Group #28363993 — subgroups

Subgroups — #28363993

Order 13952.

300 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20294896	C₂	1	1	yes	maximal, characteristic
3	#20294789	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	—	2	1	no	—
6	#16812699	—	2	2	no	—
7	#16812699	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	—	2	2	no	—
16	C₂×Dic₂₁₈	—	2	4	no	—
17	C₂×Dic₂₁₈	—	4	2	no	—
18	C₂×Dic₂₁₈	—	4	2	no	—
19	C₂²×Dic₁₀₉	—	2	2	no	—
20	C₂²×Dic₁₀₉	—	2	2	no	—
21	C₄⋊Dic₁₀₉	—	2	4	no	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	—	2	4	no	—
24	C₄⋊Dic₁₀₉	—	4	2	no	—
25	C₄⋊Dic₁₀₉	—	4	2	no	—
26	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	Q₈	1	4	yes	complemented
33	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄²×C₁₀₉	D₄	1	2	yes	—
35	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
39	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
40	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₀₉	#41	1	2	yes	—
46	C₂×Dic₁₀₉	—	2	4	no	—
47	C₂×Dic₁₀₉	—	4	2	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	Dic₂₁₈	—	4	8	no	—
53	C₈×C₁₀₉	—	2	4	no	—
54	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₉	#124	1	1	yes	characteristic
56	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₀₉	#102	1	4	yes	—
62	C₄×C₁₀₉	#103	1	4	yes	—
63	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
64	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
65	Dic₁₀₉	—	4	4	no	—
66	Dic₁₀₉	—	8	2	no	—
67	C₂×C₁₀₉	#391	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
69	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
70	C₂×C₁₀₉	#486	1	1	yes	characteristic
71	C₂×C₁₀₉	#487	1	1	yes	characteristic
72	C₂×C₁₀₉	#498	1	1	yes	characteristic
73	C₂×C₁₀₉	#500	1	1	yes	characteristic
74	#1990	—	109	1	no	maximal, Sylow
75	C₁₀₉	#1990	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	—	109	1	no	—
77	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
78	#384	—	109	1	no	—
79	#340	—	109	4	no	—
80	C₂²×Q₈	—	218	1	no	—
81	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
82	C₄⋊Q₈	—	109	4	no	—
83	C₄⋊Q₈	—	218	2	no	—
84	C₂×C₄⋊C₄	—	109	2	no	—
85	C₂×C₄⋊C₄	—	109	2	no	—
86	C₂×C₄⋊C₄	—	218	1	no	—
87	C₂×C₄⋊C₄	—	218	1	no	—
88	C₂×C₄⋊C₄	—	218	1	no	—
89	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
90	C₄×C₈	—	2	2	no	—
91	C₂×Q₈	—	436	2	no	—
92	C₂×Q₈	—	436	2	no	—
93	C₂×Q₈	—	218	4	no	—
94	C₂²×C₄	—	218	2	no	—
95	C₂²×C₄	—	218	2	no	—
96	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
97	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₂₁₈	1	4	yes	complemented
102	C₄⋊C₄	—	109	4	no	—
103	C₄⋊C₄	—	436	2	no	—
104	C₄⋊C₄	—	436	2	no	—
105	C₄⋊C₄	—	218	4	no	—
106	C₄⋊C₄	—	218	4	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₄²	C₄×D₁₀₉	1	2	yes	—
109	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
110	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
111	Q₈	—	436	8	no	—
112	C₂×C₄	—	436	2	no	—
113	C₂×C₄	—	436	2	no	—
114	C₂×C₄	—	436	2	no	—
115	C₂×C₄	—	436	2	no	—
116	C₂×C₄	—	436	2	no	—
117	C₂×C₄	—	218	4	no	—
118	C₂×C₄	#13182006	1	2	yes	—
119	C₂×C₄	#13182006	1	2	yes	—
120	C₂×C₄	D₈₇₂	1	2	yes	—
121	C₂×C₄	#13182018	1	2	yes	—
122	C₂×C₄	#13182018	1	2	yes	—
123	C₂×C₄	Dic₄₃₆	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
127	C₂×C₄	#13182029	1	2	yes	—
128	C₈	—	2	4	no	—
129	C₂²	C₂×#13182006	1	1	yes	characteristic
130	C₂²	#16812687	1	1	yes	characteristic
131	C₂²	#16812699	1	1	yes	characteristic
132	C₂²	C₂×D₈₇₂	1	1	yes	characteristic
133	C₂²	C₂×#13182018	1	1	yes	characteristic
134	C₂²	C₂×#13182018	1	1	yes	characteristic
135	C₂²	C₂×Dic₄₃₆	1	1	yes	characteristic
136	C₄	—	872	2	no	—
137	C₄	—	436	4	no	—
138	C₄	#16812621	1	4	yes	—
139	C₄	#16812624	1	4	yes	—
140	C₄	C₄×Dic₂₁₈	1	2	yes	—
141	C₄	#16812678	1	2	yes	—
142	C₂	#20294932	1	1	yes	characteristic
143	C₂	C₂×#16812621	1	1	yes	characteristic
144	C₂	C₂×#16812624	1	1	yes	characteristic
145	C₂	#20295373	1	1	yes	characteristic
146	C₂	#20295377	1	1	yes	characteristic
147	C₂	#20295424	1	1	yes	characteristic
148	C₂	#20295431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20294896

C₂

yes

maximal, characteristic

#20294789

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

#16812699

—

#16812699

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

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

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

D₈

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

#124

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₄×C₁₀₉

#102

yes

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

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

#1990

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

#384

—

109

—

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₂₁₈

yes

complemented

102

C₄⋊C₄

—

109

—

103

C₄⋊C₄

—

436

—

104

C₄⋊C₄

—

436

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄²

C₄×D₁₀₉

yes

—

109

C₄²

C₂²⋊D₁₀₉

yes

—

110

C₂³

C₂×D₄₃₆

yes

characteristic

111

Q₈

—

436

—

112

C₂×C₄

—

436

—

113

C₂×C₄

—

436

—

114

C₂×C₄

—

436

—

115

C₂×C₄

—

436

—

116

C₂×C₄

—

436

—

117

C₂×C₄

—

218

—

118

C₂×C₄

#13182006

yes

—

119

C₂×C₄

#13182006

yes

—

120

C₂×C₄

D₈₇₂

yes

—

121

C₂×C₄

#13182018

yes

—

122

C₂×C₄

#13182018

yes

—

123

C₂×C₄

Dic₄₃₆

yes

—

124

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

126

C₂×C₄

C₂×Dic₂₁₈

yes

—

127

C₂×C₄

#13182029

yes

—

128

C₈

—

129

C₂²

C₂×#13182006

yes

characteristic

130

C₂²

#16812687

yes

characteristic

131

C₂²

#16812699

yes

characteristic

132

C₂²

C₂×D₈₇₂

yes

characteristic

133

C₂²

C₂×#13182018

yes

characteristic

134

C₂²

C₂×#13182018

yes

characteristic

135

C₂²

C₂×Dic₄₃₆

yes

characteristic

136

C₄

—

872

—

137

C₄

—

436

—

138

C₄

#16812621

yes

—

139

C₄

#16812624

yes

—

140

C₄

C₄×Dic₂₁₈

yes

—

141

C₄

#16812678

yes

—

142

C₂

#20294932

yes

characteristic

143

C₂

C₂×#16812621

yes

characteristic

144

C₂

C₂×#16812624

yes

characteristic

145

C₂

#20295373

yes

characteristic

146

C₂

#20295377

yes

characteristic

147

C₂

#20295424

yes

characteristic

148

C₂

#20295431

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#1990
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄×C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₉

hypocenter

C₁₀₉

#1990

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #16812699	C₄	semidirect
32. C₂×C₈×C₁₀₉	Q₈	semidirect
75. C₁₀₉	#1990	semidirect
101. C₂×C₈	Dic₂₁₈	semidirect

H ≅ G/N

type

7. #16812699

C₄

semidirect

32. C₂×C₈×C₁₀₉

Q₈

semidirect

75. C₁₀₉

#1990

semidirect

101. 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
#20294896	2
#20294789	2
C₂×C₄×C₈×C₁₀₉	2
#1990	109

subgroup

index

C₂×C₄×C₈×C₁₀₉

109

Subgroups — #28363993

Conjugacy classes of subgroups

Complements

Maximal subgroups