Group #28365071 — subgroups

Subgroups — #28365071

Order 13952.

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

normal subgroups only

This group has 174 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, complemented
2	C₂×#16812643	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295401	C₂	1	1	yes	maximal, characteristic
4	#20295120	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294826	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
10	C₂×#13182014	—	2	1	no	—
11	#16812695	—	2	1	no	—
12	#16812664	—	2	1	no	—
13	#16812644	—	2	2	no	—
14	#16812643	—	2	2	no	—
15	C₂×#13182004	C₂²	1	1	yes	characteristic
16	#16812622	—	2	1	no	—
17	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×D₈×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×C₂²⋊D₁₀₉	—	4	1	no	—
24	C₂×C₂²⋊D₁₀₉	—	4	1	no	—
25	Dic₄₃₆	—	2	1	no	—
26	Dic₄₃₆	D₄	1	2	yes	complemented
27	D₈₇₂	—	2	1	no	—
28	D₈₇₂	—	2	1	no	—
29	#13182014	—	4	2	no	—
30	C₂×#11724784	D₄	1	1	yes	characteristic
31	#13182008	—	4	1	no	—
32	C₄×Dic₁₀₉	—	4	1	no	—
33	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
34	#13182004	—	2	1	no	—
35	C₂×D₄×C₁₀₉	—	2	1	no	—
36	D₈×C₁₀₉	—	2	1	no	—
37	D₈×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	C₂³×C₁₀₉	—	4	1	no	—
42	C₂²×D₁₀₉	—	4	1	no	—
43	D₄×C₁₀₉	—	4	1	no	—
44	D₄×C₁₀₉	—	4	1	no	—
45	C₂×C₄×C₁₀₉	—	4	1	no	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₁₀₉	—	8	2	no	—
48	C₂²⋊D₁₀₉	—	8	2	no	—
49	C₂×Dic₁₀₉	—	4	1	no	—
50	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
51	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
52	D₄₃₆	—	4	1	no	—
53	D₄₃₆	—	4	1	no	—
54	Dic₂₁₈	—	2	2	no	—
55	Dic₂₁₈	—	4	1	no	—
56	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₈×C₁₀₉	#41	1	1	yes	characteristic
58	#11724784	—	2	1	no	—
59	C₂²×C₁₀₉	—	8	1	no	—
60	C₂²×C₁₀₉	—	8	1	no	—
61	C₂²×C₁₀₉	—	8	1	no	—
62	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
63	C₂×D₁₀₉	—	8	1	no	—
64	C₂×D₁₀₉	—	8	1	no	—
65	C₂×D₁₀₉	—	8	1	no	—
66	C₄×C₁₀₉	—	8	1	no	—
67	C₄×C₁₀₉	#121	1	1	yes	characteristic
68	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₁₀₉	—	2	1	no	—
70	Dic₁₀₉	—	4	2	no	—
71	Dic₁₀₉	SD₃₂	1	2	yes	complemented
72	C₂×C₁₀₉	—	16	1	no	—
73	C₂×C₁₀₉	#459	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×SD₃₂	1	1	yes	characteristic
75	C₂×C₁₀₉	#509	1	1	yes	characteristic
76	D₁₀₉	—	16	1	no	—
77	#2207	—	109	1	no	maximal, Sylow
78	C₁₀₉	#2207	1	1	yes	characteristic, complemented, Sylow
79	C₂×SD₃₂	—	109	1	no	—
80	C₂×SD₃₂	—	109	1	no	—
81	#493	—	109	1	no	—
82	C₄×Q₁₆	—	109	1	no	—
83	C₄⋊C₁₆	—	109	1	no	—
84	#357	—	109	1	no	—
85	#357	D₁₀₉	1	1	yes	characteristic, complemented
86	C₂×Q₁₆	—	109	1	no	—
87	C₂×D₈	—	109	1	no	—
88	C₂×D₈	—	218	1	no	—
89	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
90	#127	—	218	1	no	—
91	C₄×Q₈	—	218	1	no	—
92	SD₃₂	—	218	2	no	—
93	SD₃₂	—	218	2	no	—
94	C₂×C₁₆	—	109	1	no	—
95	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
96	#107	C₂×D₁₀₉	1	1	yes	characteristic
97	#103	—	218	1	no	—
98	C₄×C₈	—	109	1	no	—
99	C₂×Q₈	—	218	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	436	1	no	—
102	C₂×D₄	—	436	1	no	—
103	C₂×D₄	—	218	1	no	—
104	Q₁₆	—	109	2	no	—
105	Q₁₆	—	218	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	436	2	no	—
109	D₈	—	218	1	no	—
110	D₈	—	218	1	no	—
111	C₂×C₈	—	109	1	no	—
112	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	436	1	no	—
115	C₄²	—	109	1	no	—
116	C₄²	—	436	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	218	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	436	1	no	—
121	Q₈	—	436	1	no	—
122	Q₈	—	218	2	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	872	2	no	—
126	D₄	—	872	2	no	—
127	D₄	—	436	1	no	—
128	D₄	—	436	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	109	1	no	—
132	C₂×C₄	—	436	1	no	—
133	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
134	C₈	—	218	1	no	—
135	C₈	D₄×D₁₀₉	1	1	yes	characteristic
136	C₈	#13182029	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	872	1	no	—
141	C₂²	—	872	1	no	—
142	C₂²	—	872	1	no	—
143	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	109	2	no	—
146	C₄	—	436	2	no	—
147	C₄	—	218	1	no	—
148	C₄	#16812672	1	1	yes	characteristic
149	C₄	#16812726	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	1744	1	no	—
152	C₂	#20295220	1	1	yes	characteristic
153	C₂	SD₃₂×D₁₀₉	1	1	yes	characteristic
154	C₂	#20295460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812644

C₂

yes

maximal, characteristic, complemented

C₂×#16812643

C₂

yes

maximal, characteristic, complemented

#20295401

C₂

yes

maximal, characteristic

#20295120

C₂

yes

maximal, characteristic, complemented

#20294844

C₂

yes

maximal, characteristic, complemented

#20294826

C₂

yes

maximal, characteristic, complemented

#357×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₃₆

C₂²

yes

characteristic

C₂×D₈₇₂

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#16812622

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×D₄₃₆

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₂²⋊D₁₀₉

—

Dic₄₃₆

—

Dic₄₃₆

D₄

yes

complemented

D₈₇₂

—

D₈₇₂

—

#13182014

—

C₂×#11724784

D₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

D₈

yes

characteristic

D₄₃₆

—

D₄₃₆

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×D₄

yes

C₈×C₁₀₉

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

SD₃₂

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

#459

yes

characteristic

C₂×C₁₀₉

C₂×SD₃₂

yes

characteristic

C₂×C₁₀₉

#509

yes

characteristic

D₁₀₉

—

#2207

—

109

maximal, Sylow

C₁₀₉

#2207

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

109

—

C₂×SD₃₂

—

109

—

#493

—

109

—

C₄×Q₁₆

—

109

—

C₄⋊C₁₆

—

109

—

#357

—

109

—

#357

D₁₀₉

yes

characteristic, complemented

C₂×Q₁₆

—

109

—

C₂×D₈

—

109

—

C₂×D₈

—

218

—

C₂×D₈

C₂×D₁₀₉

yes

characteristic

#127

—

218

—

C₄×Q₈

—

218

—

SD₃₂

—

218

—

SD₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#107

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

218

—

100

C₂×D₄

—

101

C₂×D₄

—

436

—

102

C₂×D₄

—

436

—

103

C₂×D₄

—

218

—

104

Q₁₆

—

109

—

105

Q₁₆

—

218

—

106

D₈

—

107

D₈

—

108

D₈

—

436

—

109

D₈

—

218

—

110

D₈

—

218

—

111

C₂×C₈

—

109

—

112

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

436

—

115

C₄²

—

109

—

116

C₄²

—

436

—

117

C₁₆

—

118

C₁₆

—

218

—

119

C₂³

—

120

C₂³

—

436

—

121

Q₈

—

436

—

122

Q₈

—

218

—

123

D₄

—

124

D₄

—

125

D₄

—

872

—

126

D₄

—

872

—

127

D₄

—

436

—

128

D₄

—

436

—

129

C₂×C₄

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

436

—

133

C₂×C₄

D₄×D₁₀₉

yes

characteristic

134

C₈

—

218

—

135

C₈

D₄×D₁₀₉

yes

characteristic

136

C₈

#13182029

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

872

—

141

C₂²

—

872

—

142

C₂²

—

872

—

143

C₂²

D₈×D₁₀₉

yes

characteristic

144

C₄

—

145

C₄

—

109

—

146

C₄

—

436

—

147

C₄

—

218

—

148

C₄

#16812672

yes

characteristic

149

C₄

#16812726

yes

characteristic

150

C₂

—

151

C₂

—

1744

—

152

C₂

#20295220

yes

characteristic

153

C₂

SD₃₂×D₁₀₉

yes

characteristic

154

C₂

#20295460

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#357×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₈
prime radical	#20295401	C₂
hypercenter	#357	D₁₀₉
hypocenter	C₁₀₉	#2207
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#357×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₈

C₂

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#16812644	C₂	semidirect
2. C₂×#16812643	C₂	semidirect
4. #20295120	C₂	semidirect
5. #20294844	C₂	semidirect
6. #20294826	C₂	semidirect
7. #357×C₁₀₉	C₂	semidirect
26. Dic₄₃₆	D₄	semidirect
71. Dic₁₀₉	SD₃₂	semidirect
78. C₁₀₉	#2207	semidirect
85. #357	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812644

C₂

semidirect

2. C₂×#16812643

C₂

semidirect

4. #20295120

C₂

semidirect

5. #20294844

C₂

semidirect

6. #20294826

C₂

semidirect

7. #357×C₁₀₉

C₂

semidirect

26. Dic₄₃₆

D₄

semidirect

71. Dic₁₀₉

SD₃₂

semidirect

78. C₁₀₉

#2207

semidirect

85. #357

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×#16812643	2
#20295401	2
#20295120	2
#20294844	2
#20294826	2
#357×C₁₀₉	2
#2207	109

subgroup

index

#357×C₁₀₉

109

Subgroups — #28365071

Conjugacy classes of subgroups

Complements

Maximal subgroups