Group #28364939 — subgroups

Subgroups — #28364939

Order 13952.

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

normal subgroups only

This group has 126 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	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20294844	C₂	1	1	yes	maximal, characteristic
3	#20294844	C₂	1	1	yes	maximal, characteristic
4	#20294802	C₂	1	1	yes	maximal, characteristic
5	#20294801	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
9	C₂×#13182004	C₂²	1	1	yes	characteristic
10	C₂×#13182004	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₄	1	1	yes	characteristic
12	#16812628	C₄	1	1	yes	characteristic
13	#16812628	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₁₀₉	C₈	1	1	yes	characteristic
23	#13182008	C₈	1	1	yes	characteristic
24	#13182008	C₈	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	#13182004	—	2	1	no	—
28	#13182004	Q₈	1	2	yes	complemented
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₁₀₉	—	2	1	no	—
33	C₁₆×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
41	Dic₂₁₈	—	2	2	no	—
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₁₀₉	#41	1	1	yes	characteristic
45	#11724784	—	2	1	no	—
46	#11724784	—	2	1	no	—
47	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₁₀₉	#131	1	1	yes	characteristic
52	Dic₁₀₉	—	2	1	no	—
53	Dic₁₀₉	—	2	1	no	—
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	M₃₂	1	2	yes	—
56	C₂×C₁₀₉	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₀₉	#504	1	1	yes	characteristic
59	#2183	—	109	1	no	maximal, Sylow
60	C₁₀₉	#2183	1	1	yes	characteristic, complemented, Sylow
61	C₈×Q₈	—	109	1	no	—
62	C₄⋊C₁₆	—	109	1	no	—
63	C₄⋊C₁₆	—	109	1	no	—
64	C₄⋊C₁₆	D₁₀₉	1	1	yes	characteristic
65	#346	—	109	1	no	—
66	#346	—	109	1	no	—
67	C₄×C₁₆	—	109	1	no	—
68	C₄×Q₈	—	109	1	no	—
69	C₂×C₁₆	—	109	1	no	—
70	C₂×C₁₆	—	109	1	no	—
71	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
73	C₄⋊C₈	—	109	1	no	—
74	C₄⋊C₈	—	109	1	no	—
75	C₄⋊C₈	—	109	1	no	—
76	C₄×C₈	—	109	1	no	—
77	C₄×C₈	—	109	1	no	—
78	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂×Q₈	—	109	1	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₄⋊C₄	—	109	1	no	—
85	C₄⋊C₄	—	109	1	no	—
86	C₄⋊C₄	—	109	1	no	—
87	C₄²	—	109	1	no	—
88	C₄²	—	109	1	no	—
89	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	109	2	no	—
93	C₁₆	—	218	1	no	—
94	Q₈	—	218	2	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	218	1	no	—
104	C₈	—	218	1	no	—
105	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
106	C₈	#13182028	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	109	2	no	—
111	C₄	—	218	1	no	—
112	C₄	—	218	1	no	—
113	C₄	—	218	1	no	—
114	C₄	#16812664	1	1	yes	characteristic
115	C₄	#16812707	1	1	yes	characteristic
116	C₂	#20295150	1	1	yes	characteristic
117	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic
118	C₂	#20295442	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₁₈

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, characteristic

#20294802

C₂

yes

maximal, characteristic

#20294801

C₂

yes

maximal, characteristic

C₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₄×Dic₂₁₈

C₄

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812628

C₄

yes

characteristic

#16812628

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

C₈

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

C₈

yes

characteristic

#13182008

C₈

yes

characteristic

#13182008

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

#13182004

Q₈

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×Q₈

yes

C₈×C₁₀₉

yes

—

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

M₃₂

yes

—

C₂×C₁₀₉

C₈×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2183

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

D₁₀₉

yes

characteristic

#346

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₄×Q₈

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

109

—

C₁₆

—

218

—

Q₈

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₈×D₁₀₉

yes

characteristic

100

C₂×C₄

C₈×D₁₀₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

102

C₈

—

103

C₈

—

218

—

104

C₈

—

218

—

105

C₈

Q₈×D₁₀₉

yes

characteristic

106

C₈

#13182028

yes

characteristic

107

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

109

—

111

C₄

—

218

—

112

C₄

—

218

—

113

C₄

—

218

—

114

C₄

#16812664

yes

characteristic

115

C₄

#16812707

yes

characteristic

116

C₂

#20295150

yes

characteristic

117

C₂

M₃₂×D₁₀₉

yes

characteristic

118

C₂

#20295442

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle, prime radical	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₄⋊C₁₆×C₁₀₉	C₂
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₄⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2183
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle, prime radical

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₄⋊C₁₆×C₁₀₉

C₂

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₄⋊C₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
28. #13182004	Q₈	semidirect
60. C₁₀₉	#2183	semidirect

H ≅ G/N

type

28. #13182004

Q₈

semidirect

60. C₁₀₉

#2183

semidirect

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

subgroup	index
C₈×Dic₂₁₈	2
#20294844	2
#20294802	2
#20294801	2
C₁₆×Dic₁₀₉	2
C₄⋊C₁₆×C₁₀₉	2
#2183	109

subgroup

index

C₈×Dic₂₁₈

#20294844

#20294802

#20294801

C₁₆×Dic₁₀₉

C₄⋊C₁₆×C₁₀₉

#2183

109

Subgroups — #28364939

Conjugacy classes of subgroups

Complements

Maximal subgroups