Group #28363943 — subgroups

Subgroups — #28363943

Order 13952.

316 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16812635	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295031	C₂	1	2	yes	maximal, complemented
3	C₂×#16812616	C₂	1	2	yes	maximal, complemented
4	#20294981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×#13182013	C₂²	1	1	yes	characteristic
8	#16812658	C₂²	1	1	yes	characteristic
9	#16812635	—	2	2	no	—
10	#16812620	—	2	4	no	—
11	#16812616	—	2	4	no	—
12	#16812609	—	2	1	no	—
13	C₈×Dic₁₀₉	—	2	1	no	—
14	C₂²×D₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×D₈×C₁₀₉	—	2	1	no	—
16	C₂×D₈×C₁₀₉	—	2	1	no	—
17	C₂×D₈×C₁₀₉	—	2	2	no	—
18	C₂×D₈×C₁₀₉	C₄	1	4	yes	complemented
19	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#13182013	—	2	1	no	—
21	#13182013	—	2	2	no	—
22	#13182013	—	2	2	no	—
23	#13182013	D₄	1	2	yes	—
24	C₂×#11724784	—	2	1	no	—
25	C₂×#11724784	—	2	2	no	—
26	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
27	C₂²⋊Dic₁₀₉	—	2	1	no	—
28	C₄×Dic₁₀₉	—	2	1	no	—
29	C₂⁴×C₁₀₉	—	2	2	no	—
30	C₂×D₄×C₁₀₉	—	2	2	no	—
31	C₂×D₄×C₁₀₉	—	2	2	no	—
32	C₂×D₄×C₁₀₉	—	2	2	no	—
33	C₂×D₄×C₁₀₉	—	2	2	no	—
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	D₈×C₁₀₉	—	2	4	no	—
38	D₈×C₁₀₉	—	2	4	no	—
39	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
43	C₂³×C₁₀₉	—	2	4	no	—
44	C₂³×C₁₀₉	—	4	2	no	—
45	C₂³×C₁₀₉	—	4	2	no	—
46	C₂³×C₁₀₉	—	8	4	no	—
47	C₂³×C₁₀₉	#41	1	1	yes	characteristic
48	D₄×C₁₀₉	—	2	4	no	—
49	D₄×C₁₀₉	—	2	4	no	—
50	D₄×C₁₀₉	—	4	2	no	—
51	D₄×C₁₀₉	—	4	2	no	—
52	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
58	C₂×Dic₁₀₉	—	2	2	no	—
59	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
60	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
61	#11724784	—	2	2	no	—
62	#11724784	—	4	2	no	—
63	C₂²×C₁₀₉	—	2	2	no	—
64	C₂²×C₁₀₉	—	4	4	no	—
65	C₂²×C₁₀₉	—	8	4	no	—
66	C₂²×C₁₀₉	—	8	4	no	—
67	C₂²×C₁₀₉	—	8	4	no	—
68	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
69	C₂²×C₁₀₉	#121	1	1	yes	characteristic
70	C₂²×C₁₀₉	#124	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₁₀₉	#121	1	1	yes	characteristic
74	C₄×C₁₀₉	#127	1	1	yes	characteristic
75	Dic₁₀₉	—	4	2	no	—
76	C₂×C₁₀₉	—	2	1	no	—
77	C₂×C₁₀₉	—	2	1	no	—
78	C₂×C₁₀₉	—	8	4	no	—
79	C₂×C₁₀₉	#390	1	1	yes	characteristic
80	C₂×C₁₀₉	#472	1	2	yes	—
81	#1978	—	109	1	no	maximal, Sylow
82	C₁₀₉	#1978	1	1	yes	characteristic, complemented, Sylow
83	C₂²×D₈	D₁₀₉	1	1	yes	characteristic
84	C₂×#108	—	109	1	no	—
85	#418	—	109	2	no	—
86	C₂×#100	—	109	2	no	—
87	#405	—	109	1	no	—
88	C₂²×D₄	C₂×D₁₀₉	1	2	yes	—
89	C₂×D₈	—	2	1	no	—
90	C₂×D₈	—	2	1	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	Dic₁₀₉	1	4	yes	complemented
93	C₂×M₁₆	—	109	1	no	—
94	C₂×M₁₆	—	109	2	no	—
95	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
96	#117	—	109	1	no	—
97	#108	—	218	2	no	—
98	#102	—	218	4	no	—
99	#100	—	218	4	no	—
100	#97	—	218	1	no	—
101	C₄×C₈	—	218	1	no	—
102	C₂⁴	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
109	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
110	D₈	—	2	4	no	—
111	D₈	—	2	4	no	—
112	M₁₆	—	109	2	no	—
113	M₁₆	—	218	2	no	—
114	M₁₆	—	218	2	no	—
115	M₁₆	—	218	1	no	—
116	C₂×C₈	—	218	2	no	—
117	C₂×C₈	—	218	1	no	—
118	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
119	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
120	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
122	C₄⋊C₄	—	109	2	no	—
123	C₂²⋊C₄	—	218	1	no	—
124	C₄²	—	218	1	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	4	2	no	—
128	C₂³	—	8	4	no	—
129	C₂³	#13182028	1	1	yes	characteristic
130	D₄	—	2	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	C₂×C₄	—	218	2	no	—
135	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
137	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
140	C₂×C₄	#13182028	1	1	yes	characteristic
141	C₈	—	436	2	no	—
142	C₈	—	218	2	no	—
143	C₈	C₂²⋊Dic₁₀₉	1	2	yes	—
144	C₈	C₂²⋊Dic₁₀₉	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	—	8	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	4	no	—
150	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
151	C₂²	#16812670	1	1	yes	characteristic
152	C₂²	#16812686	1	1	yes	characteristic
153	C₄	—	436	2	no	—
154	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
155	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
156	C₄	#16812670	1	1	yes	characteristic
157	C₄	#16812695	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	8	4	no	—
161	C₂	#20294929	1	1	yes	characteristic
162	C₂	#20295293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic, complemented

#20295031

C₂

yes

maximal, complemented

C₂×#16812616

C₂

yes

maximal, complemented

#20294981

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

—

—

—

—

C₈×Dic₁₀₉

—

C₂²×D₄×C₁₀₉

C₂²

yes

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂⁴×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₈×C₁₀₉

C₂²⋊C₄

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

#472

yes

—

#1978

—

109

maximal, Sylow

C₁₀₉

#1978

yes

characteristic, complemented, Sylow

C₂²×D₈

D₁₀₉

yes

characteristic

C₂×#108

—

109

—

#418

—

109

—

C₂×#100

—

109

—

#405

—

109

—

C₂²×D₄

C₂×D₁₀₉

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

Dic₁₀₉

yes

complemented

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#108

—

218

—

#102

—

218

—

#100

—

218

—

100

#97

—

218

—

101

C₄×C₈

—

218

—

102

C₂⁴

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

C₂×Dic₁₀₉

yes

—

109

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

110

D₈

—

111

D₈

—

112

M₁₆

—

109

—

113

M₁₆

—

218

—

114

M₁₆

—

218

—

115

M₁₆

—

218

—

116

C₂×C₈

—

218

—

117

C₂×C₈

—

218

—

118

C₂×C₈

C₂×Dic₁₀₉

yes

—

119

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

120

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

121

C₂×C₈

C₂²⋊D₁₀₉

yes

—

122

C₄⋊C₄

—

109

—

123

C₂²⋊C₄

—

218

—

124

C₄²

—

218

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#13182028

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

218

—

135

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

136

C₂×C₄

D₄×D₁₀₉

yes

characteristic

137

C₂×C₄

D₄×D₁₀₉

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

140

C₂×C₄

#13182028

yes

characteristic

141

C₈

—

436

—

142

C₈

—

218

—

143

C₈

C₂²⋊Dic₁₀₉

yes

—

144

C₈

C₂²⋊Dic₁₀₉

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

D₄×Dic₁₀₉

yes

characteristic

151

C₂²

#16812670

yes

characteristic

152

C₂²

#16812686

yes

characteristic

153

C₄

—

436

—

154

C₄

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

155

C₄

D₄×Dic₁₀₉

yes

characteristic

156

C₄

#16812670

yes

characteristic

157

C₄

#16812695

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

#20294929

yes

characteristic

162

C₂

#20295293

yes

—

	subgroup	G/H
center	C₂²	#16812670
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting, prime radical	C₂²×D₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂²×D₈	D₁₀₉
hypocenter	C₁₀₉	#1978
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#16812670

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting, prime radical

C₂²×D₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#121

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂²×D₈

D₁₀₉

hypocenter

C₁₀₉

#1978

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#16812635	C₂	semidirect
2. #20295031	C₂	semidirect
3. C₂×#16812616	C₂	semidirect
4. #20294981	C₂	semidirect
18. C₂×D₈×C₁₀₉	C₄	semidirect
82. C₁₀₉	#1978	semidirect
92. C₂×D₈	Dic₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812635

C₂

semidirect

2. #20295031

C₂

semidirect

3. C₂×#16812616

C₂

semidirect

4. #20294981

C₂

semidirect

18. C₂×D₈×C₁₀₉

C₄

semidirect

82. C₁₀₉

#1978

semidirect

92. C₂×D₈

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812635	2
#20295031	2
C₂×#16812616	2
#20294981	2
C₂²×D₈×C₁₀₉	2
#1978	109

subgroup

index

C₂²×D₈×C₁₀₉

109

Subgroups — #28363943

Conjugacy classes of subgroups

Complements

Maximal subgroups