁₈	—	2	1	no	—
11	Dic₈₇₂	—	2	4	no	—
12	#16812633	C₂²	1	1	yes	characteristic
13	#16812624	—	2	1	no	—
14	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₈	—	2	1	no	—
17	C₂×Dic₂₁₈	—	2	2	no	—
18	Dic₄₃₆	—	2	2	no	—
19	Dic₄₃₆	—	2	1	no	—
20	Dic₄₃₆	—	2	2	no	—
21	Dic₄₃₆	—	4	2	no	—
22	Dic₄₃₆	D₄	1	2	yes	—
23	C₄⋊Dic₁₀₉	—	2	1	no	—
24	C₄⋊Dic₁₀₉	—	4	1	no	—
25	C₄⋊Dic₁₀₉	—	4	1	no	—
26	#13182008	—	4	1	no	—
27	C₄×Dic₁₀₉	—	4	1	no	—
28	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	2	no	—
32	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	4	1	no	—
36	C₂×Dic₁₀₉	—	4	1	no	—
37	C₂×Dic₁₀₉	—	4	2	no	—
38	Dic₂₁₈	—	2	2	no	—
39	Dic₂₁₈	—	4	1	no	—
40	Dic₂₁₈	—	4	2	no	—
41	Dic₂₁₈	—	4	2	no	—
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₀₉	#41	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	Q₃₂	1	2	yes	complemented
48	C₄×C₁₀₉	#121	1	1	yes	characteristic
49	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₀₉	—	4	2	no	—
51	Dic₁₀₉	—	8	2	no	—
52	Dic₁₀₉	—	8	1	no	—
53	C₂×C₁₀₉	#459	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₀₉	#510	1	1	yes	characteristic
56	#2209	—	109	1	no	maximal, Sylow
57	C₁₀₉	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	109	2	no	—
59	#494	—	109	1	no	—
60	C₄×Q₁₆	—	109	1	no	—
61	C₄⋊C₁₆	D₁₀₉	1	1	yes	characteristic
62	#358	—	109	2	no	—
63	C₂×Q₁₆	—	109	1	no	—
64	C₂×Q₁₆	—	109	2	no	—
65	C₂×Q₁₆	—	218	1	no	—
66	C₄⋊Q₈	—	218	1	no	—
67	C₄×Q₈	—	218	1	no	—
68	Q₃₂	—	218	4	no	—
69	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
70	#107	—	109	1	no	—
71	#103	—	218	1	no	—
72	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₂×Q₈	—	218	2	no	—
74	C₂×Q₈	—	218	1	no	—
75	Q₁₆	—	109	2	no	—
76	Q₁₆	—	436	2	no	—
77	Q₁₆	—	218	2	no	—
78	Q₁₆	—	218	2	no	—
79	Q₁₆	—	218	1	no	—
80	C₂×C₈	D₄₃₆	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₄⋊C₄	—	436	1	no	—
83	C₄⋊C₄	—	436	1	no	—
84	C₄⋊C₄	—	436	1	no	—
85	C₄⋊C₄	—	218	1	no	—
86	C₄²	—	436	1	no	—
87	C₄²	D₄₃₆	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	436	2	no	—
90	Q₈	—	436	2	no	—
91	Q₈	—	436	1	no	—
92	Q₈	—	218	2	no	—
93	C₂×C₄	—	436	1	no	—
94	C₂×C₄	—	436	2	no	—
95	C₂×C₄	—	436	1	no	—
96	C₂×C₄	D₈₇₂	1	1	yes	characteristic
97	C₂×C₄	D₈₇₂	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₀₉	1	1	yes	characteristic
101	C₈	#13182030	1	1	yes	characteristic
102	C₂²	C₂×D₈₇₂	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	872	1	no	—
105	C₄	—	872	2	no	—
106	C₄	—	436	2	no	—
107	C₄	Dic₈₇₂	1	2	yes	complemented
108	C₄	#16812674	1	1	yes	characteristic
109	C₄	#16812727	1	1	yes	characteristic
110	C₂	#20295222	1	1	yes	characteristic
111	C₂	C₂×Dic₈₇₂	1	1	yes	characteristic
112	C₂	#20295469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₇₂

C₂

yes

maximal

#20295406

C₂

yes

maximal, characteristic

C₄×Dic₄₃₆

C₂

yes

maximal, characteristic

#20294831

C₂

yes

maximal

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

—

C₂×Dic₄₃₆

C₂²

yes

—

C₂×Dic₄₃₆

C₂²

yes

characteristic

#16812699

—

C₄×Dic₂₁₈

—

Dic₈₇₂

—

#16812633

C₂²

yes

characteristic

#16812624

—

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₄×Dic₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

#41

yes

characteristic

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

Q₃₂

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#459

yes

characteristic

C₂×C₁₀₉

C₂×Q₃₂

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

109

—

#494

—

109

—

C₄×Q₁₆

—

109

—

C₄⋊C₁₆

D₁₀₉

yes

characteristic

#358

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

—

218

—

C₄×Q₈

—

218

—

Q₃₂

—

218

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

#107

—

109

—

#103

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

Q₁₆

—

109

—

Q₁₆

—

436

—

Q₁₆

—

218

—

Q₁₆

—

218

—

Q₁₆

—

218

—

C₂×C₈

D₄₃₆

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

—

436

—

C₄²

D₄₃₆

yes

characteristic

C₁₆

—

Q₈

—

436

—

Q₈

—

436

—

Q₈

—

436

—

Q₈

—

218

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

D₈₇₂

yes

characteristic

C₂×C₄

D₈₇₂

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₈

—

100

C₈

D₄×D₁₀₉

yes

characteristic

101

C₈

#13182030

yes

characteristic

102

C₂²

C₂×D₈₇₂

yes

characteristic

103

C₄

—

104

C₄

—

872

—

105

C₄

—

872

—

106

C₄

—

436

—

107

C₄

Dic₈₇₂

yes

complemented

108

C₄

#16812674

yes

characteristic

109

C₄

#16812727

yes

characteristic

110

C₂

#20295222

yes

characteristic

111

C₂

C₂×Dic₈₇₂

yes

characteristic

112

C₂

#20295469

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₇₂
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	C₄⋊C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2209
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₈₇₂

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

C₄⋊C₁₆×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₀₉

hypocenter

C₁₀₉

#2209

LCS

G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₀₉	Q₃₂	semidirect
57. C₁₀₉	#2209	semidirect
107. C₄	Dic₈₇₂	semidirect

H ≅ G/N

type

47. C₄×C₁₀₉

Q₃₂

semidirect

57. C₁₀₉

#2209

semidirect

107. C₄

Dic₈₇₂

semidirect

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

subgroup	index
C₂×Dic₈₇₂	2
#20295406	2
C₄×Dic₄₃₆	2
#20294831	2
C₄⋊C₁₆×C₁₀₉	2
#2209	109

subgroup

index

C₂×Dic₈₇₂

#20295406

C₄×Dic₄₃₆

#20294831

C₄⋊C₁₆×C₁₀₉

#2209

109

Subgroups — #28365083

Conjugacy classes of subgroups

Complements

Maximal subgroups