Group #28363060 — subgroups

Subgroups — #28363060

Order 13952.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812632	C₂	1	2	yes	maximal, complemented
3	#20294883	C₂	1	2	yes	maximal
4	#20294789	C₂	1	1	yes	maximal, characteristic, complemented
5	#340×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₁₀₉	—	2	4	no	—
9	C₂×C₈×D₁₀₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₂×#13182006	—	2	2	no	—
12	#16812632	—	2	4	no	—
13	#16812606	—	2	2	no	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
16	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	—	2	2	no	—
18	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₁₀₉	—	2	2	no	—
20	C₂×C₄×D₁₀₉	—	2	2	no	—
21	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
23	C₂×#11724784	C₂×C₄	1	2	yes	—
24	C₂×#11724784	D₄	1	4	yes	—
25	C₈×D₁₀₉	—	2	8	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	—	2	2	no	—
28	#13182006	—	4	4	no	—
29	C₂²×C₄×C₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₄⋊C₄×C₁₀₉	—	2	2	no	—
35	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₁₀₉	—	2	2	no	—
37	C₂²×D₁₀₉	—	2	2	no	—
38	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
39	C₂²×D₁₀₉	D₈	1	2	yes	—
40	C₂²×D₁₀₉	Q₁₆	1	2	yes	—
41	C₂×C₄×C₁₀₉	—	4	2	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	#41	1	2	yes	—
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	—	4	2	no	—
50	C₂×Dic₁₀₉	—	4	2	no	—
51	C₂×Dic₁₀₉	—	4	2	no	—
52	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
53	C₄×D₁₀₉	—	2	2	no	—
54	C₄×D₁₀₉	—	2	2	no	—
55	C₄×D₁₀₉	—	2	4	no	—
56	C₈×C₁₀₉	—	2	2	no	—
57	#11724784	C₂²⋊C₄	1	4	yes	complemented
58	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₉	#121	1	2	yes	—
60	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₉	#135	1	2	yes	—
63	C₂×D₁₀₉	—	2	2	no	—
64	C₂×D₁₀₉	—	2	2	no	—
65	C₂×D₁₀₉	—	2	4	no	—
66	C₂×D₁₀₉	—	2	4	no	—
67	C₂×D₁₀₉	#107	1	4	yes	complemented
68	C₄×C₁₀₉	—	8	2	no	—
69	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₁₀₉	#117	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
72	Dic₁₀₉	—	2	2	no	—
73	Dic₁₀₉	—	8	2	no	—
74	C₂×C₁₀₉	#380	1	1	yes	characteristic
75	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
76	C₂×C₁₀₉	#427	1	1	yes	characteristic
77	C₂×C₁₀₉	#466	1	2	yes	—
78	C₂×C₁₀₉	#467	1	2	yes	—
79	D₁₀₉	—	2	4	no	—
80	#1808	—	109	1	no	maximal, Sylow
81	C₁₀₉	#1808	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	109	1	no	—
83	C₂×#107	—	109	2	no	—
84	#380	—	109	2	no	—
85	#340	—	109	1	no	—
86	#340	D₁₀₉	1	1	yes	characteristic, complemented
87	C₂³×C₄	—	109	1	no	—
88	C₂²×C₈	—	109	1	no	—
89	C₂²×C₈	—	109	4	no	—
90	C₂²×C₈	—	218	4	no	—
91	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	109	2	no	—
93	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
94	C₂×C₂²⋊C₄	—	218	2	no	—
95	#107	—	218	4	no	—
96	#95	—	218	2	no	—
97	C₂⁴	—	109	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	2	no	—
101	C₂²×C₄	—	109	2	no	—
102	C₂²×C₄	—	218	2	no	—
103	C₂²×C₄	—	218	2	no	—
104	C₂²×C₄	—	218	2	no	—
105	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	109	2	no	—
108	C₂×C₈	—	109	4	no	—
109	C₂×C₈	—	218	8	no	—
110	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	218	2	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₂²⋊C₄	—	436	4	no	—
116	C₂³	—	109	2	no	—
117	C₂³	—	109	2	no	—
118	C₂³	—	109	2	no	—
119	C₂³	—	218	2	no	—
120	C₂³	—	218	2	no	—
121	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	109	2	no	—
126	C₂×C₄	—	436	2	no	—
127	C₂×C₄	—	436	2	no	—
128	C₂×C₄	—	436	2	no	—
129	C₂×C₄	—	218	2	no	—
130	C₂×C₄	—	218	2	no	—
131	C₂×C₄	—	218	2	no	—
132	C₂×C₄	—	218	4	no	—
133	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
134	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
135	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
136	C₂×C₄	#13182029	1	2	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	109	4	no	—
139	C₂²	—	109	4	no	—
140	C₂²	—	218	4	no	—
141	C₂²	—	218	4	no	—
142	C₂²	—	218	2	no	—
143	C₂²	—	218	2	no	—
144	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
145	C₂²	#16812672	1	2	yes	—
146	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
147	C₂²	Q₁₆×D₁₀₉	1	1	yes	characteristic
148	C₂²	#16812720	1	2	yes	—
149	C₄	—	8	2	no	—
150	C₄	—	872	2	no	—
151	C₄	—	218	2	no	—
152	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
153	C₄	#16812656	1	1	yes	characteristic
154	C₄	#16812677	1	2	yes	—
155	C₂	—	218	4	no	—
156	C₂	#20294885	1	1	yes	characteristic
157	C₂	#107×D₁₀₉	1	1	yes	characteristic
158	C₂	#20295061	1	1	yes	characteristic
159	C₂	#20295261	1	2	yes	—
160	C₂	#20295266	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#16812632

C₂

yes

maximal, complemented

#20294883

C₂

yes

maximal

#20294789

C₂

yes

maximal, characteristic, complemented

#340×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

—

C₂×C₈×D₁₀₉

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182006

—

#16812632

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×D₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

—

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₄⋊C₄

yes

—

C₂²×D₁₀₉

D₈

yes

—

C₂²×D₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

C₂²⋊C₄

yes

complemented

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₉

#135

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#107

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#467

yes

—

D₁₀₉

—

#1808

—

109

maximal, Sylow

C₁₀₉

#1808

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂×#107

—

109

—

#380

—

109

—

#340

—

109

—

#340

D₁₀₉

yes

characteristic, complemented

C₂³×C₄

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₂²⋊C₄

—

218

—

#107

—

218

—

#95

—

218

—

C₂⁴

—

109

—

C₂²×C₄

—

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

218

—

103

C₂²×C₄

—

218

—

104

C₂²×C₄

—

218

—

105

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

218

—

110

C₂×C₈

C₄×D₁₀₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₂²⋊C₄

—

436

—

116

C₂³

—

109

—

117

C₂³

—

109

—

118

C₂³

—

109

—

119

C₂³

—

218

—

120

C₂³

—

218

—

121

C₂³

D₄×D₁₀₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

436

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

436

—

129

C₂×C₄

—

218

—

130

C₂×C₄

—

218

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

134

C₂×C₄

D₄×D₁₀₉

yes

—

135

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

136

C₂×C₄

#13182029

yes

—

137

C₈

—

138

C₈

—

109

—

139

C₂²

—

109

—

140

C₂²

—

218

—

141

C₂²

—

218

—

142

C₂²

—

218

—

143

C₂²

—

218

—

144

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

145

C₂²

#16812672

yes

—

146

C₂²

D₈×D₁₀₉

yes

characteristic

147

C₂²

Q₁₆×D₁₀₉

yes

characteristic

148

C₂²

#16812720

yes

—

149

C₄

—

150

C₄

—

872

—

151

C₄

—

218

—

152

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

153

C₄

#16812656

yes

characteristic

154

C₄

#16812677

yes

—

155

C₂

—

218

—

156

C₂

#20294885

yes

characteristic

157

C₂

#107×D₁₀₉

yes

characteristic

158

C₂

#20295061

yes

characteristic

159

C₂

#20295261

yes

—

160

C₂

#20295266

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#340×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#340	D₁₀₉
hypocenter	C₁₀₉	#1808
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#340×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂³×D₁₀₉

D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#16812632	C₂	semidirect
4. #20294789	C₂	semidirect
5. #340×C₁₀₉	C₂	semidirect
9. C₂×C₈×D₁₀₉	C₄	semidirect
57. #11724784	C₂²⋊C₄	semidirect
67. C₂×D₁₀₉	#107	semidirect
81. C₁₀₉	#1808	semidirect
86. #340	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#16812632

C₂

semidirect

4. #20294789

C₂

semidirect

5. #340×C₁₀₉

C₂

semidirect

9. C₂×C₈×D₁₀₉

C₄

semidirect

57. #11724784

C₂²⋊C₄

semidirect

67. C₂×D₁₀₉

#107

semidirect

81. C₁₀₉

#1808

semidirect

86. #340

D₁₀₉

semidirect

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

subgroup	index
C₂²×C₈×D₁₀₉	2
C₂×#16812632	2
#20294883	2
#20294789	2
#340×C₁₀₉	2
#1808	109

subgroup

index

C₂²×C₈×D₁₀₉

#340×C₁₀₉

109

Subgroups — #28363060

Conjugacy classes of subgroups

Complements

Maximal subgroups