Group #28365107 — subgroups

Subgroups — #28365107

Order 13952.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 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	#20295464	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295462	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295401	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295142	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294846	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294838	C₂	1	1	yes	maximal, characteristic, complemented
7	#361×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#16812721	C₂²	1	1	yes	characteristic
9	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
10	C₂×#13182014	—	2	1	no	—
11	#16812707	—	2	1	no	—
12	#16812695	—	2	1	no	—
13	#16812644	—	2	1	no	—
14	#16812643	—	2	1	no	—
15	D₁₇₄₄	—	2	1	no	—
16	#16812640	—	2	1	no	—
17	#16812639	C₂²	1	1	yes	characteristic
18	#16812625	—	2	1	no	—
19	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
22	#108×C₁₀₉	C₂²	1	1	yes	characteristic
23	#13182029	—	2	1	no	—
24	C₂×D₄₃₆	—	2	1	no	—
25	C₂×C₂²⋊D₁₀₉	—	4	2	no	—
26	Dic₄₃₆	D₄	1	1	yes	characteristic, complemented
27	#13182018	—	2	1	no	—
28	D₈₇₂	—	2	1	no	—
29	D₈₇₂	—	2	1	no	—
30	D₈₇₂	D₄	1	1	yes	characteristic, complemented
31	#13182014	—	4	1	no	—
32	#13182014	—	4	1	no	—
33	#13182012	—	4	1	no	—
34	C₂×#11724784	D₄	1	1	yes	characteristic
35	C₈×D₁₀₉	—	4	1	no	—
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	#13182004	—	2	1	no	—
38	C₂×D₄×C₁₀₉	—	2	1	no	—
39	D₈×C₁₀₉	—	2	1	no	—
40	D₈×C₁₀₉	—	2	1	no	—
41	M₁₆×C₁₀₉	—	2	1	no	—
42	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₁₆×C₁₀₉	—	2	1	no	—
44	C₂³×C₁₀₉	—	4	1	no	—
45	C₂²×D₁₀₉	—	4	1	no	—
46	D₄×C₁₀₉	—	4	1	no	—
47	D₄×C₁₀₉	—	4	1	no	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊D₁₀₉	—	4	1	no	—
50	C₂²⋊D₁₀₉	—	8	4	no	—
51	C₂×Dic₁₀₉	—	2	1	no	—
52	D₄₃₆	—	2	1	no	—
53	D₄₃₆	—	4	1	no	—
54	D₄₃₆	—	4	1	no	—
55	C₄×D₁₀₉	—	4	1	no	—
56	Dic₂₁₈	—	2	1	no	—
57	C₈×C₁₀₉	—	4	1	no	—
58	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₈×C₁₀₉	#41	1	1	yes	characteristic
60	#11724784	D₈	1	1	yes	characteristic
61	#11724784	D₈	1	1	yes	characteristic
62	C₂²×C₁₀₉	—	8	2	no	—
63	C₂²×C₁₀₉	—	8	1	no	—
64	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×D₁₀₉	—	4	1	no	—
66	C₂×D₁₀₉	—	8	2	no	—
67	C₂×D₁₀₉	—	8	1	no	—
68	C₄×C₁₀₉	#121	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
70	Dic₁₀₉	—	2	2	no	—
71	Dic₁₀₉	—	4	1	no	—
72	C₂×C₁₀₉	—	2	1	no	—
73	C₂×C₁₀₉	—	16	1	no	—
74	C₂×C₁₀₉	#459	1	1	yes	characteristic
75	D₁₀₉	—	8	1	no	—
76	D₁₀₉	—	16	1	no	—
77	#2213	—	109	1	no	maximal, Sylow
78	C₁₀₉	#2213	1	1	yes	characteristic, complemented, Sylow
79	#509	—	109	1	no	—
80	#509	—	109	1	no	—
81	#493	—	109	1	no	—
82	#443	—	109	1	no	—
83	#364	—	109	1	no	—
84	#361	—	109	1	no	—
85	#361	D₁₀₉	1	1	yes	characteristic, complemented
86	#135	—	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	#131	—	218	1	no	—
91	#127	—	218	1	no	—
92	SD₃₂	—	218	1	no	—
93	SD₃₂	—	218	1	no	—
94	D₁₆	—	218	1	no	—
95	D₁₆	—	218	1	no	—
96	M₃₂	—	109	1	no	—
97	M₃₂	C₂×D₁₀₉	1	1	yes	characteristic
98	#108	C₂×D₁₀₉	1	1	yes	characteristic
99	#104	—	218	1	no	—
100	C₄×C₈	—	109	1	no	—
101	#41	—	218	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	436	2	no	—
104	C₂×D₄	—	218	1	no	—
105	Q₁₆	—	109	1	no	—
106	SD₁₆	—	218	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	109	1	no	—
110	D₈	—	436	1	no	—
111	D₈	—	436	1	no	—
112	D₈	—	218	1	no	—
113	D₈	—	218	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	436	1	no	—
116	C₂×C₈	—	109	1	no	—
117	C₂×C₈	—	436	1	no	—
118	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
119	C₄²	—	109	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	218	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	436	1	no	—
124	Q₈	—	218	1	no	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	872	4	no	—
128	D₄	—	436	1	no	—
129	D₄	—	436	1	no	—
130	D₄	—	436	1	no	—
131	D₄	—	218	1	no	—
132	C₂×C₄	—	436	1	no	—
133	C₂×C₄	—	218	1	no	—
134	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	109	1	no	—
137	C₈	—	109	1	no	—
138	C₈	D₄×D₁₀₉	1	1	yes	characteristic
139	C₈	#13182029	1	1	yes	characteristic
140	C₂²	—	8	2	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	872	2	no	—
143	C₂²	—	872	1	no	—
144	C₂²	—	436	1	no	—
145	C₂²	#16812726	1	1	yes	characteristic
146	C₄	—	436	1	no	—
147	C₄	—	218	2	no	—
148	C₄	#16812672	1	1	yes	characteristic
149	C₄	D₈×D₁₀₉	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	16	1	no	—
152	C₂	—	872	1	no	—
153	C₂	—	1744	1	no	—
154	C₂	#20295220	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295464

C₂

yes

maximal, characteristic, complemented

#20295462

C₂

yes

maximal, characteristic, complemented

#20295401

C₂

yes

maximal, characteristic, complemented

#20295142

C₂

yes

maximal, characteristic, complemented

#20294846

C₂

yes

maximal, characteristic, complemented

#20294838

C₂

yes

maximal, characteristic, complemented

#361×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812721

C₂²

yes

characteristic

C₂×D₈₇₂

C₂²

yes

—

—

—

—

—

D₁₇₄₄

—

#16812640

—

#16812639

C₂²

yes

characteristic

#16812625

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

C₂²

yes

characteristic

M₃₂×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

C₂²

yes

characteristic

#13182029

—

C₂×D₄₃₆

—

C₂×C₂²⋊D₁₀₉

—

Dic₄₃₆

D₄

yes

characteristic, complemented

#13182018

—

D₈₇₂

—

D₈₇₂

—

D₈₇₂

D₄

yes

characteristic, complemented

—

—

—

D₄

yes

characteristic

C₈×D₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

C₈×C₁₀₉

yes

D₈

yes

characteristic

#11724784

D₈

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#459

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#2213

—

109

maximal, Sylow

C₁₀₉

#2213

yes

characteristic, complemented, Sylow

#509

—

109

—

#509

—

109

—

#493

—

109

—

#443

—

109

—

#364

—

109

—

#361

—

109

—

#361

D₁₀₉

yes

characteristic, complemented

#135

—

109

—

C₂×D₈

—

109

—

C₂×D₈

—

218

—

C₂×D₈

C₂×D₁₀₉

yes

characteristic

#131

—

218

—

#127

—

218

—

SD₃₂

—

218

—

SD₃₂

—

218

—

D₁₆

—

218

—

D₁₆

—

218

—

M₃₂

—

109

—

M₃₂

C₂×D₁₀₉

yes

characteristic

#108

C₂×D₁₀₉

yes

characteristic

#104

—

218

—

100

C₄×C₈

—

109

—

101

#41

—

218

—

102

C₂×D₄

—

103

C₂×D₄

—

436

—

104

C₂×D₄

—

218

—

105

Q₁₆

—

109

—

106

SD₁₆

—

218

—

107

D₈

—

108

D₈

—

109

D₈

—

109

—

110

D₈

—

436

—

111

D₈

—

436

—

112

D₈

—

218

—

113

D₈

—

218

—

114

M₁₆

—

115

M₁₆

—

436

—

116

C₂×C₈

—

109

—

117

C₂×C₈

—

436

—

118

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

119

C₄²

—

109

—

120

C₁₆

—

121

C₁₆

—

218

—

122

C₂³

—

123

C₂³

—

436

—

124

Q₈

—

218

—

125

D₄

—

126

D₄

—

127

D₄

—

872

—

128

D₄

—

436

—

129

D₄

—

436

—

130

D₄

—

436

—

131

D₄

—

218

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

218

—

134

C₂×C₄

D₄×D₁₀₉

yes

characteristic

135

C₈

—

136

C₈

—

109

—

137

C₈

—

109

—

138

C₈

D₄×D₁₀₉

yes

characteristic

139

C₈

#13182029

yes

characteristic

140

C₂²

—

141

C₂²

—

142

C₂²

—

872

—

143

C₂²

—

872

—

144

C₂²

—

436

—

145

C₂²

#16812726

yes

characteristic

146

C₄

—

436

—

147

C₄

—

218

—

148

C₄

#16812672

yes

characteristic

149

C₄

D₈×D₁₀₉

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

—

872

—

153

C₂

—

1744

—

154

C₂

#20295220

yes

characteristic

	subgroup	G/H
center	C₂	#20295220
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#361×C₁₀₉	C₂
socle	C₂×C₁₀₉	#459
hypercenter	#361	D₁₀₉
hypocenter	C₁₀₉	#2213
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#20295220

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#361×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
1. #20295464	C₂	semidirect
2. #20295462	C₂	semidirect
3. #20295401	C₂	semidirect
4. #20295142	C₂	semidirect
5. #20294846	C₂	semidirect
6. #20294838	C₂	semidirect
7. #361×C₁₀₉	C₂	semidirect
26. Dic₄₃₆	D₄	semidirect
30. D₈₇₂	D₄	semidirect
78. C₁₀₉	#2213	semidirect
85. #361	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295464

C₂

semidirect

2. #20295462

C₂

semidirect

3. #20295401

C₂

semidirect

4. #20295142

C₂

semidirect

5. #20294846

C₂

semidirect

6. #20294838

C₂

semidirect

7. #361×C₁₀₉

C₂

semidirect

26. Dic₄₃₆

D₄

semidirect

30. D₈₇₂

D₄

semidirect

78. C₁₀₉

#2213

semidirect

85. #361

D₁₀₉

semidirect

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

subgroup	index
#20295464	2
#20295462	2
#20295401	2
#20295142	2
#20294846	2
#20294838	2
#361×C₁₀₉	2
#2213	109

subgroup

index

#361×C₁₀₉

109

Subgroups — #28365107

Conjugacy classes of subgroups

Complements

Maximal subgroups