Group #28365076 — subgroups

Subgroups — #28365076

Order 13952.

166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 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×#16812644	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812642	C₂	1	1	yes	maximal, characteristic
3	#20295404	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295108	C₂	1	1	yes	maximal, characteristic
5	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294829	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×#13182019	—	2	1	no	—
10	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
11	#16812697	—	2	1	no	—
12	#16812661	—	2	1	no	—
13	#16812644	—	2	2	no	—
14	#16812642	—	2	2	no	—
15	C₂×#13182004	C₂²	1	1	yes	characteristic
16	#16812619	—	2	1	no	—
17	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
20	#107×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₈	—	2	1	no	—
22	C₂×D₄₃₆	—	2	1	no	—
23	C₂×C₄×D₁₀₉	—	4	1	no	—
24	Dic₄₃₆	—	2	1	no	—
25	Dic₄₃₆	—	2	1	no	—
26	#13182019	—	4	2	no	—
27	D₈₇₂	—	2	1	no	—
28	D₈₇₂	D₄	1	2	yes	—
29	C₂×#11724784	D₄	1	1	yes	characteristic
30	#13182008	—	4	1	no	—
31	#13182008	—	4	1	no	—
32	#13182006	—	4	1	no	—
33	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
34	#13182004	—	2	1	no	—
35	C₂×Q₈×C₁₀₉	—	2	1	no	—
36	Q₁₆×C₁₀₉	—	2	1	no	—
37	Q₁₆×C₁₀₉	—	2	1	no	—
38	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	2	1	no	—
40	C₁₆×C₁₀₉	—	2	1	no	—
41	C₂²×D₁₀₉	—	4	1	no	—
42	Q₈×C₁₀₉	—	4	1	no	—
43	Q₈×C₁₀₉	—	4	1	no	—
44	C₂×C₄×C₁₀₉	—	4	1	no	—
45	C₂×C₄×C₁₀₉	—	4	1	no	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₀₉	—	4	1	no	—
48	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
49	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
50	D₄₃₆	—	2	2	no	—
51	D₄₃₆	—	4	1	no	—
52	C₄×D₁₀₉	—	8	2	no	—
53	Dic₂₁₈	—	4	1	no	—
54	Dic₂₁₈	—	4	1	no	—
55	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₀₉	#41	1	1	yes	characteristic
57	#11724784	—	2	1	no	—
58	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
59	C₂×D₁₀₉	—	4	2	no	—
60	C₂×D₁₀₉	—	8	1	no	—
61	C₂×D₁₀₉	—	8	1	no	—
62	C₄×C₁₀₉	—	8	1	no	—
63	C₄×C₁₀₉	—	8	1	no	—
64	C₄×C₁₀₉	#121	1	1	yes	characteristic
65	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₁₀₉	—	2	1	no	—
67	Dic₁₀₉	—	8	1	no	—
68	Dic₁₀₉	SD₃₂	1	2	yes	complemented
69	C₂×C₁₀₉	#459	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₁₀₉	#510	1	1	yes	characteristic
72	D₁₀₉	—	8	2	no	—
73	#2208	—	109	1	no	maximal, Sylow
74	C₁₀₉	#2208	1	1	yes	characteristic, complemented, Sylow
75	C₂×SD₃₂	—	109	1	no	—
76	C₂×SD₃₂	—	109	1	no	—
77	#494	—	109	1	no	—
78	C₄×D₈	—	109	1	no	—
79	C₄⋊C₁₆	—	109	1	no	—
80	#358	—	109	1	no	—
81	#358	D₁₀₉	1	1	yes	characteristic, complemented
82	C₂×Q₁₆	—	109	1	no	—
83	C₂×Q₁₆	—	218	1	no	—
84	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
85	C₂×D₈	—	109	1	no	—
86	C₄⋊Q₈	—	218	1	no	—
87	C₄×D₄	—	218	1	no	—
88	SD₃₂	—	218	2	no	—
89	SD₃₂	—	218	2	no	—
90	C₂×C₁₆	—	109	1	no	—
91	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
92	#107	C₂×D₁₀₉	1	1	yes	characteristic
93	#102	—	218	1	no	—
94	C₄×C₈	—	109	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	218	1	no	—
97	C₂×D₄	—	218	1	no	—
98	C₂²×C₄	—	436	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	436	2	no	—
102	Q₁₆	—	218	1	no	—
103	Q₁₆	—	218	1	no	—
104	D₈	—	109	2	no	—
105	D₈	—	218	1	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	436	1	no	—
110	C₄⋊C₄	—	436	1	no	—
111	C₂²⋊C₄	—	436	1	no	—
112	C₄²	—	109	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	218	1	no	—
115	C₂³	—	436	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	436	1	no	—
119	Q₈	—	436	1	no	—
120	D₄	—	436	1	no	—
121	D₄	—	218	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	872	2	no	—
125	C₂×C₄	—	109	1	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	436	1	no	—
128	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
129	C₈	—	218	1	no	—
130	C₈	D₄×D₁₀₉	1	1	yes	characteristic
131	C₈	#13182029	1	1	yes	characteristic
132	C₂²	—	872	1	no	—
133	C₂²	—	872	1	no	—
134	C₂²	—	436	2	no	—
135	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	872	1	no	—
139	C₄	—	109	2	no	—
140	C₄	—	218	1	no	—
141	C₄	#16812672	1	1	yes	characteristic
142	C₄	#16812726	1	1	yes	characteristic
143	C₂	—	872	2	no	—
144	C₂	#20295220	1	1	yes	characteristic
145	C₂	SD₃₂×D₁₀₉	1	1	yes	characteristic
146	C₂	#20295468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812644

C₂

yes

maximal, characteristic

C₂×#16812642

C₂

yes

maximal, characteristic

#20295404

C₂

yes

maximal, characteristic, complemented

#20295108

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, characteristic, complemented

#20294829

C₂

yes

maximal, characteristic, complemented

#358×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×#13182019

—

C₂×D₈₇₂

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#16812619

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×D₄₃₆

—

C₂×C₄×D₁₀₉

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182019

—

D₈₇₂

—

D₈₇₂

D₄

yes

—

C₂×#11724784

D₄

yes

—

—

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

D₈

yes

characteristic

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

C₈×C₁₀₉

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

SD₃₂

yes

complemented

C₂×C₁₀₉

#459

yes

characteristic

C₂×C₁₀₉

C₂×SD₃₂

yes

characteristic

C₂×C₁₀₉

#510

yes

characteristic

D₁₀₉

—

#2208

—

109

maximal, Sylow

C₁₀₉

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

109

—

C₂×SD₃₂

—

109

—

#494

—

109

—

C₄×D₈

—

109

—

C₄⋊C₁₆

—

109

—

#358

—

109

—

#358

D₁₀₉

yes

characteristic, complemented

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

218

—

C₂×Q₁₆

C₂×D₁₀₉

yes

characteristic

C₂×D₈

—

109

—

C₄⋊Q₈

—

218

—

C₄×D₄

—

218

—

SD₃₂

—

218

—

SD₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#107

C₂×D₁₀₉

yes

characteristic

#102

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

436

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

436

—

102

Q₁₆

—

218

—

103

Q₁₆

—

218

—

104

D₈

—

109

—

105

D₈

—

218

—

106

C₂×C₈

—

109

—

107

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

436

—

110

C₄⋊C₄

—

436

—

111

C₂²⋊C₄

—

436

—

112

C₄²

—

109

—

113

C₁₆

—

114

C₁₆

—

218

—

115

C₂³

—

436

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

436

—

119

Q₈

—

436

—

120

D₄

—

436

—

121

D₄

—

218

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

872

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

436

—

128

C₂×C₄

D₄×D₁₀₉

yes

characteristic

129

C₈

—

218

—

130

C₈

D₄×D₁₀₉

yes

characteristic

131

C₈

#13182029

yes

characteristic

132

C₂²

—

872

—

133

C₂²

—

872

—

134

C₂²

—

436

—

135

C₂²

D₈×D₁₀₉

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

872

—

139

C₄

—

109

—

140

C₄

—

218

—

141

C₄

#16812672

yes

characteristic

142

C₄

#16812726

yes

characteristic

143

C₂

—

872

—

144

C₂

#20295220

yes

characteristic

145

C₂

SD₃₂×D₁₀₉

yes

characteristic

146

C₂

#20295468

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#358×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	C₂×D₈₇₂	C₂²
hypercenter	#358	D₁₀₉
hypocenter	C₁₀₉	#2208
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#358×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

prime radical

C₂×D₈₇₂

C₂²

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. #20295404	C₂	semidirect
5. #20294844	C₂	semidirect
6. #20294829	C₂	semidirect
7. #358×C₁₀₉	C₂	semidirect
68. Dic₁₀₉	SD₃₂	semidirect
74. C₁₀₉	#2208	semidirect
81. #358	D₁₀₉	semidirect

H ≅ G/N

type

3. #20295404

C₂

semidirect

5. #20294844

C₂

semidirect

6. #20294829

C₂

semidirect

7. #358×C₁₀₉

C₂

semidirect

68. Dic₁₀₉

SD₃₂

semidirect

74. C₁₀₉

#2208

semidirect

81. #358

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812644	2
C₂×#16812642	2
#20295404	2
#20295108	2
#20294844	2
#20294829	2
#358×C₁₀₉	2
#2208	109

subgroup

index

#358×C₁₀₉

109

Subgroups — #28365076

Conjugacy classes of subgroups

Complements

Maximal subgroups