Group #28378089 — subgroups

Subgroups — #28378089

Order 13952.

152 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 60 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×D₁₀₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812768	C₂	1	1	yes	maximal, characteristic
3	#107×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#89137	—	2	1	no	—
5	D₁₀₉⋊C₁₆	—	2	1	no	—
6	D₁₀₉⋊C₁₆	C₄	1	2	yes	complemented
7	C₂×#13182032	—	2	1	no	—
8	#16812768	—	2	1	no	—
9	#16812768	C₄	1	2	yes	complemented
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×D₁₀₉	—	2	1	no	—
12	#16812633	C₄	1	1	yes	characteristic, complemented
13	#16812632	—	2	1	no	—
14	#107×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	4	1	no	—
16	C₄⋊#89137	—	2	2	no	—
17	C₄⋊#89137	—	4	1	no	—
18	#13182032	—	2	2	no	—
19	C₂×C₄×D₁₀₉	—	4	1	no	—
20	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₂×#11724784	D₄	1	1	yes	characteristic
22	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₀₉	D₄	1	1	yes	characteristic
25	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	#13182008	—	4	1	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
31	C₂×#89137	—	4	2	no	—
32	C₂×#89137	—	8	1	no	—
33	C₂×#89137	—	8	1	no	—
34	C₂×C₄×C₁₀₉	—	4	1	no	—
35	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	—	4	1	no	—
37	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
38	C₄×D₁₀₉	—	8	1	no	—
39	C₄×D₁₀₉	—	8	1	no	—
40	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
43	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
44	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
45	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	#11724784	C₂²⋊C₄	1	1	yes	characteristic
47	#11724784	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₀₉	#102	1	1	yes	characteristic
49	C₂×D₁₀₉	#102	1	1	yes	characteristic
50	C₂×D₁₀₉	#107	1	1	yes	characteristic
51	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
52	C₂×D₁₀₉	SD₃₂	1	1	yes	characteristic
53	C₂×D₁₀₉	SD₃₂	1	1	yes	characteristic
54	C₂×D₁₀₉	Q₃₂	1	1	yes	characteristic
55	#89137	—	8	2	no	—
56	C₄×C₁₀₉	—	8	1	no	—
57	C₄×C₁₀₉	#95	1	1	yes	characteristic
58	C₄×C₁₀₉	#103	1	1	yes	characteristic
59	Dic₁₀₉	—	8	1	no	—
60	Dic₁₀₉	#103	1	1	yes	characteristic
61	Dic₁₀₉	#106	1	1	yes	characteristic
62	C₂×C₁₀₉	#340	1	1	yes	characteristic
63	C₂×C₁₀₉	#357	1	1	yes	characteristic
64	C₂×C₁₀₉	#358	1	1	yes	characteristic
65	D₁₀₉	#357	1	1	yes	characteristic
66	D₁₀₉	#358	1	1	yes	characteristic
67	D₁₀₉	#366	1	1	yes	characteristic
68	D₁₀₉	#367	1	1	yes	characteristic
69	#1380	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1380	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	—	109	1	no	—
72	C₂×#107	—	109	1	no	—
73	C₂×#107	—	109	1	no	—
74	C₂²×C₈	—	109	1	no	—
75	C₂×C₄⋊C₄	—	218	1	no	—
76	C₂×C₄⋊C₄	—	218	1	no	—
77	C₂×C₁₆	—	109	2	no	—
78	C₂×C₁₆	—	218	1	no	—
79	C₂×C₁₆	—	218	1	no	—
80	#107	—	109	2	no	—
81	#107	—	109	1	no	—
82	#107	—	218	1	no	—
83	#107	—	218	1	no	—
84	#107	#89137	1	1	yes	characteristic, complemented
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	436	1	no	—
87	C₂²×C₄	—	436	1	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	1	no	—
90	C₂×C₈	—	109	1	no	—
91	C₂×C₈	—	109	1	no	—
92	C₂×C₈	—	109	1	no	—
93	C₂×C₈	C₂×#89137	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	436	1	no	—
96	C₄⋊C₄	—	436	1	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	—	218	1	no	—
99	C₁₆	—	218	2	no	—
100	C₂³	—	109	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	872	1	no	—
103	C₂×C₄	—	872	1	no	—
104	C₂×C₄	—	872	1	no	—
105	C₂×C₄	—	872	1	no	—
106	C₂×C₄	—	109	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	—	436	2	no	—
112	C₂×C₄	—	436	1	no	—
113	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
114	C₈	—	109	1	no	—
115	C₈	—	109	1	no	—
116	C₈	C₄×#89137	1	1	yes	characteristic
117	C₈	C₄⋊#89137	1	1	yes	characteristic
118	C₂²	—	109	1	no	—
119	C₂²	—	109	1	no	—
120	C₂²	—	109	1	no	—
121	C₂²	—	109	1	no	—
122	C₂²	—	109	1	no	—
123	C₂²	—	109	1	no	—
124	C₂²	#16812761	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	872	2	no	—
127	C₄	—	872	1	no	—
128	C₄	—	109	1	no	—
129	C₄	—	109	1	no	—
130	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
131	C₄	#16812762	1	1	yes	characteristic
132	C₂	—	109	1	no	—
133	C₂	—	109	1	no	—
134	C₂	—	109	1	no	—
135	C₂	—	109	1	no	—
136	C₂	#20295899	1	1	yes	characteristic
137	C₂	#20295938	1	1	yes	characteristic
138	C₂	#20295939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#16812768

C₂

yes

maximal, characteristic

#107×D₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

—

D₁₀₉⋊C₁₆

—

D₁₀₉⋊C₁₆

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×D₁₀₉

—

#16812633

C₄

yes

characteristic, complemented

#16812632

—

#107×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

C₄⋊#89137

—

C₄⋊#89137

—

#13182032

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

C₄⋊Dic₁₀₉

—

#13182008

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²×D₁₀₉

D₈

yes

—

—

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

C₂²×C₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

D₁₆

yes

characteristic

C₂×D₁₀₉

SD₃₂

yes

characteristic

C₂×D₁₀₉

SD₃₂

yes

characteristic

C₂×D₁₀₉

Q₃₂

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

D₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1380

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

109

—

C₂×#107

—

109

—

C₂×#107

—

109

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

—

218

—

#107

—

109

—

#107

—

109

—

#107

—

218

—

#107

—

218

—

#107

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

C₂²×C₄

—

436

—

C₂²×C₄

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₁₆

—

218

—

100

C₂³

—

109

—

101

C₂×C₄

—

102

C₂×C₄

—

872

—

103

C₂×C₄

—

872

—

104

C₂×C₄

—

872

—

105

C₂×C₄

—

872

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

436

—

113

C₂×C₄

C₂²⋊#89137

yes

characteristic

114

C₈

—

109

—

115

C₈

—

109

—

116

C₈

C₄×#89137

yes

characteristic

117

C₈

C₄⋊#89137

yes

characteristic

118

C₂²

—

109

—

119

C₂²

—

109

—

120

C₂²

—

109

—

121

C₂²

—

109

—

122

C₂²

—

109

—

123

C₂²

—

109

—

124

C₂²

#16812761

yes

characteristic

125

C₄

—

126

C₄

—

872

—

127

C₄

—

872

—

128

C₄

—

109

—

129

C₄

—

109

—

130

C₄

(C₂×#89137)⋊C₄

yes

characteristic

131

C₄

#16812762

yes

characteristic

132

C₂

—

109

—

133

C₂

—

109

—

134

C₂

—

109

—

135

C₂

—

109

—

136

C₂

#20295899

yes

characteristic

137

C₂

#20295938

yes

characteristic

138

C₂

#20295939

yes

characteristic

	subgroup	G/H
center	C₂²	#16812761
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂×C₈	C₂×#89137
Fitting	#107×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#102
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	#107	#89137
hypocenter	C₁₀₉	#1380
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

subgroup

G/H

center

C₂²

#16812761

commutator

C₈×C₁₀₉

C₄²

Frattini

C₂×C₈

C₂×#89137

Fitting

#107×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#102

prime radical

C₂²×D₁₀₉

D₈

cosocle

C₂×C₈×D₁₀₉

C₂²

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

N	H ≅ G/N	type
6. D₁₀₉⋊C₁₆	C₄	semidirect
9. #16812768	C₄	semidirect
12. #16812633	C₄	semidirect
14. #107×C₁₀₉	C₄	semidirect
70. C₁₀₉	#1380	semidirect
84. #107	#89137	semidirect

H ≅ G/N

type

6. D₁₀₉⋊C₁₆

C₄

semidirect

9. #16812768

C₄

semidirect

12. #16812633

C₄

semidirect

14. #107×C₁₀₉

C₄

semidirect

70. C₁₀₉

#1380

semidirect

84. #107

#89137

semidirect

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

subgroup	index
C₂×D₁₀₉⋊C₁₆	2
C₂×#16812768	2
#107×D₁₀₉	2
#1380	109

subgroup

index

C₂×D₁₀₉⋊C₁₆

C₂×#16812768

#107×D₁₀₉

#1380

109

Subgroups — #28378089

Conjugacy classes of subgroups

Complements

Maximal subgroups