Finite Group Databasebeta

← C2×C89×#2916 #36437960

Subgroups — C2×C89×#2916 #36437960

Order 22784.

2586 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#568×C89C212yesmaximal
2C22×#131×C89C211yesmaximal, characteristic
3C89×#2916C2116yesmaximal, direct product factor
4C2×#430×C89C216yesmaximal, complemented
5C22×#108×C89C216yesmaximal, complemented
6C22×#41×C89C2211yescharacteristic
7#568×C89—28no—
8#568×C89C22116yes—
9C2×#131×C89—28no—
10C2×#131×C89C418yes—
11C2×#131×C89C418yes—
12C2×#131×C89C2218yes—
13C22×M16×C89C22112yes—
14C22×M16×C89C2213yes—
15C23×C8×C89C2213yes—
16#430×C89C22148yescomplemented
17C2×#108×C89C2214yes—
18C2×#108×C89C22112yes—
19C2×#108×C89C22148yescomplemented
20C2×#41×C89C2×C414yes—
21C2×#41×C89C2×C414yes—
22C2×#41×C89D414yes—
23C2×#41×C89D414yes—
24C2×#41×C89Q814yes—
25C2×#41×C89C2314yes—
26C22×Q8×C89Q811yescharacteristic
27C22×D4×C89Q813yes—
28C23×C4×C89C2313yes—
29#131×C89—264no—
30#131×C89C2×C4164yes—
31C2×M16×C89—212no—
32C2×M16×C89—224no—
33C2×M16×C89C2×C4112yes—
34C2×M16×C89C2×C4112yes—
35C2×M16×C89C2318yes—
36C2×M16×C89C23148yes—
37C2×M16×C89C23112yes—
38C22×C8×C89—212no—
39C22×C8×C89C2×C4112yes—
40C22×C8×C89C2×C4112yes—
41C22×C8×C89C2311yescharacteristic
42C22×C8×C89C2313yes—
43C22×C8×C89C23112yes—
44#108×C89C23116yes—
45#108×C89C23148yes—
46C24×C89C2×Q813yes—
47#41×C89C4⋊C4116yes—
48#41×C89C4⋊C4116yes—
49#41×C89C22×C4116yes—
50#41×C89C2×D4116yes—
51C2×Q8×C89C4⋊C414yes—
52C2×Q8×C89C4⋊C414yes—
53C2×Q8×C89C2×Q814yes—
54C2×D4×C89C4⋊C4112yes—
55C2×D4×C89C4⋊C4112yes—
56C2×D4×C89C2×Q8112yes—
57C22×C4×C89C22×C416yes—
58C22×C4×C89C22×C416yes—
59C22×C4×C89C2×D416yes—
60C22×C4×C89C2×D416yes—
61C22×C4×C89C2×Q813yes—
62C22×C4×C89C2×Q816yes—
63C22×C4×C89C2416yes—
64C22×C4×C89C2411yescharacteristic
65M16×C89—28no—
66M16×C89—248no—
67M16×C89C22×C4148yes—
68M16×C89C24116yes—
69C2×C8×C89—248no—
70C2×C8×C89—28no—
71C2×C8×C89C22×C412yes—
72C2×C8×C89C22×C412yes—
73C2×C8×C89C22×C416yes—
74C2×C8×C89C22×C416yes—
75C2×C8×C89C22×C4148yes—
76C2×C8×C89C2412yes—
77C2×C8×C89C2416yes—
78C23×C89—46no—
79C23×C89C2×C4⋊C416yes—
80C23×C89C2×C4⋊C416yes—
81C23×C89C22×Q811yescharacteristic
82C23×C89C22×Q816yes—
83Q8×C89C2×C4⋊C4116yes—
84D4×C89C2×C4⋊C4148yes—
85C2×C4×C89C2×C4⋊C416yes—
86C2×C4×C89C2×C4⋊C416yes—
87C2×C4×C89C2×C4⋊C4112yes—
88C2×C4×C89C2×C4⋊C4112yes—
89C2×C4×C89C23×C411yescharacteristic
90C2×C4×C89C23×C411yescharacteristic
91C2×C4×C89C23×C4112yes—
92C2×C4×C89C22×D4112yes—
93C2×C4×C89C22×D411yescharacteristic
94C2×C4×C89C22×D411yescharacteristic
95C2×C4×C89C22×Q811yescharacteristic
96C2×C4×C89C22×Q816yes—
97C2×C4×C89C2511yescharacteristic
98C8×C89—216no—
99C8×C89C23×C414yes—
100C8×C89C23×C4112yes—
101C22×C89—22no—
102C22×C89—212no—
103C22×C89—46no—
104C22×C89—46no—
105C22×C89C22×C4⋊C411yescharacteristic
106C22×C89C22×C4⋊C4112yes—
107C22×C89C22×C4⋊C411yescharacteristic
108C22×C89C23×Q811yescharacteristic
109C4×C89C22×C4⋊C411yescharacteristic
110C4×C89C22×C4⋊C411yescharacteristic
111C4×C89C22×C4⋊C4112yes—
112C4×C89C24×C411yescharacteristic
113C4×C89C23×D411yescharacteristic
114C2×#2916C8911yesmaximal, characteristic, direct product factor, Sylow
115C2×C89—21no—
116C2×C89—21no—
117C2×C89—212no—
118C2×C89#291612yesdirect product factor
119C2×C89C23×C4⋊C411yescharacteristic
120C2×#568C2×C8912yes—
121C22×#131C2×C8911yescharacteristic
122#2916C2×C89116yesdirect product factor
123C2×#430C2×C8916yescomplemented
124C22×#108C2×C8916yescomplemented
125C89C2×#291611yescharacteristic, direct product factor, Sylow
126C22×#41C22×C8911yescharacteristic
127#568—28no—
128#568C22×C89116yes—
129C2×#131—28no—
130C2×#131C4×C8918yes—
131C2×#131C4×C8918yes—
132C2×#131C22×C8918yes—
133C22×M16C22×C89112yes—
134C22×M16C22×C8913yes—
135C23×C8C22×C8913yes—
136#430C22×C89148yescomplemented
137C2×#108C22×C8914yes—
138C2×#108C22×C89112yes—
139C2×#108C22×C89148yescomplemented
140C2×#41C2×C4×C8914yes—
141C2×#41C2×C4×C8914yes—
142C2×#41D4×C8914yes—
143C2×#41D4×C8914yes—
144C2×#41Q8×C8914yes—
145C2×#41C23×C8914yes—
146C22×Q8Q8×C8911yescharacteristic
147C22×D4Q8×C8913yes—
148C23×C4C23×C8913yes—
149#131—264no—
150#131C2×C4×C89164yes—
151C2×M16—212no—
152C2×M16—224no—
153C2×M16C2×C4×C89112yes—
154C2×M16C2×C4×C89112yes—
155C2×M16C23×C8918yes—
156C2×M16C23×C89148yes—
157C2×M16C23×C89112yes—
158C22×C8—212no—
159C22×C8C2×C4×C89112yes—
160C22×C8C2×C4×C89112yes—
161C22×C8C23×C8911yescharacteristic
162C22×C8C23×C8913yes—
163C22×C8C23×C89112yes—
164#108C23×C89116yes—
165#108C23×C89148yes—
166C24C2×Q8×C8913yes—
167#41C4⋊C4×C89116yes—
168#41C4⋊C4×C89116yes—
169#41C22×C4×C89116yes—
170#41C2×D4×C89116yes—
171C2×Q8C4⋊C4×C8914yes—
172C2×Q8C4⋊C4×C8914yes—
173C2×Q8C2×Q8×C8914yes—
174C2×D4C4⋊C4×C89112yes—
175C2×D4C4⋊C4×C89112yes—
176C2×D4C2×Q8×C89112yes—
177C22×C4C22×C4×C8916yes—
178C22×C4C22×C4×C8916yes—
179C22×C4C2×D4×C8916yes—
180C22×C4C2×D4×C8916yes—
181C22×C4C2×Q8×C8913yes—
182C22×C4C2×Q8×C8916yes—
183C22×C4C24×C8911yescharacteristic
184C22×C4C24×C8916yes—
185M16—28no—
186M16—248no—
187M16C22×C4×C89148yes—
188M16C24×C89116yes—
189C2×C8—28no—
190C2×C8—248no—
191C2×C8C22×C4×C8912yes—
192C2×C8C22×C4×C8912yes—
193C2×C8C22×C4×C8916yes—
194C2×C8C22×C4×C8916yes—
195C2×C8C22×C4×C89148yes—
196C2×C8C24×C8912yes—
197C2×C8C24×C8916yes—
198C23—46no—
199C23C2×C4⋊C4×C8916yes—
200C23C2×C4⋊C4×C8916yes—
201C23C22×Q8×C8911yescharacteristic
202C23C22×Q8×C8916yes—
203Q8C2×C4⋊C4×C89116yes—
204D4C2×C4⋊C4×C89148yes—
205C2×C4C2×C4⋊C4×C8916yes—
206C2×C4C2×C4⋊C4×C8916yes—
207C2×C4C2×C4⋊C4×C89112yes—
208C2×C4C2×C4⋊C4×C89112yes—
209C2×C4C23×C4×C8911yescharacteristic
210C2×C4C23×C4×C8911yescharacteristic
211C2×C4C23×C4×C89112yes—
212C2×C4C22×D4×C89112yes—
213C2×C4C22×D4×C8911yescharacteristic
214C2×C4C22×D4×C8911yescharacteristic
215C2×C4C22×Q8×C8911yescharacteristic
216C2×C4C22×Q8×C8916yes—
217C2×C4C25×C8911yescharacteristic
218C8—216no—
219C8C23×C4×C8914yes—
220C8C23×C4×C89112yes—
221C22—22no—
222C22—212no—
223C22—46no—
224C22—46no—
225C22C22×C4⋊C4×C8911yescharacteristic
226C22C22×C4⋊C4×C89112yes—
227C22C22×C4⋊C4×C8911yescharacteristic
228C22C23×Q8×C8911yescharacteristic
229C4C22×C4⋊C4×C8911yescharacteristic
230C4C22×C4⋊C4×C8911yescharacteristic
231C4C22×C4⋊C4×C89112yes—
232C4C24×C4×C8911yescharacteristic
233C4C23×D4×C8911yescharacteristic
234C2—21no—
235C2—21no—
236C2—212no—
237C2C89×#291612yesdirect product factor
238C2C23×C4⋊C4×C8911yescharacteristic
subgroupG/H
centerC2×C4×C89C22×D4
commutatorC4C24×C4×C89
Frattini, cosocleC2×C4C25×C89
socleC22×C89C22×C4⋊C4
prime radicalC22×#41×C89C22
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C89 ◁ C22×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
3. C89×#2916C2direct (N × H)
4. C2×#430×C89C2semidirect
5. C22×#108×C89C2semidirect
16. #430×C89C22semidirect
19. C2×#108×C89C22semidirect
114. C2×#2916C89direct (N × H)
118. C2×C89#2916direct (N × H)
122. #2916C2×C89direct (N × H)
123. C2×#430C2×C89semidirect
124. C22×#108C2×C89semidirect
125. C89C2×#2916direct (N × H)
136. #430C22×C89semidirect
139. C2×#108C22×C89semidirect
237. C2C89×#2916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#568×C892
C22×#131×C892
C89×#29162
C2×#430×C892
C22×#108×C892
C2×#291689