Group #28361819 — subgroups

Subgroups — #28361819

Order 13952.

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

normal subgroups only

This group has 158 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	#20295078	C₂	1	1	yes	maximal, characteristic
2	#20295077	C₂	1	1	yes	maximal, characteristic
3	#20295048	C₂	1	1	yes	maximal, characteristic
4	#20294779	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294778	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294776	C₂	1	1	yes	maximal, characteristic, complemented
7	#334×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182012	—	2	1	no	—
9	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₄⋊#11724784	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₂²	1	1	yes	characteristic
12	#16812628	—	2	1	no	—
13	#16812628	C₄	1	1	yes	characteristic
14	#16812628	C₄	1	1	yes	characteristic
15	#16812611	—	2	1	no	—
16	#16812611	C₄	1	1	yes	characteristic
17	#16812611	C₄	1	1	yes	characteristic
18	C₄×#11724784	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₀₉	—	2	1	no	—
20	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
26	#13182012	—	4	2	no	—
27	C₂×#11724784	—	2	1	no	—
28	C₂×#11724784	—	2	1	no	—
29	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
30	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
33	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	—	2	1	no	—
35	C₂×C₈×C₁₀₉	—	2	1	no	—
36	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄×D₁₀₉	—	2	1	no	—
47	C₄×D₁₀₉	—	2	1	no	—
48	C₄×D₁₀₉	—	2	1	no	—
49	C₄×D₁₀₉	—	2	1	no	—
50	C₄×D₁₀₉	—	2	2	no	—
51	C₈×C₁₀₉	—	2	1	no	—
52	C₈×C₁₀₉	—	2	1	no	—
53	C₈×C₁₀₉	—	4	1	no	—
54	C₈×C₁₀₉	—	4	1	no	—
55	#11724784	—	2	1	no	—
56	#11724784	—	2	1	no	—
57	#11724784	—	4	1	no	—
58	#11724784	—	4	1	no	—
59	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×D₁₀₉	—	2	1	no	—
61	C₂×D₁₀₉	—	2	1	no	—
62	C₂×D₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	#131	1	1	yes	characteristic
65	C₄×C₁₀₉	#131	1	1	yes	characteristic
66	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₄×C₁₀₉	#137	1	1	yes	characteristic
68	Dic₁₀₉	—	2	1	no	—
69	Dic₁₀₉	—	2	1	no	—
70	Dic₁₀₉	#108	1	2	yes	—
71	C₂×C₁₀₉	#424	1	1	yes	characteristic
72	C₂×C₁₀₉	#428	1	1	yes	characteristic
73	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
74	D₁₀₉	—	4	1	no	—
75	#1574	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1574	1	1	yes	characteristic, complemented, Sylow
77	#431	—	109	1	no	—
78	#431	—	109	1	no	—
79	#423	—	109	1	no	—
80	#335	—	109	1	no	—
81	#335	—	109	1	no	—
82	#334	—	109	1	no	—
83	#334	D₁₀₉	1	1	yes	characteristic, complemented
84	C₂×M₁₆	—	218	1	no	—
85	C₂×C₄²	—	109	1	no	—
86	C₄⋊C₈	—	109	1	no	—
87	C₄⋊C₈	—	109	1	no	—
88	C₄⋊C₈	—	109	1	no	—
89	C₄⋊C₈	—	109	1	no	—
90	C₄⋊C₈	—	218	1	no	—
91	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
93	C₂²⋊C₈	—	109	1	no	—
94	C₂²⋊C₈	—	109	1	no	—
95	C₂²⋊C₈	—	218	1	no	—
96	C₄×C₈	—	109	1	no	—
97	C₄×C₈	—	218	1	no	—
98	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	1	no	—
102	M₁₆	—	436	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	109	1	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	218	1	no	—
108	C₂×C₈	—	218	1	no	—
109	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
111	C₄²	—	109	1	no	—
112	C₄²	—	109	1	no	—
113	C₄²	—	109	1	no	—
114	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
115	C₂³	—	109	1	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	109	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	—	218	1	no	—
123	C₂×C₄	—	218	1	no	—
124	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	4	1	no	—
131	C₈	—	436	1	no	—
132	C₈	—	436	1	no	—
133	C₈	—	218	1	no	—
134	C₈	—	218	1	no	—
135	C₂²	—	218	1	no	—
136	C₂²	—	218	1	no	—
137	C₂²	—	218	1	no	—
138	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	109	2	no	—
141	C₄	—	218	1	no	—
142	C₄	—	218	1	no	—
143	C₄	#16812705	1	1	yes	characteristic
144	C₄	#16812707	1	1	yes	characteristic
145	C₄	#16812726	1	1	yes	characteristic
146	C₄	#16812728	1	1	yes	characteristic
147	C₂	—	436	1	no	—
148	C₂	#20295051	1	1	yes	characteristic
149	C₂	#20295065	1	1	yes	characteristic
150	C₂	#108×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295078

C₂

yes

maximal, characteristic

#20295077

C₂

yes

maximal, characteristic

#20295048

C₂

yes

maximal, characteristic

#20294779

C₂

yes

maximal, characteristic, complemented

#20294778

C₂

yes

maximal, characteristic, complemented

#20294776

C₂

yes

maximal, characteristic, complemented

#334×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

characteristic

#16812628

—

#16812628

C₄

yes

characteristic

#16812628

C₄

yes

characteristic

#16812611

—

#16812611

C₄

yes

characteristic

#16812611

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

—

—

—

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#108

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1574

—

109

maximal, Sylow

C₁₀₉

#1574

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

—

109

—

#423

—

109

—

#335

—

109

—

#335

—

109

—

#334

—

109

—

#334

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

M₁₆

—

436

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

109

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

218

—

108

C₂×C₈

—

218

—

109

C₂×C₈

C₄×D₁₀₉

yes

characteristic

110

C₂×C₈

C₄×D₁₀₉

yes

characteristic

111

C₄²

—

109

—

112

C₄²

—

109

—

113

C₄²

—

109

—

114

C₄²

C₂²×D₁₀₉

yes

characteristic

115

C₂³

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

218

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

125

C₂×C₄

D₄×D₁₀₉

yes

characteristic

126

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

436

—

132

C₈

—

436

—

133

C₈

—

218

—

134

C₈

—

218

—

135

C₂²

—

218

—

136

C₂²

—

218

—

137

C₂²

—

218

—

138

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

139

C₄

—

140

C₄

—

109

—

141

C₄

—

218

—

142

C₄

—

218

—

143

C₄

#16812705

yes

characteristic

144

C₄

#16812707

yes

characteristic

145

C₄

#16812726

yes

characteristic

146

C₄

#16812728

yes

characteristic

147

C₂

—

436

—

148

C₂

#20295051

yes

characteristic

149

C₂

#20295065

yes

characteristic

150

C₂

#108×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#334×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#334	D₁₀₉
hypocenter	C₁₀₉	#1574
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#334×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
4. #20294779	C₂	semidirect
5. #20294778	C₂	semidirect
6. #20294776	C₂	semidirect
7. #334×C₁₀₉	C₂	semidirect
76. C₁₀₉	#1574	semidirect
83. #334	D₁₀₉	semidirect

H ≅ G/N

type

4. #20294779

C₂

semidirect

5. #20294778

C₂

semidirect

6. #20294776

C₂

semidirect

7. #334×C₁₀₉

C₂

semidirect

76. C₁₀₉

#1574

semidirect

83. #334

D₁₀₉

semidirect

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

subgroup	index
#20295078	2
#20295077	2
#20295048	2
#20294779	2
#20294778	2
#20294776	2
#334×C₁₀₉	2
#1574	109

subgroup

index

#334×C₁₀₉

109

Subgroups — #28361819

Conjugacy classes of subgroups

Complements

Maximal subgroups