Group #28364736 — subgroups

Subgroups — #28364736

Order 13952.

228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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×#16812718	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295063	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294831	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294829	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294827	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182028	—	2	1	no	—
9	#16812718	—	2	2	no	—
10	#16812718	C₄	1	4	yes	complemented
11	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×#13182017	—	2	1	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812657	—	2	1	no	—
15	C₂×#13182004	C₂²	1	1	yes	characteristic
16	C₁₆×D₁₀₉	—	2	2	no	—
17	#16812633	C₂²	1	1	yes	characteristic
18	#16812630	—	2	1	no	—
19	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
21	#107×C₁₀₉	C₂²	1	1	yes	characteristic
22	#13182028	—	2	4	no	—
23	#13182028	—	4	2	no	—
24	C₂×Dic₂₁₈	—	2	1	no	—
25	C₂×C₂²⋊D₁₀₉	—	4	1	no	—
26	C₂²×Dic₁₀₉	—	4	1	no	—
27	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
28	Dic₄₃₆	—	2	1	no	—
29	Dic₄₃₆	C₂×C₄	1	2	yes	—
30	#13182017	—	2	4	no	—
31	C₂×#11724784	D₄	1	1	yes	characteristic
32	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
33	C₈×D₁₀₉	D₄	1	2	yes	—
34	C₄⋊Dic₁₀₉	—	2	1	no	—
35	#13182006	—	4	1	no	—
36	C₄×Dic₁₀₉	—	4	1	no	—
37	#13182004	—	2	1	no	—
38	C₂×D₄×C₁₀₉	—	2	1	no	—
39	D₈×C₁₀₉	—	2	1	no	—
40	D₈×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₀₉	—	2	1	no	—
43	C₁₆×C₁₀₉	—	2	1	no	—
44	C₂³×C₁₀₉	—	4	1	no	—
45	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
46	D₄×C₁₀₉	—	2	2	no	—
47	D₄×C₁₀₉	—	4	1	no	—
48	C₂×C₄×C₁₀₉	—	4	1	no	—
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₁₀₉	—	4	4	no	—
51	C₂×Dic₁₀₉	—	4	1	no	—
52	C₂×Dic₁₀₉	—	4	4	no	—
53	C₂×Dic₁₀₉	—	4	1	no	—
54	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
55	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
56	C₄×D₁₀₉	SD₁₆	1	2	yes	—
57	Dic₂₁₈	—	2	2	no	—
58	Dic₂₁₈	—	4	1	no	—
59	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	#11724784	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₁₀₉	—	4	2	no	—
63	C₂²×C₁₀₉	—	8	1	no	—
64	C₂²×C₁₀₉	—	8	1	no	—
65	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
66	C₂×D₁₀₉	—	2	1	no	—
67	C₂×D₁₀₉	—	2	1	no	—
68	C₂×D₁₀₉	#102	1	2	yes	—
69	C₄×C₁₀₉	—	8	1	no	—
70	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₁₀₉	—	4	2	no	—
73	Dic₁₀₉	—	8	1	no	—
74	Dic₁₀₉	#102	1	2	yes	—
75	C₂×C₁₀₉	—	8	2	no	—
76	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
77	C₂×C₁₀₉	#508	1	1	yes	characteristic
78	C₂×C₁₀₉	#508	1	1	yes	characteristic
79	D₁₀₉	—	2	2	no	—
80	#2138	—	109	1	no	maximal, Sylow
81	C₁₀₉	#2138	1	1	yes	characteristic, complemented, Sylow
82	C₂×#135	—	109	1	no	—
83	C₂²×C₁₆	—	109	1	no	—
84	#427	—	109	1	no	—
85	#358	—	109	1	no	—
86	#358	—	109	1	no	—
87	#357	—	109	1	no	—
88	#357	D₁₀₉	1	1	yes	characteristic, complemented
89	C₂×#41	—	218	1	no	—
90	#135	—	109	4	no	—
91	#135	—	218	2	no	—
92	C₂×Q₁₆	—	109	1	no	—
93	C₂×SD₁₆	—	218	1	no	—
94	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
95	C₂²×C₈	—	109	1	no	—
96	#117	—	218	1	no	—
97	C₂×C₁₆	—	109	1	no	—
98	C₂×C₁₆	—	218	2	no	—
99	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
100	#107	—	109	1	no	—
101	#107	C₂×D₁₀₉	1	1	yes	characteristic
102	#106	—	218	1	no	—
103	#41	—	436	2	no	—
104	#41	—	218	4	no	—
105	C₂×Q₈	—	218	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	436	1	no	—
108	C₂²×C₄	—	109	1	no	—
109	C₂²×C₄	—	436	1	no	—
110	Q₁₆	—	109	2	no	—
111	Q₁₆	—	218	1	no	—
112	SD₁₆	—	218	4	no	—
113	D₈	—	2	1	no	—
114	D₈	C₄×D₁₀₉	1	2	yes	—
115	C₂×C₈	—	109	1	no	—
116	C₂×C₈	—	109	2	no	—
117	C₂×C₈	—	109	2	no	—
118	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	218	1	no	—
121	C₂²⋊C₄	—	436	1	no	—
122	C₄²	—	436	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	218	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	109	1	no	—
127	Q₈	—	436	1	no	—
128	Q₈	—	218	2	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	1	no	—
131	D₄	—	436	4	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	109	1	no	—
134	C₂×C₄	—	109	2	no	—
135	C₂×C₄	—	109	2	no	—
136	C₂×C₄	—	436	1	no	—
137	C₂×C₄	—	436	4	no	—
138	C₂×C₄	—	436	1	no	—
139	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
140	C₈	—	109	2	no	—
141	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
142	C₈	D₄×D₁₀₉	1	1	yes	characteristic
143	C₂²	—	4	2	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	109	2	no	—
147	C₂²	—	218	1	no	—
148	C₂²	—	218	1	no	—
149	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
150	C₄	—	8	1	no	—
151	C₄	—	872	1	no	—
152	C₄	—	109	2	no	—
153	C₄	—	436	2	no	—
154	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
155	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
156	C₂	—	8	2	no	—
157	C₂	—	218	2	no	—
158	C₂	#102×D₁₀₉	1	1	yes	characteristic
159	C₂	#20295455	1	1	yes	characteristic
160	C₂	#20295457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812718

C₂

yes

maximal, characteristic

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#20295063

C₂

yes

maximal, characteristic, complemented

#20294831

C₂

yes

maximal, characteristic, complemented

#20294829

C₂

yes

maximal, characteristic, complemented

#20294827

C₂

yes

maximal, characteristic, complemented

#357×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182028

—

#16812718

—

#16812718

C₄

yes

complemented

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×#13182017

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812657

—

C₂×#13182004

C₂²

yes

characteristic

C₁₆×D₁₀₉

—

#16812633

C₂²

yes

characteristic

#16812630

—

C₂×D₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

#13182028

—

#13182028

—

C₂×Dic₂₁₈

—

C₂×C₂²⋊D₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

Dic₄₃₆

—

Dic₄₃₆

C₂×C₄

yes

—

#13182017

—

C₂×#11724784

D₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₄×Dic₁₀₉

—

#13182004

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂²×D₁₀₉

D₈

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

—

C₄×D₁₀₉

SD₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₈×C₁₀₉

C₂×D₄

yes

characteristic

#11724784

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#102

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#102

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#2138

—

109

maximal, Sylow

C₁₀₉

#2138

yes

characteristic, complemented, Sylow

C₂×#135

—

109

—

C₂²×C₁₆

—

109

—

#427

—

109

—

#358

—

109

—

#358

—

109

—

#357

—

109

—

#357

D₁₀₉

yes

characteristic, complemented

C₂×#41

—

218

—

#135

—

109

—

#135

—

218

—

C₂×Q₁₆

—

109

—

C₂×SD₁₆

—

218

—

C₂×D₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

#117

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

100

#107

—

109

—

101

#107

C₂×D₁₀₉

yes

characteristic

102

#106

—

218

—

103

#41

—

436

—

104

#41

—

218

—

105

C₂×Q₈

—

218

—

106

C₂×D₄

—

107

C₂×D₄

—

436

—

108

C₂²×C₄

—

109

—

109

C₂²×C₄

—

436

—

110

Q₁₆

—

109

—

111

Q₁₆

—

218

—

112

SD₁₆

—

218

—

113

D₈

—

114

D₈

C₄×D₁₀₉

yes

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

109

—

117

C₂×C₈

—

109

—

118

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

218

—

121

C₂²⋊C₄

—

436

—

122

C₄²

—

436

—

123

C₁₆

—

124

C₁₆

—

218

—

125

C₂³

—

126

C₂³

—

109

—

127

Q₈

—

436

—

128

Q₈

—

218

—

129

D₄

—

130

D₄

—

131

D₄

—

436

—

132

C₂×C₄

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

109

—

135

C₂×C₄

—

109

—

136

C₂×C₄

—

436

—

137

C₂×C₄

—

436

—

138

C₂×C₄

—

436

—

139

C₂×C₄

D₄×D₁₀₉

yes

characteristic

140

C₈

—

109

—

141

C₈

C₂×C₄×D₁₀₉

yes

characteristic

142

C₈

D₄×D₁₀₉

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

109

—

147

C₂²

—

218

—

148

C₂²

—

218

—

149

C₂²

D₈×D₁₀₉

yes

characteristic

150

C₄

—

151

C₄

—

872

—

152

C₄

—

109

—

153

C₄

—

436

—

154

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

155

C₄

SD₁₆×D₁₀₉

yes

characteristic

156

C₂

—

157

C₂

—

218

—

158

C₂

#102×D₁₀₉

yes

characteristic

159

C₂

#20295455

yes

characteristic

160

C₂

#20295457

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#357×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂×#16812718	C₂
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#357	D₁₀₉
hypocenter	C₁₀₉	#2138
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#357×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

prime radical

C₂×#16812718

C₂

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. C₂×C₁₆×D₁₀₉	C₂	semidirect
3. #20295063	C₂	semidirect
4. #20294831	C₂	semidirect
5. #20294829	C₂	semidirect
6. #20294827	C₂	semidirect
7. #357×C₁₀₉	C₂	semidirect
10. #16812718	C₄	semidirect
81. C₁₀₉	#2138	semidirect
88. #357	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×C₁₆×D₁₀₉

C₂

semidirect

3. #20295063

C₂

semidirect

4. #20294831

C₂

semidirect

5. #20294829

C₂

semidirect

6. #20294827

C₂

semidirect

7. #357×C₁₀₉

C₂

semidirect

10. #16812718

C₄

semidirect

81. C₁₀₉

#2138

semidirect

88. #357

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812718	2
C₂×C₁₆×D₁₀₉	2
#20295063	2
#20294831	2
#20294829	2
#20294827	2
#357×C₁₀₉	2
#2138	109

subgroup

index

C₂×C₁₆×D₁₀₉

#357×C₁₀₉

109

Subgroups — #28364736

Conjugacy classes of subgroups

Complements

Maximal subgroups