Group #34203649 — subgroups

Subgroups — #34203649

Order 20224.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#23765174	C₂	1	2	yes	maximal
2	M₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23763282	C₂	1	2	yes	maximal
4	#23761502	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	1	yes	characteristic
7	#19098601	C₂²	1	2	yes	—
8	#19098521	—	2	2	no	—
9	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098387	C₄	1	8	yes	complemented
11	#19098338	C₄	1	2	yes	complemented
12	C₁₆×Dic₇₉	C₄	1	2	yes	complemented
13	#19098324	—	2	2	no	—
14	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₉	C₂²	1	2	yes	—
16	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
17	#14579120	—	2	2	no	—
18	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
19	#14579101	C₂×C₄	1	4	yes	—
20	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14579095	—	2	2	no	—
22	#14579093	C₂×C₄	1	4	yes	—
23	#14579071	—	4	2	no	—
24	C₈×Dic₇₉	—	4	2	no	—
25	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₉	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₉	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₇₉	—	2	2	no	—
30	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
31	M₃₂×C₇₉	—	2	2	no	—
32	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
33	#108×C₇₉	—	2	2	no	—
34	#108×C₇₉	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₉	—	2	1	no	—
36	C₂×C₇₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇₉⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
39	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₂²⋊Dic₇₉	—	4	2	no	—
42	C₄×Dic₇₉	—	2	1	no	—
43	C₄×Dic₇₉	—	4	2	no	—
44	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₇₉⋊C₁₆	C₄²	1	4	yes	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₉	—	2	4	no	—
49	M₁₆×C₇₉	—	4	2	no	—
50	C₂×C₈×C₇₉	—	4	2	no	—
51	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	2	yes	—
55	C₁₆×C₇₉	—	2	2	no	—
56	C₂³×C₇₉	#135	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
61	C₂×Dic₇₉	—	2	1	no	—
62	C₂×Dic₇₉	—	2	1	no	—
63	C₂×Dic₇₉	—	4	2	no	—
64	C₂×Dic₇₉	—	4	4	no	—
65	C₈×C₇₉	—	4	4	no	—
66	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₇₉	#117	1	1	yes	characteristic
68	C₈×C₇₉	C₄×Q₈	1	2	yes	—
69	C₇₉⋊C₈	#106	1	1	yes	characteristic
70	C₇₉⋊C₈	#106	1	2	yes	—
71	C₇₉⋊C₈	#106	1	1	yes	characteristic
72	C₂²×C₇₉	—	4	1	no	—
73	C₂²×C₇₉	#427	1	1	yes	characteristic
74	C₂²×C₇₉	C₄×SD₁₆	1	2	yes	—
75	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
77	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
78	Dic₇₉	—	2	2	no	—
79	Dic₇₉	—	4	1	no	—
80	Dic₇₉	—	8	4	no	—
81	#12668	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	2	1	no	—
83	C₂×C₇₉	—	2	2	no	—
84	C₂×C₇₉	C₄×#106	1	1	yes	characteristic
85	#2155	—	79	2	no	—
86	C₄×M₃₂	—	79	1	no	—
87	#1772	—	79	2	no	—
88	#1385	D₇₉	1	1	yes	characteristic
89	#1384	—	79	1	no	—
90	C₇₉	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	79	1	no	—
92	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
93	C₂×#108	C₂×D₇₉	1	2	yes	—
94	#427	—	79	2	no	—
95	#405	—	158	2	no	—
96	C₂×C₄×C₈	—	79	1	no	—
97	#365	—	79	8	no	—
98	#346	—	79	2	no	—
99	C₄×C₁₆	—	79	2	no	—
100	#339	—	158	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	79	1	no	—
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	#117	—	158	2	no	—
105	C₂×C₄²	—	79	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	79	4	no	—
108	C₂×C₁₆	—	79	2	no	—
109	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₄×D₇₉	1	4	yes	—
112	#107	—	158	2	no	—
113	#106	—	79	4	no	—
114	#97	—	316	2	no	—
115	C₄×C₈	—	316	2	no	—
116	C₄×C₈	—	79	1	no	—
117	C₄×C₈	—	79	1	no	—
118	C₄×C₈	—	79	1	no	—
119	C₄×C₈	—	79	1	no	—
120	C₂²×C₄	—	158	1	no	—
121	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	79	1	no	—
126	C₂×C₈	—	79	1	no	—
127	C₂×C₈	—	79	2	no	—
128	C₂×C₈	—	79	2	no	—
129	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
131	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
132	C₂×C₈	#11929623	1	2	yes	—
133	C₄⋊C₄	—	158	4	no	—
134	C₂²⋊C₄	—	316	2	no	—
135	C₄²	—	316	2	no	—
136	C₄²	—	79	1	no	—
137	C₄²	—	79	1	no	—
138	C₄²	—	158	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	79	4	no	—
141	C₂³	#14579184	1	1	yes	characteristic
142	C₂×C₄	—	316	2	no	—
143	C₂×C₄	—	316	4	no	—
144	C₂×C₄	—	158	1	no	—
145	C₂×C₄	—	158	1	no	—
146	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
148	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
149	C₂×C₄	#14579180	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	—	79	1	no	—
152	C₈	—	79	2	no	—
153	C₈	—	79	1	no	—
154	C₈	C₄²×D₇₉	1	1	yes	characteristic
155	C₈	#14579117	1	1	yes	characteristic
156	C₈	#14579126	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#19098600	1	1	yes	characteristic
159	C₂²	#19098648	1	2	yes	—
160	C₄	—	316	1	no	—
161	C₄	—	632	4	no	—
162	C₄	—	158	2	no	—
163	C₄	#19098414	1	1	yes	characteristic
164	C₄	#106×D₇₉	1	1	yes	characteristic
165	C₄	#19098654	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#23763288	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765174

C₂

yes

maximal

M₃₂×Dic₇₉

C₂

yes

maximal, characteristic

#23763282

C₂

yes

maximal

#23761502

C₂

yes

maximal, characteristic

C₇₉×#1385

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

characteristic

#19098601

C₂²

yes

—

#19098521

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098387

C₄

yes

complemented

#19098338

C₄

yes

complemented

C₁₆×Dic₇₉

C₄

yes

complemented

#19098324

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

C₂×#108×C₇₉

C₂²

yes

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

#14579120

—

C₂×C₄×Dic₇₉

D₄

yes

characteristic

#14579101

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

#14579095

—

#14579093

C₂×C₄

yes

—

#14579071

—

C₈×Dic₇₉

—

C₈×Dic₇₉

C₂×C₄

yes

characteristic

C₈×Dic₇₉

C₂×C₄

yes

characteristic

C₈×Dic₇₉

Q₈

yes

characteristic

C₈×Dic₇₉

Q₈

yes

characteristic

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

#108×C₇₉

—

#108×C₇₉

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

SD₁₆

yes

—

C₂×C₇₉⋊C₈

SD₁₆

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

characteristic

C₄×Dic₇₉

C₄⋊C₄

yes

characteristic

C₇₉⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂×C₄²

yes

characteristic

C₈×C₇₉

#117

yes

characteristic

C₈×C₇₉

C₄×Q₈

yes

—

C₇₉⋊C₈

#106

yes

characteristic

C₇₉⋊C₈

#106

yes

—

C₇₉⋊C₈

#106

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#427

yes

characteristic

C₂²×C₇₉

C₄×SD₁₆

yes

—

C₄×C₇₉

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

C₄×SD₁₆

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

#12668

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

C₄×#106

yes

characteristic

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₇₉

yes

characteristic

#1384

—

C₇₉

#12668

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂×#108

C₂×D₇₉

yes

—

#427

—

#405

—

158

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

100

#339

—

158

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₇₉

yes

characteristic

104

#117

—

158

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₇₉

yes

—

110

#108

—

111

#108

C₄×D₇₉

yes

—

112

#107

—

158

—

113

#106

—

114

#97

—

316

—

115

C₄×C₈

—

316

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₄

—

158

—

121

C₂²×C₄

D₄×D₇₉

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

130

C₂×C₈

C₂×C₄×D₇₉

yes

—

131

C₂×C₈

Q₈×D₇₉

yes

characteristic

132

C₂×C₈

#11929623

yes

—

133

C₄⋊C₄

—

158

—

134

C₂²⋊C₄

—

316

—

135

C₄²

—

316

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

158

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#14579184

yes

characteristic

142

C₂×C₄

—

316

—

143

C₂×C₄

—

316

—

144

C₂×C₄

—

158

—

145

C₂×C₄

—

158

—

146

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

147

C₂×C₄

(C₂²⋊D₇₉)⋊C₄

yes

—

148

C₂×C₄

SD₁₆×D₇₉

yes

—

149

C₂×C₄

#14579180

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₇₉

yes

characteristic

155

C₈

#14579117

yes

characteristic

156

C₈

#14579126

yes

—

157

C₂²

—

158

C₂²

#19098600

yes

characteristic

159

C₂²

#19098648

yes

—

160

C₄

—

316

—

161

C₄

—

632

—

162

C₄

—

158

—

163

C₄

#19098414

yes

characteristic

164

C₄

#106×D₇₉

yes

characteristic

165

C₄

#19098654

yes

—

166

C₂

—

167

C₂

—

168

C₂

#23763288

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1385	C₂
socle	C₂×C₇₉	C₄×#106
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1385	D₇₉
hypocenter	C₇₉	#12668
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#106×D₇₉

commutator

C₈×C₇₉

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1385

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
10. #19098387	C₄	semidirect
11. #19098338	C₄	semidirect
12. C₁₆×Dic₇₉	C₄	semidirect
90. C₇₉	#12668	semidirect

H ≅ G/N

type

10. #19098387

C₄

semidirect

11. #19098338

C₄

semidirect

12. C₁₆×Dic₇₉

C₄

semidirect

90. C₇₉

#12668

semidirect

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

subgroup	index
#23765174	2
M₃₂×Dic₇₉	2
#23763282	2
#23761502	2
C₇₉×#1385	2
#12668	79

subgroup

index

M₃₂×Dic₇₉

C₇₉×#1385

Subgroups — #34203649

Conjugacy classes of subgroups

Complements

Maximal subgroups