Group #28377928 — subgroups

Subgroups — #28377928

Order 13952.

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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	#20296044	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	#20295048	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#89137	—	2	1	no	—
5	C₂×C₄×#89137	—	2	1	no	—
6	#16812766	C₄	1	2	yes	complemented
7	#16812756	—	2	1	no	—
8	C₄×#11724786	—	2	1	no	—
9	(C₂×#89137)⋊C₄	C₄	1	2	yes	—
10	C₂×#13182012	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₄	1	1	yes	characteristic, complemented
13	#16812628	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₄⋊#89137	—	2	4	no	—
18	C₄×#89137	—	4	1	no	—
19	C₄×#89137	—	4	1	no	—
20	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
23	#13182012	—	4	1	no	—
24	#13182012	—	4	1	no	—
25	C₂×#11724784	—	2	1	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	1	no	—
30	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×#89137	—	2	4	no	—
33	C₂×#89137	—	4	2	no	—
34	C₂×#89137	—	4	2	no	—
35	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄×D₁₀₉	—	2	2	no	—
42	C₄×D₁₀₉	—	4	1	no	—
43	C₄×D₁₀₉	D₈	1	1	yes	characteristic
44	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
45	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
46	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
47	#11724786	—	4	2	no	—
48	C₈×C₁₀₉	—	4	1	no	—
49	#11724784	—	4	1	no	—
50	C₂²×C₁₀₉	#95	1	1	yes	characteristic
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	—	2	1	no	—
53	C₂×D₁₀₉	#102	1	1	yes	characteristic
54	C₂×D₁₀₉	#103	1	1	yes	characteristic
55	#89137	—	4	4	no	—
56	C₄×C₁₀₉	—	2	1	no	—
57	C₄×C₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	#102	1	1	yes	characteristic
59	C₄×C₁₀₉	#103	1	1	yes	characteristic
60	Dic₁₀₉	—	2	1	no	—
61	Dic₁₀₉	#104	1	2	yes	—
62	Dic₁₀₉	#106	1	1	yes	characteristic
63	Dic₁₀₉	#107	1	1	yes	characteristic
64	C₂×C₁₀₉	#339	1	1	yes	characteristic
65	C₂×C₁₀₉	#340	1	1	yes	characteristic
66	C₂×C₁₀₉	#344	1	1	yes	characteristic
67	D₁₀₉	—	2	1	no	—
68	D₁₀₉	—	2	1	no	—
69	#1303	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1303	1	1	yes	characteristic, complemented, Sylow
71	#431	—	109	1	no	—
72	#423	—	109	1	no	—
73	C₄×C₄⋊C₄	—	109	1	no	—
74	C₂×M₁₆	—	218	1	no	—
75	C₂×C₄⋊C₄	—	218	1	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₂×C₄²	—	218	1	no	—
78	C₄⋊C₈	—	109	2	no	—
79	C₄⋊C₈	—	109	1	no	—
80	C₄⋊C₈	—	218	1	no	—
81	C₄⋊C₈	#89137	1	1	yes	characteristic, complemented
82	C₂²⋊C₈	—	218	1	no	—
83	C₄×C₈	—	218	1	no	—
84	#95	—	109	2	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂²×C₄	—	218	2	no	—
89	M₁₆	—	436	1	no	—
90	M₁₆	—	436	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	1	no	—
94	C₄⋊C₄	—	218	4	no	—
95	C₄²	—	109	1	no	—
96	C₄²	—	109	1	no	—
97	C₄²	—	109	1	no	—
98	C₄²	—	436	1	no	—
99	C₄²	—	436	1	no	—
100	C₄²	C₂×#89137	1	1	yes	characteristic
101	C₂³	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	109	1	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	2	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	—	218	4	no	—
114	C₂×C₄	C₄×#89137	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
116	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
117	C₈	—	4	1	no	—
118	C₈	—	436	2	no	—
119	C₈	—	436	1	no	—
120	C₂²	—	109	1	no	—
121	C₂²	—	109	1	no	—
122	C₂²	—	218	1	no	—
123	C₂²	—	218	1	no	—
124	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	109	2	no	—
128	C₄	—	109	1	no	—
129	C₄	—	109	1	no	—
130	C₄	—	436	4	no	—
131	C₄	—	218	1	no	—
132	C₄	#16812761	1	1	yes	characteristic
133	C₄	#16812762	1	1	yes	characteristic
134	C₂	—	218	1	no	—
135	C₂	—	218	1	no	—
136	C₂	#20295898	1	1	yes	characteristic
137	C₂	#20295899	1	1	yes	characteristic
138	C₂	#20295908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296044

C₂

yes

maximal, characteristic

C₄×C₄⋊#89137

C₂

yes

maximal, characteristic

#20295048

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

—

C₂×C₄×#89137

—

#16812766

C₄

yes

—

—

C₄

yes

—

C₂×#13182012

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic, complemented

#16812628

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

—

—

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

#104

yes

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1303

—

109

maximal, Sylow

C₁₀₉

#1303

yes

characteristic, complemented, Sylow

#431

—

109

—

#423

—

109

—

C₄×C₄⋊C₄

—

109

—

C₂×M₁₆

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic, complemented

C₂²⋊C₈

—

218

—

C₄×C₈

—

218

—

#95

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

436

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

436

—

C₄²

—

436

—

100

C₄²

C₂×#89137

yes

characteristic

101

C₂³

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

C₄×#89137

yes

characteristic

115

C₂×C₄

C₂²⋊#89137

yes

characteristic

116

C₂×C₄

C₄⋊#89137

yes

characteristic

117

C₈

—

118

C₈

—

436

—

119

C₈

—

436

—

120

C₂²

—

109

—

121

C₂²

—

109

—

122

C₂²

—

218

—

123

C₂²

—

218

—

124

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

109

—

128

C₄

—

109

—

129

C₄

—

109

—

130

C₄

—

436

—

131

C₄

—

218

—

132

C₄

#16812761

yes

characteristic

133

C₄

#16812762

yes

characteristic

134

C₂

—

218

—

135

C₂

—

218

—

136

C₂

#20295898

yes

characteristic

137

C₂

#20295899

yes

characteristic

138

C₂

#20295908

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1303
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
6. #16812766	C₄	semidirect
12. C₄⋊#11724784	C₄	semidirect
14. C₄⋊C₈×C₁₀₉	C₄	semidirect
70. C₁₀₉	#1303	semidirect
81. C₄⋊C₈	#89137	semidirect

H ≅ G/N

type

6. #16812766

C₄

semidirect

12. C₄⋊#11724784

C₄

semidirect

14. C₄⋊C₈×C₁₀₉

C₄

semidirect

70. C₁₀₉

#1303

semidirect

81. C₄⋊C₈

#89137

semidirect

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

subgroup	index
#20296044	2
C₄×C₄⋊#89137	2
#20295048	2
#1303	109

subgroup

index

#20296044

C₄×C₄⋊#89137

#20295048

#1303

109

Subgroups — #28377928

Conjugacy classes of subgroups

Complements

Maximal subgroups