Finite Group Databasebeta

← C2×#22609427 #33068502

Subgroups — C2×#22609427 #33068502

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988874C211yesmaximal, characteristic
2C2×#17988789C211yesmaximal, characteristic
3C2×#17988666C211yesmaximal, characteristic
4#22609427C218yesmaximal, direct product factor
5C2×#17988340C211yesmaximal, characteristic, complemented
6C2×#17988336C211yesmaximal, characteristic, complemented
7C2×#17988333C211yesmaximal, characteristic, complemented
8C2×#498×C73C211yesmaximal, characteristic, complemented
9C22×#11924015—21no—
10C22×#11924011—21no—
11C2×C4×D292C2211yescharacteristic
12#17988874—22no—
13#17988874C2214yes—
14#17988789—22no—
15#17988789C2214yes—
16#17988666—22no—
17#17988666C2214yes—
18C2×#14556522—21no—
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556515—21no—
23C2×#14556511—21no—
24C2×#14556503—21no—
25#17988340C2214yescomplemented
26#17988336C2214yescomplemented
27#17988333C2214yescomplemented
28C2×C4⋊Q8×C73C2211yescharacteristic
29C2×C4⋊Q8×C73C2211yescharacteristic
30#498×C73—22no—
31#498×C73C2214yescomplemented
32C2×#106×C73—21no—
33C2×#106×C73—21no—
34C2×C4×C8×C73C2211yescharacteristic
35C22×D292D411yescharacteristic
36C22×C4×D73—21no—
37C2×#11924015—24no—
38C2×#11924015—44no—
39C2×#11924011—24no—
40C2×#11924011—44no—
41C22×#10596831—21no—
42C22×#10596831—21no—
43C22×#10596831—21no—
44C4×D292—21no—
45C4×D292—21no—
46C4×D292D412yes—
47C4×D292C2312yes—
48C2×C4⋊Dic73D411yescharacteristic
49C2×#11924005—21no—
50#14556522—24no—
51C4⋊#10596831—21no—
52C4⋊#10596831—21no—
53C4⋊#10596831—21no—
54C4⋊#10596831C2312yes—
55C4⋊#10596831C2312yes—
56C4⋊#10596831C2312yes—
57#14556515—24no—
58#14556511—24no—
59#14556503—24no—
60C22×Q8×C73—21no—
61C22×Q8×C73—21no—
62C22×C8×C73D411yescharacteristic
63C22×C8×C73D411yescharacteristic
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73—21no—
70C4⋊Q8×C73C2312yes—
71C4⋊Q8×C73C2312yes—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73—21no—
76C2×C42×C73C2311yescharacteristic
77#106×C73—24no—
78#106×C73—24no—
79C4×C8×C73D412yes—
80C4×C8×C73C2312yes—
81C23×D73—21no—
82C2×D292—22no—
83C2×D292—22no—
84C2×D292—21no—
85C2×D292C2×D412yes—
86C22×Dic73—21no—
87C2×C4×D73—22no—
88C2×C4×D73—22no—
89C2×C4×D73—42no—
90C2×C4×D73—42no—
91#11924015—48no—
92#11924011—48no—
93C2×#10596831—22no—
94C2×#10596831—22no—
95C2×#10596831—22no—
96C2×#10596831—42no—
97C2×#10596831—42no—
98C2×#10596831—42no—
99C4⋊Dic73—21no—
100C4⋊Dic73C2×D412yes—
101#11924005—22no—
102#11924005—22no—
103C2×Q8×C73—22no—
104C2×Q8×C73—22no—
105C2×Q8×C73—41no—
106C2×Q8×C73—41no—
107C2×Q8×C73—42no—
108C2×Q8×C73—42no—
109C2×Q8×C73—82no—
110C22×C4×C73—41no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73#4111yescharacteristic
115C2×C8×C73—22no—
116C2×C8×C73SD1614yescomplemented
117C2×C8×C73C2×D412yes—
118C2×C8×C73C2×D412yes—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—22no—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C42×C73C2×D411yescharacteristic
128C42×C73C2×D411yescharacteristic
129C42×C73#4111yescharacteristic
130C42×C73C2411yescharacteristic
131C23×C73#12111yescharacteristic
132C22×D73—22no—
133C22×D73—42no—
134C22×D73—42no—
135C22×D73—42no—
136Q8×C73—42no—
137Q8×C73—42no—
138Q8×C73—82no—
139Q8×C73—82no—
140Q8×C73—82no—
141C2×C4×C73—22no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—82no—
145C2×C4×C73—82no—
146C2×C4×C73#12111yescharacteristic
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73C2×SD1612yes—
150C2×C4×C73AGL(1,ℤ/8)12yes—
151C2×C4×C73#13712yes—
152C2×C4×C73#13712yes—
153C2×C4×C73C22×D411yescharacteristic
154C2×C4×C73C22×D411yescharacteristic
155C2×C4×C73C2×#4111yescharacteristic
156C2×Dic73—22no—
157C2×Dic73—42no—
158D292—22no—
159D292—22no—
160D292—42no—
161C4×D73—42no—
162C4×D73—42no—
163C4×D73—44no—
164C8×C73—22no—
165C8×C73C2×SD1614yescomplemented
166#10596831—42no—
167#10596831—42no—
168#10596831—42no—
169C22×C73#46112yes—
170C22×C73#46812yes—
171C22×C73#47112yes—
172C22×C73C2×#12111yescharacteristic
173C2×D73—42no—
174C2×D73—42no—
175C2×D73—42no—
176C2×D73—84no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—82no—
180C4×C73—82no—
181C4×C73#46112yes—
182C4×C73#46812yes—
183C4×C73C22×SD1611yescharacteristic
184C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
185C4×C73C2×#13711yescharacteristic
186C4×C73C2×#13711yescharacteristic
187Dic73—42no—
188C2×#1691—731nomaximal, Sylow
189C2×C73#169114yescomplemented
190C2×C73C2×#46111yescharacteristic
191C2×C73C2×#46811yescharacteristic
192C2×C73C2×#47111yescharacteristic
193D73—82no—
194C2×#498D7311yescharacteristic, complemented
195C2×#476—731no—
196C2×#461—731no—
197C2×#436—731no—
198#1691—738no—
199C2×#335—731no—
200C2×#333—731no—
201C2×#332—731no—
202C73C2×#169111yescharacteristic, complemented, Sylow
203C22×SD16—1461no—
204C22×M16—1461no—
205C2×C4⋊Q8C2×D7311yescharacteristic
206C2×C4⋊Q8C2×D7311yescharacteristic
207C2×C4×D4—731no—
208#498—22no—
209#498C2×D7314yescomplemented
210#476—734no—
211#476—1462no—
212#461—734no—
213#461—1462no—
214#436—734no—
215#436—1462no—
216C2×#106—21no—
217C2×#106—21no—
218C2×#106—1461no—
219C2×C4⋊C8—731no—
220C2×C4⋊C8—731no—
221C2×C4⋊C8—731no—
222C2×#103—1461no—
223C2×#102—1461no—
224C2×C22⋊C8—1461no—
225C2×C4×C8C2×D7311yescharacteristic
226#335—734no—
227#333—734no—
228#332—734no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4—731no—
232C23×C4—1461no—
233C2×SD16—2924no—
234C2×SD16—1464no—
235C2×M16—2924no—
236C2×M16—1464no—
237C22×C8—1461no—
238C22×C8—1461no—
239C22×C8—1461no—
240C22×C8C22⋊D7311yescharacteristic
241C22×C8C22⋊D7311yescharacteristic
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8C22×D7312yes—
249C4⋊Q8C22×D7312yes—
250C4×D4—732no—
251C4×D4—732no—
252C4×D4—1461no—
253C4×D4—1461no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—731no—
259C2×C22⋊C4—1461no—
260C2×C42C22×D7311yescharacteristic
261#106—24no—
262#106—24no—
263#106—1464no—
264C4⋊C8—732no—
265C4⋊C8—732no—
266C4⋊C8—732no—
267C4⋊C8—1461no—
268C4⋊C8—1461no—
269C4⋊C8—1461no—
270#103—1464no—
271#102—1464no—
272C22⋊C8—1464no—
273C4×C8C22⋊D7312yes—
274C4×C8C22×D7312yes—
275C24—1461no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—82no—
283C2×D4—732no—
284C2×D4—1461no—
285C2×D4—1462no—
286C2×D4—1462no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—2922no—
290C22×C4—2922no—
291C22×C4—1461no—
292C22×C4—1462no—
293C22×C4—1462no—
294C22×C4D4×D7311yescharacteristic
295C22×C4C2×C22⋊D7311yescharacteristic
296C22×C4#1192402711yescharacteristic
297SD16—2928no—
298M16—2928no—
299C2×C8—22no—
300C2×C8—2922no—
301C2×C8—2922no—
302C2×C8—2922no—
303C2×C8—1462no—
304C2×C8—1462no—
305C2×C8—1462no—
306C2×C8#1192401514yescomplemented
307C2×C8C2×C22⋊D7312yes—
308C2×C8C2×C22⋊D7312yes—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—732no—
318C4⋊C4—1461no—
319C22⋊C4—1462no—
320C22⋊C4—1462no—
321C42D4×D7311yescharacteristic
322C42C2×C22⋊D7311yescharacteristic
323C42#1192402711yescharacteristic
324C42C23×D7311yescharacteristic
325C23—2922no—
326C23—2922no—
327C23—2922no—
328C23—1462no—
329C23#1455656211yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—82no—
335D4—2922no—
336D4—1462no—
337D4—1462no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—2924no—
346C2×C4—2922no—
347C2×C4—1462no—
348C2×C4#1455656211yescharacteristic
349C2×C4#1455656211yescharacteristic
350C2×C4#1455656211yescharacteristic
351C2×C4C2×#1192401512yes—
352C2×C4#1455661812yes—
353C2×C4#1455662012yes—
354C2×C4#1455662212yes—
355C2×C4C2×D4×D7311yescharacteristic
356C2×C4C22×C22⋊D7311yescharacteristic
357C2×C4C2×#1192402711yescharacteristic
358C8—22no—
359C8—2922no—
360C8—2922no—
361C8—2922no—
362C8C2×#1192401514yescomplemented
363C22—2922no—
364C22—2922no—
365C22—2922no—
366C22—5844no—
367C22#1798878912yes—
368C22#1798883012yes—
369C22#1798884712yes—
370C22C2×#1455656211yescharacteristic
371C4—21no—
372C4—21no—
373C4—82no—
374C4—82no—
375C4—2922no—
376C4#1798878912yes—
377C4#1798883012yes—
378C4C22×#1192401511yescharacteristic
379C4C2×#1455661811yescharacteristic
380C4C2×#1455662011yescharacteristic
381C4C2×#1455662211yescharacteristic
382C2—5842no—
383C2#2260942714yesdirect product factor
384C2C2×#1798878911yescharacteristic
385C2C2×#1798883011yescharacteristic
386C2C2×#1798884711yescharacteristic
subgroupG/H
centerC23#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#498×C73C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenterC2×#498D73
hypocenterC73C2×#1691
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. #22609427C2direct (N × H)
5. C2×#17988340C2semidirect
6. C2×#17988336C2semidirect
7. C2×#17988333C2semidirect
8. C2×#498×C73C2semidirect
25. #17988340C22semidirect
26. #17988336C22semidirect
27. #17988333C22semidirect
31. #498×C73C22semidirect
116. C2×C8×C73SD16semidirect
165. C8×C73C2×SD16semidirect
189. C2×C73#1691semidirect
194. C2×#498D73semidirect
202. C73C2×#1691semidirect
209. #498C2×D73semidirect
306. C2×C8#11924015semidirect
362. C8C2×#11924015semidirect
383. C2#22609427direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179888742
C2×#179887892
C2×#179886662
#226094272
C2×#179883402
C2×#179883362
C2×#179883332
C2×#498×C732
C2×#169173