Group #28364830 — subgroups

Subgroups — #28364830

Order 13952.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294805	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₄²×D₁₀₉	C₄	1	1	yes	characteristic
9	C₂×#13182004	C₂²	1	2	yes	—
10	C₄×#11724784	C₄	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₀₉	—	2	1	no	—
21	C₈×D₁₀₉	—	2	1	no	—
22	C₈×D₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
24	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	#13182004	—	2	2	no	—
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₁₀₉	—	2	2	no	—
31	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	2	1	no	—
40	C₄×D₁₀₉	—	2	2	no	—
41	C₄×D₁₀₉	C₁₆	1	4	yes	complemented
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	#41	1	1	yes	characteristic
44	C₈×C₁₀₉	#41	1	1	yes	characteristic
45	#11724784	—	2	1	no	—
46	#11724784	M₁₆	1	2	yes	—
47	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₁₀₉	—	2	1	no	—
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	C₂×C₁₆	1	2	yes	—
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
53	C₄×C₁₀₉	#117	1	1	yes	characteristic
54	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
55	Dic₁₀₉	—	2	1	no	—
56	Dic₁₀₉	C₂×C₁₆	1	2	yes	—
57	Dic₁₀₉	M₃₂	1	2	yes	—
58	C₂×C₁₀₉	#431	1	1	yes	characteristic
59	C₂×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂×M₃₂	1	1	yes	characteristic
61	D₁₀₉	—	2	2	no	—
62	#2162	—	109	1	no	maximal, Sylow
63	C₁₀₉	#2162	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×C₈	—	109	1	no	—
65	C₄⋊C₁₆	—	109	1	no	—
66	C₄⋊C₁₆	D₁₀₉	1	1	yes	characteristic, complemented
67	C₂²⋊C₁₆	—	109	2	no	—
68	C₄×C₁₆	—	109	2	no	—
69	C₂²×C₈	—	109	1	no	—
70	C₂²×C₈	—	109	1	no	—
71	C₂×C₄²	—	109	1	no	—
72	C₂×C₁₆	—	109	2	no	—
73	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
74	C₄×C₈	—	109	1	no	—
75	C₄×C₈	—	109	1	no	—
76	C₄×C₈	—	109	1	no	—
77	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂²×C₄	—	109	1	no	—
79	C₂²×C₄	—	109	1	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	1	no	—
85	C₂×C₈	—	218	1	no	—
86	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
88	C₄²	—	109	1	no	—
89	C₄²	—	109	1	no	—
90	C₄²	—	109	1	no	—
91	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	—	218	2	no	—
94	C₂³	—	109	1	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	4	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	218	2	no	—
100	C₂×C₄	—	218	1	no	—
101	C₂×C₄	—	218	1	no	—
102	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	109	2	no	—
107	C₈	—	218	1	no	—
108	C₈	#13182028	1	1	yes	characteristic
109	C₈	#13182030	1	1	yes	characteristic
110	C₂²	—	109	2	no	—
111	C₂²	—	218	1	no	—
112	C₂²	—	218	1	no	—
113	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	109	2	no	—
116	C₄	—	109	2	no	—
117	C₄	—	218	1	no	—
118	C₄	C₁₆×D₁₀₉	1	2	yes	—
119	C₄	#16812657	1	1	yes	characteristic
120	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
121	C₂	—	218	2	no	—
122	C₂	#20295078	1	1	yes	characteristic
123	C₂	C₂×C₁₆×D₁₀₉	1	1	yes	characteristic
124	C₂	M₃₂×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#13182004

C₂

yes

maximal, characteristic, complemented

#20294805

C₂

yes

maximal

C₁₆×Dic₁₀₉

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄²×D₁₀₉

C₄

yes

characteristic

C₂×#13182004

C₂²

yes

—

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

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

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₁₆

yes

complemented

C₈×C₁₀₉

—

C₈×C₁₀₉

yes

C₈×C₁₀₉

yes

—

M₁₆

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₁₆

yes

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×C₁₆

yes

—

Dic₁₀₉

M₃₂

yes

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

C₂²×C₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

characteristic

D₁₀₉

—

#2162

—

109

maximal, Sylow

C₁₀₉

#2162

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

D₁₀₉

yes

characteristic, complemented

C₂²⋊C₁₆

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₄×D₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₈×D₁₀₉

yes

characteristic

103

C₂×C₄

C₈×D₁₀₉

yes

characteristic

104

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

105

C₈

—

106

C₈

—

109

—

107

C₈

—

218

—

108

C₈

#13182028

yes

characteristic

109

C₈

#13182030

yes

characteristic

110

C₂²

—

109

—

111

C₂²

—

218

—

112

C₂²

—

218

—

113

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

114

C₄

—

115

C₄

—

109

—

116

C₄

—

109

—

117

C₄

—

218

—

118

C₄

C₁₆×D₁₀₉

yes

—

119

C₄

#16812657

yes

characteristic

120

C₄

M₁₆×D₁₀₉

yes

characteristic

121

C₂

—

218

—

122

C₂

#20295078

yes

characteristic

123

C₂

C₂×C₁₆×D₁₀₉

yes

characteristic

124

C₂

M₃₂×D₁₀₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₁₆
Fitting	C₄⋊C₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₄⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2162
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator

C₂×C₁₀₉

C₂²×C₁₆

Fitting

C₄⋊C₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₄⋊C₁₆

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. C₄⋊#13182004	C₂	semidirect
4. C₁₆×Dic₁₀₉	C₂	semidirect
5. C₄⋊C₁₆×C₁₀₉	C₂	semidirect
41. C₄×D₁₀₉	C₁₆	semidirect
63. C₁₀₉	#2162	semidirect
66. C₄⋊C₁₆	D₁₀₉	semidirect

H ≅ G/N

type

2. C₄⋊#13182004

C₂

semidirect

4. C₁₆×Dic₁₀₉

C₂

semidirect

5. C₄⋊C₁₆×C₁₀₉

C₂

semidirect

41. C₄×D₁₀₉

C₁₆

semidirect

63. C₁₀₉

#2162

semidirect

66. C₄⋊C₁₆

D₁₀₉

semidirect

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

subgroup	index
C₄×C₈×D₁₀₉	2
C₄⋊#13182004	2
#20294805	2
C₁₆×Dic₁₀₉	2
C₄⋊C₁₆×C₁₀₉	2
#2162	109

subgroup

index

C₄×C₈×D₁₀₉

C₄⋊#13182004

#20294805

C₁₆×Dic₁₀₉

C₄⋊C₁₆×C₁₀₉

#2162

109

Subgroups — #28364830

Conjugacy classes of subgroups

Complements

Maximal subgroups