Group #34208403 — subgroups

Subgroups — #34208403

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	#23761514	C₂	1	2	yes	maximal
3	#23761511	C₂	1	2	yes	maximal
4	#23761480	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579097	C₂²	1	1	yes	characteristic
8	#19098601	C₂²	1	1	yes	characteristic
9	#19098589	—	2	1	no	—
10	#19098323	—	2	1	no	—
11	C₂×#108×C₇₉	—	2	1	no	—
12	C₂×#108×C₇₉	C₂²	1	2	yes	—
13	#423×C₇₉	—	2	1	no	—
14	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#11929608	—	2	1	no	—
16	#14579120	—	2	1	no	—
17	#14579101	—	2	2	no	—
18	#14579101	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	2	2	no	—
20	#14579097	—	2	2	no	—
21	#14579095	—	2	1	no	—
22	#14579093	Q₈	1	2	yes	—
23	Dic₇₉⋊C₈	—	4	2	no	—
24	C₂×M₁₆×C₇₉	—	2	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	#108×C₇₉	—	2	4	no	—
29	#108×C₇₉	—	4	2	no	—
30	C₄⋊C₈×C₇₉	—	4	2	no	—
31	#97×C₇₉	—	2	2	no	—
32	#11929608	—	4	1	no	—
33	#11929608	—	4	1	no	—
34	C₂×C₇₉⋊C₈	—	4	1	no	—
35	C₄⋊Dic₇₉	—	2	2	no	—
36	C₂²⋊Dic₇₉	—	4	1	no	—
37	C₄×Dic₇₉	—	4	1	no	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₇₉	—	4	2	no	—
42	M₁₆×C₇₉	—	4	2	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	—	4	2	no	—
46	C₂×C₈×C₇₉	D₈	1	2	yes	—
47	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
49	C₂×C₈×C₇₉	#41	1	2	yes	—
50	C₂×C₈×C₇₉	#41	1	2	yes	—
51	C₄²×C₇₉	—	2	1	no	—
52	C₄²×C₇₉	SD₁₆	1	2	yes	—
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	—	8	2	no	—
65	C₈×C₇₉	#125	1	2	yes	—
66	C₈×C₇₉	#126	1	2	yes	—
67	C₇₉⋊C₈	—	8	1	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	D₇₉	1	1	yes	characteristic
85	#1389	—	79	2	no	—
86	#1388	—	79	2	no	—
87	#1375	—	79	1	no	—
88	C₇₉	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	79	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	79	1	no	—
92	C₂×#108	C₂×D₇₉	1	2	yes	—
93	#427	—	79	1	no	—
94	#424	—	158	1	no	—
95	#423	—	2	1	no	—
96	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
97	#339	—	158	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	158	1	no	—
100	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
102	#117	—	158	1	no	—
103	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
104	M₃₂	—	158	2	no	—
105	M₃₂	—	158	2	no	—
106	C₂×C₁₆	—	158	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	158	2	no	—
110	#107	—	158	1	no	—
111	#106	—	79	2	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	316	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	M₁₆	—	4	2	no	—
119	M₁₆	—	316	1	no	—
120	M₁₆	—	316	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	316	1	no	—
125	C₂×C₈	#11929609	1	2	yes	—
126	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
127	C₂×C₈	#11929623	1	2	yes	—
128	C₂×C₈	#11929624	1	2	yes	—
129	C₂×C₈	#11929625	1	1	yes	characteristic
130	C₄⋊C₄	—	158	2	no	—
131	C₂²⋊C₄	—	316	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	316	1	no	—
134	C₄²	#11929612	1	2	yes	—
135	C₁₆	—	316	2	no	—
136	C₂³	#14579192	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	#14579137	1	1	yes	characteristic
141	C₂×C₄	#14579144	1	1	yes	characteristic
142	C₂×C₄	#14579148	1	1	yes	characteristic
143	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
144	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
145	C₂×C₄	#14579192	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	8	2	no	—
148	C₈	—	632	1	no	—
149	C₈	#14579151	1	2	yes	—
150	C₈	#14579156	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#19098840	1	1	yes	characteristic
153	C₂²	#19098893	1	1	yes	characteristic
154	C₂²	#19098931	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	632	2	no	—
157	C₄	#19098501	1	1	yes	characteristic
158	C₄	#19098845	1	1	yes	characteristic
159	C₄	#19098876	1	1	yes	characteristic
160	C₄	#19098909	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#23764957	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764153

C₂

yes

maximal, characteristic

#23761514

C₂

yes

maximal

#23761511

C₂

yes

maximal

#23761480

C₂

yes

maximal, characteristic

C₇₉×#1849

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579097

C₂²

yes

characteristic

#19098601

C₂²

yes

characteristic

#19098589

—

#19098323

—

C₂×#108×C₇₉

—

C₂×#108×C₇₉

C₂²

yes

—

#423×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

—

—

—

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579095

—

#14579093

Q₈

yes

—

Dic₇₉⋊C₈

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#108×C₇₉

—

#108×C₇₉

—

C₄⋊C₈×C₇₉

—

#97×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

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

—

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

—

C₈×C₇₉

#125

yes

—

C₈×C₇₉

#126

yes

—

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

—

—

—

C₇₉

#13540

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#108

C₂×D₇₉

yes

—

#427

—

#424

—

158

—

#423

—

C₂×#97

C₂×D₇₉

yes

characteristic

#339

—

158

—

C₂×M₁₆

—

C₂×M₁₆

—

158

—

100

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

101

C₂²×C₈

C₂²×D₇₉

yes

characteristic

102

#117

—

158

—

103

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

104

M₃₂

—

158

—

105

M₃₂

—

158

—

106

C₂×C₁₆

—

158

—

107

#108

—

108

#108

—

109

#108

—

158

—

110

#107

—

158

—

111

#106

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

316

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

M₁₆

—

316

—

120

M₁₆

—

316

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

316

—

125

C₂×C₈

#11929609

yes

—

126

C₂×C₈

Q₈×D₇₉

yes

characteristic

127

C₂×C₈

#11929623

yes

—

128

C₂×C₈

#11929624

yes

—

129

C₂×C₈

#11929625

yes

characteristic

130

C₄⋊C₄

—

158

—

131

C₂²⋊C₄

—

316

—

132

C₄²

—

133

C₄²

—

316

—

134

C₄²

#11929612

yes

—

135

C₁₆

—

316

—

136

C₂³

#14579192

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

316

—

140

C₂×C₄

#14579137

yes

characteristic

141

C₂×C₄

#14579144

yes

characteristic

142

C₂×C₄

#14579148

yes

characteristic

143

C₂×C₄

C₂×#11929609

yes

characteristic

144

C₂×C₄

C₂×#11929612

yes

characteristic

145

C₂×C₄

#14579192

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

632

—

149

C₈

#14579151

yes

—

150

C₈

#14579156

yes

—

151

C₂²

—

152

C₂²

#19098840

yes

characteristic

153

C₂²

#19098893

yes

characteristic

154

C₂²

#19098931

yes

characteristic

155

C₄

—

156

C₄

—

632

—

157

C₄

#19098501

yes

characteristic

158

C₄

#19098845

yes

characteristic

159

C₄

#19098876

yes

characteristic

160

C₄

#19098909

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#23764957

yes

characteristic

	subgroup	G/H
center	C₄	#19098909
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1849	C₂
socle	C₂×C₇₉	#2099
prime radical	C₂³×C₇₉	#137
hypercenter	#1849	D₇₉
hypocenter	C₇₉	#13540
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₄

#19098909

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1849

C₂

C₂×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

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
#23761514	2
#23761511	2
#23761480	2
C₇₉×#1849	2
#13540	79

subgroup

index

C₇₉×#1849

Subgroups — #34208403

Conjugacy classes of subgroups

Complements

Maximal subgroups