Finite Group Databasebeta

← C113×#20347 #43288367

Subgroups — C113×#20347 #43288367

Order 28928.

2270 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C113C211yesmaximal, characteristic
2C22×#117×C113C212yesmaximal
3C113×#1736C218yesmaximal, complemented
4C113×#1735C214yesmaximal, complemented
5C24×C4×C113C2211yescharacteristic
6C22×M16×C113—28no—
7C22×M16×C113C418yescomplemented
8C22×M16×C113C2214yes—
9C23×C8×C113C2212yes—
10C2×#117×C113—28no—
11C2×#117×C113—22no—
12C2×#117×C113—22no—
13C2×#117×C113C4116yescomplemented
14C2×#117×C113C2214yes—
15C2×#117×C113C2214yes—
16C22×C4⋊C4×C113C2214yes—
17C22×C22⋊C4×C113—22no—
18C22×C42×C113—22no—
19#340×C113C22116yescomplemented
20C25×C113D411yescharacteristic
21C23×C4×C113—24no—
22C23×C4×C113C2×C414yes—
23C23×C4×C113C2×C418yes—
24C23×C4×C113D411yescharacteristic
25C23×C4×C113D412yes—
26C23×C4×C113D418yes—
27C23×C4×C113Q814yes—
28C23×C4×C113C2312yes—
29C23×C4×C113C2311yescharacteristic
30C2×M16×C113—232no—
31C2×M16×C113—28no—
32C2×M16×C113—28no—
33C2×M16×C113C2×C4116yescomplemented
34C22×C8×C113—28no—
35C22×C8×C113C2×C418yescomplemented
36C22×C8×C113C2314yes—
37#117×C113—216no—
38#117×C113—216no—
39#117×C113—28no—
40#117×C113—28no—
41#117×C113C2×C4116yes—
42#117×C113C2×C4116yescomplemented
43C2×C4⋊C4×C113—24no—
44C2×C4⋊C4×C113—28no—
45C2×C4⋊C4×C113C2×C4116yescomplemented
46C2×C4⋊C4×C113C2318yes—
47C2×C22⋊C4×C113—216no—
48C2×C22⋊C4×C113—24no—
49C2×C22⋊C4×C113—24no—
50C2×C42×C113—216no—
51C2×C42×C113—24no—
52C2×C42×C113—24no—
53C24×C113—44no—
54C24×C113C22⋊C418yes—
55C24×C113C4⋊C414yes—
56C24×C113C2×D411yescharacteristic
57C24×C113C2×D412yes—
58C22×C4×C113—216no—
59C22×C4×C113—28no—
60C22×C4×C113—416no—
61C22×C4×C113C42116yes—
62C22×C4×C113C22⋊C418yes—
63C22×C4×C113C22⋊C418yes—
64C22×C4×C113C22⋊C418yes—
65C22×C4×C113C22⋊C4116yes—
66C22×C4×C113C4⋊C414yes—
67C22×C4×C113C4⋊C418yes—
68C22×C4×C113C4⋊C4132yes—
69C22×C4×C113C22×C412yes—
70C22×C4×C113C22×C414yes—
71C22×C4×C113C22×C414yes—
72C22×C4×C113C22×C414yes—
73C22×C4×C113C22×C414yes—
74C22×C4×C113C2×D412yes—
75C22×C4×C113C2×D414yes—
76C22×C4×C113C2×D414yes—
77C22×C4×C113C2×D414yes—
78C22×C4×C113C2×D411yescharacteristic
79C22×C4×C113C2×Q814yes—
80C22×C4×C113C2×Q812yes—
81C22×C4×C113C2411yescharacteristic
82M16×C113—232no—
83C2×C8×C113—216no—
84C2×C8×C113—28no—
85C2×C8×C113C22×C418yes—
86C4⋊C4×C113—216no—
87C4⋊C4×C113—28no—
88C4⋊C4×C113C22×C4116yes—
89C22⋊C4×C113—216no—
90C22⋊C4×C113—216no—
91C42×C113—216no—
92C42×C113—216no—
93C23×C113—28no—
94C23×C113—24no—
95C23×C113—416no—
96C23×C113—48no—
97C23×C113#95116yes—
98C23×C113C2×C22⋊C414yes—
99C23×C113C2×C22⋊C414yes—
100C23×C113C2×C22⋊C414yes—
101C23×C113C2×C4⋊C412yes—
102C23×C113C2×C4⋊C414yes—
103C23×C113C22×D411yescharacteristic
104C2×C4×C113—216no—
105C2×C4×C113—432no—
106C2×C4×C113—416no—
107C2×C4×C113#95132yes—
108C2×C4×C113#95116yes—
109C2×C4×C113C2×C4214yes—
110C2×C4×C113C2×C4218yes—
111C2×C4×C113C2×C22⋊C414yes—
112C2×C4×C113C2×C22⋊C418yes—
113C2×C4×C113C2×C22⋊C414yes—
114C2×C4×C113C2×C22⋊C414yes—
115C2×C4×C113C2×C22⋊C414yes—
116C2×C4×C113C2×C4⋊C418yes—
117C2×C4×C113C2×C4⋊C418yes—
118C2×C4×C113C2×C4⋊C418yes—
119C2×C4×C113C2×C4⋊C414yes—
120C2×C4×C113C2×C4⋊C412yes—
121C2×C4×C113C23×C411yescharacteristic
122C2×C4×C113C23×C412yes—
123C2×C4×C113C22×D412yes—
124C2×C4×C113C22×Q811yescharacteristic
125C8×C113—28no—
126C22×C113—28no—
127C22×C113—28no—
128C22×C113—28no—
129C22×C113—28no—
130C22×C113—24no—
131C22×C113—416no—
132C22×C113C2×#9514yes—
133C22×C113C2×#9518yes—
134C22×C113C22×C22⋊C412yes—
135C22×C113C22×C4⋊C411yescharacteristic
136C22×C113#57514yes—
137C4×C113—416no—
138C4×C113C2×#9514yes—
139C4×C113C2×#9518yes—
140C4×C113C22×C4211yescharacteristic
141C4×C113C22×C22⋊C411yescharacteristic
142C4×C113C22×C4⋊C412yes—
143#20347C11311yesmaximal, characteristic, direct product factor, Sylow
144C2×C113—28no—
145C2×C113—24no—
146C2×C113C22×#9511yescharacteristic
147C2×C113#289614yes—
148C2×C113#291112yes—
149C23×M16C2×C11311yescharacteristic
150C22×#117C2×C11312yes—
151#1736C2×C11318yescomplemented
152#1735C2×C11314yescomplemented
153C113#2034711yescharacteristic, direct product factor, Sylow
154C24×C4C22×C11311yescharacteristic
155C22×M16—28no—
156C22×M16C4×C11318yescomplemented
157C22×M16C22×C11314yes—
158C23×C8C22×C11312yes—
159C2×#117—28no—
160C2×#117—22no—
161C2×#117—22no—
162C2×#117C4×C113116yescomplemented
163C2×#117C22×C11314yes—
164C2×#117C22×C11314yes—
165C22×C4⋊C4C22×C11314yes—
166C22×C22⋊C4—22no—
167C22×C42—22no—
168#340C22×C113116yescomplemented
169C25D4×C11311yescharacteristic
170C23×C4—24no—
171C23×C4C2×C4×C11314yes—
172C23×C4C2×C4×C11318yes—
173C23×C4D4×C11311yescharacteristic
174C23×C4D4×C11312yes—
175C23×C4D4×C11318yes—
176C23×C4Q8×C11314yes—
177C23×C4C23×C11312yes—
178C23×C4C23×C11311yescharacteristic
179C2×M16—232no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16C2×C4×C113116yescomplemented
183C22×C8—28no—
184C22×C8C2×C4×C11318yescomplemented
185C22×C8C23×C11314yes—
186#117—216no—
187#117—216no—
188#117—28no—
189#117—28no—
190#117C2×C4×C113116yes—
191#117C2×C4×C113116yescomplemented
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4C2×C4×C113116yescomplemented
195C2×C4⋊C4C23×C11318yes—
196C2×C22⋊C4—216no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C42—216no—
200C2×C42—24no—
201C2×C42—24no—
202C24—44no—
203C24C22⋊C4×C11318yes—
204C24C4⋊C4×C11314yes—
205C24C2×D4×C11311yescharacteristic
206C24C2×D4×C11312yes—
207C22×C4—216no—
208C22×C4—28no—
209C22×C4—416no—
210C22×C4C42×C113116yes—
211C22×C4C22⋊C4×C11318yes—
212C22×C4C22⋊C4×C11318yes—
213C22×C4C22⋊C4×C11318yes—
214C22×C4C22⋊C4×C113116yes—
215C22×C4C4⋊C4×C11314yes—
216C22×C4C4⋊C4×C11318yes—
217C22×C4C4⋊C4×C113132yes—
218C22×C4C22×C4×C11312yes—
219C22×C4C22×C4×C11314yes—
220C22×C4C22×C4×C11314yes—
221C22×C4C22×C4×C11314yes—
222C22×C4C22×C4×C11314yes—
223C22×C4C2×D4×C11311yescharacteristic
224C22×C4C2×D4×C11312yes—
225C22×C4C2×D4×C11314yes—
226C22×C4C2×D4×C11314yes—
227C22×C4C2×D4×C11314yes—
228C22×C4C2×Q8×C11314yes—
229C22×C4C2×Q8×C11312yes—
230C22×C4C24×C11311yescharacteristic
231M16—232no—
232C2×C8—216no—
233C2×C8—28no—
234C2×C8C22×C4×C11318yes—
235C4⋊C4—216no—
236C4⋊C4—28no—
237C4⋊C4C22×C4×C113116yes—
238C22⋊C4—216no—
239C22⋊C4—216no—
240C42—216no—
241C42—216no—
242C23—28no—
243C23—24no—
244C23—416no—
245C23—48no—
246C23#95×C113116yes—
247C23C2×C22⋊C4×C11314yes—
248C23C2×C22⋊C4×C11314yes—
249C23C2×C22⋊C4×C11314yes—
250C23C2×C4⋊C4×C11312yes—
251C23C2×C4⋊C4×C11314yes—
252C23C22×D4×C11311yescharacteristic
253C2×C4—216no—
254C2×C4—432no—
255C2×C4—416no—
256C2×C4#95×C113132yes—
257C2×C4#95×C113116yes—
258C2×C4C2×C42×C11314yes—
259C2×C4C2×C42×C11318yes—
260C2×C4C2×C22⋊C4×C11314yes—
261C2×C4C2×C22⋊C4×C11318yes—
262C2×C4C2×C22⋊C4×C11314yes—
263C2×C4C2×C22⋊C4×C11314yes—
264C2×C4C2×C22⋊C4×C11314yes—
265C2×C4C2×C4⋊C4×C11318yes—
266C2×C4C2×C4⋊C4×C11318yes—
267C2×C4C2×C4⋊C4×C11318yes—
268C2×C4C2×C4⋊C4×C11312yes—
269C2×C4C2×C4⋊C4×C11314yes—
270C2×C4C23×C4×C11311yescharacteristic
271C2×C4C23×C4×C11312yes—
272C2×C4C22×D4×C11312yes—
273C2×C4C22×Q8×C11311yescharacteristic
274C8—28no—
275C22—28no—
276C22—28no—
277C22—28no—
278C22—28no—
279C22—24no—
280C22—416no—
281C22C2×#95×C11314yes—
282C22C2×#95×C11318yes—
283C22C22×C22⋊C4×C11312yes—
284C22C22×C4⋊C4×C11311yescharacteristic
285C22#575×C11314yes—
286C4—416no—
287C4C2×#95×C11314yes—
288C4C2×#95×C11318yes—
289C4C22×C42×C11311yescharacteristic
290C4C22×C22⋊C4×C11311yescharacteristic
291C4C22×C4⋊C4×C11312yes—
292C2—28no—
293C2—24no—
294C2C22×#95×C11311yescharacteristic
295C2C113×#289614yes—
296C2C113×#291112yes—
subgroupG/H
centerC22×C4×C113C2×D4
commutatorC4C22×C42×C113
Frattini, cosocleC22×C4C24×C113
socleC23×C113C22×D4
prime radicalC25×C113D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C113 ◁ C24×C4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
3. C113×#1736C2semidirect
4. C113×#1735C2semidirect
7. C22×M16×C113C4semidirect
13. C2×#117×C113C4semidirect
19. #340×C113C22semidirect
33. C2×M16×C113C2×C4semidirect
35. C22×C8×C113C2×C4semidirect
42. #117×C113C2×C4semidirect
45. C2×C4⋊C4×C113C2×C4semidirect
143. #20347C113direct (N × H)
151. #1736C2×C113semidirect
152. #1735C2×C113semidirect
153. C113#20347direct (N × H)
156. C22×M16C4×C113semidirect
162. C2×#117C4×C113semidirect
168. #340C22×C113semidirect
182. C2×M16C2×C4×C113semidirect
184. C22×C8C2×C4×C113semidirect
191. #117C2×C4×C113semidirect
194. C2×C4⋊C4C2×C4×C113semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C1132
C22×#117×C1132
C113×#17362
C113×#17352
#20347113