Group #34174861 — subgroups

Subgroups — #34174861

Order 20224.

292 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 131 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	#23763517	C₂	1	2	yes	maximal
2	C₇₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
4	#19098318	—	2	2	no	—
5	#19098318	C₄	1	4	yes	—
6	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
7	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
8	C₄³×C₇₉	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	—	2	4	no	—
10	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
11	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₇₉	—	2	2	no	—
13	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇₉	D₄	1	2	yes	—
16	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
18	C₄⋊C₈×C₇₉	—	4	2	no	—
19	C₄⋊C₈×C₇₉	—	4	2	no	—
20	C₄⋊C₈×C₇₉	—	4	2	no	—
21	C₂×C₇₉⋊C₈	—	4	4	no	—
22	C₂×C₇₉⋊C₈	—	4	4	no	—
23	C₂×C₇₉⋊C₈	—	4	4	no	—
24	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₉	—	4	2	no	—
30	C₂×C₈×C₇₉	—	4	4	no	—
31	C₄²×C₇₉	—	2	1	no	—
32	C₄²×C₇₉	—	2	1	no	—
33	C₄²×C₇₉	—	2	2	no	—
34	C₄²×C₇₉	—	4	2	no	—
35	C₄²×C₇₉	—	4	1	no	—
36	C₄²×C₇₉	D₈	1	2	yes	—
37	C₄²×C₇₉	SD₁₆	1	2	yes	—
38	C₄²×C₇₉	SD₁₆	1	2	yes	—
39	C₄²×C₇₉	Q₁₆	1	2	yes	—
40	C₂³×C₇₉	#95	1	1	yes	characteristic
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	1	no	—
43	C₂×C₄×C₇₉	—	2	2	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	2	1	no	—
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
51	C₂×C₄×C₇₉	#102	1	2	yes	—
52	C₂×C₄×C₇₉	#102	1	2	yes	—
53	C₂×C₄×C₇₉	#103	1	2	yes	—
54	C₂×C₄×C₇₉	#103	1	2	yes	—
55	C₂×C₄×C₇₉	#106	1	2	yes	—
56	C₂×C₄×C₇₉	#107	1	2	yes	—
57	C₂×C₄×C₇₉	#108	1	2	yes	—
58	C₈×C₇₉	—	8	2	no	—
59	C₇₉⋊C₈	—	8	4	no	—
60	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
61	C₂²×C₇₉	#340	1	2	yes	—
62	C₂²×C₇₉	#341	1	1	yes	characteristic
63	C₂²×C₇₉	#343	1	1	yes	characteristic
64	C₂²×C₇₉	#344	1	2	yes	—
65	C₄×C₇₉	—	2	2	no	—
66	C₄×C₇₉	—	2	2	no	—
67	C₄×C₇₉	—	2	2	no	—
68	C₄×C₇₉	—	2	2	no	—
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	4	1	no	—
71	C₄×C₇₉	#331	1	1	yes	characteristic
72	C₄×C₇₉	#332	1	2	yes	—
73	C₄×C₇₉	#333	1	1	yes	characteristic
74	#7040	—	79	1	no	maximal, Sylow
75	C₂×C₇₉	#1286	1	2	yes	—
76	C₂×C₇₉	#1290	1	1	yes	characteristic
77	C₂×C₇₉	#1296	1	1	yes	characteristic
78	C₂×C₇₉	#1301	1	2	yes	—
79	C₂×C₇₉	#1307	1	1	yes	characteristic
80	#1842	D₇₉	1	1	yes	characteristic
81	#1831	—	79	2	no	—
82	C₇₉	#7040	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	2	2	no	—
84	C₂×C₄⋊C₈	—	158	2	no	—
85	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
86	C₄³	C₂×D₇₉	1	1	yes	characteristic
87	#336	—	79	4	no	—
88	#336	—	158	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	158	4	no	—
91	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₇₉	1	2	yes	—
93	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	316	2	no	—
100	C₄⋊C₈	—	316	2	no	—
101	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
102	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
103	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
104	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
105	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	—	316	4	no	—
109	C₂×C₈	—	316	4	no	—
110	C₂×C₈	—	316	4	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	—	4	1	no	—
116	C₄²	#11929609	1	2	yes	—
117	C₄²	#11929611	1	2	yes	—
118	C₄²	#11929612	1	2	yes	—
119	C₄²	#11929614	1	2	yes	—
120	C₂³	C₇₉⋊#95	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#14579078	1	1	yes	characteristic
130	C₂×C₄	#14579080	1	1	yes	characteristic
131	C₂×C₄	#14579081	1	2	yes	—
132	C₂×C₄	#14579082	1	2	yes	—
133	C₂×C₄	#14579084	1	2	yes	—
134	C₂×C₄	#14579085	1	2	yes	—
135	C₂×C₄	#14579092	1	2	yes	—
136	C₂×C₄	#14579094	1	2	yes	—
137	C₂×C₄	#14579097	1	2	yes	—
138	C₈	—	8	2	no	—
139	C₈	—	632	4	no	—
140	C₂²	#19098320	1	1	yes	characteristic
141	C₂²	#19098325	1	2	yes	—
142	C₂²	#19098328	1	1	yes	characteristic
143	C₂²	#19098332	1	1	yes	characteristic
144	C₂²	#19098335	1	2	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	#19098307	1	1	yes	characteristic
152	C₄	#19098309	1	2	yes	—
153	C₄	#19098312	1	1	yes	characteristic
154	C₂	#23761260	1	2	yes	—
155	C₂	#23761270	1	1	yes	characteristic
156	C₂	#23761283	1	1	yes	characteristic
157	C₂	#23761297	1	2	yes	—
158	C₂	#23761311	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763517

C₂

yes

maximal

C₇₉×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

#19098318

—

#19098318

C₄

yes

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄⋊C₈×C₇₉

C₄

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

Q₈

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄²

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

D₈

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₂³×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

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

#102

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

#103

yes

—

C₂×C₄×C₇₉

#106

yes

—

C₂×C₄×C₇₉

#107

yes

—

C₂×C₄×C₇₉

#108

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

H(3,ℤ/4)

yes

characteristic

C₂²×C₇₉

#340

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#331

yes

characteristic

C₄×C₇₉

#332

yes

—

C₄×C₇₉

yes

—

C₂×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1307

yes

characteristic

#1842

D₇₉

yes

characteristic

#1831

—

C₇₉

#7040

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

158

—

C₂×C₄⋊C₈

Dic₇₉

yes

—

C₄³

C₂×D₇₉

yes

characteristic

#336

—

#336

—

158

—

C₂²×C₈

—

C₂²×C₈

—

158

—

C₂×C₄²

Dic₁₅₈

yes

characteristic

C₂×C₄²

C₄×D₇₉

yes

—

C₂×C₄²

D₃₁₆

yes

characteristic

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

C₂×C₄²

C₂²⋊D₇₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

316

—

100

C₄⋊C₈

—

316

—

101

C₂²×C₄

C₄×Dic₇₉

yes

characteristic

102

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

103

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

104

C₂²×C₄

Dic₇₉⋊C₄

yes

—

105

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

316

—

109

C₂×C₈

—

316

—

110

C₂×C₈

—

316

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

#11929609

yes

—

117

C₄²

#11929611

yes

—

118

C₄²

#11929612

yes

—

119

C₄²

#11929614

yes

—

120

C₂³

C₇₉⋊#95

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#14579078

yes

characteristic

130

C₂×C₄

#14579080

yes

characteristic

131

C₂×C₄

#14579081

yes

—

132

C₂×C₄

#14579082

yes

—

133

C₂×C₄

#14579084

yes

—

134

C₂×C₄

#14579085

yes

—

135

C₂×C₄

#14579092

yes

—

136

C₂×C₄

#14579094

yes

—

137

C₂×C₄

#14579097

yes

—

138

C₈

—

139

C₈

—

632

—

140

C₂²

#19098320

yes

characteristic

141

C₂²

#19098325

yes

—

142

C₂²

#19098328

yes

characteristic

143

C₂²

#19098332

yes

characteristic

144

C₂²

#19098335

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#19098307

yes

characteristic

152

C₄

#19098309

yes

—

153

C₄

#19098312

yes

characteristic

154

C₂

#23761260

yes

—

155

C₂

#23761270

yes

characteristic

156

C₂

#23761283

yes

characteristic

157

C₂

#23761297

yes

—

158

C₂

#23761311

yes

characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂²×C₄×C₇₉	C₄²
Frattini	C₄³	C₂×D₇₉
Fitting	C₇₉×#1842	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	#1842	D₇₉
hypocenter	C₇₉	#7040
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂³

C₇₉⋊#95

commutator

C₂²×C₄×C₇₉

C₄²

Frattini

C₄³

C₂×D₇₉

Fitting

C₇₉×#1842

C₂

socle, prime radical

C₂³×C₇₉

#95

cosocle

C₄³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
82. C₇₉	#7040	semidirect

H ≅ G/N

type

82. C₇₉

#7040

semidirect

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

subgroup	index
#23763517	2
C₇₉×#1842	2
#7040	79

subgroup

index

#23763517

C₇₉×#1842

#7040

Subgroups — #34174861

Conjugacy classes of subgroups

Complements

Maximal subgroups