Finite Group Databasebeta

← C22×#19098904 #35156014

Subgroups — C22×#19098904 #35156014

Order 20224.

2358 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579140C211yesmaximal, characteristic
2C2×#19098904C2124yesmaximal, direct product factor
3C22×#14579094C211yesmaximal, characteristic, complemented
4C22×#14579092C211yesmaximal, characteristic, complemented
5C22×#14579084C211yesmaximal, characteristic, complemented
6C22×#14579084C211yesmaximal, characteristic, complemented
7C22×C22⋊(C79⋊C8)C211yesmaximal, characteristic
8C22×#117×C79C211yesmaximal, characteristic
9C23×Dic158C2211yescharacteristic
10C23×C79⋊C8C2211yescharacteristic
11C23×C79⋊C8C2211yescharacteristic
12C2×#14579140—26no—
13C2×#14579140C22112yes—
14C22×C4⋊Dic79C2211yescharacteristic
15C22×Dic79⋊C4—21no—
16C22×C22⋊Dic79—21no—
17#19098904C22164yesdirect product factor
18C2×#14579094C22112yescomplemented
19C2×#14579092C22112yescomplemented
20C2×#14579084C22112yescomplemented
21C2×#14579084C22112yescomplemented
22C2×C22⋊(C79⋊C8)C22112yes—
23C24×C4×C79C2211yescharacteristic
24C2×#117×C79—26no—
25C2×#117×C79C22112yes—
26C22×C4⋊C4×C79C2211yescharacteristic
27C22×C4⋊C4×C79C2211yescharacteristic
28C22×C22⋊C4×C79—21no—
29C22×C42×C79—21no—
30C22×Dic158—23no—
31C22×Dic158—24no—
32C22×Dic158—24no—
33C22×Dic158C2316yes—
34C23×Dic79—21no—
35C23×Dic79—21no—
36C22×C79⋊C8—24no—
37C22×C79⋊C8—24no—
38C22×C79⋊C8C2316yes—
39C22×C79⋊C8C2316yes—
40#14579140—224no—
41#14579140C23116yes—
42C2×C4⋊Dic79—23no—
43C2×C4⋊Dic79C2316yes—
44C2×Dic79⋊C4—212no—
45C2×C22⋊Dic79—212no—
46#14579094C23116yescomplemented
47#14579092C23116yescomplemented
48#14579084C23116yescomplemented
49#14579084C23116yescomplemented
50C22⋊(C79⋊C8)C23116yes—
51C25×C79D411yescharacteristic
52C23×C4×C79—24no—
53C23×C4×C79—24no—
54C23×C4×C79—21no—
55C23×C4×C79—21no—
56C23×C4×C79D411yescharacteristic
57C23×C4×C79D416yes—
58C23×C4×C79C2316yes—
59C23×C4×C79C2311yescharacteristic
60#117×C79—224no—
61#117×C79C23116yes—
62C2×C4⋊C4×C79—23no—
63C2×C4⋊C4×C79—23no—
64C2×C4⋊C4×C79C2316yes—
65C2×C4⋊C4×C79C2316yes—
66C2×C22⋊C4×C79—212no—
67C2×C42×C79—212no—
68C2×Dic158—26no—
69C2×Dic158—212no—
70C2×Dic158—212no—
71C2×Dic158—412no—
72C2×Dic158C2414yes—
73C22×Dic79—26no—
74C22×Dic79—26no—
75C22×Dic79—44no—
76C22×Dic79—44no—
77C2×C79⋊C8—212no—
78C2×C79⋊C8—212no—
79C2×C79⋊C8C2414yes—
80C2×C79⋊C8C2414yes—
81C4⋊Dic79—26no—
82C4⋊Dic79C2414yes—
83Dic79⋊C4—216no—
84C22⋊Dic79—216no—
85C24×C79—24no—
86C24×C79—24no—
87C24×C79—24no—
88C24×C79C2×D416yes—
89C24×C79C2×D411yescharacteristic
90C22×C4×C79—24no—
91C22×C4×C79—224no—
92C22×C4×C79—212no—
93C22×C4×C79—212no—
94C22×C4×C79—26no—
95C22×C4×C79—26no—
96C22×C4×C79—44no—
97C22×C4×C79—44no—
98C22×C4×C79C2×D416yes—
99C22×C4×C79C2×D4112yes—
100C22×C4×C79C2×D413yes—
101C22×C4×C79#4114yes—
102C22×C4×C79#4114yes—
103C22×C4×C79C2414yes—
104C22×C4×C79C2413yes—
105C4⋊C4×C79—26no—
106C4⋊C4×C79—26no—
107C4⋊C4×C79C2414yes—
108C4⋊C4×C79C2414yes—
109C22⋊C4×C79—216no—
110C42×C79—216no—
111C23×C79—212no—
112C23×C79—212no—
113C23×C79—212no—
114C23×C79—416no—
115C23×C79#12314yes—
116C23×C79#13514yes—
117C23×C79#13714yes—
118C23×C79C22×D414yes—
119C23×C79C22×D413yes—
120C2×C4×C79—212no—
121C2×C4×C79—224no—
122C2×C4×C79—24no—
123C2×C4×C79—24no—
124C2×C4×C79—24no—
125C2×C4×C79—24no—
126C2×C4×C79—412no—
127C2×C4×C79—412no—
128C2×C4×C79#123112yes—
129C2×C4×C79C22×D414yes—
130C2×C4×C79C22×D413yes—
131C2×C4×C79C2×#4116yes—
132C2×C4×C79C2×#4116yes—
133C2×C4×C79C2511yescharacteristic
134C2×Dic79—24no—
135C2×Dic79—24no—
136C2×Dic79—412no—
137C2×Dic79—412no—
138Dic158—24no—
139Dic158—24no—
140Dic158—412no—
141C79⋊C8—24no—
142C79⋊C8—24no—
143C22×C79—24no—
144C22×C79—24no—
145C22×C79—24no—
146C22×C79—424no—
147C22×C79#485116yescomplemented
148C22×C79C2×#12316yes—
149C22×C79C2×#13516yes—
150C22×C79C2×#13716yes—
151C22×C79C23×D411yescharacteristic
152C4×C79—24no—
153C4×C79—44no—
154C4×C79—44no—
155C4×C79C2×#12316yes—
156C4×C79C22×#4111yescharacteristic
157C4×C79C22×#4111yescharacteristic
158Dic79—44no—
159Dic79—44no—
160C22×#485—791nomaximal, Sylow
161C2×C79—44no—
162C2×C79C2×#485112yescomplemented
163C2×C79C22×#12311yescharacteristic
164C2×C79C22×#13511yescharacteristic
165C2×C79C22×#13711yescharacteristic
166C22×C22⋊Q8—791no—
167C22×#117D7911yescharacteristic
168C2×#485—7924no—
169C22×#107—791no—
170C22×#106—791no—
171C22×#103—791no—
172C22×#103—791no—
173C22×C22⋊C8—791no—
174C79C22×#48511yescharacteristic, complemented, Sylow
175C23×Q8—791no—
176C24×C4C2×D7911yescharacteristic
177C23×C8—791no—
178C23×C8—791no—
179C2×C22⋊Q8—7912no—
180C2×C22⋊Q8—1586no—
181C2×#117—26no—
182C2×#117C2×D79112yes—
183C22×C4⋊C4—791no—
184C22×C4⋊C4—1581no—
185C22×C4⋊C4C2×D7911yescharacteristic
186C22×C4⋊C4C2×D7911yescharacteristic
187C22×C22⋊C4—21no—
188C22×C22⋊C4—1581no—
189C22×C42—21no—
190#485—7964no—
191C2×#107—7912no—
192C2×#106—7912no—
193C2×#103—7912no—
194C2×#103—7912no—
195C2×C22⋊C8—7912no—
196C25C22⋊D7911yescharacteristic
197C22×Q8—796no—
198C22×Q8—1584no—
199C22×Q8—1584no—
200C22×Q8—1583no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—1581no—
206C23×C4—1581no—
207C23×C4C22⋊D7916yes—
208C23×C4C22⋊D7911yescharacteristic
209C23×C4C22×D7916yes—
210C23×C4C22×D7911yescharacteristic
211C22×C8—796no—
212C22×C8—796no—
213C22×C8—1584no—
214C22×C8—1584no—
215C22⋊Q8—7916no—
216C22⋊Q8—15824no—
217#117—224no—
218#117C22×D79116yes—
219C2×C4⋊C4—23no—
220C2×C4⋊C4—23no—
221C2×C4⋊C4—796no—
222C2×C4⋊C4—1583no—
223C2×C4⋊C4—15812no—
224C2×C4⋊C4C22×D7916yes—
225C2×C4⋊C4C22×D7916yes—
226C2×C22⋊C4—212no—
227C2×C22⋊C4—15812no—
228C2×C42—212no—
229#107—7916no—
230#106—7916no—
231#103—7916no—
232#103—7916no—
233C22⋊C8—7916no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24C2×C22⋊D7916yes—
238C24C2×C22⋊D7911yescharacteristic
239C2×Q8—31612no—
240C2×Q8—794no—
241C2×Q8—15812no—
242C2×Q8—15812no—
243C2×Q8—1586no—
244C22×C4—24no—
245C22×C4—212no—
246C22×C4—212no—
247C22×C4—224no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—3164no—
253C22×C4—3164no—
254C22×C4—1586no—
255C22×C4—1586no—
256C22×C4C2×C22⋊D7916yes—
257C22×C4C2×C22⋊D79112yes—
258C22×C4C2×C22⋊D7913yes—
259C22×C4#1192962414yes—
260C22×C4#1192962414yes—
261C22×C4C23×D7914yes—
262C22×C4C23×D7913yes—
263C2×C8—794no—
264C2×C8—794no—
265C2×C8—15812no—
266C2×C8—15812no—
267C4⋊C4—26no—
268C4⋊C4—26no—
269C4⋊C4—794no—
270C4⋊C4—15816no—
271C4⋊C4—1586no—
272C4⋊C4C23×D7914yes—
273C4⋊C4C23×D7914yes—
274C22⋊C4—216no—
275C22⋊C4—15816no—
276C42—216no—
277C23—212no—
278C23—212no—
279C23—212no—
280C23—416no—
281C23#1457914414yes—
282C23#1457918114yes—
283C23#1457919114yes—
284C23C22×C22⋊D7914yes—
285C23C22×C22⋊D7913yes—
286Q8—31612no—
287Q8—1584no—
288Q8—1584no—
289C2×C4—212no—
290C2×C4—224no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—412no—
296C2×C4—412no—
297C2×C4—31612no—
298C2×C4—31612no—
299C2×C4—1584no—
300C2×C4—1584no—
301C2×C4#14579144112yes—
302C2×C4C22×C22⋊D7914yes—
303C2×C4C22×C22⋊D7913yes—
304C2×C4C2×#1192962416yes—
305C2×C4C2×#1192962416yes—
306C2×C4C24×D7911yescharacteristic
307C8—1584no—
308C8—1584no—
309C22—24no—
310C22—24no—
311C22—24no—
312C22—424no—
313C22#19098904116yesdirect product factor
314C22C2×#1457914416yes—
315C22C2×#1457918116yes—
316C22C2×#1457919116yes—
317C22C23×C22⋊D7911yescharacteristic
318C4—24no—
319C4—44no—
320C4—44no—
321C4—3164no—
322C4—3164no—
323C4C2×#1457914416yes—
324C4C22×#1192962411yescharacteristic
325C4C22×#1192962411yescharacteristic
326C2—44no—
327C2C2×#19098904112yesdirect product factor
328C2C22×#1457914411yescharacteristic
329C2C22×#1457918111yescharacteristic
330C2C22×#1457919111yescharacteristic
subgroupG/H
centerC24C2×C22⋊D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC22×#117×C79C2
socleC24×C79C2×D4
prime radicalC25×C79D4
hypercenterC22×#117D79
hypocenterC79C22×#485
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19098904C2direct (N × H)
3. C22×#14579094C2semidirect
4. C22×#14579092C2semidirect
5. C22×#14579084C2semidirect
6. C22×#14579084C2semidirect
17. #19098904C22direct (N × H)
18. C2×#14579094C22semidirect
19. C2×#14579092C22semidirect
20. C2×#14579084C22semidirect
21. C2×#14579084C22semidirect
46. #14579094C23semidirect
47. #14579092C23semidirect
48. #14579084C23semidirect
49. #14579084C23semidirect
147. C22×C79#485semidirect
162. C2×C79C2×#485semidirect
174. C79C22×#485semidirect
313. C22#19098904direct (N × H)
327. C2C2×#19098904direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145791402
C2×#190989042
C22×#145790942
C22×#145790922
C22×#145790842
C22×C22⋊(C79⋊C8)2
C22×#117×C792
C22×#48579