Group #42125109 — subgroups

Subgroups — C₁₀₉×#7115 #42125109

Order 27904.

248 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₁₀₉×#1963	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1274	C₂	1	2	yes	maximal, complemented
3	#532×C₁₀₉	C₄	1	1	yes	characteristic
4	#431×C₁₀₉	—	2	2	no	—
5	#423×C₁₀₉	C₄	1	1	yes	characteristic
6	C₂×#104×C₁₀₉	—	2	2	no	—
7	#376×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#41×C₁₀₉	—	2	1	no	—
9	C₂×M₁₆×C₁₀₉	—	2	1	no	—
10	C₄⋊Q₈×C₁₀₉	C₈	1	1	yes	characteristic, complemented
11	#127×C₁₀₉	C₈	1	1	yes	characteristic, complemented
12	#124×C₁₀₉	—	2	1	no	—
13	#121×C₁₀₉	—	4	1	no	—
14	C₄×D₄×C₁₀₉	—	4	1	no	—
15	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
17	C₄⋊C₈×C₁₀₉	—	4	1	no	—
18	C₄⋊C₈×C₁₀₉	—	4	2	no	—
19	#104×C₁₀₉	—	4	2	no	—
20	#104×C₁₀₉	—	4	2	no	—
21	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
22	C₄×C₈×C₁₀₉	—	2	4	no	—
23	#95×C₁₀₉	—	4	1	no	—
24	#41×C₁₀₉	—	4	2	no	—
25	#41×C₁₀₉	—	4	2	no	—
26	C₂×Q₈×C₁₀₉	—	2	1	no	—
27	C₂×D₄×C₁₀₉	—	2	1	no	—
28	C₂×D₄×C₁₀₉	—	4	1	no	—
29	C₂×D₄×C₁₀₉	—	4	1	no	—
30	C₂²×C₄×C₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	—	2	1	no	—
32	C₂²×C₄×C₁₀₉	—	4	1	no	—
33	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	4	2	no	—
35	C₂×C₈×C₁₀₉	—	4	4	no	—
36	C₂×C₈×C₁₀₉	—	4	1	no	—
37	C₄⋊C₄×C₁₀₉	—	4	1	no	—
38	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
39	C₄²×C₁₀₉	—	2	1	no	—
40	C₄²×C₁₀₉	—	2	2	no	—
41	C₄²×C₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₀₉	M₁₆	1	1	yes	characteristic
43	C₄²×C₁₀₉	D₈	1	2	yes	—
44	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
45	C₂³×C₁₀₉	—	4	1	no	—
46	C₂³×C₁₀₉	#99	1	1	yes	characteristic
47	Q₈×C₁₀₉	—	4	2	no	—
48	D₄×C₁₀₉	—	4	2	no	—
49	D₄×C₁₀₉	—	8	2	no	—
50	D₄×C₁₀₉	—	8	2	no	—
51	C₂×C₄×C₁₀₉	—	2	1	no	—
52	C₂×C₄×C₁₀₉	—	2	1	no	—
53	C₂×C₄×C₁₀₉	—	2	2	no	—
54	C₂×C₄×C₁₀₉	—	2	2	no	—
55	C₂×C₄×C₁₀₉	—	4	1	no	—
56	C₂×C₄×C₁₀₉	—	8	2	no	—
57	C₂×C₄×C₁₀₉	—	8	1	no	—
58	C₂×C₄×C₁₀₉	—	8	1	no	—
59	C₂×C₄×C₁₀₉	—	8	1	no	—
60	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₀₉	#100	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	#102	1	2	yes	—
63	C₂×C₄×C₁₀₉	#104	1	2	yes	—
64	C₈×C₁₀₉	—	8	4	no	—
65	C₈×C₁₀₉	—	8	1	no	—
66	C₂²×C₁₀₉	—	4	1	no	—
67	C₂²×C₁₀₉	—	8	2	no	—
68	C₂²×C₁₀₉	—	8	1	no	—
69	C₂²×C₁₀₉	#323	1	1	yes	characteristic
70	C₂²×C₁₀₉	#327	1	2	yes	—
71	C₄×C₁₀₉	—	4	2	no	—
72	C₄×C₁₀₉	—	4	1	no	—
73	C₄×C₁₀₉	—	4	2	no	—
74	C₄×C₁₀₉	—	4	1	no	—
75	C₄×C₁₀₉	—	8	1	no	—
76	C₄×C₁₀₉	D₄⋊C₈	1	2	yes	—
77	C₄×C₁₀₉	#329	1	1	yes	characteristic
78	C₄×C₁₀₉	#331	1	1	yes	characteristic
79	#7115	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₀₉	—	2	1	no	—
81	C₂×C₁₀₉	—	2	2	no	—
82	C₂×C₁₀₉	—	16	1	no	—
83	C₂×C₁₀₉	#1270	1	1	yes	characteristic
84	#1963	C₂×C₁₀₉	1	1	yes	characteristic
85	#1274	C₂×C₁₀₉	1	2	yes	complemented
86	C₁₀₉	#7115	1	1	yes	characteristic, direct product factor, Sylow
87	#532	C₄×C₁₀₉	1	1	yes	characteristic
88	#431	—	2	2	no	—
89	#423	C₄×C₁₀₉	1	1	yes	characteristic
90	C₂×#104	—	2	2	no	—
91	#376	C₂²×C₁₀₉	1	1	yes	characteristic
92	C₂×#41	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₄⋊Q₈	C₈×C₁₀₉	1	1	yes	characteristic, complemented
95	#127	C₈×C₁₀₉	1	1	yes	characteristic, complemented
96	#124	—	2	1	no	—
97	#121	—	4	1	no	—
98	C₄×D₄	—	4	1	no	—
99	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	4	2	no	—
103	#104	—	4	2	no	—
104	#104	—	4	2	no	—
105	C₂²⋊C₈	—	4	2	no	—
106	C₄×C₈	—	2	4	no	—
107	#95	—	4	1	no	—
108	#41	—	4	2	no	—
109	#41	—	4	2	no	—
110	C₂×Q₈	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	4	1	no	—
113	C₂×D₄	—	4	1	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	4	4	no	—
120	C₂×C₈	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×C₈×C₁₀₉	1	1	yes	characteristic
126	C₄²	M₁₆×C₁₀₉	1	1	yes	characteristic
127	C₄²	D₈×C₁₀₉	1	2	yes	—
128	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
129	C₂³	—	4	1	no	—
130	C₂³	#99×C₁₀₉	1	1	yes	characteristic
131	Q₈	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	2	no	—
134	D₄	—	8	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	1	yes	characteristic
145	C₂×C₄	#100×C₁₀₉	1	1	yes	characteristic
146	C₂×C₄	#102×C₁₀₉	1	2	yes	—
147	C₂×C₄	#104×C₁₀₉	1	2	yes	—
148	C₈	—	8	4	no	—
149	C₈	—	8	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	8	1	no	—
153	C₂²	#323×C₁₀₉	1	1	yes	characteristic
154	C₂²	#327×C₁₀₉	1	2	yes	—
155	C₄	—	4	2	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	8	1	no	—
160	C₄	D₄⋊C₈×C₁₀₉	1	2	yes	—
161	C₄	#329×C₁₀₉	1	1	yes	characteristic
162	C₄	#331×C₁₀₉	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	16	1	no	—
166	C₂	C₁₀₉×#1270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#1963

C₂

yes

maximal, characteristic

C₁₀₉×#1274

C₂

yes

maximal, complemented

#532×C₁₀₉

C₄

yes

characteristic

#431×C₁₀₉

—

#423×C₁₀₉

C₄

yes

characteristic

C₂×#104×C₁₀₉

—

#376×C₁₀₉

C₂²

yes

characteristic

C₂×#41×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₈

yes

characteristic, complemented

#127×C₁₀₉

C₈

yes

characteristic, complemented

#124×C₁₀₉

—

#121×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₀₉

D₄

yes

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#104×C₁₀₉

—

#104×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₈

yes

characteristic

C₄²×C₁₀₉

M₁₆

yes

characteristic

C₄²×C₁₀₉

D₈

yes

—

C₄²×C₁₀₉

SD₁₆

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#99

yes

characteristic

Q₈×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×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

—

C₂×C₄×C₁₀₉

#104

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#323

yes

characteristic

C₂²×C₁₀₉

#327

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

D₄⋊C₈

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#1270

yes

characteristic

#1963

C₂×C₁₀₉

yes

characteristic

#1274

C₂×C₁₀₉

yes

complemented

C₁₀₉

#7115

yes

characteristic, direct product factor, Sylow

#532

C₄×C₁₀₉

yes

characteristic

#431

—

#423

C₄×C₁₀₉

yes

characteristic

C₂×#104

—

#376

C₂²×C₁₀₉

yes

characteristic

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

C₈×C₁₀₉

yes

characteristic, complemented

#127

C₈×C₁₀₉

yes

characteristic, complemented

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×C₄×C₁₀₉

yes

characteristic

100

C₂×C₄²

D₄×C₁₀₉

yes

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#104

—

104

#104

—

105

C₂²⋊C₈

—

106

C₄×C₈

—

107

#95

—

108

#41

—

109

#41

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×C₈×C₁₀₉

yes

characteristic

126

C₄²

M₁₆×C₁₀₉

yes

characteristic

127

C₄²

D₈×C₁₀₉

yes

—

128

C₄²

SD₁₆×C₁₀₉

yes

—

129

C₂³

—

130

C₂³

#99×C₁₀₉

yes

characteristic

131

Q₈

—

132

D₄

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊C₈×C₁₀₉

yes

characteristic

145

C₂×C₄

#100×C₁₀₉

yes

characteristic

146

C₂×C₄

#102×C₁₀₉

yes

—

147

C₂×C₄

#104×C₁₀₉

yes

—

148

C₈

—

149

C₈

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#323×C₁₀₉

yes

characteristic

154

C₂²

#327×C₁₀₉

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

D₄⋊C₈×C₁₀₉

yes

—

161

C₄

#329×C₁₀₉

yes

characteristic

162

C₄

#331×C₁₀₉

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

C₁₀₉×#1270

yes

characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#329
commutator	C₄²	C₂×C₈×C₁₀₉
Frattini, cosocle	#376	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#1270
prime radical	#532×C₁₀₉	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #376×C₁₀₉ ◁ G

subgroup

G/H

center

C₄×C₁₀₉

#329

commutator

C₄²

C₂×C₈×C₁₀₉

C₂²×C₁₀₉

C₂×C₁₀₉

#532×C₁₀₉

C₄

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #376×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₁₀₉×#1274	C₂	semidirect
10. C₄⋊Q₈×C₁₀₉	C₈	semidirect
11. #127×C₁₀₉	C₈	semidirect
79. #7115	C₁₀₉	direct (N × H)
85. #1274	C₂×C₁₀₉	semidirect
86. C₁₀₉	#7115	direct (N × H)
94. C₄⋊Q₈	C₈×C₁₀₉	semidirect
95. #127	C₈×C₁₀₉	semidirect

H ≅ G/N

type

2. C₁₀₉×#1274

C₂

semidirect

10. C₄⋊Q₈×C₁₀₉

C₈

semidirect

11. #127×C₁₀₉

C₈

semidirect

79. #7115

C₁₀₉

direct (N × H)

85. #1274

C₂×C₁₀₉

semidirect

86. C₁₀₉

#7115

direct (N × H)

94. C₄⋊Q₈

C₈×C₁₀₉

semidirect

95. #127

C₈×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#1963	2
C₁₀₉×#1274	2
#7115	109

subgroup

index

C₁₀₉×#1963

C₁₀₉×#1274

#7115

109

Subgroups — C₁₀₉×#7115 #42125109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#7115 #42125109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7115 #42125109