Group #28364361 — subgroups

Subgroups — #28364361

Order 13952.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812633	C₂	1	2	yes	maximal
3	#20294924	C₂	1	1	yes	maximal, characteristic
4	#20294789	C₂	1	2	yes	maximal
5	#336×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₈	—	2	1	no	—
7	#16812699	—	2	4	no	—
8	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×#13182008	—	2	1	no	—
13	#16812633	—	2	4	no	—
14	#16812606	—	2	1	no	—
15	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₈	—	2	4	no	—
18	C₂×Dic₂₁₈	—	4	2	no	—
19	C₂×Dic₂₁₈	—	4	2	no	—
20	C₂²×Dic₁₀₉	—	2	1	no	—
21	C₂²×Dic₁₀₉	—	2	2	no	—
22	C₂²×Dic₁₀₉	—	2	1	no	—
23	C₄⋊Dic₁₀₉	—	2	4	no	—
24	C₄⋊Dic₁₀₉	—	2	2	no	—
25	C₄⋊Dic₁₀₉	—	2	1	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	D₄	1	4	yes	—
28	C₄⋊Dic₁₀₉	Q₈	1	2	yes	complemented
29	#13182008	—	4	2	no	—
30	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
36	C₄²×C₁₀₉	—	2	2	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	1	no	—
42	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
43	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	2	4	no	—
49	C₂×Dic₁₀₉	—	2	2	no	—
50	C₂×Dic₁₀₉	—	4	1	no	—
51	C₂×Dic₁₀₉	—	4	2	no	—
52	C₂×Dic₁₀₉	—	4	2	no	—
53	C₂×Dic₁₀₉	—	4	1	no	—
54	C₂×Dic₁₀₉	—	4	1	no	—
55	C₂×Dic₁₀₉	—	4	1	no	—
56	C₂×Dic₁₀₉	—	4	1	no	—
57	Dic₂₁₈	—	4	8	no	—
58	C₈×C₁₀₉	—	2	4	no	—
59	C₂²×C₁₀₉	#120	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₀₉	#137	1	1	yes	characteristic
66	C₄×C₁₀₉	—	2	2	no	—
67	C₄×C₁₀₉	—	2	2	no	—
68	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
69	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
70	Dic₁₀₉	—	4	4	no	—
71	Dic₁₀₉	—	4	2	no	—
72	Dic₁₀₉	—	8	1	no	—
73	C₂×C₁₀₉	#395	1	1	yes	characteristic
74	C₂×C₁₀₉	#447	1	1	yes	characteristic
75	C₂×C₁₀₉	#451	1	1	yes	characteristic
76	C₂×C₁₀₉	#474	1	1	yes	characteristic
77	C₂×C₁₀₉	#477	1	1	yes	characteristic
78	C₂×C₁₀₉	#500	1	1	yes	characteristic
79	C₂×C₁₀₉	#501	1	1	yes	characteristic
80	#2057	—	109	1	no	maximal, Sylow
81	C₁₀₉	#2057	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	109	1	no	—
83	C₂×#107	—	109	2	no	—
84	#389	—	109	1	no	—
85	#340	—	109	2	no	—
86	#336	D₁₀₉	1	1	yes	characteristic
87	C₂²×Q₈	—	218	1	no	—
88	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
89	C₄⋊Q₈	—	218	4	no	—
90	C₂×C₄⋊C₄	—	109	2	no	—
91	C₂×C₄⋊C₄	—	109	1	no	—
92	C₂×C₄⋊C₄	—	109	1	no	—
93	C₂×C₄⋊C₄	—	218	1	no	—
94	C₂×C₄⋊C₄	—	218	1	no	—
95	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
96	#107	—	218	4	no	—
97	#95	—	218	1	no	—
98	C₂×Q₈	—	436	2	no	—
99	C₂×Q₈	—	436	2	no	—
100	C₂×Q₈	—	218	4	no	—
101	C₂²×C₄	—	218	1	no	—
102	C₂²×C₄	—	218	1	no	—
103	C₂²×C₄	—	218	2	no	—
104	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
105	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
110	C₄⋊C₄	—	109	4	no	—
111	C₄⋊C₄	—	109	2	no	—
112	C₄⋊C₄	—	436	2	no	—
113	C₄⋊C₄	—	218	4	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₄⋊C₄	—	218	2	no	—
116	C₄⋊C₄	—	218	1	no	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
119	Q₈	—	436	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	436	1	no	—
125	C₂×C₄	—	436	1	no	—
126	C₂×C₄	—	436	1	no	—
127	C₂×C₄	—	436	1	no	—
128	C₂×C₄	—	436	2	no	—
129	C₂×C₄	—	436	2	no	—
130	C₂×C₄	—	436	1	no	—
131	C₂×C₄	—	218	4	no	—
132	C₂×C₄	—	218	2	no	—
133	C₂×C₄	D₈₇₂	1	2	yes	—
134	C₂×C₄	Dic₄₃₆	1	2	yes	—
135	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
136	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
138	C₂×C₄	#13182028	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₂²	#16812669	1	1	yes	characteristic
141	C₂²	#16812679	1	1	yes	characteristic
142	C₂²	#16812699	1	1	yes	characteristic
143	C₂²	C₂×D₈₇₂	1	1	yes	characteristic
144	C₂²	C₂×Dic₄₃₆	1	1	yes	characteristic
145	C₂²	#16812727	1	1	yes	characteristic
146	C₂²	#16812732	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	872	1	no	—
150	C₄	—	436	2	no	—
151	C₄	—	436	4	no	—
152	C₄	#16812676	1	2	yes	—
153	C₄	#16812698	1	2	yes	—
154	C₂	#20294947	1	1	yes	characteristic
155	C₂	#20295162	1	1	yes	characteristic
156	C₂	#20295184	1	1	yes	characteristic
157	C₂	#20295307	1	1	yes	characteristic
158	C₂	#20295322	1	1	yes	characteristic
159	C₂	#20295431	1	1	yes	characteristic
160	C₂	#20295435	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₂×#16812633

C₂

yes

maximal

#20294924

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal

#336×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

#16812699

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

Q₈

yes

complemented

#13182008

—

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

Q₈

yes

complemented

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2057

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#107

—

109

—

#389

—

109

—

#340

—

109

—

#336

D₁₀₉

yes

characteristic

C₂²×Q₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#95

—

218

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

100

C₂×Q₈

—

218

—

101

C₂²×C₄

—

218

—

102

C₂²×C₄

—

218

—

103

C₂²×C₄

—

218

—

104

C₂²×C₄

D₄₃₆

yes

characteristic

105

C₂²×C₄

D₄₃₆

yes

characteristic

106

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

Dic₂₁₈

yes

complemented

110

C₄⋊C₄

—

109

—

111

C₄⋊C₄

—

109

—

112

C₄⋊C₄

—

436

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

—

218

—

116

C₄⋊C₄

—

218

—

117

C₄²

—

118

C₂³

C₂×D₄₃₆

yes

characteristic

119

Q₈

—

436

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

436

—

125

C₂×C₄

—

436

—

126

C₂×C₄

—

436

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

436

—

129

C₂×C₄

—

436

—

130

C₂×C₄

—

436

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

D₈₇₂

yes

—

134

C₂×C₄

Dic₄₃₆

yes

—

135

C₂×C₄

D₄×D₁₀₉

yes

—

136

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

137

C₂×C₄

C₂×Dic₂₁₈

yes

—

138

C₂×C₄

#13182028

yes

characteristic

139

C₈

—

140

C₂²

#16812669

yes

characteristic

141

C₂²

#16812679

yes

characteristic

142

C₂²

#16812699

yes

characteristic

143

C₂²

C₂×D₈₇₂

yes

characteristic

144

C₂²

C₂×Dic₄₃₆

yes

characteristic

145

C₂²

#16812727

yes

characteristic

146

C₂²

#16812732

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

872

—

150

C₄

—

436

—

151

C₄

—

436

—

152

C₄

#16812676

yes

—

153

C₄

#16812698

yes

—

154

C₂

#20294947

yes

characteristic

155

C₂

#20295162

yes

characteristic

156

C₂

#20295184

yes

characteristic

157

C₂

#20295307

yes

characteristic

158

C₂

#20295322

yes

characteristic

159

C₂

#20295431

yes

characteristic

160

C₂

#20295435

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#336×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#336	D₁₀₉
hypocenter	C₁₀₉	#2057
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#336×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
28. C₄⋊Dic₁₀₉	Q₈	semidirect
35. C₂×C₈×C₁₀₉	Q₈	semidirect
81. C₁₀₉	#2057	semidirect
109. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

28. C₄⋊Dic₁₀₉

Q₈

semidirect

35. C₂×C₈×C₁₀₉

Q₈

semidirect

81. C₁₀₉

#2057

semidirect

109. C₂×C₈

Dic₂₁₈

semidirect

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

subgroup	index
C₂×#16812699	2
C₂×#16812633	2
#20294924	2
#20294789	2
#336×C₁₀₉	2
#2057	109

subgroup

index

#336×C₁₀₉

109

Subgroups — #28364361

Conjugacy classes of subgroups

Complements

Maximal subgroups