Group #28360613 — subgroups

Subgroups — C₁₀₉×#3359 #28360613

Order 13952.

370 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 192 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	#557×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#487×C₁₀₉	C₂	1	3	yes	maximal, complemented
4	#475×C₁₀₉	C₂	1	3	yes	maximal, complemented
5	#461×C₁₀₉	C₂	1	3	yes	maximal
6	C₈×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₁₀₉	C₂	1	3	yes	maximal
8	C₂²×Q₈×C₁₀₉	—	2	1	no	—
9	C₂×SD₁₆×C₁₀₉	C₂²	1	3	yes	—
10	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
11	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
12	#125×C₁₀₉	—	2	3	no	—
13	C₂²⋊Q₈×C₁₀₉	—	2	3	no	—
14	C₄×Q₈×C₁₀₉	—	2	1	no	—
15	C₄×Q₈×C₁₀₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
18	#106×C₁₀₉	C₂²	1	3	yes	—
19	C₄⋊C₈×C₁₀₉	C₂²	1	3	yes	—
20	#103×C₁₀₉	C₂²	1	3	yes	—
21	#103×C₁₀₉	C₂²	1	6	yes	—
22	#102×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₀₉	—	2	3	no	—
25	C₂×Q₈×C₁₀₉	—	4	4	no	—
26	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₁₀₉	C₂³	1	3	yes	—
28	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	—	2	3	no	—
30	SD₁₆×C₁₀₉	—	2	6	no	—
31	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂³	1	3	yes	—
33	C₄⋊C₄×C₁₀₉	—	2	3	no	—
34	C₄⋊C₄×C₁₀₉	—	2	3	no	—
35	C₄⋊C₄×C₁₀₉	—	2	3	no	—
36	C₄⋊C₄×C₁₀₉	—	2	3	no	—
37	C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
38	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
40	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
41	C₄²×C₁₀₉	—	2	3	no	—
42	C₄²×C₁₀₉	C₂³	1	3	yes	—
43	C₂³×C₁₀₉	—	2	1	no	—
44	Q₈×C₁₀₉	—	2	3	no	—
45	Q₈×C₁₀₉	—	2	3	no	—
46	Q₈×C₁₀₉	—	4	6	no	—
47	Q₈×C₁₀₉	SD₁₆	1	4	yes	complemented
48	D₄×C₁₀₉	—	2	1	no	—
49	D₄×C₁₀₉	#41	1	2	yes	—
50	C₂×C₄×C₁₀₉	—	2	1	no	—
51	C₂×C₄×C₁₀₉	—	2	3	no	—
52	C₂×C₄×C₁₀₉	—	2	3	no	—
53	C₂×C₄×C₁₀₉	—	4	6	no	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₀₉	—	2	3	no	—
58	C₈×C₁₀₉	—	2	1	no	—
59	C₂²×C₁₀₉	—	2	2	no	—
60	C₂²×C₁₀₉	—	4	1	no	—
61	C₂²×C₁₀₉	—	4	1	no	—
62	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₉	—	2	3	no	—
64	C₄×C₁₀₉	—	4	3	no	—
65	C₄×C₁₀₉	—	4	3	no	—
66	C₄×C₁₀₉	—	4	1	no	—
67	C₄×C₁₀₉	C₂×SD₁₆	1	6	yes	—
68	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
69	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂×C₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	#548	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₀₉	#578	1	1	yes	characteristic
74	#3359	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₀₉	#3359	1	1	yes	characteristic, direct product factor, Sylow
76	#557	C₂×C₁₀₉	1	1	yes	characteristic, complemented
77	D₄×Q₈	C₂×C₁₀₉	1	1	yes	characteristic
78	#487	C₂×C₁₀₉	1	3	yes	complemented
79	#475	C₂×C₁₀₉	1	3	yes	complemented
80	#461	C₂×C₁₀₉	1	3	yes	—
81	C₈×Q₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
82	C₄×SD₁₆	C₂×C₁₀₉	1	3	yes	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	C₂²×C₁₀₉	1	3	yes	—
85	C₄⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
86	C₄⋊Q₈	C₂²×C₁₀₉	1	3	yes	—
87	#125	—	2	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	C₂²×C₁₀₉	1	3	yes	—
91	C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
92	C₄×D₄	C₂²×C₁₀₉	1	3	yes	—
93	#106	C₂²×C₁₀₉	1	3	yes	—
94	C₄⋊C₈	C₂²×C₁₀₉	1	3	yes	—
95	#103	C₂²×C₁₀₉	1	3	yes	—
96	#103	C₂²×C₁₀₉	1	6	yes	—
97	#102	C₂²×C₁₀₉	1	1	yes	characteristic
98	C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
102	C₂×Q₈	C₂³×C₁₀₉	1	3	yes	—
103	C₂×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	2	6	no	—
106	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
107	C₂×C₈	C₂³×C₁₀₉	1	3	yes	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	2	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	D₄×C₁₀₉	1	3	yes	—
113	C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
114	C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	2	3	no	—
117	C₄²	C₂³×C₁₀₉	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	2	3	no	—
121	Q₈	—	4	6	no	—
122	Q₈	SD₁₆×C₁₀₉	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#41×C₁₀₉	1	2	yes	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	4	6	no	—
129	C₂×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
130	C₂×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
131	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
132	C₈	—	2	3	no	—
133	C₈	—	2	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	4	3	no	—
141	C₄	—	4	1	no	—
142	C₄	C₂×SD₁₆×C₁₀₉	1	6	yes	—
143	C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
144	C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#548×C₁₀₉	1	1	yes	characteristic
147	C₂	C₂²×SD₁₆×C₁₀₉	1	1	yes	characteristic
148	C₂	#578×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#557×C₁₀₉

C₂

yes

maximal, characteristic, complemented

D₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

#487×C₁₀₉

C₂

yes

maximal, complemented

#475×C₁₀₉

C₂

yes

maximal, complemented

#461×C₁₀₉

C₂

yes

maximal

C₈×Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₁₀₉

C₂

yes

maximal

C₂²×Q₈×C₁₀₉

—

C₂×SD₁₆×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#103×C₁₀₉

C₂²

yes

—

#103×C₁₀₉

C₂²

yes

—

#102×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

SD₁₆

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

#41

yes

—

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

C₂⁴

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#548

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#3359

yes

characteristic, direct product factor, Sylow

#557

C₂×C₁₀₉

yes

characteristic, complemented

D₄×Q₈

C₂×C₁₀₉

yes

characteristic

#487

C₂×C₁₀₉

yes

complemented

#475

C₂×C₁₀₉

yes

complemented

#461

C₂×C₁₀₉

yes

—

C₈×Q₈

C₂×C₁₀₉

yes

characteristic, complemented

C₄×SD₁₆

C₂×C₁₀₉

yes

—

C₂²×Q₈

—

C₂×SD₁₆

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

C₂²×C₁₀₉

yes

—

C₄⋊Q₈

C₂²×C₁₀₉

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₀₉

yes

—

C₄×Q₈

C₂²×C₁₀₉

yes

characteristic

C₄×D₄

C₂²×C₁₀₉

yes

—

#106

C₂²×C₁₀₉

yes

—

C₄⋊C₈

C₂²×C₁₀₉

yes

—

#103

C₂²×C₁₀₉

yes

—

#103

C₂²×C₁₀₉

yes

—

#102

C₂²×C₁₀₉

yes

characteristic

C₄×C₈

C₂²×C₁₀₉

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

D₄×C₁₀₉

yes

characteristic

102

C₂×Q₈

C₂³×C₁₀₉

yes

—

103

C₂×D₄

C₂³×C₁₀₉

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

106

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

107

C₂×C₈

C₂³×C₁₀₉

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

D₄×C₁₀₉

yes

—

113

C₄⋊C₄

C₂³×C₁₀₉

yes

characteristic

114

C₄⋊C₄

C₂³×C₁₀₉

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₂³×C₁₀₉

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

SD₁₆×C₁₀₉

yes

complemented

123

D₄

—

124

D₄

#41×C₁₀₉

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

130

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

131

C₂×C₄

C₂⁴×C₁₀₉

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×D₄×C₁₀₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₂×SD₁₆×C₁₀₉

yes

—

143

C₄

C₂×#41×C₁₀₉

yes

characteristic

144

C₄

2₋¹⁺⁴×C₁₀₉

yes

characteristic

145

C₂

—

146

C₂

#548×C₁₀₉

yes

characteristic

147

C₂

C₂²×SD₁₆×C₁₀₉

yes

characteristic

148

C₂

#578×C₁₀₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₀₉

prime radical

C₂×D₄×C₁₀₉

C₂³

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄×Q₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. #557×C₁₀₉	C₂	semidirect
3. #487×C₁₀₉	C₂	semidirect
4. #475×C₁₀₉	C₂	semidirect
6. C₈×Q₈×C₁₀₉	C₂	semidirect
47. Q₈×C₁₀₉	SD₁₆	semidirect
74. #3359	C₁₀₉	direct (N × H)
75. C₁₀₉	#3359	direct (N × H)
76. #557	C₂×C₁₀₉	semidirect
78. #487	C₂×C₁₀₉	semidirect
79. #475	C₂×C₁₀₉	semidirect
81. C₈×Q₈	C₂×C₁₀₉	semidirect
122. Q₈	SD₁₆×C₁₀₉	semidirect

H ≅ G/N

type

1. #557×C₁₀₉

C₂

semidirect

3. #487×C₁₀₉

C₂

semidirect

4. #475×C₁₀₉

C₂

semidirect

6. C₈×Q₈×C₁₀₉

C₂

semidirect

47. Q₈×C₁₀₉

SD₁₆

semidirect

74. #3359

C₁₀₉

direct (N × H)

75. C₁₀₉

#3359

direct (N × H)

76. #557

C₂×C₁₀₉

semidirect

78. #487

C₂×C₁₀₉

semidirect

79. #475

C₂×C₁₀₉

semidirect

81. C₈×Q₈

C₂×C₁₀₉

semidirect

122. Q₈

SD₁₆×C₁₀₉

semidirect

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

subgroup	index
#557×C₁₀₉	2
D₄×Q₈×C₁₀₉	2
#487×C₁₀₉	2
#475×C₁₀₉	2
#461×C₁₀₉	2
C₈×Q₈×C₁₀₉	2
C₄×SD₁₆×C₁₀₉	2
#3359	109

subgroup

index

#557×C₁₀₉

D₄×Q₈×C₁₀₉

#487×C₁₀₉

#475×C₁₀₉

#461×C₁₀₉

C₈×Q₈×C₁₀₉

C₄×SD₁₆×C₁₀₉

#3359

109

Subgroups — C₁₀₉×#3359 #28360613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#3359 #28360613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3359 #28360613