Group #34181998 — subgroups

Subgroups — #34181998

Order 20224.

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

normal subgroups only

This group has 258 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	#23763149	C₂	1	1	yes	maximal, characteristic
2	#23761244	C₂	1	2	yes	maximal
3	#23761233	C₂	1	1	yes	maximal, characteristic
4	#23761230	C₂	1	2	yes	maximal
5	C₇₉×#1276	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	C₂×#14579072	C₂²	1	1	yes	characteristic
8	C₂×#14579072	C₂²	1	1	yes	characteristic
9	#19098318	C₄	1	2	yes	—
10	#19098318	C₄	1	2	yes	—
11	#431×C₇₉	C₂²	1	2	yes	—
12	C₄³×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
18	#14579072	—	2	1	no	—
19	#14579072	—	2	2	no	—
20	#14579072	—	2	1	no	—
21	C₄×C₇₉⋊C₈	—	2	2	no	—
22	C₂×C₄²×C₇₉	—	2	2	no	—
23	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₇₉	—	2	2	no	—
26	C₂²⋊C₈×C₇₉	—	2	2	no	—
27	C₄×C₈×C₇₉	—	2	2	no	—
28	C₂×C₇₉⋊C₈	—	2	1	no	—
29	C₂×C₇₉⋊C₈	—	2	1	no	—
30	C₂×C₇₉⋊C₈	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	2	2	no	—
32	C₂×C₇₉⋊C₈	—	2	2	no	—
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	—	2	2	no	—
37	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	#41	1	2	yes	—
39	C₂×C₈×C₇₉	—	4	2	no	—
40	C₄²×C₇₉	—	2	2	no	—
41	C₄²×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	—	4	4	no	—
43	C₄²×C₇₉	#41	1	1	yes	characteristic
44	C₄²×C₇₉	#41	1	2	yes	—
45	C₄²×C₇₉	#41	1	1	yes	characteristic
46	C₂³×C₇₉	#117	1	1	yes	characteristic
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	—	4	2	no	—
49	C₂×C₄×C₇₉	—	4	2	no	—
50	C₂×C₄×C₇₉	—	4	2	no	—
51	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
52	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#126	1	2	yes	—
54	C₂×C₄×C₇₉	#126	1	2	yes	—
55	C₂×C₄×C₇₉	#131	1	2	yes	—
56	C₂×C₄×C₇₉	#131	1	2	yes	—
57	C₈×C₇₉	—	8	2	no	—
58	C₇₉⋊C₈	—	4	1	no	—
59	C₇₉⋊C₈	—	4	1	no	—
60	C₇₉⋊C₈	—	4	2	no	—
61	C₇₉⋊C₈	—	8	2	no	—
62	C₂²×C₇₉	—	2	2	no	—
63	C₂²×C₇₉	#383	1	1	yes	characteristic
64	C₂²×C₇₉	#433	1	2	yes	—
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	—	2	2	no	—
67	C₄×C₇₉	—	4	4	no	—
68	C₄×C₇₉	#433	1	1	yes	characteristic
69	C₄×C₇₉	#433	1	1	yes	characteristic
70	C₄×C₇₉	#433	1	2	yes	—
71	#8802	—	79	1	no	maximal, Sylow
72	C₂×C₇₉	—	2	2	no	—
73	C₂×C₇₉	#1840	1	1	yes	characteristic
74	C₂×C₇₉	#1841	1	2	yes	—
75	#1745	—	79	1	no	—
76	#1281	—	79	2	no	—
77	#1276	—	79	1	no	—
78	#1276	D₇₉	1	1	yes	characteristic
79	#1275	—	79	2	no	—
80	C₇₉	#8802	1	1	yes	characteristic, complemented, Sylow
81	#431	—	79	2	no	—
82	#431	C₂×D₇₉	1	2	yes	—
83	C₂×#97	—	79	1	no	—
84	C₂×#97	—	79	1	no	—
85	C₄³	C₂×D₇₉	1	1	yes	characteristic
86	#336	—	79	2	no	—
87	#336	—	79	2	no	—
88	C₂²×C₈	—	79	1	no	—
89	C₂²×C₈	—	79	1	no	—
90	C₂²×C₈	—	79	2	no	—
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₄×D₇₉	1	2	yes	—
93	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	158	2	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	158	2	no	—
98	#97	—	158	1	no	—
99	#97	—	158	2	no	—
100	#97	—	158	1	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	158	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
105	C₂²×C₄	#11929623	1	2	yes	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	316	2	no	—
108	C₂×C₈	—	158	1	no	—
109	C₂×C₈	—	158	1	no	—
110	C₂×C₈	—	158	2	no	—
111	C₂×C₈	—	158	2	no	—
112	C₂×C₈	—	158	2	no	—
113	C₂×C₈	—	158	2	no	—
114	C₂×C₈	—	158	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#11929623	1	1	yes	characteristic
119	C₄²	#11929624	1	2	yes	—
120	C₄²	#11929625	1	1	yes	characteristic
121	C₂³	#14579117	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#14579118	1	1	yes	characteristic
127	C₂×C₄	#14579119	1	1	yes	characteristic
128	C₂×C₄	#14579154	1	2	yes	—
129	C₂×C₄	#14579156	1	2	yes	—
130	C₂×C₄	#14579167	1	2	yes	—
131	C₂×C₄	#14579169	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	316	1	no	—
134	C₈	—	316	1	no	—
135	C₈	—	316	2	no	—
136	C₈	—	632	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#19098430	1	1	yes	characteristic
139	C₂²	#19098621	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	4	no	—
143	C₄	#19098622	1	1	yes	characteristic
144	C₄	#19098623	1	2	yes	—
145	C₄	#19098625	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#23763544	1	1	yes	characteristic
148	C₂	#23763546	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763149

C₂

yes

maximal, characteristic

#23761244

C₂

yes

maximal

#23761233

C₂

yes

maximal, characteristic

#23761230

C₂

yes

maximal

C₇₉×#1276

C₂

yes

maximal, characteristic

#19098616

C₂²

yes

—

C₂×#14579072

C₂²

yes

characteristic

C₂×#14579072

C₂²

yes

characteristic

#19098318

C₄

yes

—

#19098318

C₄

yes

—

#431×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

#41

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

—

C₂×C₄×C₇₉

#126

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#383

yes

characteristic

C₂²×C₇₉

#433

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#8802

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

#1840

yes

characteristic

C₂×C₇₉

#1841

yes

—

—

—

—

D₇₉

yes

characteristic

#1275

—

C₇₉

#8802

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₉

yes

—

C₂×#97

—

C₂×#97

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

158

—

C₂²⋊C₈

—

C₂²⋊C₈

—

158

—

#97

—

158

—

#97

—

158

—

100

#97

—

158

—

101

C₄×C₈

—

102

C₄×C₈

—

158

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

105

C₂²×C₄

#11929623

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

316

—

108

C₂×C₈

—

158

—

109

C₂×C₈

—

158

—

110

C₂×C₈

—

158

—

111

C₂×C₈

—

158

—

112

C₂×C₈

—

158

—

113

C₂×C₈

—

158

—

114

C₂×C₈

—

158

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

#11929623

yes

characteristic

119

C₄²

#11929624

yes

—

120

C₄²

#11929625

yes

characteristic

121

C₂³

#14579117

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#14579118

yes

characteristic

127

C₂×C₄

#14579119

yes

characteristic

128

C₂×C₄

#14579154

yes

—

129

C₂×C₄

#14579156

yes

—

130

C₂×C₄

#14579167

yes

—

131

C₂×C₄

#14579169

yes

—

132

C₈

—

133

C₈

—

316

—

134

C₈

—

316

—

135

C₈

—

316

—

136

C₈

—

632

—

137

C₂²

—

138

C₂²

#19098430

yes

characteristic

139

C₂²

#19098621

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#19098622

yes

characteristic

144

C₄

#19098623

yes

—

145

C₄

#19098625

yes

characteristic

146

C₂

—

147

C₂

#23763544

yes

characteristic

148

C₂

#23763546

yes

—

	subgroup	G/H
center	C₂×C₄	#14579119
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1276	C₂
socle	C₂²×C₇₉	#383
prime radical	C₂³×C₇₉	#117
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1276	D₇₉
hypocenter	C₇₉	#8802
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂×C₄

#14579119

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1276

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
80. C₇₉	#8802	semidirect

H ≅ G/N

type

80. C₇₉

#8802

semidirect

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

subgroup	index
#23763149	2
#23761244	2
#23761233	2
#23761230	2
C₇₉×#1276	2
#8802	79

subgroup

index

C₇₉×#1276

Subgroups — #34181998

Conjugacy classes of subgroups

Complements

Maximal subgroups