Group #34208405 — subgroups

Subgroups — #34208405

Order 20224.

234 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23764153	C₂	1	1	yes	maximal, characteristic
2	#23763575	C₂	1	1	yes	maximal, characteristic
3	#23761515	C₂	1	2	yes	maximal
4	#23761512	C₂	1	2	yes	maximal
5	C₇₉×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	—	2	1	no	—
7	C₂×#14579097	C₂²	1	1	yes	characteristic
8	C₂×#14579097	C₂²	1	2	yes	—
9	#19098601	C₂²	1	1	yes	characteristic
10	#19098589	—	2	1	no	—
11	#19098587	—	2	1	no	—
12	#19098323	—	2	1	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
14	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	1	no	—
16	C₂×#11929608	—	2	2	no	—
17	#14579120	—	2	1	no	—
18	#14579097	—	2	2	no	—
19	#14579097	—	2	4	no	—
20	#14579097	—	4	2	no	—
21	#14579095	—	2	1	no	—
22	#14579093	Q₈	1	2	yes	—
23	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
24	Dic₇₉⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
28	M₃₂×C₇₉	—	2	2	no	—
29	M₃₂×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	—	2	2	no	—
31	#97×C₇₉	—	2	2	no	—
32	#11929608	—	4	1	no	—
33	#11929608	—	4	2	no	—
34	#11929608	—	4	2	no	—
35	#11929608	—	4	1	no	—
36	C₂×C₇₉⋊C₈	—	4	1	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₄⋊Dic₇₉	—	2	2	no	—
39	C₂²⋊Dic₇₉	—	4	1	no	—
40	C₄×Dic₇₉	—	4	1	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	D₈	1	2	yes	—
46	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	2	yes	—
49	C₂×C₈×C₇₉	#41	1	2	yes	—
50	C₄²×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	SD₁₆	1	2	yes	—
52	C₁₆×C₇₉	—	4	2	no	—
53	C₂³×C₇₉	#137	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	8	1	no	—
56	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
62	C₂×Dic₇₉	—	4	2	no	—
63	C₈×C₇₉	—	2	2	no	—
64	C₈×C₇₉	#125	1	2	yes	—
65	C₈×C₇₉	#126	1	2	yes	—
66	C₇₉⋊C₈	—	8	1	no	—
67	C₇₉⋊C₈	—	8	2	no	—
68	C₂²×C₇₉	—	4	1	no	—
69	C₂²×C₇₉	#474	1	1	yes	characteristic
70	C₂²×C₇₉	#483	1	1	yes	characteristic
71	C₂²×C₇₉	#491	1	1	yes	characteristic
72	C₄×C₇₉	—	4	2	no	—
73	C₄×C₇₉	#400	1	1	yes	characteristic
74	C₄×C₇₉	#475	1	1	yes	characteristic
75	C₄×C₇₉	#480	1	1	yes	characteristic
76	C₄×C₇₉	#486	1	1	yes	characteristic
77	Dic₇₉	—	8	2	no	—
78	#13540	—	79	1	no	maximal, Sylow
79	C₂×C₇₉	—	2	1	no	—
80	C₂×C₇₉	—	2	1	no	—
81	C₂×C₇₉	—	2	1	no	—
82	C₂×C₇₉	#2099	1	1	yes	characteristic
83	#1946	—	79	1	no	—
84	#1849	—	79	1	no	—
85	#1389	—	79	2	no	—
86	#1388	—	79	2	no	—
87	#1375	D₇₉	1	1	yes	characteristic
88	C₇₉	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
90	C₂×#108	—	79	2	no	—
91	C₂×#108	—	79	1	no	—
92	C₂×#108	—	158	1	no	—
93	#427	—	79	1	no	—
94	#424	—	158	1	no	—
95	#423	—	158	1	no	—
96	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
97	#339	—	158	1	no	—
98	C₂×M₁₆	—	158	1	no	—
99	C₂×M₁₆	—	158	2	no	—
100	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
102	#117	—	158	1	no	—
103	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	316	2	no	—
108	#108	—	158	2	no	—
109	#108	—	158	4	no	—
110	#107	—	158	1	no	—
111	#106	—	79	2	no	—
112	C₄⋊C₈	—	316	2	no	—
113	C₄⋊C₈	—	316	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
117	M₁₆	—	316	1	no	—
118	M₁₆	—	316	2	no	—
119	M₁₆	—	316	2	no	—
120	M₁₆	—	316	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	316	1	no	—
124	C₂×C₈	—	316	2	no	—
125	C₂×C₈	D₆₃₂	1	2	yes	—
126	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
127	C₂×C₈	#11929623	1	2	yes	—
128	C₂×C₈	#11929623	1	1	yes	characteristic
129	C₂×C₈	#11929624	1	2	yes	—
130	C₄⋊C₄	—	158	2	no	—
131	C₂²⋊C₄	—	316	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	316	1	no	—
134	C₄²	#11929613	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#14579194	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	316	2	no	—
140	C₂×C₄	#14579141	1	1	yes	characteristic
141	C₂×C₄	#14579145	1	1	yes	characteristic
142	C₂×C₄	#14579149	1	1	yes	characteristic
143	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
144	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
145	C₂×C₄	#14579194	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	632	2	no	—
148	C₈	—	632	1	no	—
149	C₈	#14579151	1	2	yes	—
150	C₈	#14579154	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#19098844	1	1	yes	characteristic
153	C₂²	#19098894	1	1	yes	characteristic
154	C₂²	#19098932	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	632	2	no	—
157	C₄	#19098503	1	1	yes	characteristic
158	C₄	#19098849	1	1	yes	characteristic
159	C₄	#19098877	1	1	yes	characteristic
160	C₄	#19098910	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#23764959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764153

C₂

yes

maximal, characteristic

#23763575

C₂

yes

maximal, characteristic

#23761515

C₂

yes

maximal

#23761512

C₂

yes

maximal

C₇₉×#1375

C₂

yes

maximal, characteristic

C₂×#14579097

—

C₂×#14579097

C₂²

yes

characteristic

C₂×#14579097

C₂²

yes

—

#19098601

C₂²

yes

—

—

—

C₂×M₃₂×C₇₉

C₂²

yes

—

C₂×#97×C₇₉

C₂²

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊(C₇₉⋊C₈)

—

Dic₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

#97×C₇₉

—

—

—

—

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₈

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#137

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

#137

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#125

yes

—

C₈×C₇₉

#126

yes

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

#13540

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#13540

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₉

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

—

158

—

#427

—

#424

—

158

—

#423

—

158

—

C₂×#97

C₂×D₇₉

yes

characteristic

#339

—

158

—

C₂×M₁₆

—

158

—

C₂×M₁₆

—

158

—

100

C₂²×C₈

D₃₁₆

yes

characteristic

101

C₂²×C₈

C₂²×D₇₉

yes

characteristic

102

#117

—

158

—

103

C₂×C₄²

D₃₁₆

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

316

—

108

#108

—

158

—

109

#108

—

158

—

110

#107

—

158

—

111

#106

—

112

C₄⋊C₈

—

316

—

113

C₄⋊C₈

—

316

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

117

M₁₆

—

316

—

118

M₁₆

—

316

—

119

M₁₆

—

316

—

120

M₁₆

—

316

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

316

—

124

C₂×C₈

—

316

—

125

C₂×C₈

D₆₃₂

yes

—

126

C₂×C₈

Q₈×D₇₉

yes

characteristic

127

C₂×C₈

#11929623

yes

—

128

C₂×C₈

#11929623

yes

characteristic

129

C₂×C₈

#11929624

yes

—

130

C₄⋊C₄

—

158

—

131

C₂²⋊C₄

—

316

—

132

C₄²

—

133

C₄²

—

316

—

134

C₄²

#11929613

yes

—

135

C₁₆

—

136

C₂³

#14579194

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

316

—

140

C₂×C₄

#14579141

yes

characteristic

141

C₂×C₄

#14579145

yes

characteristic

142

C₂×C₄

#14579149

yes

characteristic

143

C₂×C₄

C₂×D₆₃₂

yes

characteristic

144

C₂×C₄

C₂×#11929613

yes

characteristic

145

C₂×C₄

#14579194

yes

characteristic

146

C₈

—

147

C₈

—

632

—

148

C₈

—

632

—

149

C₈

#14579151

yes

—

150

C₈

#14579154

yes

—

151

C₂²

—

152

C₂²

#19098844

yes

characteristic

153

C₂²

#19098894

yes

characteristic

154

C₂²

#19098932

yes

characteristic

155

C₄

—

156

C₄

—

632

—

157

C₄

#19098503

yes

characteristic

158

C₄

#19098849

yes

characteristic

159

C₄

#19098877

yes

characteristic

160

C₄

#19098910

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#23764959

yes

characteristic

	subgroup	G/H
center	C₄	#19098910
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1375	C₂
socle	C₂×C₇₉	#2099
prime radical	C₂³×C₇₉	#137
hypercenter	#1375	D₇₉
hypocenter	C₇₉	#13540
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#19098910

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1375

C₂

C₂×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

N	H ≅ G/N	type
88. C₇₉	#13540	semidirect

H ≅ G/N

type

88. C₇₉

#13540

semidirect

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

subgroup	index
#23764153	2
#23763575	2
#23761515	2
#23761512	2
C₇₉×#1375	2
#13540	79

subgroup

index

C₇₉×#1375

Subgroups — #34208405

Conjugacy classes of subgroups

Complements

Maximal subgroups