Group #34175341 — subgroups

Subgroups — #34175341

Order 20224.

244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 79 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	#23763450	C₂	1	2	yes	maximal
2	C₇₉×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579097	—	2	2	no	—
4	#19098547	—	2	2	no	—
5	C₇₉⋊#338	C₄	1	4	yes	—
6	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
9	C₂²⋊C₁₆×C₇₉	—	2	2	no	—
10	C₂×#11929608	—	2	4	no	—
11	#14579097	—	4	2	no	—
12	#14579097	—	4	2	no	—
13	#14579080	—	4	2	no	—
14	#14579078	—	4	2	no	—
15	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
16	#131×C₇₉	—	4	1	no	—
17	#131×C₇₉	—	4	1	no	—
18	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₇₉	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	D₄	1	2	yes	—
22	M₃₂×C₇₉	—	4	1	no	—
23	C₂×C₁₆×C₇₉	—	2	1	no	—
24	C₂×C₁₆×C₇₉	—	2	1	no	—
25	C₂×C₁₆×C₇₉	—	4	1	no	—
26	#11929608	—	4	4	no	—
27	#11929608	—	4	4	no	—
28	C₂×C₇₉⋊C₈	—	4	4	no	—
29	#41×C₇₉	—	4	1	no	—
30	#41×C₇₉	—	4	1	no	—
31	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
32	C₂×D₄×C₇₉	SD₁₆	1	2	yes	—
33	C₂×D₄×C₇₉	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	M₁₆×C₇₉	—	4	2	no	—
37	M₁₆×C₇₉	—	4	1	no	—
38	C₂×C₈×C₇₉	—	2	1	no	—
39	C₂×C₈×C₇₉	—	2	1	no	—
40	C₂×C₈×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	—	2	2	no	—
42	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₁₆×C₇₉	—	4	1	no	—
46	C₁₆×C₇₉	—	4	1	no	—
47	C₂³×C₇₉	#103	1	2	yes	—
48	C₂³×C₇₉	#106	1	1	yes	characteristic
49	Q₈×C₇₉	—	4	1	no	—
50	D₄×C₇₉	—	4	2	no	—
51	D₄×C₇₉	—	4	1	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	2	1	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#102	1	2	yes	—
58	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	2	2	no	—
61	C₈×C₇₉	#99	1	1	yes	characteristic
62	C₈×C₇₉	#99	1	1	yes	characteristic
63	C₇₉⋊C₈	—	8	4	no	—
64	C₂²×C₇₉	—	2	2	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	—	4	2	no	—
67	C₂²×C₇₉	—	4	1	no	—
68	C₂²×C₇₉	#340	1	1	yes	characteristic
69	C₄×C₇₉	—	2	2	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₇₉	#342	1	1	yes	characteristic
73	#7229	—	79	1	no	maximal, Sylow
74	C₂×C₇₉	—	2	1	no	—
75	C₂×C₇₉	—	4	2	no	—
76	C₂×C₇₉	—	4	1	no	—
77	C₂×C₇₉	#1289	1	1	yes	characteristic
78	#2113	D₇₉	1	1	yes	characteristic
79	#1813	—	79	2	no	—
80	C₇₉	#7229	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
82	C₂×M₃₂	Dic₇₉	1	1	yes	characteristic
83	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
84	C₂×#108	—	158	2	no	—
85	#413	—	158	2	no	—
86	C₂²⋊C₁₆	—	2	2	no	—
87	#338	—	79	4	no	—
88	C₂×#41	D₃₁₆	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	158	4	no	—
92	C₂×M₁₆	Dic₁₅₈	1	1	yes	characteristic
93	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
94	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	316	2	no	—
101	#108	—	316	2	no	—
102	#101	—	316	2	no	—
103	#100	—	316	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
107	C₂×D₄	#11929613	1	2	yes	—
108	C₂×D₄	Dic₃₁₆	1	1	yes	characteristic
109	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	316	4	no	—
114	M₁₆	—	316	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	316	4	no	—
120	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
122	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#14579086	1	2	yes	—
126	C₂³	#14579093	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
135	C₂×C₄	#14579083	1	2	yes	—
136	C₂×C₄	#14579095	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	632	4	no	—
140	C₈	#14579076	1	1	yes	characteristic
141	C₈	#14579076	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#19098326	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	#19098320	1	1	yes	characteristic
150	C₄	#19098330	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	#23761268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763450

C₂

yes

maximal

C₇₉×#2113

C₂

yes

maximal, characteristic

C₂×#14579097

—

#19098547

—

C₇₉⋊#338

C₄

yes

—

C₂×#131×C₇₉

C₂²

yes

characteristic

C₂×M₃₂×C₇₉

C₄

yes

characteristic

C₂²×C₁₆×C₇₉

C₄

yes

characteristic

C₂²⋊C₁₆×C₇₉

—

—

—

—

—

—

C₂×#41×C₇₉

D₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×M₁₆×C₇₉

Q₈

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

D₄

yes

—

M₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

D₈

yes

characteristic

C₂×D₄×C₇₉

SD₁₆

yes

—

C₂×D₄×C₇₉

Q₁₆

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄²

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₁₆×C₇₉

—

C₁₆×C₇₉

—

C₂³×C₇₉

#103

yes

—

C₂³×C₇₉

#106

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#95

yes

characteristic

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

#107

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

yes

C₈×C₇₉

yes

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#340

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1289

yes

characteristic

#2113

D₇₉

yes

characteristic

#1813

—

C₇₉

#7229

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₉

yes

characteristic

C₂×M₃₂

Dic₇₉

yes

characteristic

C₂²×C₁₆

Dic₇₉

yes

characteristic

C₂×#108

—

158

—

#413

—

158

—

C₂²⋊C₁₆

—

#338

—

C₂×#41

D₃₁₆

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

158

—

C₂×M₁₆

Dic₁₅₈

yes

characteristic

C₂×M₁₆

C₄×D₇₉

yes

—

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

C₂²×C₈

C₂²⋊D₇₉

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

316

—

101

#108

—

316

—

102

#101

—

316

—

103

#100

—

316

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₆₃₂

yes

characteristic

107

C₂×D₄

#11929613

yes

—

108

C₂×D₄

Dic₃₁₆

yes

characteristic

109

C₂²×C₄

(C₂×D₇₉)⋊C₄

yes

—

110

C₂²×C₄

C₄⋊Dic₇₉

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

316

—

114

M₁₆

—

316

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

316

—

120

C₂×C₈

C₄×Dic₇₉

yes

characteristic

121

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

122

C₂×C₈

Dic₇₉⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#14579086

yes

—

126

C₂³

#14579093

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₇₉⋊#95

yes

characteristic

135

C₂×C₄

#14579083

yes

—

136

C₂×C₄

#14579095

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

632

—

140

C₈

#14579076

yes

characteristic

141

C₈

#14579076

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#19098326

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#19098320

yes

characteristic

150

C₄

#19098330

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#23761268

yes

characteristic

	subgroup	G/H
center	C₄	#19098320
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×#131	C₂×D₇₉
Fitting	C₇₉×#2113	C₂
socle	C₂×C₇₉	#1289
prime radical	C₂×#41×C₇₉	D₄
cosocle	C₂×#131×C₇₉	C₂²
hypercenter	#2113	D₇₉
hypocenter	C₇₉	#7229
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#19098320

commutator

C₂×C₈×C₇₉

C₄²

Frattini

C₂×#131

C₂×D₇₉

Fitting

C₇₉×#2113

C₂

socle

C₂×C₇₉

#1289

prime radical

C₂×#41×C₇₉

D₄

cosocle

C₂×#131×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
80. C₇₉	#7229	semidirect

H ≅ G/N

type

80. C₇₉

#7229

semidirect

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

subgroup	index
#23763450	2
C₇₉×#2113	2
#7229	79

subgroup

index

#23763450

C₇₉×#2113

#7229

Subgroups — #34175341

Conjugacy classes of subgroups

Complements

Maximal subgroups