Group #34182029 — subgroups

Subgroups — #34182029

Order 20224.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#19098316	C₂	1	2	yes	maximal
3	C₂×#19098314	C₂	1	2	yes	maximal
4	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
6	#19098318	—	2	4	no	—
7	#19098316	—	2	4	no	—
8	#19098314	—	2	4	no	—
9	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
11	C₄³×C₇₉	C₂²	1	1	yes	characteristic
12	#336×C₇₉	C₄	1	4	yes	—
13	C₂²×C₇₉⋊C₈	—	2	4	no	—
14	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
15	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
16	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₇₉	D₄	1	2	yes	—
20	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₇₉	D₄	1	2	yes	—
23	C₄×C₈×C₇₉	D₄	1	2	yes	—
24	C₄×C₈×C₇₉	Q₈	1	2	yes	—
25	C₄×C₈×C₇₉	Q₈	1	2	yes	—
26	C₂×C₇₉⋊C₈	—	4	4	no	—
27	C₂×C₇₉⋊C₈	—	4	8	no	—
28	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
29	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₇₉	—	2	4	no	—
33	C₂×C₈×C₇₉	—	2	4	no	—
34	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
36	C₄²×C₇₉	—	2	4	no	—
37	C₄²×C₇₉	—	2	4	no	—
38	C₄²×C₇₉	—	4	2	no	—
39	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	2	2	no	—
45	C₂×C₄×C₇₉	—	2	4	no	—
46	C₂×C₄×C₇₉	—	2	4	no	—
47	C₂×C₄×C₇₉	—	2	4	no	—
48	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#131	1	4	yes	—
55	C₂×C₄×C₇₉	#135	1	4	yes	—
56	C₈×C₇₉	—	2	4	no	—
57	C₈×C₇₉	—	2	4	no	—
58	C₇₉⋊C₈	—	8	4	no	—
59	C₂²×C₇₉	#384	1	1	yes	characteristic
60	C₂²×C₇₉	#424	1	2	yes	—
61	C₂²×C₇₉	#427	1	2	yes	—
62	C₂²×C₇₉	#430	1	2	yes	—
63	C₄×C₇₉	—	2	1	no	—
64	C₄×C₇₉	—	2	1	no	—
65	C₄×C₇₉	—	4	4	no	—
66	C₄×C₇₉	#424	1	2	yes	—
67	C₄×C₇₉	#424	1	2	yes	—
68	C₄×C₇₉	#495	1	2	yes	—
69	C₄×C₇₉	#499	1	2	yes	—
70	#8811	—	79	1	no	maximal, Sylow
71	C₂×C₇₉	#1572	1	4	yes	—
72	C₂×C₇₉	#1843	1	1	yes	characteristic
73	C₂×C₇₉	#1846	1	1	yes	characteristic
74	C₂×C₇₉	#1849	1	1	yes	characteristic
75	#1831	—	79	2	no	—
76	#1744	D₇₉	1	1	yes	characteristic
77	C₂×#335	—	79	2	no	—
78	C₂×#334	—	79	2	no	—
79	C₇₉	#8811	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄⋊C₈	—	79	4	no	—
81	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
83	C₄³	C₂×D₇₉	1	1	yes	characteristic
84	#336	—	158	4	no	—
85	#336	Dic₇₉	1	4	yes	—
86	#335	—	158	4	no	—
87	#334	—	158	4	no	—
88	C₂²×C₈	—	158	4	no	—
89	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
90	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
91	C₂×C₄²	Dic₁₅₈	1	2	yes	—
92	C₂×C₄²	D₃₁₆	1	2	yes	—
93	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
94	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
95	C₄⋊C₈	—	158	4	no	—
96	C₄⋊C₈	—	158	4	no	—
97	C₄×C₈	Dic₁₅₈	1	2	yes	—
98	C₄×C₈	Dic₁₅₈	1	2	yes	—
99	C₄×C₈	D₃₁₆	1	2	yes	—
100	C₄×C₈	D₃₁₆	1	2	yes	—
101	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
102	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
103	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
104	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	316	4	no	—
108	C₂×C₈	—	316	8	no	—
109	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
110	C₂×C₈	C₄⋊Dic₇₉	1	4	yes	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	4	2	no	—
114	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
115	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
116	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
117	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
118	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
125	C₂×C₄	#14579158	1	1	yes	characteristic
126	C₂×C₄	#14579161	1	1	yes	characteristic
127	C₂×C₄	#14579161	1	1	yes	characteristic
128	C₂×C₄	#14579161	1	1	yes	characteristic
129	C₂×C₄	#14579168	1	4	yes	—
130	C₂×C₄	#14579183	1	4	yes	—
131	C₈	—	2	4	no	—
132	C₈	—	2	4	no	—
133	C₈	—	632	4	no	—
134	C₂²	#19098433	1	1	yes	characteristic
135	C₂²	#19098590	1	2	yes	—
136	C₂²	#19098601	1	2	yes	—
137	C₂²	#19098612	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#19098590	1	2	yes	—
142	C₄	#19098590	1	2	yes	—
143	C₄	#19098947	1	2	yes	—
144	C₄	#19098965	1	2	yes	—
145	C₂	#23762180	1	4	yes	—
146	C₂	#23763554	1	1	yes	characteristic
147	C₂	#23763564	1	1	yes	characteristic
148	C₂	#23763575	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763517

C₂

yes

maximal

C₂×#19098316

C₂

yes

maximal

C₂×#19098314

C₂

yes

maximal

C₇₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

C₂²

yes

—

#19098318

—

#19098316

—

#19098314

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₄³×C₇₉

C₂²

yes

characteristic

#336×C₇₉

C₄

yes

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

Q₈

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₄×C₈×C₇₉

Q₈

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#384

yes

characteristic

C₂²×C₇₉

#424

yes

—

C₂²×C₇₉

#427

yes

—

C₂²×C₇₉

#430

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#424

yes

—

C₄×C₇₉

#424

yes

—

C₄×C₇₉

#495

yes

—

C₄×C₇₉

#499

yes

—

#8811

—

maximal, Sylow

C₂×C₇₉

#1572

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

D₇₉

yes

characteristic

C₂×#335

—

C₂×#334

—

C₇₉

#8811

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄³

C₂×D₇₉

yes

characteristic

#336

—

158

—

#336

Dic₇₉

yes

—

#335

—

158

—

#334

—

158

—

C₂²×C₈

—

158

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂²×C₈

C₂×Dic₇₉

yes

—

C₂×C₄²

Dic₁₅₈

yes

—

C₂×C₄²

D₃₁₆

yes

—

C₂×C₄²

C₂×Dic₇₉

yes

—

C₂×C₄²

C₂²×D₇₉

yes

characteristic

C₄⋊C₈

—

158

—

C₄⋊C₈

—

158

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

Dic₁₅₈

yes

—

C₄×C₈

D₃₁₆

yes

—

100

C₄×C₈

D₃₁₆

yes

—

101

C₂²×C₄

C₄⋊Dic₇₉

yes

—

102

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

103

C₂²×C₄

C₂×D₃₁₆

yes

characteristic

104

C₂²×C₄

C₂×Dic₁₅₈

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

316

—

108

C₂×C₈

—

316

—

109

C₂×C₈

C₄⋊Dic₇₉

yes

—

110

C₂×C₈

C₄⋊Dic₇₉

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×D₃₁₆

yes

characteristic

115

C₄²

C₂×D₃₁₆

yes

characteristic

116

C₄²

C₂×Dic₁₅₈

yes

characteristic

117

C₄²

C₂×Dic₁₅₈

yes

characteristic

118

C₂³

C₂×C₄⋊Dic₇₉

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

124

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

125

C₂×C₄

#14579158

yes

characteristic

126

C₂×C₄

#14579161

yes

characteristic

127

C₂×C₄

#14579161

yes

characteristic

128

C₂×C₄

#14579161

yes

characteristic

129

C₂×C₄

#14579168

yes

—

130

C₂×C₄

#14579183

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

632

—

134

C₂²

#19098433

yes

characteristic

135

C₂²

#19098590

yes

—

136

C₂²

#19098601

yes

—

137

C₂²

#19098612

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#19098590

yes

—

142

C₄

#19098590

yes

—

143

C₄

#19098947

yes

—

144

C₄

#19098965

yes

—

145

C₂

#23762180

yes

—

146

C₂

#23763554

yes

characteristic

147

C₂

#23763564

yes

characteristic

148

C₂

#23763575

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#8811
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₁₆

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1744

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
79. C₇₉	#8811	semidirect

H ≅ G/N

type

79. C₇₉

#8811

semidirect

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

subgroup	index
#23763517	2
C₂×#19098316	2
C₂×#19098314	2
C₇₉×#1744	2
#8811	79

subgroup

index

C₇₉×#1744

Subgroups — #34182029

Conjugacy classes of subgroups

Complements

Maximal subgroups