Group #28364355 — subgroups

Subgroups — #28364355

Order 13952.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812624	C₂	1	2	yes	maximal
3	#20294932	C₂	1	2	yes	maximal
4	#20294789	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
7	C₂×Dic₄₃₆	—	2	4	no	—
8	C₂×Dic₄₃₆	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	#16812624	—	2	4	no	—
11	#16812606	—	2	2	no	—
12	#16812606	—	2	2	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	—	2	2	no	—
17	C₂×Dic₂₁₈	—	2	2	no	—
18	C₂×Dic₂₁₈	—	2	2	no	—
19	C₂×Dic₂₁₈	—	2	2	no	—
20	C₂×Dic₂₁₈	—	2	2	no	—
21	C₂×Dic₂₁₈	—	2	2	no	—
22	C₂×Dic₂₁₈	D₄	1	4	yes	—
23	C₂²×Dic₁₀₉	—	2	2	no	—
24	C₂²×Dic₁₀₉	—	2	2	no	—
25	Dic₄₃₆	—	2	8	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	—	2	2	no	—
28	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₄²×C₁₀₉	—	2	1	no	—
36	C₄²×C₁₀₉	—	2	1	no	—
37	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	#41	1	2	yes	—
47	C₂×Dic₁₀₉	—	2	4	no	—
48	C₂×Dic₁₀₉	—	4	2	no	—
49	C₂×Dic₁₀₉	—	4	2	no	—
50	C₂×Dic₁₀₉	—	4	2	no	—
51	C₂×Dic₁₀₉	—	4	2	no	—
52	C₂×Dic₁₀₉	—	4	2	no	—
53	Dic₂₁₈	—	2	4	no	—
54	Dic₂₁₈	—	2	4	no	—
55	Dic₂₁₈	—	4	2	no	—
56	Dic₂₁₈	—	4	2	no	—
57	C₈×C₁₀₉	—	2	2	no	—
58	C₈×C₁₀₉	—	4	1	no	—
59	C₂²×C₁₀₉	#121	1	1	yes	characteristic
60	C₂²×C₁₀₉	#121	1	2	yes	—
61	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₉	#135	1	1	yes	characteristic
63	C₂²×C₁₀₉	#137	1	2	yes	—
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	—	4	1	no	—
66	C₄×C₁₀₉	#120	1	1	yes	characteristic
67	C₄×C₁₀₉	#123	1	2	yes	—
68	C₄×C₁₀₉	#124	1	1	yes	characteristic
69	Dic₁₀₉	—	4	4	no	—
70	Dic₁₀₉	—	8	2	no	—
71	C₂×C₁₀₉	#394	1	1	yes	characteristic
72	C₂×C₁₀₉	#462	1	2	yes	—
73	C₂×C₁₀₉	#464	1	2	yes	—
74	C₂×C₁₀₉	#467	1	1	yes	characteristic
75	C₂×C₁₀₉	#470	1	1	yes	characteristic
76	#2056	—	109	1	no	maximal, Sylow
77	C₁₀₉	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	109	1	no	—
79	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
80	C₂×#103	—	109	2	no	—
81	#391	—	109	2	no	—
82	#340	—	109	1	no	—
83	C₂²×Q₈	—	109	2	no	—
84	C₂×Q₁₆	—	218	4	no	—
85	C₂×Q₁₆	—	218	2	no	—
86	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	109	2	no	—
89	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	218	4	no	—
92	#95	—	218	2	no	—
93	#95	—	218	2	no	—
94	C₂×Q₈	—	109	4	no	—
95	C₂×Q₈	—	218	2	no	—
96	C₂×Q₈	—	218	2	no	—
97	C₂×Q₈	—	218	2	no	—
98	C₂×Q₈	—	218	2	no	—
99	C₂×Q₈	—	218	2	no	—
100	C₂²×C₄	—	218	2	no	—
101	C₂²×C₄	—	218	2	no	—
102	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
103	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
105	Q₁₆	—	218	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
110	C₄⋊C₄	—	218	2	no	—
111	C₄⋊C₄	—	218	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
115	Q₈	—	436	2	no	—
116	Q₈	—	436	2	no	—
117	Q₈	—	218	4	no	—
118	Q₈	—	218	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	436	2	no	—
124	C₂×C₄	—	436	2	no	—
125	C₂×C₄	—	436	2	no	—
126	C₂×C₄	—	436	2	no	—
127	C₂×C₄	—	436	2	no	—
128	C₂×C₄	—	218	4	no	—
129	C₂×C₄	Dic₄₃₆	1	2	yes	complemented
130	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
132	C₂×C₄	#13182029	1	1	yes	characteristic
133	C₂×C₄	#13182030	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#16812673	1	1	yes	characteristic
137	C₂²	#16812674	1	2	yes	—
138	C₂²	C₂×Dic₄₃₆	1	1	yes	characteristic
139	C₂²	#16812721	1	1	yes	characteristic
140	C₂²	#16812732	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	872	2	no	—
144	C₄	—	436	4	no	—
145	C₄	#16812667	1	1	yes	characteristic
146	C₄	#16812684	1	2	yes	—
147	C₄	#16812688	1	1	yes	characteristic
148	C₂	#20294943	1	1	yes	characteristic
149	C₂	#20295237	1	2	yes	—
150	C₂	#20295249	1	2	yes	—
151	C₂	#20295267	1	1	yes	characteristic
152	C₂	#20295284	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₃₆

C₂

yes

maximal, characteristic

C₂×#16812624

C₂

yes

maximal

#20294932

C₂

yes

maximal

#20294789

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂×Dic₄₃₆

—

C₂×Dic₄₃₆

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812624

—

#16812606

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

D₄

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

Dic₄₃₆

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

Q₁₆

yes

complemented

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#121

yes

—

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#123

yes

—

C₄×C₁₀₉

#124

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#394

yes

characteristic

C₂×C₁₀₉

#462

yes

—

C₂×C₁₀₉

#464

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

C₂×#103

—

109

—

#391

—

109

—

#340

—

109

—

C₂²×Q₈

—

109

—

C₂×Q₁₆

—

218

—

C₂×Q₁₆

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

#103

—

218

—

#95

—

218

—

#95

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

100

C₂²×C₄

—

218

—

101

C₂²×C₄

—

218

—

102

C₂²×C₄

D₄₃₆

yes

characteristic

103

C₂²×C₄

D₄₃₆

yes

characteristic

104

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

105

Q₁₆

—

218

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₁₀₉

yes

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

—

218

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₄₃₆

yes

characteristic

115

Q₈

—

436

—

116

Q₈

—

436

—

117

Q₈

—

218

—

118

Q₈

—

218

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

436

—

124

C₂×C₄

—

436

—

125

C₂×C₄

—

436

—

126

C₂×C₄

—

436

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

218

—

129

C₂×C₄

Dic₄₃₆

yes

complemented

130

C₂×C₄

D₄×D₁₀₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

132

C₂×C₄

#13182029

yes

characteristic

133

C₂×C₄

#13182030

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#16812673

yes

characteristic

137

C₂²

#16812674

yes

—

138

C₂²

C₂×Dic₄₃₆

yes

characteristic

139

C₂²

#16812721

yes

characteristic

140

C₂²

#16812732

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

872

—

144

C₄

—

436

—

145

C₄

#16812667

yes

characteristic

146

C₄

#16812684

yes

—

147

C₄

#16812688

yes

characteristic

148

C₂

#20294943

yes

characteristic

149

C₂

#20295237

yes

—

150

C₂

#20295249

yes

—

151

C₂

#20295267

yes

characteristic

152

C₂

#20295284

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#2056
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₀₉

hypocenter

C₁₀₉

#2056

LCS

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₁₀₉	Q₁₆	semidirect
77. C₁₀₉	#2056	semidirect
129. C₂×C₄	Dic₄₃₆	semidirect

H ≅ G/N

type

42. C₂×C₄×C₁₀₉

Q₁₆

semidirect

77. C₁₀₉

#2056

semidirect

129. C₂×C₄

Dic₄₃₆

semidirect

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

subgroup	index
C₂²×Dic₄₃₆	2
C₂×#16812624	2
#20294932	2
#20294789	2
C₂×C₄⋊C₈×C₁₀₉	2
#2056	109

subgroup

index

C₂²×Dic₄₃₆

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28364355

Conjugacy classes of subgroups

Complements

Maximal subgroups