Group #28359307 — subgroups

Subgroups — C₁₀₉×#2053 #28359307

Order 13952.

264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 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	#575×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#416×C₁₀₉	C₂	1	2	yes	maximal, complemented
4	#410×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₁₀₉	C₂²	1	2	yes	—
7	#137×C₁₀₉	—	2	2	no	—
8	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
9	#135×C₁₀₉	—	2	2	no	—
10	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	#117×C₁₀₉	C₂²	1	2	yes	—
13	#117×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	#103×C₁₀₉	—	2	2	no	—
16	#102×C₁₀₉	—	2	2	no	—
17	#99×C₁₀₉	—	2	4	no	—
18	#41×C₁₀₉	—	2	2	no	—
19	#41×C₁₀₉	—	2	2	no	—
20	#41×C₁₀₉	—	2	2	no	—
21	#41×C₁₀₉	—	2	2	no	—
22	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
23	C₂×D₄×C₁₀₉	—	2	2	no	—
24	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	Q₁₆×C₁₀₉	—	2	2	no	—
28	SD₁₆×C₁₀₉	—	2	4	no	—
29	D₈×C₁₀₉	—	2	2	no	—
30	M₁₆×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
38	C₄²×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	—	2	2	no	—
40	C₂³×C₁₀₉	—	2	2	no	—
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	2	2	no	—
43	Q₈×C₁₀₉	—	2	2	no	—
44	D₄×C₁₀₉	—	2	2	no	—
45	D₄×C₁₀₉	—	2	2	no	—
46	D₄×C₁₀₉	—	4	2	no	—
47	D₄×C₁₀₉	—	4	2	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	—	4	2	no	—
51	C₂×C₄×C₁₀₉	—	4	2	no	—
52	C₂×C₄×C₁₀₉	—	4	2	no	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	#41	1	2	yes	—
57	C₈×C₁₀₉	—	2	2	no	—
58	C₈×C₁₀₉	—	4	1	no	—
59	C₂²×C₁₀₉	—	4	1	no	—
60	C₂²×C₁₀₉	—	4	2	no	—
61	C₂²×C₁₀₉	—	8	2	no	—
62	C₂²×C₁₀₉	#121	1	1	yes	characteristic
63	C₂²×C₁₀₉	#121	1	2	yes	—
64	C₄×C₁₀₉	—	4	2	no	—
65	C₄×C₁₀₉	—	4	2	no	—
66	C₄×C₁₀₉	—	8	2	no	—
67	C₄×C₁₀₉	#120	1	1	yes	characteristic
68	C₄×C₁₀₉	#123	1	2	yes	—
69	C₄×C₁₀₉	#124	1	1	yes	characteristic
70	C₂×C₁₀₉	—	2	1	no	—
71	C₂×C₁₀₉	—	2	2	no	—
72	C₂×C₁₀₉	—	8	2	no	—
73	C₂×C₁₀₉	#394	1	1	yes	characteristic
74	#2053	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₀₉	#2053	1	1	yes	characteristic, direct product factor, Sylow
76	#575	C₂×C₁₀₉	1	1	yes	characteristic
77	#424	C₂×C₁₀₉	1	1	yes	characteristic, complemented
78	#416	C₂×C₁₀₉	1	2	yes	complemented
79	#410	C₂×C₁₀₉	1	2	yes	complemented
80	H(3,ℤ/4)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
81	C₂×#41	C₂²×C₁₀₉	1	2	yes	—
82	#137	—	2	2	no	—
83	AGL(1,ℤ/8)	—	2	2	no	—
84	#135	—	2	2	no	—
85	C₂×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
87	#117	C₂²×C₁₀₉	1	2	yes	—
88	#117	C₂²×C₁₀₉	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	2	2	no	—
91	#102	—	2	2	no	—
92	#99	—	2	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	D₄×C₁₀₉	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	D₄×C₁₀₉	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	SD₁₆	—	2	4	no	—
104	D₈	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	D₄×C₁₀₉	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
129	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄	#41×C₁₀₉	1	1	yes	characteristic
131	C₂×C₄	#41×C₁₀₉	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#121×C₁₀₉	1	1	yes	characteristic
138	C₂²	#121×C₁₀₉	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	4	2	no	—
141	C₄	—	8	2	no	—
142	C₄	#120×C₁₀₉	1	1	yes	characteristic
143	C₄	#123×C₁₀₉	1	2	yes	—
144	C₄	#124×C₁₀₉	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	8	2	no	—
148	C₂	#394×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#575×C₁₀₉

C₂

yes

maximal, characteristic

#424×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#416×C₁₀₉

C₂

yes

maximal, complemented

#410×C₁₀₉

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₀₉

C₂²

yes

—

#137×C₁₀₉

—

AGL(1,ℤ/8)×C₁₀₉

—

#135×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

#103×C₁₀₉

—

#102×C₁₀₉

—

#99×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#121

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

#123

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#394

yes

characteristic

#2053

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2053

yes

characteristic, direct product factor, Sylow

#575

C₂×C₁₀₉

yes

characteristic

#424

C₂×C₁₀₉

yes

characteristic, complemented

#416

C₂×C₁₀₉

yes

complemented

#410

C₂×C₁₀₉

yes

complemented

H(3,ℤ/4)

C₂×C₁₀₉

yes

characteristic, complemented

C₂×#41

C₂²×C₁₀₉

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂²×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

#117

C₂²×C₁₀₉

yes

—

#117

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₈

—

—

—

—

—

—

—

—

C₂×Q₈

D₄×C₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₁₀₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

D₄×C₁₀₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

D₄×C₁₀₉

yes

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

129

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

130

C₂×C₄

#41×C₁₀₉

yes

characteristic

131

C₂×C₄

#41×C₁₀₉

yes

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#121×C₁₀₉

yes

characteristic

138

C₂²

#121×C₁₀₉

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#120×C₁₀₉

yes

characteristic

143

C₄

#123×C₁₀₉

yes

—

144

C₄

#124×C₁₀₉

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#394×C₁₀₉

yes

characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#394
prime radical	#575×C₁₀₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

subgroup

G/H

center

C₄×C₁₀₉

#124

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

socle

C₂×C₁₀₉

#394

prime radical

#575×C₁₀₉

C₂

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. #424×C₁₀₉	C₂	semidirect
3. #416×C₁₀₉	C₂	semidirect
4. #410×C₁₀₉	C₂	semidirect
5. H(3,ℤ/4)×C₁₀₉	C₂	semidirect
74. #2053	C₁₀₉	direct (N × H)
75. C₁₀₉	#2053	direct (N × H)
77. #424	C₂×C₁₀₉	semidirect
78. #416	C₂×C₁₀₉	semidirect
79. #410	C₂×C₁₀₉	semidirect
80. H(3,ℤ/4)	C₂×C₁₀₉	semidirect

H ≅ G/N

type

2. #424×C₁₀₉

C₂

semidirect

3. #416×C₁₀₉

C₂

semidirect

4. #410×C₁₀₉

C₂

semidirect

5. H(3,ℤ/4)×C₁₀₉

C₂

semidirect

74. #2053

C₁₀₉

direct (N × H)

75. C₁₀₉

#2053

direct (N × H)

77. #424

C₂×C₁₀₉

semidirect

78. #416

C₂×C₁₀₉

semidirect

79. #410

C₂×C₁₀₉

semidirect

80. H(3,ℤ/4)

C₂×C₁₀₉

semidirect

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

subgroup	index
#575×C₁₀₉	2
#424×C₁₀₉	2
#416×C₁₀₉	2
#410×C₁₀₉	2
H(3,ℤ/4)×C₁₀₉	2
#2053	109

subgroup

index

#575×C₁₀₉

#424×C₁₀₉

#416×C₁₀₉

#410×C₁₀₉

H(3,ℤ/4)×C₁₀₉

#2053

109

Subgroups — C₁₀₉×#2053 #28359307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2053 #28359307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2053 #28359307