Group #34204692 — subgroups

Subgroups — #34204692

Order 20224.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#23765165	C₂	1	2	yes	maximal
2	#23765160	C₂	1	2	yes	maximal
3	#23765159	C₂	1	1	yes	maximal, characteristic
4	#23763155	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	1	yes	characteristic
7	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#19098518	C₂²	1	2	yes	—
9	#19098383	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	Dic₇₉⋊C₁₆	—	2	4	no	—
12	C₇₉⋊#338	C₄	1	1	yes	characteristic
13	#19098320	C₄	1	1	yes	characteristic
14	#19098319	C₄	1	2	yes	—
15	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
16	#405×C₇₉	C₂²	1	1	yes	characteristic
17	#364×C₇₉	—	2	2	no	—
18	C₂×#11929608	C₂×C₄	1	2	yes	—
19	#14579120	C₂×C₄	1	2	yes	—
20	#14579101	—	2	2	no	—
21	C₂×C₇₉⋊C₁₆	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	—	2	1	no	—
23	C₂×C₇₉⋊C₁₆	—	2	1	no	—
24	#14579071	Q₈	1	4	yes	—
25	C₈×Dic₇₉	D₄	1	4	yes	—
26	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₇₉	—	2	2	no	—
30	M₃₂×C₇₉	—	2	2	no	—
31	C₂×C₁₆×C₇₉	—	2	2	no	—
32	#97×C₇₉	Q₈	1	2	yes	—
33	C₄×C₈×C₇₉	D₄	1	2	yes	—
34	#11929608	—	2	2	no	—
35	#11929608	—	2	2	no	—
36	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₄⋊Dic₇₉	—	2	2	no	—
38	C₂²⋊Dic₇₉	—	2	2	no	—
39	C₄×Dic₇₉	C₄⋊C₄	1	4	yes	—
40	C₇₉⋊C₁₆	—	4	1	no	—
41	C₇₉⋊C₁₆	—	4	1	no	—
42	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	M₁₆×C₇₉	—	2	2	no	—
44	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₇₉	—	2	1	no	—
50	C₂²⋊C₄×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
52	C₁₆×C₇₉	—	4	2	no	—
53	C₂³×C₇₉	#131	1	1	yes	characteristic
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#131	1	2	yes	—
58	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
59	C₂×Dic₇₉	—	2	4	no	—
60	C₈×C₇₉	—	2	2	no	—
61	C₈×C₇₉	#127	1	1	yes	characteristic
62	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
64	C₇₉⋊C₈	—	2	4	no	—
65	C₂²×C₇₉	—	4	1	no	—
66	C₂²×C₇₉	#424	1	2	yes	—
67	C₂²×C₇₉	#424	1	1	yes	characteristic
68	C₄×C₇₉	—	4	2	no	—
69	C₄×C₇₉	#384	1	1	yes	characteristic
70	C₄×C₇₉	#424	1	2	yes	—
71	C₄×C₇₉	#424	1	1	yes	characteristic
72	Dic₇₉	—	4	4	no	—
73	#12902	—	79	1	no	maximal, Sylow
74	C₂×C₇₉	—	2	2	no	—
75	C₂×C₇₉	—	2	1	no	—
76	C₂×C₇₉	#1843	1	1	yes	characteristic
77	#2153	—	79	2	no	—
78	#2153	D₇₉	1	1	yes	characteristic
79	#2151	—	79	2	no	—
80	#2151	—	79	1	no	—
81	#1747	—	79	1	no	—
82	C₇₉	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	79	1	no	—
84	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
85	C₂²×C₁₆	—	79	1	no	—
86	#405	—	79	2	no	—
87	#405	C₂×D₇₉	1	1	yes	characteristic
88	#364	—	2	2	no	—
89	#364	—	158	4	no	—
90	C₄⋊C₁₆	—	158	4	no	—
91	C₄⋊C₁₆	—	158	2	no	—
92	#338	—	79	1	no	—
93	H(3,ℤ/4)	—	79	1	no	—
94	H(3,ℤ/4)	—	79	2	no	—
95	C₂×M₁₆	—	79	2	no	—
96	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
98	#117	—	79	2	no	—
99	#117	C₄×D₇₉	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	158	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	158	2	no	—
105	C₂×C₁₆	—	158	1	no	—
106	C₂×C₁₆	—	158	1	no	—
107	#97	—	79	4	no	—
108	#97	Dic₁₅₈	1	2	yes	—
109	C₄×C₈	—	79	4	no	—
110	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	158	2	no	—
114	M₁₆	—	158	2	no	—
115	C₂×C₈	—	79	4	no	—
116	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
117	C₂×C₈	D₄×D₇₉	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
119	C₂×C₈	Q₈×D₇₉	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	158	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	158	2	no	—
125	C₄²	—	79	4	no	—
126	C₄²	Dic₇₉⋊C₄	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	316	1	no	—
129	C₁₆	—	316	1	no	—
130	C₂³	#14579169	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	158	4	no	—
133	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
134	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#14579167	1	2	yes	—
136	C₂×C₄	#14579169	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	158	4	no	—
139	C₈	#14579157	1	1	yes	characteristic
140	C₈	#14579159	1	1	yes	characteristic
141	C₈	#14579160	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#19098588	1	2	yes	—
144	C₂²	#19098591	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	316	4	no	—
147	C₄	#19098432	1	1	yes	characteristic
148	C₄	#19098588	1	2	yes	—
149	C₄	#19098589	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#23763553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765165

C₂

yes

maximal

#23765160

C₂

yes

maximal

#23765159

C₂

yes

maximal, characteristic

#23763155

C₂

yes

maximal, characteristic

C₇₉×#2153

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

characteristic

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098518

C₂²

yes

—

#19098383

—

C₄⋊(C₇₉⋊C₁₆)

—

Dic₇₉⋊C₁₆

—

C₇₉⋊#338

C₄

yes

characteristic

#19098320

C₄

yes

characteristic

#19098319

C₄

yes

—

C₂×M₃₂×C₇₉

C₂²

yes

—

#405×C₇₉

C₂²

yes

characteristic

#364×C₇₉

—

C₂×#11929608

C₂×C₄

yes

—

#14579120

C₂×C₄

yes

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

#14579071

Q₈

yes

—

C₈×Dic₇₉

D₄

yes

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

C₂×C₄

yes

characteristic

M₃₂×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

#97×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

C₄⋊C₄

yes

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

—

C₂³×C₇₉

#131

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#131

yes

characteristic

C₂×Dic₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#127

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#424

yes

—

C₂²×C₇₉

#424

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#384

yes

characteristic

C₄×C₇₉

#424

yes

—

C₄×C₇₉

#424

yes

characteristic

Dic₇₉

—

#12902

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

D₇₉

yes

—

—

—

C₇₉

#12902

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₇₉

yes

characteristic

#364

—

#364

—

158

—

C₄⋊C₁₆

—

158

—

C₄⋊C₁₆

—

158

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₇₉

yes

characteristic

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

#117

C₄×D₇₉

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

M₃₂

—

158

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

158

—

105

C₂×C₁₆

—

158

—

106

C₂×C₁₆

—

158

—

107

#97

—

108

#97

Dic₁₅₈

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₇₉

yes

—

111

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

158

—

114

M₁₆

—

158

—

115

C₂×C₈

—

116

C₂×C₈

Dic₇₉⋊C₄

yes

—

117

C₂×C₈

D₄×D₇₉

yes

—

118

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

119

C₂×C₈

Q₈×D₇₉

yes

—

120

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

158

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

158

—

125

C₄²

—

126

C₄²

Dic₇₉⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

316

—

129

C₁₆

—

316

—

130

C₂³

#14579169

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

158

—

133

C₂×C₄

C₄⋊C₄×D₇₉

yes

—

134

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

135

C₂×C₄

#14579167

yes

—

136

C₂×C₄

#14579169

yes

characteristic

137

C₈

—

138

C₈

—

158

—

139

C₈

#14579157

yes

characteristic

140

C₈

#14579159

yes

characteristic

141

C₈

#14579160

yes

—

142

C₂²

—

143

C₂²

#19098588

yes

—

144

C₂²

#19098591

yes

characteristic

145

C₄

—

146

C₄

—

316

—

147

C₄

#19098432

yes

characteristic

148

C₄

#19098588

yes

—

149

C₄

#19098589

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#23763553

yes

characteristic

	subgroup	G/H
center	C₈	#14579157
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2153	C₂
socle	C₂×C₇₉	#1843
prime radical	C₂³×C₇₉	#131
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2153	D₇₉
hypocenter	C₇₉	#12902
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#14579157

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2153

C₂

C₂×C₇₉

C₂³×C₇₉

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
82. C₇₉	#12902	semidirect

H ≅ G/N

type

82. C₇₉

#12902

semidirect

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

subgroup	index
#23765165	2
#23765160	2
#23765159	2
#23763155	2
C₇₉×#2153	2
#12902	79

subgroup

index

C₇₉×#2153

Subgroups — #34204692

Conjugacy classes of subgroups

Complements

Maximal subgroups