Group #42150795 — subgroups

Subgroups — C₁₀₉×#32801 #42150795

Order 27904.

762 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 238 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2863	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2043	C₂	1	6	yes	maximal
4	C₁₀₉×#2016	C₂	1	6	yes	maximal
5	C₁₀₉×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
8	C₂×#103×C₁₀₉	C₂²	1	12	yes	—
9	#391×C₁₀₉	C₂²	1	3	yes	—
10	#389×C₁₀₉	C₂²	1	3	yes	—
11	#387×C₁₀₉	—	2	3	no	—
12	#384×C₁₀₉	—	2	3	no	—
13	#384×C₁₀₉	—	2	1	no	—
14	#383×C₁₀₉	—	2	1	no	—
15	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
16	#336×C₁₀₉	C₂²	1	6	yes	—
17	C₂²×Q₈×C₁₀₉	C₂³	1	4	yes	—
18	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×Q₁₆×C₁₀₉	—	4	12	no	—
20	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
21	C₄⋊Q₈×C₁₀₉	—	4	12	no	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	6	no	—
23	C₂×C₄⋊C₄×C₁₀₉	—	2	6	no	—
24	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
25	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
26	C₂×C₄²×C₁₀₉	D₄	1	3	yes	—
27	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₀₉	C₂³	1	3	yes	—
29	#103×C₁₀₉	—	4	12	no	—
30	#95×C₁₀₉	—	2	3	no	—
31	#95×C₁₀₉	—	2	3	no	—
32	#95×C₁₀₉	—	2	3	no	—
33	C₂×Q₈×C₁₀₉	—	2	12	no	—
34	C₂×Q₈×C₁₀₉	—	2	12	no	—
35	C₂×Q₈×C₁₀₉	—	2	6	no	—
36	C₂²×C₄×C₁₀₉	—	2	1	no	—
37	C₂²×C₄×C₁₀₉	—	2	3	no	—
38	C₂²×C₄×C₁₀₉	—	2	4	no	—
39	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	Q₁₆×C₁₀₉	—	4	16	no	—
43	C₂×C₈×C₁₀₉	—	2	12	no	—
44	C₄⋊C₄×C₁₀₉	—	2	6	no	—
45	C₄⋊C₄×C₁₀₉	—	4	12	no	—
46	C₄⋊C₄×C₁₀₉	—	4	12	no	—
47	C₄⋊C₄×C₁₀₉	—	4	6	no	—
48	C₄²×C₁₀₉	—	2	6	no	—
49	C₄²×C₁₀₉	—	2	6	no	—
50	C₄²×C₁₀₉	Q₁₆	1	4	yes	complemented
51	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
52	Q₈×C₁₀₉	—	4	4	no	—
53	Q₈×C₁₀₉	—	4	4	no	—
54	Q₈×C₁₀₉	—	4	12	no	—
55	C₂×C₄×C₁₀₉	—	2	3	no	—
56	C₂×C₄×C₁₀₉	—	2	3	no	—
57	C₂×C₄×C₁₀₉	—	2	3	no	—
58	C₂×C₄×C₁₀₉	—	2	3	no	—
59	C₂×C₄×C₁₀₉	—	4	3	no	—
60	C₂×C₄×C₁₀₉	—	4	12	no	—
61	C₂×C₄×C₁₀₉	—	4	3	no	—
62	C₂×C₄×C₁₀₉	—	4	6	no	—
63	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₁₀₉	#137	1	6	yes	—
65	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	6	yes	—
66	C₈×C₁₀₉	—	4	4	no	—
67	C₂²×C₁₀₉	#535	1	3	yes	—
68	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂²×C₁₀₉	C₂×#137	1	3	yes	—
70	C₄×C₁₀₉	—	2	6	no	—
71	C₄×C₁₀₉	—	2	6	no	—
72	C₄×C₁₀₉	—	8	4	no	—
73	C₄×C₁₀₉	—	8	1	no	—
74	C₄×C₁₀₉	—	8	3	no	—
75	C₄×C₁₀₉	#560	1	4	yes	—
76	#32801	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₁₀₉	#2818	1	1	yes	characteristic
78	C₂×C₁₀₉	#3209	1	3	yes	—
79	C₂×C₁₀₉	#3218	1	3	yes	—
80	#2868	C₂×C₁₀₉	1	1	yes	characteristic
81	#2863	C₂×C₁₀₉	1	1	yes	characteristic
82	#2043	C₂×C₁₀₉	1	6	yes	—
83	#2016	C₂×C₁₀₉	1	6	yes	—
84	#1839	C₂×C₁₀₉	1	1	yes	characteristic
85	C₁₀₉	#32801	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×Q₁₆	C₂²×C₁₀₉	1	4	yes	—
87	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
88	C₂×#103	C₂²×C₁₀₉	1	12	yes	—
89	#391	C₂²×C₁₀₉	1	3	yes	—
90	#389	C₂²×C₁₀₉	1	3	yes	—
91	#387	—	2	3	no	—
92	#384	—	2	3	no	—
93	#384	—	2	1	no	—
94	#383	—	2	1	no	—
95	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
96	#336	C₂²×C₁₀₉	1	6	yes	—
97	C₂²×Q₈	C₂³×C₁₀₉	1	4	yes	—
98	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
99	C₂×Q₁₆	—	4	12	no	—
100	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
101	C₄⋊Q₈	—	4	12	no	—
102	C₂×C₄⋊C₄	—	2	6	no	—
103	C₂×C₄⋊C₄	—	2	6	no	—
104	C₂×C₄⋊C₄	—	2	3	no	—
105	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	3	yes	—
106	C₂×C₄²	D₄×C₁₀₉	1	3	yes	—
107	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
108	C₂×C₄²	C₂³×C₁₀₉	1	3	yes	—
109	#103	—	4	12	no	—
110	#95	—	2	3	no	—
111	#95	—	2	3	no	—
112	#95	—	2	3	no	—
113	C₂×Q₈	—	2	12	no	—
114	C₂×Q₈	—	2	12	no	—
115	C₂×Q₈	—	2	6	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	3	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
120	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
121	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
122	Q₁₆	—	4	16	no	—
123	C₂×C₈	—	2	12	no	—
124	C₄⋊C₄	—	2	6	no	—
125	C₄⋊C₄	—	4	12	no	—
126	C₄⋊C₄	—	4	12	no	—
127	C₄⋊C₄	—	4	6	no	—
128	C₄²	—	2	6	no	—
129	C₄²	—	2	6	no	—
130	C₄²	Q₁₆×C₁₀₉	1	4	yes	complemented
131	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	Q₈	—	4	4	no	—
134	Q₈	—	4	12	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	4	3	no	—
140	C₂×C₄	—	4	12	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	6	yes	—
144	C₂×C₄	#137×C₁₀₉	1	6	yes	—
145	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	6	yes	—
146	C₈	—	4	4	no	—
147	C₂²	#535×C₁₀₉	1	3	yes	—
148	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
149	C₂²	C₂×#137×C₁₀₉	1	3	yes	—
150	C₄	—	2	6	no	—
151	C₄	—	2	6	no	—
152	C₄	—	8	3	no	—
153	C₄	—	8	4	no	—
154	C₄	—	8	1	no	—
155	C₄	#560×C₁₀₉	1	4	yes	—
156	C₂	C₁₀₉×#2818	1	1	yes	characteristic
157	C₂	C₁₀₉×#3209	1	3	yes	—
158	C₂	C₁₀₉×#3218	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2868

C₂

yes

maximal, characteristic

C₁₀₉×#2863

C₂

yes

maximal, characteristic

C₁₀₉×#2043

C₂

yes

maximal

C₁₀₉×#2016

C₂

yes

maximal

C₁₀₉×#1839

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×#103×C₁₀₉

C₂²

yes

—

#391×C₁₀₉

C₂²

yes

—

#389×C₁₀₉

C₂²

yes

—

#387×C₁₀₉

—

#384×C₁₀₉

—

#384×C₁₀₉

—

#383×C₁₀₉

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#336×C₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂³

yes

—

C₂²×Q₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂³

yes

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂×C₄²×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

D₄

yes

characteristic

C₂×C₄²×C₁₀₉

C₂³

yes

—

#103×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂⁴

yes

characteristic

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

Q₁₆

yes

complemented

C₂³×C₁₀₉

C₂²×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×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×Q₁₆

yes

—

C₂×C₄×C₁₀₉

#137

yes

—

C₂×C₄×C₁₀₉

H(5,𝔽₂)

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#535

yes

—

C₂²×C₁₀₉

C₂²×Q₁₆

yes

characteristic

C₂²×C₁₀₉

C₂×#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#560

yes

—

#32801

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

#2818

yes

characteristic

C₂×C₁₀₉

#3209

yes

—

C₂×C₁₀₉

#3218

yes

—

#2868

C₂×C₁₀₉

yes

characteristic

#2863

C₂×C₁₀₉

yes

characteristic

#2043

C₂×C₁₀₉

yes

—

#2016

C₂×C₁₀₉

yes

—

#1839

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#32801

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂²×C₁₀₉

yes

—

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

C₂×#103

C₂²×C₁₀₉

yes

—

#391

C₂²×C₁₀₉

yes

—

#389

C₂²×C₁₀₉

yes

—

—

—

—

—

C₄³

C₂²×C₁₀₉

yes

characteristic

#336

C₂²×C₁₀₉

yes

—

C₂²×Q₈

C₂³×C₁₀₉

yes

—

C₂²×Q₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

100

C₂²×C₈

C₂³×C₁₀₉

yes

—

101

C₄⋊Q₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

C₂³×C₁₀₉

yes

—

106

C₂×C₄²

D₄×C₁₀₉

yes

—

107

C₂×C₄²

D₄×C₁₀₉

yes

characteristic

108

C₂×C₄²

C₂³×C₁₀₉

yes

—

109

#103

—

110

#95

—

111

#95

—

112

#95

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

120

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

121

C₂²×C₄

C₂⁴×C₁₀₉

yes

characteristic

122

Q₁₆

—

123

C₂×C₈

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

Q₁₆×C₁₀₉

yes

complemented

131

C₂³

C₂²×D₄×C₁₀₉

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

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

C₂×Q₁₆×C₁₀₉

yes

—

144

C₂×C₄

#137×C₁₀₉

yes

—

145

C₂×C₄

H(5,𝔽₂)×C₁₀₉

yes

—

146

C₈

—

147

C₂²

#535×C₁₀₉

yes

—

148

C₂²

C₂²×Q₁₆×C₁₀₉

yes

characteristic

149

C₂²

C₂×#137×C₁₀₉

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#560×C₁₀₉

yes

—

156

C₂

C₁₀₉×#2818

yes

characteristic

157

C₂

C₁₀₉×#3209

yes

—

158

C₂

C₁₀₉×#3218

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₁₀₉

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

N	H ≅ G/N	type
50. C₄²×C₁₀₉	Q₁₆	semidirect
76. #32801	C₁₀₉	direct (N × H)
85. C₁₀₉	#32801	direct (N × H)
130. C₄²	Q₁₆×C₁₀₉	semidirect

H ≅ G/N

type

50. C₄²×C₁₀₉

Q₁₆

semidirect

76. #32801

C₁₀₉

direct (N × H)

85. C₁₀₉

#32801

direct (N × H)

130. C₄²

Q₁₆×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#2868	2
C₁₀₉×#2863	2
C₁₀₉×#2043	2
C₁₀₉×#2016	2
C₁₀₉×#1839	2
#32801	109

subgroup

index

C₁₀₉×#2868

C₁₀₉×#2863

C₁₀₉×#2043

C₁₀₉×#2016

C₁₀₉×#1839

#32801

109

Subgroups — C₁₀₉×#32801 #42150795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#32801 #42150795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32801 #42150795