Group #20310099 — subgroups

Subgroups — C₂₉×#15199 #20310099

Order 7424.

1606 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 558 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	C₂×C₂₉×#391	C₂	1	3	yes	maximal
2	C₂×C₂₉×#384	C₂	1	3	yes	maximal
3	C₂₉×#1444	C₂	1	8	yes	maximal
4	C₄×C₂₉×#95	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₂₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₂₉	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₂₉	C₂²	1	12	yes	—
8	C₂²×C₄²×C₂₉	C₂²	1	3	yes	—
9	C₂₉×#391	—	2	12	no	—
10	C₂₉×#384	—	2	12	no	—
11	C₂×C₂₉×#95	C₂²	1	12	yes	—
12	C₂×C₂₉×#95	C₂²	1	4	yes	—
13	C₂²×Q₈×C₂₉	—	2	6	no	—
14	C₂²×Q₈×C₂₉	—	2	6	no	—
15	C₂²×Q₈×C₂₉	Q₈	1	4	yes	—
16	C₂³×C₄×C₂₉	C₂³	1	6	yes	—
17	C₂³×C₄×C₂₉	C₂³	1	2	yes	—
18	C₂³×C₄×C₂₉	C₂³	1	6	yes	—
19	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₉	—	2	24	no	—
21	C₂×C₄⋊C₄×C₂₉	—	2	12	no	—
22	C₂×C₄⋊C₄×C₂₉	—	2	6	no	—
23	C₂×C₄⋊C₄×C₂₉	—	4	24	no	—
24	C₂×C₄²×C₂₉	—	2	12	no	—
25	C₂×C₄²×C₂₉	D₄	1	12	yes	—
26	C₂₉×#95	—	2	24	no	—
27	C₂₉×#95	Q₈	1	16	yes	complemented
28	C₂⁴×C₂₉	C₂⁴	1	1	yes	characteristic
29	C₂×Q₈×C₂₉	—	2	24	no	—
30	C₂×Q₈×C₂₉	—	4	12	no	—
31	C₂×Q₈×C₂₉	—	4	4	no	—
32	C₂²×C₄×C₂₉	—	2	12	no	—
33	C₂²×C₄×C₂₉	—	2	12	no	—
34	C₂²×C₄×C₂₉	—	2	6	no	—
35	C₂²×C₄×C₂₉	—	2	6	no	—
36	C₂²×C₄×C₂₉	—	2	12	no	—
37	C₂²×C₄×C₂₉	—	2	6	no	—
38	C₂²×C₄×C₂₉	—	2	24	no	—
39	C₂²×C₄×C₂₉	C₂×D₄	1	12	yes	—
40	C₂²×C₄×C₂₉	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₂₉	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₂₉	C₂×Q₈	1	24	yes	—
43	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₂₉	#41	1	6	yes	—
45	C₄⋊C₄×C₂₉	—	4	48	no	—
46	C₄⋊C₄×C₂₉	—	4	12	no	—
47	C₄²×C₂₉	—	2	24	no	—
48	#15199	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₉	C₂²×D₄	1	3	yes	—
50	C₂³×C₂₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂³×C₂₉	C₂²×Q₈	1	4	yes	—
52	C₂³×C₂₉	C₂×#41	1	3	yes	—
53	C₂³×C₂₉	H(5,𝔽₂)	1	4	yes	—
54	Q₈×C₂₉	—	4	16	no	—
55	C₂×C₄×C₂₉	—	2	12	no	—
56	C₂×C₄×C₂₉	—	2	12	no	—
57	C₂×C₄×C₂₉	—	2	48	no	—
58	C₂×C₄×C₂₉	—	2	12	no	—
59	C₂×C₄×C₂₉	—	4	6	no	—
60	C₂×C₄×C₂₉	—	4	2	no	—
61	C₂×C₄×C₂₉	—	4	6	no	—
62	C₂×C₄×C₂₉	—	4	24	no	—
63	C₂×C₄×C₂₉	—	4	12	no	—
64	C₂×C₄×C₂₉	C₂²⋊Q₈	1	12	yes	—
65	C₂×C₄×C₂₉	#123	1	12	yes	—
66	C₂×C₄×C₂₉	#127	1	4	yes	—
67	C₂×C₄×C₂₉	C₄⋊Q₈	1	24	yes	—
68	C₂×#391	C₂×C₂₉	1	3	yes	—
69	C₂×#384	C₂×C₂₉	1	3	yes	—
70	#1444	C₂×C₂₉	1	8	yes	—
71	C₄×#95	C₂×C₂₉	1	1	yes	characteristic
72	C₂²×C₂₉	C₂×C₂²⋊Q₈	1	3	yes	—
73	C₂²×C₂₉	C₂×#123	1	3	yes	—
74	C₂²×C₂₉	C₂×#127	1	1	yes	characteristic
75	C₂²×C₂₉	C₂×C₄⋊Q₈	1	6	yes	—
76	C₂²×C₂₉	#535	1	6	yes	—
77	C₂²×C₂₉	#554	1	12	yes	—
78	C₂²×C₂₉	Q₈²	1	4	yes	—
79	C₄×C₂₉	—	2	24	no	—
80	C₄×C₂₉	—	4	4	no	—
81	C₄×C₂₉	—	8	6	no	—
82	C₄×C₂₉	#399	1	8	yes	—
83	C₂³×Q₈	C₂²×C₂₉	1	1	yes	characteristic
84	C₂²×C₄⋊C₄	C₂²×C₂₉	1	3	yes	—
85	C₂²×C₄⋊C₄	C₂²×C₂₉	1	12	yes	—
86	C₂²×C₄²	C₂²×C₂₉	1	3	yes	—
87	#391	—	2	12	no	—
88	#384	—	2	12	no	—
89	C₂×#95	C₂²×C₂₉	1	12	yes	—
90	C₂×#95	C₂²×C₂₉	1	4	yes	—
91	C₂×C₂₉	C₂×#399	1	1	yes	characteristic
92	C₂×C₂₉	#2502	1	6	yes	—
93	C₂×C₂₉	#2567	1	6	yes	—
94	C₂×C₂₉	#2657	1	1	yes	characteristic
95	C₂×C₂₉	#2868	1	1	yes	characteristic
96	C₂²×Q₈	—	2	6	no	—
97	C₂²×Q₈	—	2	6	no	—
98	C₂²×Q₈	Q₈×C₂₉	1	4	yes	—
99	C₂³×C₄	C₂³×C₂₉	1	6	yes	—
100	C₂³×C₄	C₂³×C₂₉	1	2	yes	—
101	C₂³×C₄	C₂³×C₂₉	1	6	yes	—
102	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	24	no	—
104	C₂×C₄⋊C₄	—	2	12	no	—
105	C₂×C₄⋊C₄	—	2	6	no	—
106	C₂×C₄⋊C₄	—	4	24	no	—
107	C₂×C₄²	—	2	12	no	—
108	C₂×C₄²	D₄×C₂₉	1	12	yes	—
109	#95	—	2	24	no	—
110	#95	Q₈×C₂₉	1	16	yes	complemented
111	C₂₉	#15199	1	1	yes	characteristic, direct product factor, Sylow
112	C₂⁴	C₂⁴×C₂₉	1	1	yes	characteristic
113	C₂×Q₈	—	2	24	no	—
114	C₂×Q₈	—	4	12	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	—	2	12	no	—
118	C₂²×C₄	—	2	6	no	—
119	C₂²×C₄	—	2	6	no	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	C₂×D₄×C₂₉	1	12	yes	—
124	C₂²×C₄	C₂×D₄×C₂₉	1	6	yes	—
125	C₂²×C₄	C₂×Q₈×C₂₉	1	4	yes	—
126	C₂²×C₄	C₂×Q₈×C₂₉	1	24	yes	—
127	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
128	C₂²×C₄	#41×C₂₉	1	6	yes	—
129	C₄⋊C₄	—	4	48	no	—
130	C₄⋊C₄	—	4	12	no	—
131	C₄²	—	2	24	no	—
132	C₂³	C₂²×D₄×C₂₉	1	3	yes	—
133	C₂³	C₂²×Q₈×C₂₉	1	1	yes	characteristic
134	C₂³	C₂²×Q₈×C₂₉	1	4	yes	—
135	C₂³	C₂×#41×C₂₉	1	3	yes	—
136	C₂³	C₂₉×H(5,𝔽₂)	1	4	yes	—
137	Q₈	—	4	16	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	12	no	—
140	C₂×C₄	—	2	48	no	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	6	no	—
145	C₂×C₄	—	4	24	no	—
146	C₂×C₄	—	4	12	no	—
147	C₂×C₄	C₂₉×C₂²⋊Q₈	1	12	yes	—
148	C₂×C₄	C₂₉×#123	1	12	yes	—
149	C₂×C₄	C₂₉×#127	1	4	yes	—
150	C₂×C₄	C₂₉×C₄⋊Q₈	1	24	yes	—
151	C₂²	C₂×C₂₉×C₂²⋊Q₈	1	3	yes	—
152	C₂²	C₂×C₂₉×#123	1	3	yes	—
153	C₂²	C₂×C₂₉×#127	1	1	yes	characteristic
154	C₂²	C₂×C₂₉×C₄⋊Q₈	1	6	yes	—
155	C₂²	C₂₉×#535	1	6	yes	—
156	C₂²	C₂₉×#554	1	12	yes	—
157	C₂²	Q₈²×C₂₉	1	4	yes	—
158	C₄	—	2	24	no	—
159	C₄	—	4	4	no	—
160	C₄	—	8	6	no	—
161	C₄	C₂₉×#399	1	8	yes	—
162	C₂	C₂×C₂₉×#399	1	1	yes	characteristic
163	C₂	C₂₉×#2502	1	6	yes	—
164	C₂	C₂₉×#2567	1	6	yes	—
165	C₂	C₂₉×#2657	1	1	yes	characteristic
166	C₂	C₂₉×#2868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₉×#391

C₂

yes

maximal

C₂×C₂₉×#384

C₂

yes

maximal

C₂₉×#1444

C₂

yes

maximal

C₄×C₂₉×#95

C₂

yes

maximal, characteristic

C₂³×Q₈×C₂₉

C₂²

yes

characteristic

C₂²×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂²×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂²×C₄²×C₂₉

C₂²

yes

—

C₂₉×#391

—

C₂₉×#384

—

C₂×C₂₉×#95

C₂²

yes

—

C₂×C₂₉×#95

C₂²

yes

—

C₂²×Q₈×C₂₉

—

C₂²×Q₈×C₂₉

—

C₂²×Q₈×C₂₉

Q₈

yes

—

C₂³×C₄×C₂₉

C₂³

yes

—

C₂³×C₄×C₂₉

C₂³

yes

—

C₂³×C₄×C₂₉

C₂³

yes

—

C₂³×C₄×C₂₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄²×C₂₉

—

C₂×C₄²×C₂₉

D₄

yes

—

C₂₉×#95

—

C₂₉×#95

Q₈

yes

complemented

C₂⁴×C₂₉

C₂⁴

yes

characteristic

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂×D₄

yes

—

C₂²×C₄×C₂₉

C₂×D₄

yes

—

C₂²×C₄×C₂₉

C₂×Q₈

yes

—

C₂²×C₄×C₂₉

C₂×Q₈

yes

—

C₂²×C₄×C₂₉

C₂×Q₈

yes

—

C₂²×C₄×C₂₉

#41

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

—

#15199

C₂₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₉

C₂²×D₄

yes

—

C₂³×C₂₉

C₂²×Q₈

yes

characteristic

C₂³×C₂₉

C₂²×Q₈

yes

—

C₂³×C₂₉

C₂×#41

yes

—

C₂³×C₂₉

H(5,𝔽₂)

yes

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₉

#123

yes

—

C₂×C₄×C₂₉

#127

yes

—

C₂×C₄×C₂₉

C₄⋊Q₈

yes

—

C₂×#391

C₂×C₂₉

yes

—

C₂×#384

C₂×C₂₉

yes

—

#1444

C₂×C₂₉

yes

—

C₄×#95

C₂×C₂₉

yes

characteristic

C₂²×C₂₉

C₂×C₂²⋊Q₈

yes

—

C₂²×C₂₉

C₂×#123

yes

—

C₂²×C₂₉

C₂×#127

yes

characteristic

C₂²×C₂₉

C₂×C₄⋊Q₈

yes

—

C₂²×C₂₉

#535

yes

—

C₂²×C₂₉

#554

yes

—

C₂²×C₂₉

Q₈²

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#399

yes

—

C₂³×Q₈

C₂²×C₂₉

yes

characteristic

C₂²×C₄⋊C₄

C₂²×C₂₉

yes

—

C₂²×C₄⋊C₄

C₂²×C₂₉

yes

—

C₂²×C₄²

C₂²×C₂₉

yes

—

#391

—

#384

—

C₂×#95

C₂²×C₂₉

yes

—

C₂×#95

C₂²×C₂₉

yes

—

C₂×C₂₉

C₂×#399

yes

characteristic

C₂×C₂₉

#2502

yes

—

C₂×C₂₉

#2567

yes

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Q₈×C₂₉

yes

—

C₂³×C₄

C₂³×C₂₉

yes

—

100

C₂³×C₄

C₂³×C₂₉

yes

—

101

C₂³×C₄

C₂³×C₂₉

yes

—

102

C₂³×C₄

C₂³×C₂₉

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

—

108

C₂×C₄²

D₄×C₂₉

yes

—

109

#95

—

110

#95

Q₈×C₂₉

yes

complemented

111

C₂₉

#15199

yes

characteristic, direct product factor, Sylow

112

C₂⁴

C₂⁴×C₂₉

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₄×C₂₉

yes

—

124

C₂²×C₄

C₂×D₄×C₂₉

yes

—

125

C₂²×C₄

C₂×Q₈×C₂₉

yes

—

126

C₂²×C₄

C₂×Q₈×C₂₉

yes

—

127

C₂²×C₄

C₂×Q₈×C₂₉

yes

—

128

C₂²×C₄

#41×C₂₉

yes

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄²

—

132

C₂³

C₂²×D₄×C₂₉

yes

—

133

C₂³

C₂²×Q₈×C₂₉

yes

characteristic

134

C₂³

C₂²×Q₈×C₂₉

yes

—

135

C₂³

C₂×#41×C₂₉

yes

—

136

C₂³

C₂₉×H(5,𝔽₂)

yes

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂₉×C₂²⋊Q₈

yes

—

148

C₂×C₄

C₂₉×#123

yes

—

149

C₂×C₄

C₂₉×#127

yes

—

150

C₂×C₄

C₂₉×C₄⋊Q₈

yes

—

151

C₂²

C₂×C₂₉×C₂²⋊Q₈

yes

—

152

C₂²

C₂×C₂₉×#123

yes

—

153

C₂²

C₂×C₂₉×#127

yes

characteristic

154

C₂²

C₂×C₂₉×C₄⋊Q₈

yes

—

155

C₂²

C₂₉×#535

yes

—

156

C₂²

C₂₉×#554

yes

—

157

C₂²

Q₈²×C₂₉

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂₉×#399

yes

—

162

C₂

C₂×C₂₉×#399

yes

characteristic

163

C₂

C₂₉×#2502

yes

—

164

C₂

C₂₉×#2567

yes

—

165

C₂

C₂₉×#2657

yes

characteristic

166

C₂

C₂₉×#2868

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₂₉
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₂₉

C₂⁴

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₂₉

LCS

G ▷ C₂⁴ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₂₉ ◁ G

N	H ≅ G/N	type
27. C₂₉×#95	Q₈	semidirect
48. #15199	C₂₉	direct (N × H)
110. #95	Q₈×C₂₉	semidirect
111. C₂₉	#15199	direct (N × H)

H ≅ G/N

type

27. C₂₉×#95

Q₈

semidirect

48. #15199

C₂₉

direct (N × H)

110. #95

Q₈×C₂₉

semidirect

111. C₂₉

#15199

direct (N × H)

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

subgroup	index
C₂×C₂₉×#391	2
C₂×C₂₉×#384	2
C₂₉×#1444	2
C₄×C₂₉×#95	2
#15199	29

subgroup

index

C₂×C₂₉×#391

C₂×C₂₉×#384

C₂₉×#1444

C₄×C₂₉×#95

#15199

Subgroups — C₂₉×#15199 #20310099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C29×#15199 #20310099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#15199 #20310099