Group #28364836 — subgroups

Subgroups — #28364836

Order 13952.

138 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20294981	C₂	1	1	yes	maximal, characteristic
2	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#13182004	C₂	1	1	yes	maximal, characteristic
5	#20294801	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₄	1	1	yes	characteristic
9	#16812658	C₄	1	1	yes	characteristic
10	C₂×#13182004	C₂²	1	1	yes	characteristic
11	C₂×#13182004	C₂²	1	1	yes	characteristic
12	#16812609	C₄	1	1	yes	characteristic
13	#16812609	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	#13182013	—	2	2	no	—
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	C₄⋊Dic₁₀₉	C₈	1	1	yes	characteristic
24	C₂²⋊Dic₁₀₉	C₈	1	1	yes	characteristic
25	C₂²⋊Dic₁₀₉	C₈	1	1	yes	characteristic
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	#13182004	—	2	1	no	—
29	#13182004	—	2	1	no	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	1	no	—
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₁₆×C₁₀₉	—	2	1	no	—
36	C₁₆×C₁₀₉	—	2	1	no	—
37	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₈×C₁₀₉	—	2	1	no	—
47	C₈×C₁₀₉	#41	1	1	yes	characteristic
48	C₈×C₁₀₉	#41	1	1	yes	characteristic
49	#11724784	—	2	1	no	—
50	#11724784	M₁₆	1	2	yes	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	1	no	—
53	C₂²×C₁₀₉	—	2	1	no	—
54	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	#117	1	1	yes	characteristic
57	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	1	no	—
60	Dic₁₀₉	—	2	1	no	—
61	Dic₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	—	4	1	no	—
63	C₂×C₁₀₉	#431	1	1	yes	characteristic
64	C₂×C₁₀₉	#504	1	1	yes	characteristic
65	C₂×C₁₀₉	#504	1	1	yes	characteristic
66	#2164	—	109	1	no	maximal, Sylow
67	C₁₀₉	#2164	1	1	yes	characteristic, complemented, Sylow
68	#405	—	109	1	no	—
69	C₄⋊C₁₆	—	109	1	no	—
70	C₄⋊C₁₆	—	109	1	no	—
71	C₂²⋊C₁₆	—	109	1	no	—
72	C₂²⋊C₁₆	D₁₀₉	1	1	yes	characteristic
73	#346	—	109	1	no	—
74	C₄×C₁₆	—	109	1	no	—
75	C₂×M₁₆	—	109	1	no	—
76	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
77	#117	—	109	1	no	—
78	C₂×C₁₆	—	109	1	no	—
79	C₂×C₁₆	—	109	1	no	—
80	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
82	#97	—	109	1	no	—
83	#97	—	109	1	no	—
84	C₄×C₈	—	109	1	no	—
85	C₄×C₈	—	109	1	no	—
86	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
87	M₁₆	—	218	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	109	1	no	—
91	C₂×C₈	—	109	1	no	—
92	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₄⋊C₄	—	109	1	no	—
95	C₄⋊C₄	—	109	1	no	—
96	C₂²⋊C₄	—	109	1	no	—
97	C₂²⋊C₄	—	109	1	no	—
98	C₄²	—	109	1	no	—
99	C₄²	—	109	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	2	1	no	—
102	C₁₆	—	218	1	no	—
103	C₁₆	—	218	1	no	—
104	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	1	no	—
111	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	109	2	no	—
115	C₈	—	218	1	no	—
116	C₈	#13182028	1	1	yes	characteristic
117	C₈	#13182028	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	218	1	no	—
124	C₄	—	218	1	no	—
125	C₄	—	218	1	no	—
126	C₄	—	218	1	no	—
127	C₄	#16812655	1	1	yes	characteristic
128	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
129	C₂	—	4	1	no	—
130	C₂	#20295076	1	1	yes	characteristic
131	C₂	#20295442	1	1	yes	characteristic
132	C₂	#20295444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20294981

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, characteristic, complemented

#20294844

C₂

yes

maximal, characteristic, complemented

C₂²⋊#13182004

C₂

yes

maximal, characteristic

#20294801

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₄

yes

characteristic

#16812658

C₄

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

#16812609

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#13182013

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

C₈

yes

characteristic

C₄⋊Dic₁₀₉

C₈

yes

characteristic

C₂²⋊Dic₁₀₉

C₈

yes

characteristic

C₂²⋊Dic₁₀₉

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

#13182004

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

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

C₈×C₁₀₉

—

C₈×C₁₀₉

yes

C₈×C₁₀₉

yes

—

M₁₆

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2164

yes

characteristic, complemented, Sylow

#405

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

C₂²⋊C₁₆

D₁₀₉

yes

characteristic

#346

—

109

—

C₄×C₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

218

—

103

C₁₆

—

218

—

104

C₂³

C₈×D₁₀₉

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

C₈×D₁₀₉

yes

characteristic

112

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

113

C₈

—

114

C₈

—

109

—

115

C₈

—

218

—

116

C₈

#13182028

yes

characteristic

117

C₈

#13182028

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

122

C₄

—

123

C₄

—

218

—

124

C₄

—

218

—

125

C₄

—

218

—

126

C₄

—

218

—

127

C₄

#16812655

yes

characteristic

128

C₄

M₁₆×D₁₀₉

yes

characteristic

129

C₂

—

130

C₂

#20295076

yes

characteristic

131

C₂

#20295442

yes

characteristic

132

C₂

#20295444

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂²⋊C₁₆×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂²⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2164
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₂²⋊C₁₆×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₂²⋊C₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. #20294844	C₂	semidirect
3. #20294844	C₂	semidirect
5. #20294801	C₂	semidirect
6. C₁₆×Dic₁₀₉	C₂	semidirect
67. C₁₀₉	#2164	semidirect

H ≅ G/N

type

2. #20294844

C₂

semidirect

3. #20294844

C₂

semidirect

5. #20294801

C₂

semidirect

6. C₁₆×Dic₁₀₉

C₂

semidirect

67. C₁₀₉

#2164

semidirect

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

subgroup	index
#20294981	2
#20294844	2
C₂²⋊#13182004	2
#20294801	2
C₁₆×Dic₁₀₉	2
C₂²⋊C₁₆×C₁₀₉	2
#2164	109

subgroup

index

C₁₆×Dic₁₀₉

C₂²⋊C₁₆×C₁₀₉

#2164

109

Subgroups — #28364836

Conjugacy classes of subgroups

Complements

Maximal subgroups