Group #28373492 — subgroups

Subgroups — #28373492

Order 13952.

552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16812729	C₂	1	4	yes	maximal, complemented
2	C₂×#16812699	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295421	C₂	1	4	yes	maximal, complemented
4	#20295405	C₂	1	2	yes	maximal, complemented
5	#20295398	C₂	1	2	yes	maximal, complemented
6	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
9	#16812729	—	2	16	no	—
10	C₂×#13182019	C₂²	1	8	yes	—
11	C₂×#13182017	C₂²	1	8	yes	—
12	C₂×#13182013	C₂²	1	2	yes	—
13	#16812699	—	2	2	no	—
14	#16812699	C₂²	1	2	yes	—
15	#16812699	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
17	#16812610	C₂²	1	2	yes	—
18	C₄×#11724784	C₂²	1	2	yes	—
19	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#127×C₁₀₉	C₂²	1	1	yes	characteristic
22	#124×C₁₀₉	C₂²	1	2	yes	—
23	#121×C₁₀₉	—	2	2	no	—
24	C₄×D₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×Dic₂₁₈	—	2	8	no	—
27	C₂×Dic₂₁₈	—	2	2	no	—
28	C₂×Dic₂₁₈	C₂³	1	4	yes	—
29	C₂²×Dic₁₀₉	—	2	2	no	—
30	#13182019	—	2	16	no	—
31	#13182017	—	2	16	no	—
32	#13182013	D₄	1	8	yes	complemented
33	C₂×#11724784	C₂³	1	4	yes	—
34	C₄⋊Dic₁₀₉	—	2	4	no	—
35	C₄⋊Dic₁₀₉	—	2	4	no	—
36	#41×C₁₀₉	—	2	8	no	—
37	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
38	C₂×D₄×C₁₀₉	—	2	2	no	—
39	C₂×D₄×C₁₀₉	—	2	2	no	—
40	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₉	—	2	2	no	—
42	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
46	C₄²×C₁₀₉	D₄	1	2	yes	—
47	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₂³×C₁₀₉	—	2	2	no	—
50	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₀₉	—	2	4	no	—
52	D₄×C₁₀₉	—	2	4	no	—
53	D₄×C₁₀₉	—	4	4	no	—
54	D₄×C₁₀₉	—	4	4	no	—
55	C₂×C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄×C₁₀₉	—	2	2	no	—
57	C₂×C₄×C₁₀₉	—	4	4	no	—
58	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₁₀₉	—	2	4	no	—
64	C₂×Dic₁₀₉	—	4	4	no	—
65	Dic₂₁₈	—	2	8	no	—
66	Dic₂₁₈	—	4	4	no	—
67	#11724784	C₂×D₄	1	8	yes	—
68	C₂²×C₁₀₉	—	2	2	no	—
69	C₂²×C₁₀₉	—	4	2	no	—
70	C₂²×C₁₀₉	—	4	4	no	—
71	C₂²×C₁₀₉	#127	1	2	yes	—
72	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₀₉	—	2	2	no	—
74	C₄×C₁₀₉	—	4	2	no	—
75	C₄×C₁₀₉	#127	1	2	yes	—
76	C₄×C₁₀₉	#137	1	4	yes	complemented
77	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
78	Dic₁₀₉	—	4	4	no	—
79	C₂×C₁₀₉	—	2	2	no	—
80	C₂×C₁₀₉	—	8	2	no	—
81	C₂×C₁₀₉	C₂×#127	1	1	yes	characteristic
82	C₂×C₁₀₉	C₂×#137	1	2	yes	—
83	#3152	—	109	1	no	maximal, Sylow
84	C₁₀₉	#3152	1	1	yes	characteristic, complemented, Sylow
85	C₂×#137	—	109	4	no	—
86	#532	D₁₀₉	1	1	yes	characteristic
87	C₂×C₄⋊Q₈	—	109	1	no	—
88	#497	—	109	4	no	—
89	#494	—	109	2	no	—
90	#492	—	109	2	no	—
91	C₄×M₁₆	—	109	1	no	—
92	C₂×#41	C₂×D₁₀₉	1	2	yes	—
93	C₂²×Q₈	—	109	2	no	—
94	#137	—	218	16	no	—
95	C₂×Q₁₆	—	109	8	no	—
96	C₂×SD₁₆	—	109	8	no	—
97	C₂×M₁₆	—	109	2	no	—
98	C₄⋊Q₈	—	109	2	no	—
99	C₄⋊Q₈	—	109	2	no	—
100	C₄⋊Q₈	—	218	2	no	—
101	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
102	#127	C₂×D₁₀₉	1	1	yes	characteristic
103	#124	C₂×D₁₀₉	1	2	yes	—
104	#121	—	2	2	no	—
105	C₄×D₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	218	2	no	—
107	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
108	#97	—	109	2	no	—
109	C₄×C₈	—	109	2	no	—
110	#41	—	2	8	no	—
111	C₂×Q₈	—	109	4	no	—
112	C₂×Q₈	—	218	8	no	—
113	C₂×Q₈	—	218	2	no	—
114	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	218	2	no	—
120	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
121	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
122	Q₁₆	—	218	16	no	—
123	SD₁₆	—	218	16	no	—
124	M₁₆	—	109	8	no	—
125	C₂×C₈	—	109	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	218	4	no	—
128	C₄⋊C₄	—	218	4	no	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
131	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
132	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	436	4	no	—
137	Q₈	—	218	8	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	436	4	no	—
145	C₂×C₄	—	218	4	no	—
146	C₂×C₄	D₄×D₁₀₉	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
150	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
151	C₈	—	109	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#16812695	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	436	4	no	—
160	C₄	#16812695	1	2	yes	—
161	C₄	#16812729	1	4	yes	complemented
162	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#16812695	1	1	yes	characteristic
166	C₂	C₂×#16812729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812729

C₂

yes

maximal, complemented

C₂×#16812699

C₂

yes

maximal, characteristic, complemented

#20295421

C₂

yes

maximal, complemented

#20295405

C₂

yes

maximal, complemented

#20295398

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, characteristic, complemented

#532×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

#16812729

—

C₂×#13182019

C₂²

yes

—

C₂×#13182017

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

#16812699

—

#16812699

C₂²

yes

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₂×#41×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#127×C₁₀₉

C₂²

yes

characteristic

#124×C₁₀₉

C₂²

yes

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

—

#13182019

—

#13182017

—

#13182013

D₄

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#127

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#127

yes

—

C₄×C₁₀₉

#137

yes

complemented

C₄×C₁₀₉

C₂²×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#127

yes

characteristic

C₂×C₁₀₉

C₂×#137

yes

—

#3152

—

109

maximal, Sylow

C₁₀₉

#3152

yes

characteristic, complemented, Sylow

C₂×#137

—

109

—

#532

D₁₀₉

yes

characteristic

C₂×C₄⋊Q₈

—

109

—

#497

—

109

—

#494

—

109

—

#492

—

109

—

C₄×M₁₆

—

109

—

C₂×#41

C₂×D₁₀₉

yes

—

C₂²×Q₈

—

109

—

#137

—

218

—

C₂×Q₁₆

—

109

—

C₂×SD₁₆

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

100

C₄⋊Q₈

—

218

—

101

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

102

#127

C₂×D₁₀₉

yes

characteristic

103

#124

C₂×D₁₀₉

yes

—

104

#121

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

218

—

107

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

108

#97

—

109

—

109

C₄×C₈

—

109

—

110

#41

—

111

C₂×Q₈

—

109

—

112

C₂×Q₈

—

218

—

113

C₂×Q₈

—

218

—

114

C₂×Q₈

C₂²×D₁₀₉

yes

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

C₂²×D₁₀₉

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

218

—

120

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

121

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

122

Q₁₆

—

218

—

123

SD₁₆

—

218

—

124

M₁₆

—

109

—

125

C₂×C₈

—

109

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

218

—

128

C₄⋊C₄

—

218

—

129

C₂²⋊C₄

—

130

C₄²

C₂²⋊D₁₀₉

yes

—

131

C₄²

C₂²×D₁₀₉

yes

characteristic

132

C₄²

C₂²×D₁₀₉

yes

characteristic

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

135

Q₈

—

136

Q₈

—

436

—

137

Q₈

—

218

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

436

—

145

C₂×C₄

—

218

—

146

C₂×C₄

D₄×D₁₀₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

148

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

150

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

151

C₈

—

109

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#16812695

yes

—

156

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

436

—

160

C₄

#16812695

yes

—

161

C₄

#16812729

yes

complemented

162

C₄

C₂×D₄×D₁₀₉

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂×#16812695

yes

characteristic

166

C₂

C₂×#16812729

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting, prime radical	#532×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#532	D₁₀₉
hypocenter	C₁₀₉	#3152
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting, prime radical

#532×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#16812729	C₂	semidirect
2. C₂×#16812699	C₂	semidirect
3. #20295421	C₂	semidirect
4. #20295405	C₂	semidirect
5. #20295398	C₂	semidirect
6. C₄×#13182013	C₂	semidirect
32. #13182013	D₄	semidirect
76. C₄×C₁₀₉	#137	semidirect
84. C₁₀₉	#3152	semidirect
161. C₄	#16812729	semidirect

H ≅ G/N

type

1. C₂×#16812729

C₂

semidirect

2. C₂×#16812699

C₂

semidirect

3. #20295421

C₂

semidirect

4. #20295405

C₂

semidirect

5. #20295398

C₂

semidirect

6. C₄×#13182013

C₂

semidirect

32. #13182013

D₄

semidirect

76. C₄×C₁₀₉

#137

semidirect

84. C₁₀₉

#3152

semidirect

161. C₄

#16812729

semidirect

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

subgroup	index
C₂×#16812729	2
C₂×#16812699	2
#20295421	2
#20295405	2
#20295398	2
C₄×#13182013	2
#532×C₁₀₉	2
#3152	109

subgroup

index

#532×C₁₀₉

109

Subgroups — #28373492

Conjugacy classes of subgroups

Complements

Maximal subgroups