Group #42150762 — subgroups

Subgroups — C₁₀₉×#32768 #42150762

Order 27904.

736 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 206 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×#2621	C₂	1	2	yes	maximal
2	C₁₀₉×#2074	C₂	1	4	yes	maximal, complemented
3	C₁₀₉×#2016	C₂	1	4	yes	maximal, complemented
4	C₁₀₉×#2001	C₂	1	4	yes	maximal
5	C₁₀₉×#1818	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
8	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	4	yes	—
9	C₂×#117×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#103×C₁₀₉	C₂²	1	8	yes	—
11	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
12	#398×C₁₀₉	—	2	4	no	—
13	#397×C₁₀₉	—	2	4	no	—
14	#391×C₁₀₉	C₂²	1	4	yes	—
15	#382×C₁₀₉	C₂²	1	2	yes	—
16	#340×C₁₀₉	C₂²	1	4	yes	—
17	#336×C₁₀₉	C₂²	1	4	yes	—
18	C₂²×Q₈×C₁₀₉	C₂³	1	4	yes	—
19	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×Q₁₆×C₁₀₉	—	4	4	no	—
21	C₂×Q₁₆×C₁₀₉	—	4	8	no	—
22	C₂²×C₈×C₁₀₉	C₂³	1	4	yes	—
23	C₄⋊Q₈×C₁₀₉	—	4	4	no	—
24	C₂²⋊Q₈×C₁₀₉	—	4	8	no	—
25	#117×C₁₀₉	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
27	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
31	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
33	#103×C₁₀₉	—	4	8	no	—
34	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
35	#95×C₁₀₉	—	2	4	no	—
36	#95×C₁₀₉	—	2	8	no	—
37	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Q₈×C₁₀₉	—	2	4	no	—
39	C₂×Q₈×C₁₀₉	—	2	8	no	—
40	C₂×Q₈×C₁₀₉	—	2	8	no	—
41	C₂×Q₈×C₁₀₉	—	2	4	no	—
42	C₂²×C₄×C₁₀₉	—	2	2	no	—
43	C₂²×C₄×C₁₀₉	—	2	4	no	—
44	C₂²×C₄×C₁₀₉	—	2	4	no	—
45	C₂²×C₄×C₁₀₉	—	2	4	no	—
46	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
49	Q₁₆×C₁₀₉	—	4	16	no	—
50	C₂×C₈×C₁₀₉	—	2	4	no	—
51	C₂×C₈×C₁₀₉	—	2	8	no	—
52	C₄⋊C₄×C₁₀₉	—	2	8	no	—
53	C₄⋊C₄×C₁₀₉	—	4	2	no	—
54	C₄⋊C₄×C₁₀₉	—	4	8	no	—
55	C₄⋊C₄×C₁₀₉	—	4	8	no	—
56	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
57	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
58	C₄²×C₁₀₉	—	2	4	no	—
59	C₂³×C₁₀₉	—	2	1	no	—
60	C₂³×C₁₀₉	—	2	2	no	—
61	C₂³×C₁₀₉	—	2	4	no	—
62	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₀₉	—	4	8	no	—
64	Q₈×C₁₀₉	—	4	8	no	—
65	C₂×C₄×C₁₀₉	—	2	4	no	—
66	C₂×C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄×C₁₀₉	—	2	8	no	—
69	C₂×C₄×C₁₀₉	—	4	4	no	—
70	C₂×C₄×C₁₀₉	—	4	4	no	—
71	C₂×C₄×C₁₀₉	—	4	4	no	—
72	C₂×C₄×C₁₀₉	—	4	4	no	—
73	C₂×C₄×C₁₀₉	—	4	4	no	—
74	C₂×C₄×C₁₀₉	—	4	8	no	—
75	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
76	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	4	yes	—
77	C₈×C₁₀₉	—	4	4	no	—
78	C₂²×C₁₀₉	—	4	2	no	—
79	C₂²×C₁₀₉	—	4	4	no	—
80	C₂²×C₁₀₉	—	4	1	no	—
81	C₂²×C₁₀₉	#534	1	1	yes	characteristic
82	C₂²×C₁₀₉	#535	1	2	yes	—
83	C₂²×C₁₀₉	#578	1	4	yes	—
84	C₄×C₁₀₉	—	4	4	no	—
85	C₄×C₁₀₉	—	4	1	no	—
86	C₄×C₁₀₉	—	8	4	no	—
87	C₄×C₁₀₉	—	8	4	no	—
88	C₄×C₁₀₉	#560	1	4	yes	—
89	#32768	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×C₁₀₉	—	8	1	no	—
91	C₂×C₁₀₉	#2815	1	1	yes	characteristic
92	C₂×C₁₀₉	#3202	1	2	yes	—
93	C₂×C₁₀₉	#3223	1	4	yes	—
94	#2621	C₂×C₁₀₉	1	2	yes	—
95	#2074	C₂×C₁₀₉	1	4	yes	complemented
96	#2016	C₂×C₁₀₉	1	4	yes	complemented
97	#2001	C₂×C₁₀₉	1	4	yes	—
98	#1818	C₂×C₁₀₉	1	1	yes	characteristic
99	C₁₀₉	#32768	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×Q₁₆	C₂²×C₁₀₉	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
102	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	4	yes	—
103	C₂×#117	C₂²×C₁₀₉	1	1	yes	characteristic
104	C₂×#103	C₂²×C₁₀₉	1	8	yes	—
105	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
106	#398	—	2	4	no	—
107	#397	—	2	4	no	—
108	#391	C₂²×C₁₀₉	1	4	yes	—
109	#382	C₂²×C₁₀₉	1	2	yes	—
110	#340	C₂²×C₁₀₉	1	4	yes	—
111	#336	C₂²×C₁₀₉	1	4	yes	—
112	C₂²×Q₈	C₂³×C₁₀₉	1	4	yes	—
113	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
114	C₂×Q₁₆	—	4	4	no	—
115	C₂×Q₁₆	—	4	8	no	—
116	C₂²×C₈	C₂³×C₁₀₉	1	4	yes	—
117	C₄⋊Q₈	—	4	4	no	—
118	C₂²⋊Q₈	—	4	8	no	—
119	#117	—	4	2	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
123	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	4	yes	—
124	C₂×C₂²⋊C₄	—	2	4	no	—
125	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
126	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
127	#103	—	4	8	no	—
128	C₂²⋊C₈	—	4	2	no	—
129	#95	—	2	4	no	—
130	#95	—	2	8	no	—
131	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	—	2	8	no	—
135	C₂×Q₈	—	2	4	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
141	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
143	Q₁₆	—	4	16	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	8	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄⋊C₄	—	4	8	no	—
150	C₂²⋊C₄	—	4	2	no	—
151	C₂²⋊C₄	—	4	8	no	—
152	C₄²	—	2	4	no	—
153	C₂³	—	2	1	no	—
154	C₂³	—	2	2	no	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
157	Q₈	—	4	8	no	—
158	Q₈	—	4	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	2	yes	—
170	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	4	yes	—
171	C₈	—	4	4	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	1	no	—
175	C₂²	#534×C₁₀₉	1	1	yes	characteristic
176	C₂²	#535×C₁₀₉	1	2	yes	—
177	C₂²	#578×C₁₀₉	1	4	yes	—
178	C₄	—	4	4	no	—
179	C₄	—	4	1	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	#560×C₁₀₉	1	4	yes	—
183	C₂	—	8	1	no	—
184	C₂	C₁₀₉×#2815	1	1	yes	characteristic
185	C₂	C₁₀₉×#3202	1	2	yes	—
186	C₂	C₁₀₉×#3223	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2621

C₂

yes

maximal

C₁₀₉×#2074

C₂

yes

maximal, complemented

C₁₀₉×#2016

C₂

yes

maximal, complemented

C₁₀₉×#2001

C₂

yes

maximal

C₁₀₉×#1818

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×#117×C₁₀₉

C₂²

yes

characteristic

C₂×#103×C₁₀₉

C₂²

yes

—

C₂×C₂²⋊C₈×C₁₀₉

C₂²

yes

—

#398×C₁₀₉

—

#397×C₁₀₉

—

#391×C₁₀₉

C₂²

yes

—

#382×C₁₀₉

C₂²

yes

—

#340×C₁₀₉

C₂²

yes

—

#336×C₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂³

yes

—

C₂³×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂³

yes

—

C₄⋊Q₈×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂×C₂²⋊C₄×C₁₀₉

—

C₂×C₂²⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

—

#103×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂⁴×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂⁴

yes

characteristic

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²×D₄

yes

characteristic

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×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

H(5,𝔽₂)

yes

—

C₂×C₄×C₁₀₉

H(5,𝔽₂)

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#534

yes

characteristic

C₂²×C₁₀₉

#535

yes

—

C₂²×C₁₀₉

#578

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#560

yes

—

#32768

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#2815

yes

characteristic

C₂×C₁₀₉

#3202

yes

—

C₂×C₁₀₉

#3223

yes

—

#2621

C₂×C₁₀₉

yes

—

#2074

C₂×C₁₀₉

yes

complemented

#2016

C₂×C₁₀₉

yes

complemented

#2001

C₂×C₁₀₉

yes

—

#1818

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#32768

yes

characteristic, direct product factor, Sylow

100

C₂²×Q₁₆

C₂²×C₁₀₉

yes

—

101

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

—

102

C₂×C₂²⋊Q₈

C₂²×C₁₀₉

yes

—

103

C₂×#117

C₂²×C₁₀₉

yes

characteristic

104

C₂×#103

C₂²×C₁₀₉

yes

—

105

C₂×C₂²⋊C₈

C₂²×C₁₀₉

yes

—

106

#398

—

107

#397

—

108

#391

C₂²×C₁₀₉

yes

—

109

#382

C₂²×C₁₀₉

yes

—

110

#340

C₂²×C₁₀₉

yes

—

111

#336

C₂²×C₁₀₉

yes

—

112

C₂²×Q₈

C₂³×C₁₀₉

yes

—

113

C₂³×C₄

C₂³×C₁₀₉

yes

characteristic

114

C₂×Q₁₆

—

115

C₂×Q₁₆

—

116

C₂²×C₈

C₂³×C₁₀₉

yes

—

117

C₄⋊Q₈

—

118

C₂²⋊Q₈

—

119

#117

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

123

C₂×C₄⋊C₄

C₂³×C₁₀₉

yes

—

124

C₂×C₂²⋊C₄

—

125

C₂×C₂²⋊C₄

D₄×C₁₀₉

yes

—

126

C₂×C₄²

C₂³×C₁₀₉

yes

—

127

#103

—

128

C₂²⋊C₈

—

129

#95

—

130

#95

—

131

C₂⁴

C₂×D₄×C₁₀₉

yes

characteristic

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

141

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

142

C₂²×C₄

C₂⁴×C₁₀₉

yes

characteristic

143

Q₁₆

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₂²⋊C₄

—

151

C₂²⋊C₄

—

152

C₄²

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

C₂²×D₄×C₁₀₉

yes

characteristic

157

Q₈

—

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

H(5,𝔽₂)×C₁₀₉

yes

—

170

C₂×C₄

H(5,𝔽₂)×C₁₀₉

yes

—

171

C₈

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#534×C₁₀₉

yes

characteristic

176

C₂²

#535×C₁₀₉

yes

—

177

C₂²

#578×C₁₀₉

yes

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

#560×C₁₀₉

yes

—

183

C₂

—

184

C₂

C₁₀₉×#2815

yes

characteristic

185

C₂

C₁₀₉×#3202

yes

—

186

C₂

C₁₀₉×#3223

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂⁴×C₁₀₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₉

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₁₀₉

prime radical

C₂⁴×C₁₀₉

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×#117×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₁₀₉×#2074	C₂	semidirect
3. C₁₀₉×#2016	C₂	semidirect
89. #32768	C₁₀₉	direct (N × H)
95. #2074	C₂×C₁₀₉	semidirect
96. #2016	C₂×C₁₀₉	semidirect
99. C₁₀₉	#32768	direct (N × H)

H ≅ G/N

type

2. C₁₀₉×#2074

C₂

semidirect

3. C₁₀₉×#2016

C₂

semidirect

89. #32768

C₁₀₉

direct (N × H)

95. #2074

C₂×C₁₀₉

semidirect

96. #2016

C₂×C₁₀₉

semidirect

99. C₁₀₉

#32768

direct (N × H)

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

subgroup	index
C₁₀₉×#2621	2
C₁₀₉×#2074	2
C₁₀₉×#2016	2
C₁₀₉×#2001	2
C₁₀₉×#1818	2
#32768	109

subgroup

index

C₁₀₉×#2621

C₁₀₉×#2074

C₁₀₉×#2016

C₁₀₉×#2001

C₁₀₉×#1818

#32768

109

Subgroups — C₁₀₉×#32768 #42150762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#32768 #42150762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#32768 #42150762