Group #28363894 — subgroups

Subgroups — #28363894

Order 13952.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16812633	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812623	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294933	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812633	—	2	2	no	—
10	#16812623	—	2	4	no	—
11	C₈×Dic₁₀₉	—	2	2	no	—
12	#16812606	—	2	4	no	—
13	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
15	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
16	C₂×Q₁₆×C₁₀₉	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	—	2	2	no	—
19	C₂²×Dic₁₀₉	D₄	1	2	yes	—
20	C₂×#11724784	—	2	1	no	—
21	C₂×#11724784	—	2	1	no	—
22	C₂×#11724784	D₄	1	2	yes	—
23	C₄⋊Dic₁₀₉	—	2	2	no	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	—	2	1	no	—
27	C₂×Q₈×C₁₀₉	—	2	2	no	—
28	C₂×Q₈×C₁₀₉	—	2	2	no	—
29	C₂×Q₈×C₁₀₉	—	2	2	no	—
30	C₂×Q₈×C₁₀₉	—	2	2	no	—
31	C₂×Q₈×C₁₀₉	—	2	2	no	—
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₀₉	—	2	2	no	—
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	Q₁₆×C₁₀₉	—	2	8	no	—
36	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
38	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
39	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₀₉	—	2	4	no	—
41	Q₈×C₁₀₉	—	2	4	no	—
42	Q₈×C₁₀₉	—	4	2	no	—
43	Q₈×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	4	2	no	—
46	C₂×C₄×C₁₀₉	—	4	2	no	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	2	2	no	—
54	C₂×Dic₁₀₉	—	2	2	no	—
55	C₂×Dic₁₀₉	—	4	2	no	—
56	C₂×Dic₁₀₉	—	4	4	no	—
57	C₂×Dic₁₀₉	Q₁₆	1	4	yes	complemented
58	C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
59	#11724784	—	2	2	no	—
60	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₀₉	#121	1	1	yes	characteristic
62	C₂²×C₁₀₉	#127	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₁₀₉	#135	1	2	yes	—
65	C₄×C₁₀₉	—	4	4	no	—
66	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
68	C₄×C₁₀₉	#121	1	1	yes	characteristic
69	C₄×C₁₀₉	#124	1	1	yes	characteristic
70	Dic₁₀₉	—	2	4	no	—
71	Dic₁₀₉	—	8	2	no	—
72	C₂×C₁₀₉	#390	1	1	yes	characteristic
73	C₂×C₁₀₉	C₄×Q₁₆	1	2	yes	—
74	C₂×C₁₀₉	#467	1	2	yes	—
75	C₂×C₁₀₉	#494	1	1	yes	characteristic
76	C₂×C₁₀₉	#495	1	1	yes	characteristic
77	#1969	—	109	1	no	maximal, Sylow
78	C₁₀₉	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	D₁₀₉	1	1	yes	characteristic
80	C₂×#107	—	109	1	no	—
81	C₂×#103	—	109	2	no	—
82	C₂×C₄×C₈	—	109	1	no	—
83	#391	—	109	2	no	—
84	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	Dic₁₀₉	1	4	yes	complemented
88	C₂²×C₈	—	109	1	no	—
89	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	109	2	no	—
91	C₂×C₄²	—	109	1	no	—
92	#107	—	218	2	no	—
93	#103	—	218	4	no	—
94	C₄×C₈	—	218	2	no	—
95	#95	—	218	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₁₀₉	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	109	2	no	—
104	C₂²×C₄	—	218	2	no	—
105	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
106	Q₁₆	—	2	8	no	—
107	C₂×C₈	—	109	2	no	—
108	C₂×C₈	—	218	1	no	—
109	C₂×C₈	—	218	1	no	—
110	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
111	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
112	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
113	C₄⋊C₄	—	218	2	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₄²	—	218	1	no	—
116	C₄²	—	218	1	no	—
117	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	109	4	no	—
126	C₂×C₄	—	436	2	no	—
127	C₂×C₄	—	436	4	no	—
128	C₂×C₄	—	218	2	no	—
129	C₂×C₄	—	218	2	no	—
130	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
134	C₂×C₄	#13182028	1	1	yes	characteristic
135	C₂×C₄	#13182028	1	1	yes	characteristic
136	C₈	—	218	2	no	—
137	C₈	C₂²⋊Dic₁₀₉	1	4	yes	—
138	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
139	C₂²	#16812670	1	1	yes	characteristic
140	C₂²	#16812695	1	1	yes	characteristic
141	C₂²	Q₁₆×D₁₀₉	1	2	yes	—
142	C₂²	#16812722	1	2	yes	—
143	C₄	—	4	4	no	—
144	C₄	—	872	2	no	—
145	C₄	—	218	4	no	—
146	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
147	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
148	C₄	#16812670	1	1	yes	characteristic
149	C₄	#16812686	1	1	yes	characteristic
150	C₂	#20294929	1	1	yes	characteristic
151	C₂	Q₁₆×Dic₁₀₉	1	2	yes	—
152	C₂	#20295264	1	2	yes	—
153	C₂	#20295404	1	1	yes	characteristic
154	C₂	#20295411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812633

C₂

yes

maximal, characteristic

C₂×#16812623

C₂

yes

maximal

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294933

C₂

yes

maximal

C₂²×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812633

—

#16812623

—

C₈×Dic₁₀₉

—

#16812606

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×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

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

Q₁₆

yes

complemented

C₈×C₁₀₉

C₂²⋊C₄

yes

—

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×Q₁₆

yes

—

C₂²×C₁₀₉

#135

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

C₄×Q₁₆

yes

—

C₂×C₁₀₉

#467

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₁₀₉

yes

characteristic

C₂×#107

—

109

—

C₂×#103

—

109

—

C₂×C₄×C₈

—

109

—

#391

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₁₀₉

yes

complemented

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

#107

—

218

—

#103

—

218

—

C₄×C₈

—

218

—

#95

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₁₀₉

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

218

—

105

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

218

—

109

C₂×C₈

—

218

—

110

C₂×C₈

C₂×Dic₁₀₉

yes

—

111

C₂×C₈

C₂²⋊D₁₀₉

yes

—

112

C₂×C₈

C₂²⋊D₁₀₉

yes

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₄²

—

218

—

116

C₄²

—

218

—

117

C₂³

D₄×D₁₀₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

436

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

218

—

129

C₂×C₄

—

218

—

130

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

131

C₂×C₄

D₄×D₁₀₉

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

134

C₂×C₄

#13182028

yes

characteristic

135

C₂×C₄

#13182028

yes

characteristic

136

C₈

—

218

—

137

C₈

C₂²⋊Dic₁₀₉

yes

—

138

C₂²

D₄×Dic₁₀₉

yes

characteristic

139

C₂²

#16812670

yes

characteristic

140

C₂²

#16812695

yes

characteristic

141

C₂²

Q₁₆×D₁₀₉

yes

—

142

C₂²

#16812722

yes

—

143

C₄

—

144

C₄

—

872

—

145

C₄

—

218

—

146

C₄

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

147

C₄

D₄×Dic₁₀₉

yes

characteristic

148

C₄

#16812670

yes

characteristic

149

C₄

#16812686

yes

characteristic

150

C₂

#20294929

yes

characteristic

151

C₂

Q₁₆×Dic₁₀₉

yes

—

152

C₂

#20295264

yes

—

153

C₂

#20295404

yes

characteristic

154

C₂

#20295411

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂²×Q₁₆×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂²×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#1969
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂²×Q₁₆×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂²×Q₁₆

D₁₀₉

hypocenter

C₁₀₉

#1969

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₁₀₉	C₄	semidirect
57. C₂×Dic₁₀₉	Q₁₆	semidirect
78. C₁₀₉	#1969	semidirect
87. C₂×Q₁₆	Dic₁₀₉	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₁₀₉

C₄

semidirect

57. C₂×Dic₁₀₉

Q₁₆

semidirect

78. C₁₀₉

#1969

semidirect

87. C₂×Q₁₆

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812633	2
C₂×#16812623	2
C₂×C₈×Dic₁₀₉	2
#20294933	2
C₂²×Q₁₆×C₁₀₉	2
#1969	109

subgroup

index

C₂×C₈×Dic₁₀₉

C₂²×Q₁₆×C₁₀₉

109

Subgroups — #28363894

Conjugacy classes of subgroups

Complements

Maximal subgroups