Group #34199907 — subgroups

Subgroups — #34199907

Order 20224.

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

normal subgroups only

This group has 216 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	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
2	#23762163	C₂	1	1	yes	maximal, characteristic
3	#23761465	C₂	1	4	yes	maximal, complemented
4	C₇₉×#1568	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₄	1	2	yes	—
6	#19098620	—	2	1	no	—
7	#19098339	C₂²	1	2	yes	—
8	C₄×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₇₉⋊C₁₆	C₄	1	1	yes	characteristic
10	#19098316	—	2	1	no	—
11	#19098314	C₂²	1	2	yes	—
12	#432×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#335×C₇₉	—	2	1	no	—
15	#334×C₇₉	C₂²	1	2	yes	—
16	#14579101	—	2	4	no	—
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₉⋊C₈)	—	4	1	no	—
21	#14579072	—	4	1	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₇₉	—	2	2	no	—
26	C₂²⋊C₈×C₇₉	—	4	1	no	—
27	#97×C₇₉	—	4	1	no	—
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₇₉⋊C₈	—	4	2	no	—
33	C₇₉⋊C₁₆	—	2	2	no	—
34	C₇₉⋊C₁₆	M₁₆	1	4	yes	—
35	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	—	4	2	no	—
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
47	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂³×C₇₉	#106	1	1	yes	characteristic
50	C₂×C₄×C₇₉	—	2	1	no	—
51	C₂×C₄×C₇₉	—	2	1	no	—
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
55	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
60	C₈×C₇₉	—	2	1	no	—
61	C₈×C₇₉	—	2	1	no	—
62	C₈×C₇₉	—	8	2	no	—
63	C₈×C₇₉	#108	1	2	yes	—
64	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
65	C₇₉⋊C₈	—	8	2	no	—
66	C₂²×C₇₉	—	2	1	no	—
67	C₂²×C₇₉	—	2	1	no	—
68	C₂²×C₇₉	—	2	1	no	—
69	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
70	C₄×C₇₉	—	2	1	no	—
71	C₄×C₇₉	—	2	1	no	—
72	C₄×C₇₉	—	2	1	no	—
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	#423	1	1	yes	characteristic
76	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
77	#12021	—	79	1	no	maximal, Sylow
78	C₂×C₇₉	—	4	1	no	—
79	C₂×C₇₉	#1566	1	1	yes	characteristic
80	C₂×C₇₉	#2155	1	1	yes	characteristic
81	C₂×C₇₉	#2155	1	1	yes	characteristic
82	C₄×M₃₂	—	79	1	no	—
83	#1568	—	79	1	no	—
84	#1568	D₇₉	1	1	yes	characteristic
85	#1368	—	79	4	no	—
86	C₇₉	#12021	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₃₂	—	79	2	no	—
88	#432	—	2	1	no	—
89	#432	—	158	1	no	—
90	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
91	#346	—	79	2	no	—
92	C₄×C₁₆	—	79	1	no	—
93	C₄×C₁₆	—	79	1	no	—
94	#335	—	2	1	no	—
95	#335	—	158	1	no	—
96	#334	—	79	2	no	—
97	#334	C₂×D₇₉	1	2	yes	—
98	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
99	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
101	M₃₂	—	158	4	no	—
102	C₂×C₁₆	—	79	2	no	—
103	C₂×C₁₆	—	79	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	158	2	no	—
106	C₂²⋊C₈	—	4	1	no	—
107	C₂²⋊C₈	—	316	1	no	—
108	#97	—	4	1	no	—
109	#97	—	316	1	no	—
110	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
111	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
114	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
115	C₂²×C₄	#11929611	1	1	yes	characteristic
116	C₂²×C₄	#11929612	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	316	2	no	—
121	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
122	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
124	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
125	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
126	C₄²	#11929611	1	1	yes	characteristic
127	C₄²	#11929612	1	1	yes	characteristic
128	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
129	C₁₆	—	79	4	no	—
130	C₁₆	—	158	2	no	—
131	C₂³	#14579092	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	#14579092	1	1	yes	characteristic
137	C₂×C₄	#14579092	1	1	yes	characteristic
138	C₂×C₄	#14579092	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
141	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	8	2	no	—
145	C₈	—	632	2	no	—
146	C₈	#14579096	1	2	yes	—
147	C₈	M₁₆×D₇₉	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	C₂×#14579092	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#19098586	1	1	yes	characteristic
158	C₄	C₂×#14579096	1	1	yes	characteristic
159	C₂	—	4	1	no	—
160	C₂	#23762155	1	1	yes	characteristic
161	C₂	#23765172	1	1	yes	characteristic
162	C₂	#23765174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14579101

C₂

yes

maximal, characteristic

#23762163

C₂

yes

maximal, characteristic

#23761465

C₂

yes

maximal, complemented

C₇₉×#1568

C₂

yes

maximal, characteristic

C₂×#14579101

C₄

yes

—

#19098620

—

#19098339

C₂²

yes

—

C₄×C₇₉⋊C₁₆

C₄

yes

characteristic

C₄×C₇₉⋊C₁₆

C₄

yes

characteristic

#19098316

—

#19098314

C₂²

yes

—

#432×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#335×C₇₉

—

#334×C₇₉

C₂²

yes

—

#14579101

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

Q₈

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×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

C₂×C₇₉⋊C₈

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

M₁₆

yes

—

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

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

#108

yes

—

C₈×C₇₉

C₂×M₁₆

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×#106

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

—

C₂×C₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

C₄×M₃₂

—

#1568

—

#1568

D₇₉

yes

characteristic

#1368

—

C₇₉

#12021

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#432

—

#432

—

158

—

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

#335

—

158

—

#334

—

#334

C₂×D₇₉

yes

—

C₂²×C₈

Dic₁₅₈

yes

characteristic

C₂²×C₈

C₄×D₇₉

yes

characteristic

100

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

101

M₃₂

—

158

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

158

—

106

C₂²⋊C₈

—

107

C₂²⋊C₈

—

316

—

108

#97

—

109

#97

—

316

—

110

C₄×C₈

Dic₁₅₈

yes

characteristic

111

C₄×C₈

C₄×D₇₉

yes

characteristic

112

C₄×C₈

C₂²⋊D₇₉

yes

characteristic

113

C₄×C₈

C₂²×D₇₉

yes

characteristic

114

C₂²×C₄

Dic₇₉⋊C₄

yes

characteristic

115

C₂²×C₄

#11929611

yes

characteristic

116

C₂²×C₄

#11929612

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

316

—

121

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

122

C₂×C₈

Dic₇₉⋊C₄

yes

characteristic

123

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

124

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

125

C₄²

Dic₇₉⋊C₄

yes

characteristic

126

C₄²

#11929611

yes

characteristic

127

C₄²

#11929612

yes

characteristic

128

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

129

C₁₆

—

130

C₁₆

—

158

—

131

C₂³

#14579092

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14579092

yes

characteristic

137

C₂×C₄

#14579092

yes

characteristic

138

C₂×C₄

#14579092

yes

characteristic

139

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

140

C₂×C₄

C₂×#11929611

yes

characteristic

141

C₂×C₄

C₂×#11929612

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

632

—

146

C₈

#14579096

yes

—

147

C₈

M₁₆×D₇₉

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

C₂×#14579092

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#19098586

yes

characteristic

158

C₄

C₂×#14579096

yes

characteristic

159

C₂

—

160

C₂

#23762155

yes

characteristic

161

C₂

#23765172

yes

characteristic

162

C₂

#23765174

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14579092
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₇₉×#1568	C₂
socle	C₂²×C₇₉	C₂×#106
prime radical	C₂³×C₇₉	#106
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	#1568	D₇₉
hypocenter	C₇₉	#12021
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1568

subgroup

G/H

center

C₂²

C₂×#14579092

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₇₉×#1568

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 ◁ #1568

N	H ≅ G/N	type
3. #23761465	C₂	semidirect
86. C₇₉	#12021	semidirect

H ≅ G/N

type

3. #23761465

C₂

semidirect

86. C₇₉

#12021

semidirect

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

subgroup	index
C₄×#14579101	2
#23762163	2
#23761465	2
C₇₉×#1568	2
#12021	79

subgroup

index

C₇₉×#1568

Subgroups — #34199907

Conjugacy classes of subgroups

Complements

Maximal subgroups