Group #28378749 — subgroups

Subgroups — #28378749

Order 13952.

272 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 110 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×#16812754	C₂	1	2	yes	maximal
2	#20295892	C₂	1	2	yes	maximal
3	#20295892	C₂	1	2	yes	maximal
4	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	C₂²	1	4	yes	—
6	#16812754	C₄	1	8	yes	complemented
7	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×#11724786	—	2	4	no	—
15	C₂×#11724786	—	2	4	no	—
16	C₂×#11724786	C₂×C₄	1	8	yes	—
17	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₀₉	—	2	2	no	—
22	C₄×Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	2	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	—	2	2	no	—
31	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄×C₁₀₉	—	2	1	no	—
37	C₂×C₄×C₁₀₉	—	2	1	no	—
38	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
39	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
40	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
41	C₂×Dic₁₀₉	—	2	1	no	—
42	C₂×Dic₁₀₉	—	2	1	no	—
43	C₂×Dic₁₀₉	—	2	1	no	—
44	C₂×Dic₁₀₉	—	2	1	no	—
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	C₄²	1	2	yes	—
48	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
49	C₂×Dic₁₀₉	M₁₆	1	2	yes	—
50	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
51	C₂×Dic₁₀₉	#41	1	2	yes	—
52	C₂×Dic₁₀₉	#41	1	2	yes	—
53	#11724786	—	2	8	no	—
54	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₁₀₉	#117	1	1	yes	characteristic
56	C₂²×C₁₀₉	#117	1	1	yes	characteristic
57	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
59	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	—	2	2	no	—
63	C₄×C₁₀₉	—	2	2	no	—
64	Dic₁₀₉	—	2	2	no	—
65	Dic₁₀₉	—	2	2	no	—
66	Dic₁₀₉	—	2	2	no	—
67	Dic₁₀₉	#117	1	4	yes	—
68	C₂×C₁₀₉	#376	1	1	yes	characteristic
69	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
71	C₂×C₁₀₉	#432	1	1	yes	characteristic
72	C₂×C₁₀₉	#432	1	1	yes	characteristic
73	C₂×C₁₀₉	#432	1	1	yes	characteristic
74	C₂×C₁₀₉	#432	1	1	yes	characteristic
75	#1745	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1745	1	1	yes	characteristic, complemented, Sylow
77	C₂×#97	—	109	2	no	—
78	C₄³	—	109	1	no	—
79	#336	—	109	2	no	—
80	#336	—	109	2	no	—
81	C₂²×C₈	—	109	4	no	—
82	C₂×C₄²	—	109	1	no	—
83	C₂×C₄²	—	109	1	no	—
84	C₂×C₄²	—	109	1	no	—
85	C₂×C₄²	—	109	1	no	—
86	C₂×C₄²	—	109	1	no	—
87	C₂×C₄²	—	109	1	no	—
88	C₂×C₄²	#89137	1	1	yes	characteristic
89	#97	—	109	8	no	—
90	C₂²×C₄	—	109	1	no	—
91	C₂²×C₄	—	109	1	no	—
92	C₂²×C₄	—	109	1	no	—
93	C₂²×C₄	—	109	1	no	—
94	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
95	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
96	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
97	C₂×C₈	—	109	8	no	—
98	C₂×C₈	—	218	4	no	—
99	C₂×C₈	—	218	4	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	109	4	no	—
102	C₄²	—	218	2	no	—
103	C₄²	—	218	2	no	—
104	C₄²	—	218	2	no	—
105	C₄²	—	218	2	no	—
106	C₄²	—	218	2	no	—
107	C₂³	C₂²×#89137	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	109	2	no	—
115	C₂×C₄	—	109	2	no	—
116	C₂×C₄	—	109	2	no	—
117	C₂×C₄	—	109	2	no	—
118	C₂×C₄	—	109	2	no	—
119	C₂×C₄	—	109	2	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	—	218	1	no	—
122	C₂×C₄	—	218	1	no	—
123	C₂×C₄	—	218	1	no	—
124	C₂×C₄	—	218	1	no	—
125	C₂×C₄	—	218	1	no	—
126	C₂×C₄	C₄×#89137	1	2	yes	—
127	C₂×C₄	#13182038	1	2	yes	—
128	C₂×C₄	#13182039	1	2	yes	—
129	C₈	—	218	8	no	—
130	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
131	C₂²	#16812778	1	1	yes	characteristic
132	C₂²	#16812778	1	1	yes	characteristic
133	C₂²	C₂×#13182038	1	1	yes	characteristic
134	C₂²	D₁₀₉⋊M₁₆	1	1	yes	characteristic
135	C₂²	D₁₀₉⋊M₁₆	1	1	yes	characteristic
136	C₂²	C₂×#13182039	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	—	109	4	no	—
141	C₄	—	218	2	no	—
142	C₄	—	218	2	no	—
143	C₄	—	218	2	no	—
144	C₂	#20295962	1	1	yes	characteristic
145	C₂	C₄×#13182038	1	1	yes	characteristic
146	C₂	C₄×#13182039	1	1	yes	characteristic
147	C₂	#20296046	1	1	yes	characteristic
148	C₂	#20296047	1	1	yes	characteristic
149	C₂	#20296048	1	1	yes	characteristic
150	C₂	#20296049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812754

C₂

yes

maximal

#20295892

C₂

yes

maximal

#20295892

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724786

C₂²

yes

—

#16812754

C₄

yes

complemented

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₄

yes

—

—

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

#41

yes

—

C₂×Dic₁₀₉

#41

yes

—

#11724786

—

C₂²×C₁₀₉

C₂×C₄²

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#117

yes

—

C₂×C₁₀₉

#376

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1745

yes

characteristic, complemented, Sylow

C₂×#97

—

109

—

C₄³

—

109

—

#336

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

#89137

yes

characteristic

#97

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

100

C₄²

—

101

C₄²

—

109

—

102

C₄²

—

218

—

103

C₄²

—

218

—

104

C₄²

—

218

—

105

C₄²

—

218

—

106

C₄²

—

218

—

107

C₂³

C₂²×#89137

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

109

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

—

218

—

125

C₂×C₄

—

218

—

126

C₂×C₄

C₄×#89137

yes

—

127

C₂×C₄

#13182038

yes

—

128

C₂×C₄

#13182039

yes

—

129

C₈

—

218

—

130

C₂²

C₂×C₄×#89137

yes

characteristic

131

C₂²

#16812778

yes

characteristic

132

C₂²

#16812778

yes

characteristic

133

C₂²

C₂×#13182038

yes

characteristic

134

C₂²

D₁₀₉⋊M₁₆

yes

characteristic

135

C₂²

D₁₀₉⋊M₁₆

yes

characteristic

136

C₂²

C₂×#13182039

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

109

—

141

C₄

—

218

—

142

C₄

—

218

—

143

C₄

—

218

—

144

C₂

#20295962

yes

characteristic

145

C₂

C₄×#13182038

yes

characteristic

146

C₂

C₄×#13182039

yes

characteristic

147

C₂

#20296046

yes

characteristic

148

C₂

#20296047

yes

characteristic

149

C₂

#20296048

yes

characteristic

150

C₂

#20296049

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#89137
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₂×C₄²×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×Dic₁₀₉	C₂³
hypercenter	C₂×C₄²	#89137
hypocenter	C₁₀₉	#1745
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#89137

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₂×C₄²×C₁₀₉

C₄

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×Dic₁₀₉

C₂³

C₂×C₄²

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
6. #16812754	C₄	semidirect
76. C₁₀₉	#1745	semidirect

H ≅ G/N

type

6. #16812754

C₄

semidirect

76. C₁₀₉

#1745

semidirect

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

subgroup	index
C₂×#16812754	2
#20295892	2
C₄²×Dic₁₀₉	2
#1745	109

subgroup

index

C₂×#16812754

#20295892

C₄²×Dic₁₀₉

#1745

109

Subgroups — #28378749

Conjugacy classes of subgroups

Complements

Maximal subgroups