Group #28359900 — subgroups

Subgroups — C₁₀₉×#2646 #28359900

Order 13952.

436 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 200 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	#400×C₁₀₉	C₂	1	4	yes	maximal, complemented
2	#398×C₁₀₉	C₂	1	4	yes	maximal, complemented
3	#385×C₁₀₉	C₂	1	4	yes	maximal
4	#380×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
7	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	4	yes	—
13	#95×C₁₀₉	C₂²	1	4	yes	—
14	#95×C₁₀₉	C₂²	1	4	yes	—
15	#95×C₁₀₉	C₂²	1	4	yes	—
16	#95×C₁₀₉	C₂²	1	4	yes	—
17	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	—	2	4	no	—
19	C₂²×C₄×C₁₀₉	—	4	2	no	—
20	C₂²×C₄×C₁₀₉	—	4	2	no	—
21	C₂²×C₄×C₁₀₉	—	4	2	no	—
22	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
23	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
24	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₄⋊C₄×C₁₀₉	—	4	2	no	—
28	C₄⋊C₄×C₁₀₉	—	4	2	no	—
29	C₄⋊C₄×C₁₀₉	—	4	4	no	—
30	C₄⋊C₄×C₁₀₉	—	4	4	no	—
31	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
32	C₂³×C₁₀₉	—	2	1	no	—
33	C₂³×C₁₀₉	—	2	2	no	—
34	C₂³×C₁₀₉	—	2	1	no	—
35	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
36	C₂³×C₁₀₉	#41	1	4	yes	—
37	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	2	4	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	—	2	4	no	—
49	C₂×C₄×C₁₀₉	—	4	4	no	—
50	C₂×C₄×C₁₀₉	—	4	8	no	—
51	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
52	C₂²×C₁₀₉	—	2	4	no	—
53	C₂²×C₁₀₉	—	2	4	no	—
54	C₂²×C₁₀₉	—	2	2	no	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	#125	1	4	yes	—
57	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
59	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
61	C₂²×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×C₁₀₉	—	2	4	no	—
63	C₄×C₁₀₉	—	4	4	no	—
64	C₄×C₁₀₉	—	4	4	no	—
65	C₄×C₁₀₉	—	4	4	no	—
66	C₂×C₁₀₉	—	2	4	no	—
67	C₂×C₁₀₉	C₂×#125	1	1	yes	characteristic
68	C₂×C₁₀₉	#538	1	2	yes	—
69	C₂×C₁₀₉	#539	1	2	yes	—
70	C₂×C₁₀₉	#543	1	1	yes	characteristic
71	C₂×C₁₀₉	#544	1	1	yes	characteristic
72	#2646	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₀₉	#2646	1	1	yes	characteristic, direct product factor, Sylow
74	#400	C₂×C₁₀₉	1	4	yes	complemented
75	#398	C₂×C₁₀₉	1	4	yes	complemented
76	#385	C₂×C₁₀₉	1	4	yes	—
77	#380	C₂×C₁₀₉	1	2	yes	—
78	C₂×#95	C₂×C₁₀₉	1	1	yes	characteristic
79	C₂³×C₄	C₂²×C₁₀₉	1	2	yes	—
80	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
84	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	4	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	4	yes	—
86	#95	C₂²×C₁₀₉	1	4	yes	—
87	#95	C₂²×C₁₀₉	1	4	yes	—
88	#95	C₂²×C₁₀₉	1	4	yes	—
89	#95	C₂²×C₁₀₉	1	4	yes	—
90	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
91	C₂²×C₄	—	2	4	no	—
92	C₂²×C₄	—	4	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	Q₈×C₁₀₉	1	4	yes	—
96	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
97	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
98	C₂²×C₄	C₂³×C₁₀₉	1	4	yes	—
99	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₂²⋊C₄	—	2	8	no	—
105	C₂³	—	2	1	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	2	1	no	—
108	C₂³	C₂×Q₈×C₁₀₉	1	2	yes	—
109	C₂³	#41×C₁₀₉	1	4	yes	—
110	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	4	4	no	—
123	C₂×C₄	—	4	8	no	—
124	C₂×C₄	C₂×Q₈×C₁₀₉	1	4	yes	—
125	C₂²	—	2	4	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#125×C₁₀₉	1	4	yes	—
130	C₂²	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
131	C₂²	C₂×#41×C₁₀₉	1	2	yes	—
132	C₂²	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
133	C₂²	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
134	C₂²	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	4	no	—
137	C₄	—	4	4	no	—
138	C₄	—	4	4	no	—
139	C₂	—	2	4	no	—
140	C₂	C₂×#125×C₁₀₉	1	1	yes	characteristic
141	C₂	#538×C₁₀₉	1	2	yes	—
142	C₂	#539×C₁₀₉	1	2	yes	—
143	C₂	#543×C₁₀₉	1	1	yes	characteristic
144	C₂	#544×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#400×C₁₀₉

C₂

yes

maximal, complemented

#398×C₁₀₉

C₂

yes

maximal, complemented

#385×C₁₀₉

C₂

yes

maximal

#380×C₁₀₉

C₂

yes

maximal

C₂×#95×C₁₀₉

C₂

yes

maximal, characteristic

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#125

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂²×C₁₀₉

2₋¹⁺⁴

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#125

yes

characteristic

C₂×C₁₀₉

#538

yes

—

C₂×C₁₀₉

#539

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2646

yes

characteristic, direct product factor, Sylow

#400

C₂×C₁₀₉

yes

complemented

#398

C₂×C₁₀₉

yes

complemented

#385

C₂×C₁₀₉

yes

—

#380

C₂×C₁₀₉

yes

—

C₂×#95

C₂×C₁₀₉

yes

characteristic

C₂³×C₄

C₂²×C₁₀₉

yes

—

C₂³×C₄

C₂²×C₁₀₉

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₀₉

yes

—

#95

C₂²×C₁₀₉

yes

—

#95

C₂²×C₁₀₉

yes

—

#95

C₂²×C₁₀₉

yes

—

#95

C₂²×C₁₀₉

yes

—

C₂⁴

C₂³×C₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Q₈×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×Q₈×C₁₀₉

yes

—

109

C₂³

#41×C₁₀₉

yes

—

110

C₂³

C₂⁴×C₁₀₉

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#125×C₁₀₉

yes

—

130

C₂²

C₂²×Q₈×C₁₀₉

yes

characteristic

131

C₂²

C₂×#41×C₁₀₉

yes

—

132

C₂²

H(5,𝔽₂)×C₁₀₉

yes

characteristic

133

C₂²

H(5,𝔽₂)×C₁₀₉

yes

—

134

C₂²

2₋¹⁺⁴×C₁₀₉

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₂

—

140

C₂

C₂×#125×C₁₀₉

yes

characteristic

141

C₂

#538×C₁₀₉

yes

—

142

C₂

#539×C₁₀₉

yes

—

143

C₂

#543×C₁₀₉

yes

characteristic

144

C₂

#544×C₁₀₉

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₉

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₀₉

prime radical

C₂⁴×C₁₀₉

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. #400×C₁₀₉	C₂	semidirect
2. #398×C₁₀₉	C₂	semidirect
72. #2646	C₁₀₉	direct (N × H)
73. C₁₀₉	#2646	direct (N × H)
74. #400	C₂×C₁₀₉	semidirect
75. #398	C₂×C₁₀₉	semidirect

H ≅ G/N

type

1. #400×C₁₀₉

C₂

semidirect

2. #398×C₁₀₉

C₂

semidirect

72. #2646

C₁₀₉

direct (N × H)

73. C₁₀₉

#2646

direct (N × H)

74. #400

C₂×C₁₀₉

semidirect

75. #398

C₂×C₁₀₉

semidirect

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

subgroup	index
#400×C₁₀₉	2
#398×C₁₀₉	2
#385×C₁₀₉	2
#380×C₁₀₉	2
C₂×#95×C₁₀₉	2
#2646	109

subgroup

index

#400×C₁₀₉

#398×C₁₀₉

#385×C₁₀₉

#380×C₁₀₉

C₂×#95×C₁₀₉

#2646

109

Subgroups — C₁₀₉×#2646 #28359900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2646 #28359900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2646 #28359900