Group #34215434 — subgroups

Subgroups — #34215434

Order 20224.

4390 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 4390 conjugacy classes of subgroups — too many to draw. Only its 651 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×#19098423	C₂	1	6	yes	maximal, complemented
2	#23761671	C₂	1	2	yes	maximal, complemented
3	#23761641	C₂	1	6	yes	maximal, complemented
4	C₂³×#41×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
6	C₂²×C₂²⋊Dic₇₉	C₂²	1	12	yes	—
7	#19098423	—	2	24	no	—
8	C₂×C₇₉⋊#95	C₂²	1	12	yes	—
9	C₂²×#41×C₇₉	—	2	24	no	—
10	C₂²×#41×C₇₉	C₄	1	8	yes	complemented
11	C₂³×Q₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₇₉	C₂²	1	3	yes	—
13	C₂⁴×C₄×C₇₉	C₂²	1	3	yes	—
14	C₂³×Dic₇₉	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₇₉	—	2	24	no	—
16	C₂×C₂²⋊Dic₇₉	—	2	24	no	—
17	C₂×C₂²⋊Dic₇₉	—	4	24	no	—
18	C₇₉⋊#95	—	2	24	no	—
19	C₂⁵×C₇₉	C₂³	1	3	yes	—
20	C₂×#41×C₇₉	—	2	128	no	—
21	C₂×#41×C₇₉	—	2	96	no	—
22	C₂²×Q₈×C₇₉	—	2	12	no	—
23	C₂²×Q₈×C₇₉	C₂×C₄	1	4	yes	—
24	C₂²×D₄×C₇₉	—	2	24	no	—
25	C₂²×D₄×C₇₉	—	2	12	no	—
26	C₂²×D₄×C₇₉	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₇₉	—	2	24	no	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	12	yes	—
29	C₂³×C₄×C₇₉	D₄	1	24	yes	—
30	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
31	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₇₉	—	2	24	no	—
33	C₂²×Dic₇₉	—	2	24	no	—
34	C₂²×Dic₇₉	—	2	8	no	—
35	C₄⋊Dic₇₉	—	4	16	no	—
36	C₂²⋊Dic₇₉	—	4	48	no	—
37	C₂⁴×C₇₉	—	2	12	no	—
38	C₂⁴×C₇₉	—	2	12	no	—
39	C₂⁴×C₇₉	—	2	12	no	—
40	C₂⁴×C₇₉	C₂²×C₄	1	6	yes	—
41	C₂⁴×C₇₉	C₂×D₄	1	12	yes	—
42	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
43	#41×C₇₉	—	2	256	no	—
44	C₂×Q₈×C₇₉	—	2	32	no	—
45	C₂×Q₈×C₇₉	—	2	24	no	—
46	C₂×D₄×C₇₉	—	2	96	no	—
47	C₂×D₄×C₇₉	—	2	48	no	—
48	C₂×D₄×C₇₉	—	2	24	no	—
49	C₂²×C₄×C₇₉	—	2	12	no	—
50	C₂²×C₄×C₇₉	—	2	48	no	—
51	C₂²×C₄×C₇₉	—	2	96	no	—
52	C₂²×C₄×C₇₉	C₂²⋊C₄	1	48	yes	complemented
53	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
54	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
57	C₂³×C₇₉	—	2	12	no	—
58	C₂³×C₇₉	—	2	24	no	—
59	C₂³×C₇₉	—	2	24	no	—
60	C₂³×C₇₉	—	2	12	no	—
61	C₂³×C₇₉	—	2	12	no	—
62	C₂³×C₇₉	—	2	24	no	—
63	C₂³×C₇₉	—	4	12	no	—
64	C₂³×C₇₉	—	4	12	no	—
65	C₂³×C₇₉	—	4	24	no	—
66	C₂³×C₇₉	C₂×C₂²⋊C₄	1	12	yes	—
67	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
68	C₂³×C₇₉	C₂²×D₄	1	6	yes	—
69	C₂³×C₇₉	H(5,𝔽₂)	1	4	yes	—
70	C₂³×C₇₉	2₋¹⁺⁴	1	4	yes	—
71	Q₈×C₇₉	—	2	32	no	—
72	D₄×C₇₉	—	2	96	no	—
73	C₂×C₄×C₇₉	—	2	12	no	—
74	C₂×C₄×C₇₉	—	2	24	no	—
75	C₂×C₄×C₇₉	—	2	96	no	—
76	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	12	yes	—
77	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	12	yes	—
78	C₂×C₄×C₇₉	#120	1	16	yes	—
79	C₂×Dic₇₉	—	4	8	no	—
80	C₂×Dic₇₉	—	4	24	no	—
81	C₂×Dic₇₉	—	4	24	no	—
82	C₂²×C₇₉	—	2	24	no	—
83	C₂²×C₇₉	—	2	12	no	—
84	C₂²×C₇₉	—	4	24	no	—
85	C₂²×C₇₉	—	4	24	no	—
86	C₂²×C₇₉	—	4	24	no	—
87	C₂²×C₇₉	C₂²×C₂²⋊C₄	1	3	yes	—
88	C₂²×C₇₉	#520	1	4	yes	—
89	C₂²×C₇₉	C₂×#120	1	4	yes	—
90	C₂²×C₇₉	#535	1	6	yes	—
91	C₂²×C₇₉	#536	1	6	yes	—
92	C₂²×C₇₉	#537	1	12	yes	—
93	C₄×C₇₉	—	2	12	no	—
94	C₄×C₇₉	#379	1	8	yes	—
95	Dic₇₉	—	8	8	no	—
96	#14337	—	79	1	no	maximal, Sylow
97	C₂×C₇₉	—	4	12	no	—
98	C₂×C₇₉	C₂×#379	1	1	yes	characteristic
99	C₂×C₇₉	#2311	1	6	yes	—
100	C₂×C₇₉	#2615	1	4	yes	—
101	C₂×C₇₉	#2616	1	4	yes	—
102	C₂³×#41	D₇₉	1	1	yes	characteristic
103	C₂×#380	—	79	6	no	—
104	#1446	—	79	2	no	—
105	#1438	—	79	6	no	—
106	C₇₉	#14337	1	1	yes	characteristic, complemented, Sylow
107	C₂²×#41	—	2	24	no	—
108	C₂²×#41	Dic₇₉	1	8	yes	complemented
109	C₂³×Q₈	C₂×D₇₉	1	1	yes	characteristic
110	C₂³×D₄	C₂×D₇₉	1	3	yes	—
111	C₂⁴×C₄	C₂×D₇₉	1	3	yes	—
112	C₂²×C₄⋊C₄	—	79	4	no	—
113	C₂²×C₂²⋊C₄	—	79	12	no	—
114	#380	—	158	24	no	—
115	C₂×#95	—	79	12	no	—
116	C₂⁵	C₂²×D₇₉	1	3	yes	—
117	C₂×#41	—	2	128	no	—
118	C₂×#41	—	2	96	no	—
119	C₂²×Q₈	—	2	12	no	—
120	C₂²×Q₈	C₂×Dic₇₉	1	4	yes	—
121	C₂²×D₄	—	2	24	no	—
122	C₂²×D₄	—	2	12	no	—
123	C₂²×D₄	C₂×Dic₇₉	1	12	yes	—
124	C₂³×C₄	—	2	24	no	—
125	C₂³×C₄	—	79	8	no	—
126	C₂³×C₄	C₂×Dic₇₉	1	12	yes	—
127	C₂³×C₄	C₂²⋊D₇₉	1	24	yes	—
128	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
129	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	158	24	no	—
131	C₂×C₂²⋊C₄	—	316	24	no	—
132	C₂×C₂²⋊C₄	—	158	24	no	—
133	#95	—	158	24	no	—
134	C₂⁴	—	2	12	no	—
135	C₂⁴	—	2	12	no	—
136	C₂⁴	—	2	12	no	—
137	C₂⁴	C₂²×Dic₇₉	1	6	yes	—
138	C₂⁴	C₂×C₂²⋊D₇₉	1	12	yes	—
139	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
140	#41	—	2	256	no	—
141	C₂×Q₈	—	2	32	no	—
142	C₂×Q₈	—	2	24	no	—
143	C₂×D₄	—	2	96	no	—
144	C₂×D₄	—	2	48	no	—
145	C₂×D₄	—	2	24	no	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	2	48	no	—
148	C₂²×C₄	—	2	96	no	—
149	C₂²×C₄	—	158	24	no	—
150	C₂²×C₄	—	158	8	no	—
151	C₂²×C₄	—	158	24	no	—
152	C₂²×C₄	C₂²⋊Dic₇₉	1	48	yes	complemented
153	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
154	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₇₉	1	12	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₇₉	1	12	yes	—
157	C₄⋊C₄	—	316	16	no	—
158	C₂²⋊C₄	—	316	48	no	—
159	C₂³	—	2	24	no	—
160	C₂³	—	2	12	no	—
161	C₂³	—	2	24	no	—
162	C₂³	—	2	24	no	—
163	C₂³	—	2	12	no	—
164	C₂³	—	2	12	no	—
165	C₂³	—	4	24	no	—
166	C₂³	—	4	12	no	—
167	C₂³	—	4	12	no	—
168	C₂³	C₂×C₂²⋊Dic₇₉	1	12	yes	—
169	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
170	C₂³	C₂²×C₂²⋊D₇₉	1	6	yes	—
171	C₂³	#14579207	1	4	yes	—
172	C₂³	#14579208	1	4	yes	—
173	Q₈	—	2	32	no	—
174	D₄	—	2	96	no	—
175	C₂×C₄	—	2	12	no	—
176	C₂×C₄	—	2	24	no	—
177	C₂×C₄	—	2	96	no	—
178	C₂×C₄	—	316	24	no	—
179	C₂×C₄	—	316	8	no	—
180	C₂×C₄	—	316	24	no	—
181	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	12	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	12	yes	—
183	C₂×C₄	#14579130	1	16	yes	—
184	C₂²	—	2	12	no	—
185	C₂²	—	2	24	no	—
186	C₂²	—	4	24	no	—
187	C₂²	—	4	24	no	—
188	C₂²	—	4	24	no	—
189	C₂²	C₂²×C₂²⋊Dic₇₉	1	3	yes	—
190	C₂²	#19099046	1	4	yes	—
191	C₂²	C₂×#14579130	1	4	yes	—
192	C₂²	#19099127	1	6	yes	—
193	C₂²	#19099134	1	6	yes	—
194	C₂²	#19099141	1	12	yes	—
195	C₄	—	2	12	no	—
196	C₄	—	632	8	no	—
197	C₄	#19098419	1	8	yes	—
198	C₂	—	4	12	no	—
199	C₂	C₂×#19098419	1	1	yes	characteristic
200	C₂	#23765992	1	6	yes	—
201	C₂	#23769124	1	4	yes	—
202	C₂	#23769129	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098423

C₂

yes

maximal, complemented

#23761671

C₂

yes

maximal, complemented

#23761641

C₂

yes

maximal, complemented

C₂³×#41×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₇₉

C₂²

yes

—

C₂²×C₂²⋊Dic₇₉

C₂²

yes

—

#19098423

—

C₂×C₇₉⋊#95

C₂²

yes

—

C₂²×#41×C₇₉

—

C₂²×#41×C₇₉

C₄

yes

complemented

C₂³×Q₈×C₇₉

C₂²

yes

characteristic

C₂³×D₄×C₇₉

C₂²

yes

—

C₂⁴×C₄×C₇₉

C₂²

yes

—

C₂³×Dic₇₉

C₂³

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₂²⋊Dic₇₉

—

C₂×C₂²⋊Dic₇₉

—

C₇₉⋊#95

—

C₂⁵×C₇₉

C₂³

yes

—

C₂×#41×C₇₉

—

128

—

C₂×#41×C₇₉

—

C₂²×Q₈×C₇₉

—

C₂²×Q₈×C₇₉

C₂×C₄

yes

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

—

C₂²×D₄×C₇₉

C₂×C₄

yes

—

C₂³×C₄×C₇₉

—

C₂³×C₄×C₇₉

C₂×C₄

yes

—

C₂³×C₄×C₇₉

D₄

yes

—

C₂³×C₄×C₇₉

C₂³

yes

—

C₂³×C₄×C₇₉

C₂³

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂⁴×C₇₉

—

C₂⁴×C₇₉

—

C₂⁴×C₇₉

—

C₂⁴×C₇₉

C₂²×C₄

yes

—

C₂⁴×C₇₉

C₂×D₄

yes

—

C₂⁴×C₇₉

C₂⁴

yes

characteristic

#41×C₇₉

—

256

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

complemented

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂³×C₇₉

C₂³×C₄

yes

characteristic

C₂³×C₇₉

C₂²×D₄

yes

—

C₂³×C₇₉

H(5,𝔽₂)

yes

—

C₂³×C₇₉

2₋¹⁺⁴

yes

—

Q₈×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

#120

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×C₂²⋊C₄

yes

—

C₂²×C₇₉

#520

yes

—

C₂²×C₇₉

C₂×#120

yes

—

C₂²×C₇₉

#535

yes

—

C₂²×C₇₉

#536

yes

—

C₂²×C₇₉

#537

yes

—

C₄×C₇₉

—

C₄×C₇₉

#379

yes

—

Dic₇₉

—

#14337

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

C₂×#379

yes

characteristic

C₂×C₇₉

#2311

yes

—

100

C₂×C₇₉

#2615

yes

—

101

C₂×C₇₉

#2616

yes

—

102

C₂³×#41

D₇₉

yes

characteristic

103

C₂×#380

—

104

#1446

—

105

#1438

—

106

C₇₉

#14337

yes

characteristic, complemented, Sylow

107

C₂²×#41

—

108

C₂²×#41

Dic₇₉

yes

complemented

109

C₂³×Q₈

C₂×D₇₉

yes

characteristic

110

C₂³×D₄

C₂×D₇₉

yes

—

111

C₂⁴×C₄

C₂×D₇₉

yes

—

112

C₂²×C₄⋊C₄

—

113

C₂²×C₂²⋊C₄

—

114

#380

—

158

—

115

C₂×#95

—

116

C₂⁵

C₂²×D₇₉

yes

—

117

C₂×#41

—

128

—

118

C₂×#41

—

119

C₂²×Q₈

—

120

C₂²×Q₈

C₂×Dic₇₉

yes

—

121

C₂²×D₄

—

122

C₂²×D₄

—

123

C₂²×D₄

C₂×Dic₇₉

yes

—

124

C₂³×C₄

—

125

C₂³×C₄

—

126

C₂³×C₄

C₂×Dic₇₉

yes

—

127

C₂³×C₄

C₂²⋊D₇₉

yes

—

128

C₂³×C₄

C₂²×D₇₉

yes

—

129

C₂³×C₄

C₂²×D₇₉

yes

characteristic

130

C₂×C₄⋊C₄

—

158

—

131

C₂×C₂²⋊C₄

—

316

—

132

C₂×C₂²⋊C₄

—

158

—

133

#95

—

158

—

134

C₂⁴

—

135

C₂⁴

—

136

C₂⁴

—

137

C₂⁴

C₂²×Dic₇₉

yes

—

138

C₂⁴

C₂×C₂²⋊D₇₉

yes

—

139

C₂⁴

C₂³×D₇₉

yes

characteristic

140

#41

—

256

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

158

—

150

C₂²×C₄

—

158

—

151

C₂²×C₄

—

158

—

152

C₂²×C₄

C₂²⋊Dic₇₉

yes

complemented

153

C₂²×C₄

C₂²×Dic₇₉

yes

—

154

C₂²×C₄

C₂²×Dic₇₉

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

157

C₄⋊C₄

—

316

—

158

C₂²⋊C₄

—

316

—

159

C₂³

—

160

C₂³

—

161

C₂³

—

162

C₂³

—

163

C₂³

—

164

C₂³

—

165

C₂³

—

166

C₂³

—

167

C₂³

—

168

C₂³

C₂×C₂²⋊Dic₇₉

yes

—

169

C₂³

C₂³×Dic₇₉

yes

characteristic

170

C₂³

C₂²×C₂²⋊D₇₉

yes

—

171

C₂³

#14579207

yes

—

172

C₂³

#14579208

yes

—

173

Q₈

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

316

—

179

C₂×C₄

—

316

—

180

C₂×C₄

—

316

—

181

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

—

182

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

—

183

C₂×C₄

#14579130

yes

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

C₂²×C₂²⋊Dic₇₉

yes

—

190

C₂²

#19099046

yes

—

191

C₂²

C₂×#14579130

yes

—

192

C₂²

#19099127

yes

—

193

C₂²

#19099134

yes

—

194

C₂²

#19099141

yes

—

195

C₄

—

196

C₄

—

632

—

197

C₄

#19098419

yes

—

198

C₂

—

199

C₂

C₂×#19098419

yes

characteristic

200

C₂

#23765992

yes

—

201

C₂

#23769124

yes

—

202

C₂

#23769129

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₉
commutator	C₂³×C₇₉	C₂³×C₄
Fitting, prime radical	C₂³×#41×C₇₉	C₂
socle, cosocle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂³×#41	D₇₉
hypocenter	C₇₉	#14337
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₇₉

commutator

C₂³×C₇₉

C₂³×C₄

Fitting, prime radical

C₂³×#41×C₇₉

C₂

socle, cosocle

C₂⁴×C₇₉

C₂⁴

hypercenter

C₂³×#41

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂⁴ ◁ C₂³×#41

N	H ≅ G/N	type
1. C₂×#19098423	C₂	semidirect
2. #23761671	C₂	semidirect
3. #23761641	C₂	semidirect
10. C₂²×#41×C₇₉	C₄	semidirect
52. C₂²×C₄×C₇₉	C₂²⋊C₄	semidirect
106. C₇₉	#14337	semidirect
108. C₂²×#41	Dic₇₉	semidirect
152. C₂²×C₄	C₂²⋊Dic₇₉	semidirect

H ≅ G/N

type

1. C₂×#19098423

C₂

semidirect

2. #23761671

C₂

semidirect

3. #23761641

C₂

semidirect

10. C₂²×#41×C₇₉

C₄

semidirect

52. C₂²×C₄×C₇₉

C₂²⋊C₄

semidirect

106. C₇₉

#14337

semidirect

108. C₂²×#41

Dic₇₉

semidirect

152. C₂²×C₄

C₂²⋊Dic₇₉

semidirect

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

subgroup	index
C₂×#19098423	2
#23761671	2
#23761641	2
C₂³×#41×C₇₉	2
#14337	79

subgroup

index

C₂³×#41×C₇₉

Subgroups — #34215434

Conjugacy classes of subgroups

Complements

Maximal subgroups