Group #34208284 — subgroups

Subgroups — #34208284

Order 20224.

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

normal subgroups only

This group has 252 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	#23765195	C₂	1	1	yes	maximal, characteristic
2	#23765155	C₂	1	1	yes	maximal, characteristic
3	#23764149	C₂	1	2	yes	maximal
4	#23761516	C₂	1	2	yes	maximal
5	C₇₉×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	1	yes	characteristic
7	C₂×#14579097	C₂²	1	2	yes	—
8	#19098590	—	2	2	no	—
9	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098393	—	2	2	no	—
11	Dic₇₉⋊C₁₆	—	2	2	no	—
12	#19098322	—	2	2	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	#427×C₇₉	C₂²	1	2	yes	—
15	C₂×#11929608	—	2	2	no	—
16	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
18	#14579101	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	2	1	no	—
20	#14579097	—	2	2	no	—
21	#14579097	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
23	C₈×Dic₇₉	—	2	1	no	—
24	C₈×Dic₇₉	—	2	1	no	—
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	#117×C₇₉	—	2	2	no	—
27	M₃₂×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
29	#107×C₇₉	—	2	2	no	—
30	#106×C₇₉	Q₈	1	4	yes	—
31	C₂²×Dic₇₉	—	2	1	no	—
32	#11929608	—	4	2	no	—
33	#11929608	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₂×C₇₉⋊C₈	—	2	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₄×Dic₇₉	—	2	1	no	—
38	C₄×Dic₇₉	Q₁₆	1	1	yes	characteristic
39	C₄×Dic₇₉	Q₁₆	1	1	yes	characteristic
40	C₇₉⋊C₁₆	—	4	1	no	—
41	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₇₉	#41	1	2	yes	—
46	C₄⋊C₄×C₇₉	—	2	4	no	—
47	C₂²⋊C₄×C₇₉	—	4	2	no	—
48	C₄²×C₇₉	—	4	2	no	—
49	C₁₆×C₇₉	—	2	2	no	—
50	C₂³×C₇₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₇₉	—	4	4	no	—
52	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
56	C₂×Dic₇₉	—	2	1	no	—
57	C₂×Dic₇₉	—	2	1	no	—
58	C₂×Dic₇₉	—	4	2	no	—
59	C₈×C₇₉	#123	1	1	yes	characteristic
60	C₈×C₇₉	#125	1	2	yes	—
61	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₇₉⋊C₈	—	4	1	no	—
63	C₇₉⋊C₈	—	8	2	no	—
64	C₂²×C₇₉	—	4	1	no	—
65	C₂²×C₇₉	#467	1	1	yes	characteristic
66	C₂²×C₇₉	#478	1	2	yes	—
67	C₄×C₇₉	—	8	4	no	—
68	C₄×C₇₉	#398	1	1	yes	characteristic
69	C₄×C₇₉	#469	1	1	yes	characteristic
70	C₄×C₇₉	#477	1	2	yes	—
71	Dic₇₉	—	2	2	no	—
72	Dic₇₉	—	4	1	no	—
73	#13519	—	79	1	no	maximal, Sylow
74	C₂×C₇₉	—	2	1	no	—
75	C₂×C₇₉	—	2	2	no	—
76	C₂×C₇₉	#2087	1	1	yes	characteristic
77	#2161	—	79	1	no	—
78	#2150	—	79	1	no	—
79	#1945	—	79	2	no	—
80	#1389	—	79	2	no	—
81	#1384	D₇₉	1	1	yes	characteristic
82	C₇₉	#13519	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	79	1	no	—
84	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
85	C₂×#108	—	79	2	no	—
86	#427	C₂×D₇₉	1	2	yes	—
87	#424	—	158	2	no	—
88	C₂×C₄×C₈	—	79	1	no	—
89	#368	—	158	2	no	—
90	C₄⋊C₁₆	—	158	2	no	—
91	#339	—	158	2	no	—
92	C₂×M₁₆	—	158	2	no	—
93	C₂²×C₈	—	79	1	no	—
94	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
95	#117	—	2	2	no	—
96	C₂×C₄²	—	79	1	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	158	2	no	—
99	C₂×C₁₆	—	158	1	no	—
100	C₂×C₁₆	D₃₁₆	1	2	yes	—
101	#108	—	158	2	no	—
102	#108	—	158	2	no	—
103	#107	—	2	2	no	—
104	#106	Dic₁₅₈	1	4	yes	—
105	C₄⋊C₈	—	316	4	no	—
106	C₄×C₈	—	158	1	no	—
107	C₄×C₈	—	158	1	no	—
108	C₂²×C₄	—	158	1	no	—
109	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
110	M₁₆	—	316	2	no	—
111	M₁₆	—	316	2	no	—
112	C₂×C₈	—	316	2	no	—
113	C₂×C₈	—	158	1	no	—
114	C₂×C₈	—	158	2	no	—
115	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
117	C₂×C₈	#11929623	1	2	yes	—
118	C₂×C₈	#11929625	1	1	yes	characteristic
119	C₄⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	—	4	2	no	—
122	C₄²	—	79	1	no	—
123	C₄²	—	79	1	no	—
124	C₄²	—	158	1	no	—
125	C₁₆	—	2	2	no	—
126	C₁₆	—	316	1	no	—
127	C₂³	#14579180	1	1	yes	characteristic
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	316	2	no	—
130	C₂×C₄	—	158	1	no	—
131	C₂×C₄	—	158	1	no	—
132	C₂×C₄	#14579136	1	1	yes	characteristic
133	C₂×C₄	#14579138	1	2	yes	—
134	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
135	C₂×C₄	#14579188	1	2	yes	—
136	C₈	—	316	1	no	—
137	C₈	—	632	2	no	—
138	C₈	#14579145	1	1	yes	characteristic
139	C₈	#14579152	1	2	yes	—
140	C₈	#14579161	1	1	yes	characteristic
141	C₂²	—	4	1	no	—
142	C₂²	#19098805	1	1	yes	characteristic
143	C₂²	#19098861	1	2	yes	—
144	C₄	—	8	4	no	—
145	C₄	—	316	1	no	—
146	C₄	—	158	2	no	—
147	C₄	#19098495	1	1	yes	characteristic
148	C₄	#19098817	1	1	yes	characteristic
149	C₄	#19098856	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	—	2	1	no	—
152	C₂	#23764895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765195

C₂

yes

maximal, characteristic

#23765155

C₂

yes

maximal, characteristic

#23764149

C₂

yes

maximal

#23761516

C₂

yes

maximal

C₇₉×#1384

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

characteristic

C₂×#14579097

C₂²

yes

—

#19098590

—

C₂×C₈×Dic₇₉

C₂²

yes

characteristic

#19098393

—

Dic₇₉⋊C₁₆

—

#19098322

—

C₂×M₃₂×C₇₉

C₂²

yes

characteristic

#427×C₇₉

C₂²

yes

—

C₂×#11929608

—

C₂²×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₇₉

D₄

yes

characteristic

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579097

—

#14579097

—

C₄⋊(C₇₉⋊C₈)

—

C₈×Dic₇₉

—

C₈×Dic₇₉

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

D₄

yes

—

#107×C₇₉

—

#106×C₇₉

Q₈

yes

—

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄×Dic₇₉

—

C₄×Dic₇₉

Q₁₆

yes

characteristic

C₄×Dic₇₉

Q₁₆

yes

characteristic

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#135

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#123

yes

characteristic

C₈×C₇₉

#125

yes

—

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

#467

yes

characteristic

C₂²×C₇₉

#478

yes

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

Dic₇₉

—

Dic₇₉

—

#13519

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

—

—

—

D₇₉

yes

characteristic

C₇₉

#13519

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₉

yes

characteristic

C₂×#108

—

#427

C₂×D₇₉

yes

—

#424

—

158

—

C₂×C₄×C₈

—

#368

—

158

—

C₄⋊C₁₆

—

158

—

#339

—

158

—

C₂×M₁₆

—

158

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₉

yes

characteristic

#117

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

158

—

C₂×C₁₆

—

158

—

100

C₂×C₁₆

D₃₁₆

yes

—

101

#108

—

158

—

102

#108

—

158

—

103

#107

—

104

#106

Dic₁₅₈

yes

—

105

C₄⋊C₈

—

316

—

106

C₄×C₈

—

158

—

107

C₄×C₈

—

158

—

108

C₂²×C₄

—

158

—

109

C₂²×C₄

D₄×D₇₉

yes

characteristic

110

M₁₆

—

316

—

111

M₁₆

—

316

—

112

C₂×C₈

—

316

—

113

C₂×C₈

—

158

—

114

C₂×C₈

—

158

—

115

C₂×C₈

C₂×D₃₁₆

yes

characteristic

116

C₂×C₈

C₂×Dic₁₅₈

yes

—

117

C₂×C₈

#11929623

yes

—

118

C₂×C₈

#11929625

yes

characteristic

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

158

—

125

C₁₆

—

126

C₁₆

—

316

—

127

C₂³

#14579180

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

316

—

130

C₂×C₄

—

158

—

131

C₂×C₄

—

158

—

132

C₂×C₄

#14579136

yes

characteristic

133

C₂×C₄

#14579138

yes

—

134

C₂×C₄

Q₁₆×D₇₉

yes

characteristic

135

C₂×C₄

#14579188

yes

—

136

C₈

—

316

—

137

C₈

—

632

—

138

C₈

#14579145

yes

characteristic

139

C₈

#14579152

yes

—

140

C₈

#14579161

yes

characteristic

141

C₂²

—

142

C₂²

#19098805

yes

characteristic

143

C₂²

#19098861

yes

—

144

C₄

—

145

C₄

—

316

—

146

C₄

—

158

—

147

C₄

#19098495

yes

characteristic

148

C₄

#19098817

yes

characteristic

149

C₄

#19098856

yes

—

150

C₂

—

151

C₂

—

152

C₂

#23764895

yes

characteristic

	subgroup	G/H
center	C₄	#19098817
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1384	C₂
socle	C₂×C₇₉	#2087
prime radical	C₂³×C₇₉	#135
hypercenter	#1384	D₇₉
hypocenter	C₇₉	#13519
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#19098817

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1384

C₂

C₂×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
82. C₇₉	#13519	semidirect

H ≅ G/N

type

82. C₇₉

#13519

semidirect

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

subgroup	index
#23765195	2
#23765155	2
#23764149	2
#23761516	2
C₇₉×#1384	2
#13519	79

subgroup

index

C₇₉×#1384

Subgroups — #34208284

Conjugacy classes of subgroups

Complements

Maximal subgroups