Finite Group Databasebeta

← C22×#568×C79 #34174763

Subgroups — C22×#568×C79 #34174763

Order 20224.

11118 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C79C211yesmaximal, characteristic
2C2×#568×C79C2148yesmaximal, direct product factor
3C22×#131×C79C218yesmaximal, complemented
4C23×M16×C79C216yesmaximal, complemented
5C22×#41×C79C414yes—
6C22×#41×C79C4124yes—
7C22×#41×C79C2214yes—
8C22×#41×C79C22124yes—
9C23×Q8×C79C411yescharacteristic
10C23×D4×C79C413yes—
11C24×C4×C79C2213yes—
12#568×C79C221256yesdirect product factor
13C2×#131×C79C221192yescomplemented
14C22×M16×C79C2214yes—
15C22×M16×C79C22112yescomplemented
16C22×M16×C79C221144yescomplemented
17C23×C8×C79C22112yescomplemented
18C25×C79C2×C413yes—
19C2×#41×C79C2×C4148yes—
20C2×#41×C79C2×C4196yes—
21C2×#41×C79C2×C41192yes—
22C2×#41×C79C23148yes—
23C2×#41×C79C23164yes—
24C22×Q8×C79C2×C414yes—
25C22×Q8×C79C2×C4124yes—
26C22×D4×C79C2×C4112yes—
27C22×D4×C79C2×C4172yes—
28C23×C4×C79C2×C413yes—
29C23×C4×C79C2×C416yes—
30C23×C4×C79C2×C4136yes—
31C23×C4×C79C23136yes—
32C23×C4×C79C2316yes—
33C23×C4×C79C2311yescharacteristic
34#131×C79C231512yescomplemented
35C2×M16×C79C23148yes—
36C2×M16×C79C231144yescomplemented
37C2×M16×C79C231384yescomplemented
38C22×C8×C79C2318yes—
39C22×C8×C79C231144yescomplemented
40C24×C79—412no—
41C24×C79C22×C411yescharacteristic
42C24×C79C22×C416yes—
43C24×C79C22×C4136yes—
44#41×C79C22×C41192yes—
45#41×C79C22×C41256yes—
46#41×C79C24164yes—
47C2×Q8×C79C22×C4148yes—
48C2×Q8×C79C22×C4164yes—
49C2×D4×C79C22×C41144yes—
50C2×D4×C79C22×C41192yes—
51C22×C4×C79C22×C411yescharacteristic
52C22×C4×C79C22×C416yes—
53C22×C4×C79C22×C4136yes—
54C22×C4×C79C22×C416yes—
55C22×C4×C79C22×C4172yes—
56C22×C4×C79C22×C41144yes—
57C22×C4×C79C22×C4136yes—
58C22×C4×C79C24136yes—
59C22×C4×C79C2416yes—
60C22×C4×C79C24148yes—
61C22×C4×C79C2411yescharacteristic
62M16×C79C24164yes—
63M16×C79C241192yescomplemented
64C2×C8×C79C24148yes—
65C2×C8×C79C241192yescomplemented
66C23×C79—24no—
67C23×C79—224no—
68C23×C79—472no—
69C23×C79C23×C416yes—
70C23×C79C23×C4136yes—
71C23×C79C23×C4148yes—
72C23×C79C23×C411yescharacteristic
73Q8×C79C23×C4164yes—
74D4×C79C23×C41192yes—
75C2×C4×C79C23×C4112yes—
76C2×C4×C79C23×C416yes—
77C2×C4×C79C23×C413yes—
78C2×C4×C79C23×C4148yes—
79C2×C4×C79C23×C4136yes—
80C2×C4×C79C23×C4172yes—
81C2×C4×C79C23×C4196yes—
82C2×C4×C79C2513yes—
83C2×C4×C79C25124yes—
84C2×C4×C79C2514yes—
85C8×C79C25132yes—
86C22×C79—212no—
87C22×C79—272no—
88C22×C79—448no—
89C22×C79#56814yesdirect product factor
90C22×C79C24×C414yes—
91C22×C79C24×C4124yes—
92C22×C79C24×C413yes—
93C4×C79C24×C414yes—
94C4×C79C24×C413yes—
95C4×C79C24×C4124yes—
96C4×C79C2611yescharacteristic
97C22×#568C7911yesmaximal, characteristic, direct product factor, Sylow
98C2×C79—24no—
99C2×C79—224no—
100C2×C79C2×#56816yesdirect product factor
101C2×C79C25×C411yescharacteristic
102C23×#41C2×C7911yescharacteristic
103C2×#568C2×C79148yesdirect product factor
104C22×#131C2×C7918yescomplemented
105C23×M16C2×C7916yescomplemented
106C79C22×#56811yescharacteristic, direct product factor, Sylow
107C22×#41C4×C7914yes—
108C22×#41C4×C79124yes—
109C22×#41C22×C7914yes—
110C22×#41C22×C79124yes—
111C23×Q8C4×C7911yescharacteristic
112C23×D4C4×C7913yes—
113C24×C4C22×C7913yes—
114#568C22×C791256yesdirect product factor
115C2×#131C22×C791192yescomplemented
116C22×M16C22×C7914yes—
117C22×M16C22×C79112yescomplemented
118C22×M16C22×C791144yescomplemented
119C23×C8C22×C79112yescomplemented
120C25C2×C4×C7913yes—
121C2×#41C2×C4×C79148yes—
122C2×#41C2×C4×C79196yes—
123C2×#41C2×C4×C791192yes—
124C2×#41C23×C79148yes—
125C2×#41C23×C79164yes—
126C22×Q8C2×C4×C7914yes—
127C22×Q8C2×C4×C79124yes—
128C22×D4C2×C4×C79112yes—
129C22×D4C2×C4×C79172yes—
130C23×C4C2×C4×C7913yes—
131C23×C4C2×C4×C7916yes—
132C23×C4C2×C4×C79136yes—
133C23×C4C23×C7916yes—
134C23×C4C23×C7911yescharacteristic
135C23×C4C23×C79136yes—
136#131C23×C791512yescomplemented
137C2×M16C23×C79148yes—
138C2×M16C23×C791144yescomplemented
139C2×M16C23×C791384yescomplemented
140C22×C8C23×C7918yes—
141C22×C8C23×C791144yescomplemented
142C24—412no—
143C24C22×C4×C7911yescharacteristic
144C24C22×C4×C7916yes—
145C24C22×C4×C79136yes—
146#41C22×C4×C791192yes—
147#41C22×C4×C791256yes—
148#41C24×C79164yes—
149C2×Q8C22×C4×C79148yes—
150C2×Q8C22×C4×C79164yes—
151C2×D4C22×C4×C791144yes—
152C2×D4C22×C4×C791192yes—
153C22×C4C22×C4×C7916yes—
154C22×C4C22×C4×C7911yescharacteristic
155C22×C4C22×C4×C79136yes—
156C22×C4C22×C4×C7916yes—
157C22×C4C22×C4×C79136yes—
158C22×C4C22×C4×C79172yes—
159C22×C4C22×C4×C791144yes—
160C22×C4C24×C7911yescharacteristic
161C22×C4C24×C79136yes—
162C22×C4C24×C7916yes—
163C22×C4C24×C79148yes—
164M16C24×C79164yes—
165M16C24×C791192yescomplemented
166C2×C8C24×C79148yes—
167C2×C8C24×C791192yescomplemented
168C23—24no—
169C23—224no—
170C23—472no—
171C23C23×C4×C7916yes—
172C23C23×C4×C79136yes—
173C23C23×C4×C79148yes—
174C23C23×C4×C7911yescharacteristic
175Q8C23×C4×C79164yes—
176D4C23×C4×C791192yes—
177C2×C4C23×C4×C79112yes—
178C2×C4C23×C4×C7916yes—
179C2×C4C23×C4×C7913yes—
180C2×C4C23×C4×C79148yes—
181C2×C4C23×C4×C79136yes—
182C2×C4C23×C4×C79172yes—
183C2×C4C23×C4×C79196yes—
184C2×C4C25×C7913yes—
185C2×C4C25×C79124yes—
186C2×C4C25×C7914yes—
187C8C25×C79132yes—
188C22—212no—
189C22—272no—
190C22—448no—
191C22#568×C7914yesdirect product factor
192C22C24×C4×C7914yes—
193C22C24×C4×C79124yes—
194C22C24×C4×C7913yes—
195C4C24×C4×C7913yes—
196C4C24×C4×C7914yes—
197C4C24×C4×C79124yes—
198C4C26×C7911yescharacteristic
199C2—24no—
200C2—224no—
201C2C2×#568×C7916yesdirect product factor
202C2C25×C4×C7911yescharacteristic
subgroupG/H
centerC22×C4×C79C24
commutatorC2C25×C4×C79
Frattini, cosocleC4C26×C79
socleC23×C79C23×C4
prime radicalC23×#41×C79C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C79 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#568×C79C2direct (N × H)
3. C22×#131×C79C2semidirect
4. C23×M16×C79C2semidirect
12. #568×C79C22direct (N × H)
13. C2×#131×C79C22semidirect
15. C22×M16×C79C22semidirect
16. C22×M16×C79C22semidirect
17. C23×C8×C79C22semidirect
34. #131×C79C23semidirect
36. C2×M16×C79C23semidirect
37. C2×M16×C79C23semidirect
39. C22×C8×C79C23semidirect
63. M16×C79C24semidirect
65. C2×C8×C79C24semidirect
89. C22×C79#568direct (N × H)
97. C22×#568C79direct (N × H)
100. C2×C79C2×#568direct (N × H)
103. C2×#568C2×C79direct (N × H)
104. C22×#131C2×C79semidirect
105. C23×M16C2×C79semidirect
106. C79C22×#568direct (N × H)
114. #568C22×C79direct (N × H)
115. C2×#131C22×C79semidirect
117. C22×M16C22×C79semidirect
118. C22×M16C22×C79semidirect
119. C23×C8C22×C79semidirect
136. #131C23×C79semidirect
138. C2×M16C23×C79semidirect
139. C2×M16C23×C79semidirect
141. C22×C8C23×C79semidirect
165. M16C24×C79semidirect
167. C2×C8C24×C79semidirect
191. C22#568×C79direct (N × H)
201. C2C2×#568×C79direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C792
C2×#568×C792
C22×#131×C792
C23×M16×C792
C22×#56879