Finite Group Databasebeta

← C22×#430×C89 #36437959

Subgroups — C22×#430×C89 #36437959

Order 22784.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C89C212yesmaximal
2C23×M16×C89C211yesmaximal, characteristic
3C2×#430×C89C2124yesmaximal, direct product factor
4C22×#108×C89C214yesmaximal, complemented
5C24×C4×C89C2211yescharacteristic
6C22×M16×C89—212no—
7C22×M16×C89—22no—
8C22×M16×C89C414yes—
9C22×M16×C89C4112yes—
10C22×M16×C89C2214yes—
11C22×M16×C89C22124yes—
12C22×M16×C89C22112yes—
13C23×C8×C89—22no—
14C23×C8×C89C2212yes—
15#430×C89C22164yesdirect product factor
16C2×#108×C89C22148yescomplemented
17C25×C89Q811yescharacteristic
18C23×C4×C89C2×C412yes—
19C23×C4×C89C2×C416yes—
20C23×C4×C89D416yes—
21C23×C4×C89D412yes—
22C23×C4×C89Q811yescharacteristic
23C23×C4×C89Q816yes—
24C23×C4×C89C2316yes—
25C23×C4×C89C2311yescharacteristic
26C2×M16×C89—224no—
27C2×M16×C89—212no—
28C2×M16×C89—248no—
29C2×M16×C89C2×C4148yes—
30C2×M16×C89C2×C4148yes—
31C2×M16×C89C23124yes—
32C2×M16×C89C23132yes—
33C2×M16×C89C23116yes—
34C22×C8×C89—224no—
35C22×C8×C89—24no—
36C22×C8×C89C2×C414yes—
37C22×C8×C89C2×C4112yes—
38C22×C8×C89C23112yes—
39#108×C89C23164yescomplemented
40C24×C89—44no—
41C24×C89C4⋊C412yes—
42C24×C89C4⋊C416yes—
43C24×C89C2×Q811yescharacteristic
44C24×C89C2×Q816yes—
45C22×C4×C89C4⋊C4112yes—
46C22×C4×C89C4⋊C416yes—
47C22×C4×C89C4⋊C4124yes—
48C22×C4×C89C4⋊C412yes—
49C22×C4×C89C4⋊C4112yes—
50C22×C4×C89C22×C411yescharacteristic
51C22×C4×C89C22×C413yes—
52C22×C4×C89C22×C4112yes—
53C22×C4×C89C22×C4112yes—
54C22×C4×C89C2×D4112yes—
55C22×C4×C89C2×D413yes—
56C22×C4×C89C2×D4112yes—
57C22×C4×C89C2×D411yescharacteristic
58C22×C4×C89C2×Q816yes—
59C22×C4×C89C2×Q8112yes—
60C22×C4×C89C2×Q813yes—
61C22×C4×C89C2413yes—
62C22×C4×C89C2414yes—
63M16×C89—232no—
64M16×C89—224no—
65M16×C89C22×C4164yes—
66M16×C89C24116yes—
67C2×C8×C89—224no—
68C2×C8×C89—232no—
69C2×C8×C89C22×C4124yes—
70C2×C8×C89C22×C4124yes—
71C2×C8×C89C2418yes—
72C23×C89—28no—
73C23×C89—24no—
74C23×C89—412no—
75C23×C89—412no—
76C23×C89C2×C4⋊C413yes—
77C23×C89C2×C4⋊C4112yes—
78C23×C89C2×C4⋊C4112yes—
79C23×C89C2×C4⋊C411yescharacteristic
80C23×C89C22×Q813yes—
81C23×C89C22×Q814yes—
82C2×C4×C89C2×C4⋊C416yes—
83C2×C4×C89C2×C4⋊C4124yes—
84C2×C4×C89C2×C4⋊C4124yes—
85C2×C4×C89C2×C4⋊C4112yes—
86C2×C4×C89C2×C4⋊C413yes—
87C2×C4×C89C2×C4⋊C4112yes—
88C2×C4×C89C2×C4⋊C413yes—
89C2×C4×C89C23×C413yes—
90C2×C4×C89C23×C413yes—
91C2×C4×C89C23×C418yes—
92C2×C4×C89C22×D413yes—
93C2×C4×C89C22×D418yes—
94C2×C4×C89C22×D413yes—
95C2×C4×C89C22×Q813yes—
96C2×C4×C89C22×Q814yes—
97C2×C4×C89C2511yescharacteristic
98C8×C89—216no—
99C8×C89C23×C4116yes—
100C22×C89—224no—
101C22×C89—26no—
102C22×C89—26no—
103C22×C89—412no—
104C22×C89—44no—
105C22×C89#43014yesdirect product factor
106C22×C89C22×C4⋊C413yes—
107C22×C89C22×C4⋊C418yes—
108C22×C89C22×C4⋊C413yes—
109C22×C89C23×Q811yescharacteristic
110C4×C89C22×C4⋊C413yes—
111C4×C89C22×C4⋊C418yes—
112C4×C89C22×C4⋊C413yes—
113C4×C89C24×C411yescharacteristic
114C4×C89C23×D411yescharacteristic
115C22×#430C8911yesmaximal, characteristic, direct product factor, Sylow
116C2×C89—28no—
117C2×C89—23no—
118C2×C89—21no—
119C2×C89C2×#43016yesdirect product factor
120C2×C89C23×C4⋊C411yescharacteristic
121C23×M16C2×C8912yes—
122C23×M16C2×C8911yescharacteristic
123C2×#430C2×C89124yesdirect product factor
124C22×#108C2×C8914yescomplemented
125C89C22×#43011yescharacteristic, direct product factor, Sylow
126C24×C4C22×C8911yescharacteristic
127C22×M16—22no—
128C22×M16—212no—
129C22×M16C4×C8914yes—
130C22×M16C4×C89112yes—
131C22×M16C22×C8914yes—
132C22×M16C22×C89124yes—
133C22×M16C22×C89112yes—
134C23×C8—22no—
135C23×C8C22×C8912yes—
136#430C22×C89164yesdirect product factor
137C2×#108C22×C89148yescomplemented
138C25Q8×C8911yescharacteristic
139C23×C4C2×C4×C8912yes—
140C23×C4C2×C4×C8916yes—
141C23×C4D4×C8912yes—
142C23×C4D4×C8916yes—
143C23×C4Q8×C8911yescharacteristic
144C23×C4Q8×C8916yes—
145C23×C4C23×C8911yescharacteristic
146C23×C4C23×C8916yes—
147C2×M16—224no—
148C2×M16—212no—
149C2×M16—248no—
150C2×M16C2×C4×C89148yes—
151C2×M16C2×C4×C89148yes—
152C2×M16C23×C89124yes—
153C2×M16C23×C89132yes—
154C2×M16C23×C89116yes—
155C22×C8—24no—
156C22×C8—224no—
157C22×C8C2×C4×C89112yes—
158C22×C8C2×C4×C8914yes—
159C22×C8C23×C89112yes—
160#108C23×C89164yescomplemented
161C24—44no—
162C24C4⋊C4×C8912yes—
163C24C4⋊C4×C8916yes—
164C24C2×Q8×C8911yescharacteristic
165C24C2×Q8×C8916yes—
166C22×C4C4⋊C4×C89112yes—
167C22×C4C4⋊C4×C89112yes—
168C22×C4C4⋊C4×C8916yes—
169C22×C4C4⋊C4×C89124yes—
170C22×C4C4⋊C4×C8912yes—
171C22×C4C22×C4×C8913yes—
172C22×C4C22×C4×C8911yescharacteristic
173C22×C4C22×C4×C89112yes—
174C22×C4C22×C4×C89112yes—
175C22×C4C2×D4×C8911yescharacteristic
176C22×C4C2×D4×C89112yes—
177C22×C4C2×D4×C8913yes—
178C22×C4C2×D4×C89112yes—
179C22×C4C2×Q8×C8913yes—
180C22×C4C2×Q8×C8916yes—
181C22×C4C2×Q8×C89112yes—
182C22×C4C24×C8913yes—
183C22×C4C24×C8914yes—
184M16—232no—
185M16—224no—
186M16C22×C4×C89164yes—
187M16C24×C89116yes—
188C2×C8—224no—
189C2×C8—232no—
190C2×C8C22×C4×C89124yes—
191C2×C8C22×C4×C89124yes—
192C2×C8C24×C8918yes—
193C23—28no—
194C23—24no—
195C23—412no—
196C23—412no—
197C23C2×C4⋊C4×C8913yes—
198C23C2×C4⋊C4×C89112yes—
199C23C2×C4⋊C4×C89112yes—
200C23C2×C4⋊C4×C8911yescharacteristic
201C23C22×Q8×C8913yes—
202C23C22×Q8×C8914yes—
203C2×C4C2×C4⋊C4×C8916yes—
204C2×C4C2×C4⋊C4×C89124yes—
205C2×C4C2×C4⋊C4×C89124yes—
206C2×C4C2×C4⋊C4×C89112yes—
207C2×C4C2×C4⋊C4×C8913yes—
208C2×C4C2×C4⋊C4×C89112yes—
209C2×C4C2×C4⋊C4×C8913yes—
210C2×C4C23×C4×C8913yes—
211C2×C4C23×C4×C8913yes—
212C2×C4C23×C4×C8918yes—
213C2×C4C22×D4×C8913yes—
214C2×C4C22×D4×C8918yes—
215C2×C4C22×D4×C8913yes—
216C2×C4C22×Q8×C8913yes—
217C2×C4C22×Q8×C8914yes—
218C2×C4C25×C8911yescharacteristic
219C8—216no—
220C8C23×C4×C89116yes—
221C22—224no—
222C22—26no—
223C22—26no—
224C22—412no—
225C22—44no—
226C22#430×C8914yesdirect product factor
227C22C22×C4⋊C4×C8913yes—
228C22C22×C4⋊C4×C8918yes—
229C22C22×C4⋊C4×C8913yes—
230C22C23×Q8×C8911yescharacteristic
231C4C22×C4⋊C4×C8913yes—
232C4C22×C4⋊C4×C8913yes—
233C4C22×C4⋊C4×C8918yes—
234C4C24×C4×C8911yescharacteristic
235C4C23×D4×C8911yescharacteristic
236C2—28no—
237C2—23no—
238C2—21no—
239C2C2×#430×C8916yesdirect product factor
240C2C23×C4⋊C4×C8911yescharacteristic
subgroupG/H
centerC22×C4×C89C2×D4
commutatorC4C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC23×C89C2×C4⋊C4
prime radicalC25×C89Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C89 ◁ C24×C4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#430×C89C2direct (N × H)
4. C22×#108×C89C2semidirect
15. #430×C89C22direct (N × H)
16. C2×#108×C89C22semidirect
39. #108×C89C23semidirect
105. C22×C89#430direct (N × H)
115. C22×#430C89direct (N × H)
119. C2×C89C2×#430direct (N × H)
123. C2×#430C2×C89direct (N × H)
124. C22×#108C2×C89semidirect
125. C89C22×#430direct (N × H)
136. #430C22×C89direct (N × H)
137. C2×#108C22×C89semidirect
160. #108C23×C89semidirect
226. C22#430×C89direct (N × H)
239. C2C2×#430×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C23×M16×C892
C2×#430×C892
C22×#108×C892
C22×#43089