Group #34181981 — subgroups

Subgroups — #34181981

Order 20224.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#23762163	C₂	1	1	yes	maximal, characteristic
2	#23761692	C₂	1	1	yes	maximal, characteristic
3	#23761263	C₂	1	2	yes	maximal
4	#23761261	C₂	1	2	yes	maximal
5	C₇₉×#1568	C₂	1	1	yes	maximal, characteristic
6	#19098620	C₂²	1	2	yes	—
7	#19098616	C₂²	1	1	yes	characteristic
8	#19098616	C₂²	1	1	yes	characteristic
9	#19098316	—	2	1	no	—
10	#19098314	—	2	1	no	—
11	#19098285	C₄	1	4	yes	—
12	#432×C₇₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#335×C₇₉	—	2	1	no	—
15	#334×C₇₉	—	2	1	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
22	#14579072	—	2	2	no	—
23	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
26	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	2	no	—
28	C₂²⋊C₈×C₇₉	—	2	2	no	—
29	#97×C₇₉	—	2	2	no	—
30	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	—	2	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
39	C₂×C₈×C₇₉	—	4	2	no	—
40	C₂×C₈×C₇₉	—	4	2	no	—
41	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
43	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₇₉	#41	1	1	yes	characteristic
46	C₄²×C₇₉	#41	1	1	yes	characteristic
47	C₂³×C₇₉	#125	1	1	yes	characteristic
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	4	1	no	—
50	C₂×C₄×C₇₉	—	4	1	no	—
51	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
57	C₈×C₇₉	—	2	2	no	—
58	C₈×C₇₉	—	2	2	no	—
59	C₈×C₇₉	—	8	2	no	—
60	C₇₉⋊C₈	—	4	2	no	—
61	C₇₉⋊C₈	—	4	2	no	—
62	C₇₉⋊C₈	—	8	2	no	—
63	C₂²×C₇₉	—	2	1	no	—
64	C₂²×C₇₉	—	2	1	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	#382	1	1	yes	characteristic
67	C₄×C₇₉	—	2	1	no	—
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	#430	1	2	yes	—
73	#8798	—	79	1	no	maximal, Sylow
74	C₂×C₇₉	—	4	1	no	—
75	C₂×C₇₉	#1833	1	1	yes	characteristic
76	C₂×C₇₉	#1833	1	1	yes	characteristic
77	C₂×C₇₉	#1838	1	1	yes	characteristic
78	#1568	—	79	1	no	—
79	#1568	D₇₉	1	1	yes	characteristic
80	#1452	—	79	1	no	—
81	#1287	—	79	2	no	—
82	#1287	—	79	2	no	—
83	C₇₉	#8798	1	1	yes	characteristic, complemented, Sylow
84	#432	—	79	2	no	—
85	#432	C₂×D₇₉	1	2	yes	—
86	#431	—	79	1	no	—
87	#431	—	79	1	no	—
88	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
89	#335	—	2	1	no	—
90	#335	—	158	1	no	—
91	#334	—	2	1	no	—
92	#334	—	158	1	no	—
93	#322	—	79	4	no	—
94	C₂²×C₈	Dic₁₅₈	1	2	yes	—
95	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	158	2	no	—
98	C₄⋊C₈	—	158	1	no	—
99	C₄⋊C₈	—	158	1	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	158	2	no	—
102	C₂²⋊C₈	—	158	1	no	—
103	C₂²⋊C₈	—	158	1	no	—
104	#97	—	2	2	no	—
105	#97	—	158	2	no	—
106	C₄×C₈	—	79	2	no	—
107	C₄×C₈	—	79	2	no	—
108	C₄×C₈	C₄×D₇₉	1	2	yes	—
109	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
112	C₂²×C₄	#11929623	1	1	yes	characteristic
113	C₂²×C₄	#11929625	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	316	2	no	—
117	C₂×C₈	—	158	2	no	—
118	C₂×C₈	—	158	2	no	—
119	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
120	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
121	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
122	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
123	C₄²	#11929623	1	1	yes	characteristic
124	C₄²	#11929625	1	1	yes	characteristic
125	C₂³	#14579152	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14579119	1	1	yes	characteristic
131	C₂×C₄	#14579119	1	1	yes	characteristic
132	C₂×C₄	#14579148	1	1	yes	characteristic
133	C₂×C₄	#14579150	1	1	yes	characteristic
134	C₂×C₄	#14579152	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	316	2	no	—
139	C₈	—	316	2	no	—
140	C₈	—	632	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#19098428	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#19098611	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#23763523	1	1	yes	characteristic
153	C₂	#23763525	1	1	yes	characteristic
154	C₂	#23763540	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23762163

C₂

yes

maximal, characteristic

#23761692

C₂

yes

maximal, characteristic

#23761263

C₂

yes

maximal

#23761261

C₂

yes

maximal

C₇₉×#1568

C₂

yes

maximal, characteristic

#19098620

C₂²

yes

—

#19098616

C₂²

yes

characteristic

#19098616

C₂²

yes

—

—

C₄

yes

—

#432×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

—

#334×C₇₉

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₂×Q₈

yes

C₂²×C₄×C₇₉

yes

C₂²×C₄×C₇₉

yes

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#382

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#430

yes

—

#8798

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#8798

yes

characteristic, complemented, Sylow

#432

—

#432

C₂×D₇₉

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#335

—

#335

—

158

—

#334

—

#334

—

158

—

#322

—

C₂²×C₈

Dic₁₅₈

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

158

—

102

C₂²⋊C₈

—

158

—

103

C₂²⋊C₈

—

158

—

104

#97

—

105

#97

—

158

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₄×D₇₉

yes

—

109

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

110

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

111

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

112

C₂²×C₄

#11929623

yes

characteristic

113

C₂²×C₄

#11929625

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

316

—

117

C₂×C₈

—

158

—

118

C₂×C₈

—

158

—

119

C₂×C₈

Dic₇₉⋊C₄

yes

—

120

C₂×C₈

Dic₇₉⋊C₄

yes

—

121

C₄²

C₂×C₄×D₇₉

yes

characteristic

122

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

123

C₄²

#11929623

yes

characteristic

124

C₄²

#11929625

yes

characteristic

125

C₂³

#14579152

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

130

C₂×C₄

#14579119

yes

characteristic

131

C₂×C₄

#14579119

yes

characteristic

132

C₂×C₄

#14579148

yes

characteristic

133

C₂×C₄

#14579150

yes

characteristic

134

C₂×C₄

#14579152

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

316

—

139

C₈

—

316

—

140

C₈

—

632

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#19098428

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#19098611

yes

—

151

C₂

—

152

C₂

#23763523

yes

characteristic

153

C₂

#23763525

yes

characteristic

154

C₂

#23763540

yes

characteristic

	subgroup	G/H
center	C₂²	#19098428
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1568	C₂
socle	C₂²×C₇₉	#382
prime radical	C₂³×C₇₉	#125
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1568	D₇₉
hypocenter	C₇₉	#8798
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#19098428

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1568

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
83. C₇₉	#8798	semidirect

H ≅ G/N

type

83. C₇₉

#8798

semidirect

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

subgroup	index
#23762163	2
#23761692	2
#23761263	2
#23761261	2
C₇₉×#1568	2
#8798	79

subgroup

index

C₇₉×#1568

Subgroups — #34181981

Conjugacy classes of subgroups

Complements

Maximal subgroups