Group #34199941 — subgroups

Subgroups — #34199941

Order 20224.

218 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23765203	C₂	1	1	yes	maximal, characteristic
2	#23762156	C₂	1	1	yes	maximal, characteristic
3	#23761465	C₂	1	4	yes	maximal, complemented
4	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19098587	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
8	#19098339	C₂²	1	2	yes	—
9	#19098314	—	2	1	no	—
10	#19098314	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#334×C₇₉	—	2	1	no	—
14	#334×C₇₉	C₂²	1	2	yes	—
15	C₂×#11929608	—	4	1	no	—
16	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₇₉	—	4	1	no	—
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	—	2	2	no	—
24	C₄⋊C₈×C₇₉	—	4	1	no	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	#11929608	—	8	2	no	—
30	C₂×C₇₉⋊C₈	—	4	2	no	—
31	C₇₉⋊C₁₆	—	2	4	no	—
32	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₇₉	—	2	2	no	—
36	C₂×C₈×C₇₉	—	4	1	no	—
37	C₂×C₈×C₇₉	—	4	2	no	—
38	C₂×C₈×C₇₉	—	8	2	no	—
39	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
45	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
46	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂³×C₇₉	#106	1	1	yes	characteristic
48	C₂×C₄×C₇₉	—	2	2	no	—
49	C₂×C₄×C₇₉	—	2	2	no	—
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₈×C₇₉	—	2	1	no	—
59	C₈×C₇₉	—	2	1	no	—
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	8	2	no	—
62	C₈×C₇₉	#131	1	2	yes	—
63	C₇₉⋊C₈	—	8	2	no	—
64	C₂²×C₇₉	—	2	1	no	—
65	C₂²×C₇₉	—	2	1	no	—
66	C₂²×C₇₉	#365	1	2	yes	—
67	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
68	C₄×C₇₉	—	2	1	no	—
69	C₄×C₇₉	—	2	1	no	—
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	#365	1	2	yes	—
73	C₄×C₇₉	#424	1	1	yes	characteristic
74	C₄×C₇₉	#430	1	1	yes	characteristic
75	#12030	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	—	2	2	no	—
77	C₂×C₇₉	#1570	1	1	yes	characteristic
78	C₂×C₇₉	C₂×#365	1	1	yes	characteristic
79	C₂×C₇₉	C₂×#365	1	1	yes	characteristic
80	#2163	—	79	1	no	—
81	#1566	—	79	1	no	—
82	C₂×#334	D₇₉	1	1	yes	characteristic
83	#1368	—	79	4	no	—
84	C₇₉	#12030	1	1	yes	characteristic, complemented, Sylow
85	#423	—	158	1	no	—
86	C₂×C₄⋊C₈	—	2	1	no	—
87	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
88	C₄⋊C₁₆	—	79	2	no	—
89	C₂²⋊C₁₆	—	79	2	no	—
90	#346	—	79	2	no	—
91	#334	—	2	1	no	—
92	#334	—	79	2	no	—
93	#334	—	158	1	no	—
94	#334	C₂×D₇₉	1	2	yes	—
95	C₂×M₁₆	—	316	1	no	—
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
98	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
100	C₂×C₁₆	—	79	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	316	1	no	—
104	C₄⋊C₈	—	158	2	no	—
105	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
106	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#11929611	1	1	yes	characteristic
111	C₂²×C₄	#11929612	1	1	yes	characteristic
112	M₁₆	—	632	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	316	2	no	—
118	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
119	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
122	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
123	C₄²	#11929611	1	1	yes	characteristic
124	C₄²	#11929612	1	1	yes	characteristic
125	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
126	C₁₆	—	158	4	no	—
127	C₂³	#14579092	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	#14579092	1	1	yes	characteristic
133	C₂×C₄	#14579092	1	1	yes	characteristic
134	C₂×C₄	#14579092	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
137	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	632	2	no	—
143	C₈	#14579167	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#19098387	1	2	yes	—
147	C₂²	C₂×#14579092	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#19098385	1	2	yes	—
153	C₄	#19098589	1	1	yes	characteristic
154	C₄	#19098611	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#23762170	1	1	yes	characteristic
157	C₂	C₂×#19098385	1	1	yes	characteristic
158	C₂	C₂×#19098387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765203

C₂

yes

maximal, characteristic

#23762156

C₂

yes

maximal, characteristic

#23761465

C₂

yes

maximal, complemented

C₂×#334×C₇₉

C₂

yes

maximal, characteristic

#19098587

—

C₄⋊(C₇₉⋊C₁₆)

C₄

yes

—

C₂²⋊(C₇₉⋊C₁₆)

C₄

yes

—

#19098339

C₂²

yes

—

#19098314

—

#19098314

C₂²

yes

—

C₂×C₄⋊C₈×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#334×C₇₉

—

#334×C₇₉

C₂²

yes

—

C₂×#11929608

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

D₄

yes

characteristic

C₄×C₈×C₇₉

Q₈

yes

characteristic

C₄×C₈×C₇₉

C₂³

yes

characteristic

#11929608

—

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂²×C₄×C₇₉

SD₁₆

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

C₄⋊C₄

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

SD₁₆

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₂³×C₇₉

#106

yes

characteristic

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

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#131

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#365

yes

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#365

yes

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

—

D₇₉

yes

characteristic

#1368

—

C₇₉

#12030

yes

characteristic, complemented, Sylow

#423

—

158

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

158

—

#334

C₂×D₇₉

yes

—

C₂×M₁₆

—

316

—

C₂²×C₈

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₄×D₇₉

yes

characteristic

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

100

C₂×C₁₆

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

316

—

104

C₄⋊C₈

—

158

—

105

C₄×C₈

Dic₁₅₈

yes

characteristic

106

C₄×C₈

C₄×D₇₉

yes

characteristic

107

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

108

C₄×C₈

C₂²×D₇₉

yes

characteristic

109

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

110

C₂²×C₄

#11929611

yes

characteristic

111

C₂²×C₄

#11929612

yes

characteristic

112

M₁₆

—

632

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

316

—

118

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

119

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

120

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

121

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

122

C₄²

Dic₇₉⋊C₄

yes

characteristic

123

C₄²

#11929611

yes

characteristic

124

C₄²

#11929612

yes

characteristic

125

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

126

C₁₆

—

158

—

127

C₂³

#14579092

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#14579092

yes

characteristic

133

C₂×C₄

#14579092

yes

characteristic

134

C₂×C₄

#14579092

yes

characteristic

135

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

136

C₂×C₄

C₂×#11929611

yes

characteristic

137

C₂×C₄

C₂×#11929612

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

632

—

143

C₈

#14579167

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

#19098387

yes

—

147

C₂²

C₂×#14579092

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#19098385

yes

—

153

C₄

#19098589

yes

characteristic

154

C₄

#19098611

yes

characteristic

155

C₂

—

156

C₂

#23762170

yes

characteristic

157

C₂

C₂×#19098385

yes

characteristic

158

C₂

C₂×#19098387

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579092
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#12030
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂²

C₂×#14579092

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₂×#334×C₇₉

C₂

C₂²×C₇₉

C₂³×C₇₉

C₄×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

N	H ≅ G/N	type
3. #23761465	C₂	semidirect
84. C₇₉	#12030	semidirect

H ≅ G/N

type

3. #23761465

C₂

semidirect

84. C₇₉

#12030

semidirect

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

subgroup	index
#23765203	2
#23762156	2
#23761465	2
C₂×#334×C₇₉	2
#12030	79

subgroup

index

C₂×#334×C₇₉

Subgroups — #34199941

Conjugacy classes of subgroups

Complements

Maximal subgroups