Group #28365291 — subgroups

Subgroups — #28365291

Order 13952.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16812645	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812643	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812642	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812640	C₂	1	1	yes	maximal, characteristic
5	#20295410	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13182004	C₂	1	1	yes	maximal, characteristic, complemented
7	#495×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×#13182018	—	2	1	no	—
10	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
11	#16812687	—	2	1	no	—
12	#16812645	—	2	2	no	—
13	#16812643	—	2	2	no	—
14	#16812642	—	2	2	no	—
15	#16812640	—	2	2	no	—
16	C₂×#13182004	C₂²	1	1	yes	characteristic
17	C₂×#13182004	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
20	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#124×C₁₀₉	—	2	1	no	—
22	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×D₄₃₆	—	2	1	no	—
25	Dic₄₃₆	—	2	1	no	—
26	Dic₄₃₆	—	2	1	no	—
27	#13182018	—	4	2	no	—
28	D₈₇₂	—	2	1	no	—
29	D₈₇₂	—	2	1	no	—
30	#13182006	—	4	2	no	—
31	#13182004	D₄	1	2	yes	—
32	#13182004	D₄	1	2	yes	—
33	C₂×Q₈×C₁₀₉	—	2	1	no	—
34	C₂×D₄×C₁₀₉	—	2	1	no	—
35	Q₁₆×C₁₀₉	—	2	1	no	—
36	Q₁₆×C₁₀₉	—	2	1	no	—
37	SD₁₆×C₁₀₉	—	4	2	no	—
38	D₈×C₁₀₉	—	2	1	no	—
39	D₈×C₁₀₉	—	2	1	no	—
40	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
43	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
44	C₂³×C₁₀₉	—	4	1	no	—
45	C₂²×D₁₀₉	—	4	1	no	—
46	Q₈×C₁₀₉	—	4	1	no	—
47	Q₈×C₁₀₉	—	4	1	no	—
48	D₄×C₁₀₉	—	4	1	no	—
49	D₄×C₁₀₉	—	4	1	no	—
50	C₂×C₄×C₁₀₉	—	4	1	no	—
51	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	4	1	no	—
54	D₄₃₆	—	4	1	no	—
55	D₄₃₆	—	4	1	no	—
56	Dic₂₁₈	—	4	1	no	—
57	Dic₂₁₈	—	4	1	no	—
58	C₈×C₁₀₉	D₈	1	2	yes	—
59	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₁₀₉	—	8	1	no	—
62	C₂²×C₁₀₉	—	8	2	no	—
63	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
64	C₂×D₁₀₉	—	8	1	no	—
65	C₂×D₁₀₉	—	8	2	no	—
66	C₄×C₁₀₉	—	2	2	no	—
67	C₄×C₁₀₉	—	8	1	no	—
68	C₄×C₁₀₉	#127	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
70	Dic₁₀₉	—	8	1	no	—
71	C₂×C₁₀₉	—	16	1	no	—
72	C₂×C₁₀₉	#493	1	1	yes	characteristic
73	C₂×C₁₀₉	#508	1	2	yes	—
74	D₁₀₉	—	16	1	no	—
75	#2249	—	109	1	no	maximal, Sylow
76	C₁₀₉	#2249	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₃₂	—	109	1	no	—
78	C₂×SD₃₂	—	109	1	no	—
79	C₂×SD₃₂	—	109	1	no	—
80	C₂×D₁₆	—	109	1	no	—
81	#495	—	109	1	no	—
82	#495	D₁₀₉	1	1	yes	characteristic, complemented
83	C₄×C₁₆	—	109	1	no	—
84	C₂×Q₁₆	—	109	1	no	—
85	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂×SD₁₆	—	2	1	no	—
87	C₂×SD₁₆	—	218	1	no	—
88	C₂×D₈	—	109	1	no	—
89	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
90	#124	—	2	1	no	—
91	#124	—	218	1	no	—
92	Q₃₂	—	218	2	no	—
93	SD₃₂	—	218	2	no	—
94	SD₃₂	—	218	2	no	—
95	D₁₆	—	218	2	no	—
96	C₂×C₁₆	—	109	1	no	—
97	C₂×C₁₆	—	109	1	no	—
98	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	218	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	218	1	no	—
103	Q₁₆	—	2	1	no	—
104	Q₁₆	—	2	1	no	—
105	Q₁₆	—	218	1	no	—
106	Q₁₆	—	218	1	no	—
107	SD₁₆	—	4	2	no	—
108	SD₁₆	—	436	2	no	—
109	D₈	—	2	1	no	—
110	D₈	—	2	1	no	—
111	D₈	—	218	1	no	—
112	D₈	—	218	1	no	—
113	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	436	2	no	—
117	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
118	C₁₆	—	109	2	no	—
119	C₁₆	—	109	2	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	436	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	436	1	no	—
125	Q₈	—	436	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	436	1	no	—
129	D₄	—	436	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	436	1	no	—
132	C₂×C₄	#13182014	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
134	C₈	#13182014	1	2	yes	—
135	C₈	D₄×D₁₀₉	1	1	yes	characteristic
136	C₈	D₄×D₁₀₉	1	1	yes	characteristic
137	C₂²	—	8	2	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	872	2	no	—
140	C₂²	—	872	1	no	—
141	C₂²	C₂×#13182014	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	8	1	no	—
144	C₄	—	872	1	no	—
145	C₄	#16812695	1	1	yes	characteristic
146	C₄	C₂×#13182014	1	1	yes	characteristic
147	C₂	—	16	1	no	—
148	C₂	—	1744	1	no	—
149	C₂	#20295402	1	1	yes	characteristic
150	C₂	#20295456	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812645

C₂

yes

maximal, characteristic, complemented

C₂×#16812643

C₂

yes

maximal, characteristic, complemented

C₂×#16812642

C₂

yes

maximal, characteristic, complemented

C₂×#16812640

C₂

yes

maximal, characteristic

#20295410

C₂

yes

maximal, characteristic, complemented

C₄×#13182004

C₂

yes

maximal, characteristic, complemented

#495×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×#13182018

—

C₂×D₈₇₂

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×SD₁₆×C₁₀₉

—

C₂×D₈×C₁₀₉

C₂²

yes

characteristic

#124×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×D₄₃₆

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182018

—

D₈₇₂

—

D₈₇₂

—

#13182006

—

#13182004

D₄

yes

—

#13182004

D₄

yes

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

D₈

yes

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#493

yes

characteristic

C₂×C₁₀₉

#508

yes

—

D₁₀₉

—

#2249

—

109

maximal, Sylow

C₁₀₉

#2249

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

109

—

C₂×SD₃₂

—

109

—

C₂×SD₃₂

—

109

—

C₂×D₁₆

—

109

—

#495

—

109

—

#495

D₁₀₉

yes

characteristic, complemented

C₄×C₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

C₂×D₁₀₉

yes

characteristic

C₂×SD₁₆

—

C₂×SD₁₆

—

218

—

C₂×D₈

—

109

—

C₂×D₈

C₂×D₁₀₉

yes

characteristic

#124

—

#124

—

218

—

Q₃₂

—

218

—

SD₃₂

—

218

—

SD₃₂

—

218

—

D₁₆

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

218

—

101

C₂×D₄

—

102

C₂×D₄

—

218

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

218

—

106

Q₁₆

—

218

—

107

SD₁₆

—

108

SD₁₆

—

436

—

109

D₈

—

110

D₈

—

111

D₈

—

218

—

112

D₈

—

218

—

113

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

114

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

436

—

117

C₄²

C₂²⋊D₁₀₉

yes

characteristic

118

C₁₆

—

109

—

119

C₁₆

—

109

—

120

C₂³

—

121

C₂³

—

436

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

436

—

125

Q₈

—

436

—

126

D₄

—

127

D₄

—

128

D₄

—

436

—

129

D₄

—

436

—

130

C₂×C₄

—

131

C₂×C₄

—

436

—

132

C₂×C₄

#13182014

yes

—

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

134

C₈

#13182014

yes

—

135

C₈

D₄×D₁₀₉

yes

characteristic

136

C₈

D₄×D₁₀₉

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

872

—

140

C₂²

—

872

—

141

C₂²

C₂×#13182014

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

872

—

145

C₄

#16812695

yes

characteristic

146

C₄

C₂×#13182014

yes

characteristic

147

C₂

—

148

C₂

—

1744

—

149

C₂

#20295402

yes

characteristic

150

C₂

#20295456

yes

—

	subgroup	G/H
center	C₂²	C₂×#13182014
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#495×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂×#16812640	C₂
hypercenter	#495	D₁₀₉
hypocenter	C₁₀₉	#2249
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#13182014

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#495×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

C₂

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₂×#16812645	C₂	semidirect
2. C₂×#16812643	C₂	semidirect
3. C₂×#16812642	C₂	semidirect
5. #20295410	C₂	semidirect
6. C₄×#13182004	C₂	semidirect
7. #495×C₁₀₉	C₂	semidirect
76. C₁₀₉	#2249	semidirect
82. #495	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812645

C₂

semidirect

2. C₂×#16812643

C₂

semidirect

3. C₂×#16812642

C₂

semidirect

5. #20295410

C₂

semidirect

6. C₄×#13182004

C₂

semidirect

7. #495×C₁₀₉

C₂

semidirect

76. C₁₀₉

#2249

semidirect

82. #495

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812645	2
C₂×#16812643	2
C₂×#16812642	2
C₂×#16812640	2
#20295410	2
C₄×#13182004	2
#495×C₁₀₉	2
#2249	109

subgroup

index

#495×C₁₀₉

109

Subgroups — #28365291

Conjugacy classes of subgroups

Complements

Maximal subgroups