Finite Group Databasebeta

← #33069769

Subgroups — #33069769

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619391C212yesmaximal, complemented
2C2×C8×D292C211yesmaximal, characteristic, complemented
3#22609365C214yesmaximal, complemented
4#22609317C212yesmaximal, complemented
5#22609283C212yesmaximal, complemented
6#22608690C211yesmaximal, characteristic, complemented
7#22608608C212yesmaximal, complemented
8C73×#3169C211yesmaximal, characteristic, complemented
9C2×#14556616—22no—
10C22×C8×D73—21no—
11C2×C4×D292C2211yescharacteristic
12#17988799—24no—
13#17988789C2214yes—
14#17988779C2214yes—
15C8×D292—21no—
16C8×D292—21no—
17C8×D292C2212yes—
18C8×D292C2212yes—
19#17988611C2212yes—
20C2×C4⋊#10596831C2211yescharacteristic
21#17988588—22no—
22C2×#14556503—21no—
23#17988340C2212yescomplemented
24#17988340C2212yescomplemented
25#17988333C2214yescomplemented
26#17988331C2214yescomplemented
27#532×C73C2212yes—
28#500×C73C2211yescharacteristic, complemented
29#493×C73C2211yescharacteristic, complemented
30#486×C73C2212yescomplemented
31#466×C73—21no—
32#466×C73—21no—
33C4×D8×C73—21no—
34C4×D8×C73—21no—
35C2×C4×C8×C73C2211yescharacteristic
36C22×D292D411yescharacteristic
37C22×C4×D73—21no—
38#14556616—48no—
39C2×#11924015—24no—
40C2×#11924013—24no—
41C2×#11924012—22no—
42C22×#10596831—21no—
43C2×C8×D73—22no—
44C2×C8×D73—22no—
45C2×C8×D73—44no—
46C4×D292—21no—
47C4×D292—21no—
48C4×D292D412yes—
49C4×D292C2312yes—
50C2×C4⋊Dic73D411yescharacteristic
51C2×#11924005—21no—
52C4⋊#10596831—21no—
53C4⋊#10596831—22no—
54C4⋊#10596831C2312yes—
55C4⋊#10596831C2314yes—
56#14556515—24no—
57#14556512—24no—
58#14556503—22no—
59#14556503—22no—
60C2×#41×C73—22no—
61C2×D8×C73—21no—
62C2×D8×C73—21no—
63C22×C8×C73D411yescharacteristic
64C22×C8×C73D411yescharacteristic
65C4⋊Q8×C73C2312yes—
66#127×C73C2312yes—
67#124×C73—22no—
68#121×C73—22no—
69#121×C73—22no—
70C4×D4×C73—22no—
71C4×D4×C73—22no—
72C2×C42×C73C2311yescharacteristic
73#107×C73—21no—
74#107×C73—21no—
75#102×C73—22no—
76#102×C73—22no—
77C4×C8×C73D411yescharacteristic
78C4×C8×C73D411yescharacteristic
79C4×C8×C73C2311yescharacteristic
80C4×C8×C73C2311yescharacteristic
81C23×D73—21no—
82C2×D292—21no—
83C2×D292—22no—
84C2×D292D814yescomplemented
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#11924013—48no—
93#11924012—44no—
94C2×#10596831—22no—
95C2×#10596831—24no—
96C2×#10596831—42no—
97C8×D73—44no—
98C8×D73—44no—
99C4⋊Dic73—21no—
100C4⋊Dic73C2×D412yes—
101#11924005—22no—
102#11924005—22no—
103#41×C73—44no—
104#41×C73—44no—
105C2×Q8×C73—22no—
106C2×D4×C73—22no—
107C2×D4×C73—22no—
108C2×D4×C73—22no—
109C2×D4×C73—42no—
110C22×C4×C73—42no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4111yescharacteristic
114D8×C73—42no—
115D8×C73—42no—
116C2×C8×C73—22no—
117C2×C8×C73—41no—
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C4⋊C4×C73—22no—
123C4⋊C4×C73—22no—
124C22⋊C4×C73—42no—
125C22⋊C4×C73—42no—
126C42×C73C2×D411yescharacteristic
127C42×C73C2×D411yescharacteristic
128C42×C73#4111yescharacteristic
129C42×C73C2411yescharacteristic
130C23×C73—42no—
131C23×C73#12111yescharacteristic
132C22×D73—22no—
133C22×D73—24no—
134C22×D73—42no—
135C22×D73—42no—
136Q8×C73—42no—
137Q8×C73—42no—
138D4×C73—42no—
139D4×C73—42no—
140D4×C73—42no—
141D4×C73—42no—
142D4×C73—82no—
143D4×C73—84no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—84no—
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73C2×D812yes—
153C2×C4×C73#13512yes—
154C2×C4×C73C22×D411yescharacteristic
155C2×C4×C73C22×D411yescharacteristic
156C2×C4×C73C2×#4111yescharacteristic
157C2×Dic73—22no—
158C2×Dic73—42no—
159D292—22no—
160D292—24no—
161D292C2×D814yes—
162C4×D73—44no—
163C4×D73—42no—
164C4×D73—42no—
165C8×C73—21no—
166C8×C73—21no—
167C8×C73—21no—
168C8×C73AGL(1,ℤ/8)12yescomplemented
169#10596831—44no—
170#10596831—42no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—82no—
174C22×C73—82no—
175C22×C73—82no—
176C22×C73#46612yes—
177C22×C73C2×#12111yescharacteristic
178C2×D73—24no—
179C2×D73—48no—
180C2×D73—42no—
181C2×D73—42no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—82no—
186C4×C73#46612yes—
187C4×C73C22×D811yescharacteristic
188C4×C73C2×#13511yescharacteristic
189C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
190C4×C73#57511yescharacteristic
191Dic73—42no—
192#19255—731nomaximal, Sylow
193C2×C73—22no—
194C2×C73—162no—
195C2×C73#303611yescharacteristic
196C2×C73C2×#46611yescharacteristic
197C2×C73#306711yescharacteristic
198D73—44no—
199#3169D7311yescharacteristic, complemented
200#3036—732no—
201C2×C8×D4—731no—
202#1681—734no—
203#1673—732no—
204#1667—732no—
205#1567—731no—
206#1547—732no—
207C73#1925511yescharacteristic, complemented, Sylow
208C2×AGL(1,ℤ/8)—1462no—
209C23×C8—1461no—
210#532C2×D7312yes—
211C2×C4×D4—731no—
212#500C2×D7311yescharacteristic, complemented
213#493C2×D7311yescharacteristic, complemented
214#486C2×D7312yescomplemented
215#466—21no—
216#466—21no—
217#463—1464no—
218#461—734no—
219#459—734no—
220C4×D8—21no—
221C4×D8—21no—
222C8×D4—732no—
223C8×D4—732no—
224C8×D4—1461no—
225C8×D4—1461no—
226#423—732no—
227C2×C4⋊C8—731no—
228#417—1462no—
229C2×C22⋊C8—1461no—
230C2×C4×C8C2×D7311yescharacteristic
231#335—732no—
232#335—732no—
233#332—734no—
234#331—734no—
235C2×#41—22no—
236C22×D4—731no—
237C23×C4—1461no—
238AGL(1,ℤ/8)—2928no—
239C2×SD16—1464no—
240C2×D8—21no—
241C2×D8—21no—
242C2×D8—1464no—
243C2×M16—1462no—
244C22×C8—2924no—
245C22×C8—1461no—
246C22×C8—1462no—
247C22×C8—1462no—
248C22×C8C22⋊D7311yescharacteristic
249C22×C8C22⋊D7311yescharacteristic
250C4⋊Q8C22×D7312yes—
251#127C22×D7312yes—
252#124—22no—
253#121—22no—
254#121—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—732no—
258C4×D4—732no—
259C4×D4—1461no—
260C4×D4—1461no—
261C2×C4⋊C4—731no—
262C2×C22⋊C4—1461no—
263C2×C42C22×D7311yescharacteristic
264#107—21no—
265#107—21no—
266C4⋊C8—732no—
267C4⋊C8—734no—
268C4⋊C8—1462no—
269C4⋊C8—1461no—
270#103—1464no—
271#102—22no—
272#102—22no—
273#102—1464no—
274C22⋊C8—1462no—
275C22⋊C8—1462no—
276C4×C8C22⋊D7311yescharacteristic
277C4×C8C22⋊D7311yescharacteristic
278C4×C8C22×D7311yescharacteristic
279C4×C8C22×D7311yescharacteristic
280C24—1461no—
281#41—44no—
282#41—44no—
283C2×Q8—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—734no—
289C2×D4—732no—
290C2×D4—1461no—
291C2×D4—1462no—
292C22×C4—42no—
293C22×C4—2922no—
294C22×C4—2922no—
295C22×C4—1461no—
296C22×C4—1462no—
297C22×C4—1462no—
298C22×C4D4×D7311yescharacteristic
299C22×C4C2×C22⋊D7311yescharacteristic
300C22×C4#1192402711yescharacteristic
301SD16—2928no—
302D8—42no—
303D8—42no—
304D8—2928no—
305M16—2924no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—2924no—
309C2×C8—2922no—
310C2×C8—2924no—
311C2×C8—1464no—
312C2×C8—1462no—
313C2×C8C2×C22⋊D7311yescharacteristic
314C2×C8C2×C22⋊D7311yescharacteristic
315C2×C8C2×C22⋊D7311yescharacteristic
316C2×C8C2×C22⋊D7311yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—732no—
320C4⋊C4—1461no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—1462no—
324C22⋊C4—1462no—
325C42D4×D7311yescharacteristic
326C42C2×C22⋊D7311yescharacteristic
327C42#1192402711yescharacteristic
328C42C23×D7311yescharacteristic
329C23—42no—
330C23—2922no—
331C23—2922no—
332C23—1464no—
333C23—1462no—
334C23#1455656211yescharacteristic
335Q8—42no—
336Q8—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—42no—
341D4—82no—
342D4—84no—
343D4—734no—
344D4—1464no—
345D4—1462no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4—2924no—
352C2×C4—2922no—
353C2×C4—2922no—
354C2×C4—2922no—
355C2×C4—1462no—
356C2×C4#1455656211yescharacteristic
357C2×C4#1455656211yescharacteristic
358C2×C4#1455656211yescharacteristic
359C2×C4D8×D7312yes—
360C2×C4#1455661012yes—
361C2×C4C2×D4×D7311yescharacteristic
362C2×C4C22×C22⋊D7311yescharacteristic
363C2×C4C2×#1192402711yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—2924no—
368C8—2922no—
369C8#1455661612yescomplemented
370C22—21no—
371C22—21no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—2928no—
376C22—2922no—
377C22—2922no—
378C22—1464no—
379C22#1798882012yes—
380C22C2×#1455656211yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—2922no—
386C4#1798882012yes—
387C4C2×D8×D7311yescharacteristic
388C4C2×#1455661011yescharacteristic
389C4C2×#1455661611yescharacteristic
390C4#1798938511yescharacteristic
391C2—22no—
392C2—162no—
393C2—2924no—
394C2#2261939111yescharacteristic
395C2C2×#1798882011yescharacteristic
396C2#2261964711yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3169C2
socleC22×C73C2×#121
hypercenter#3169D73
hypocenterC73#19255
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #22619391C2semidirect
2. C2×C8×D292C2semidirect
3. #22609365C2semidirect
4. #22609317C2semidirect
5. #22609283C2semidirect
6. #22608690C2semidirect
7. #22608608C2semidirect
8. C73×#3169C2semidirect
23. #17988340C22semidirect
24. #17988340C22semidirect
25. #17988333C22semidirect
26. #17988331C22semidirect
28. #500×C73C22semidirect
29. #493×C73C22semidirect
30. #486×C73C22semidirect
84. C2×D292D8semidirect
168. C8×C73AGL(1,ℤ/8)semidirect
199. #3169D73semidirect
207. C73#19255semidirect
212. #500C2×D73semidirect
213. #493C2×D73semidirect
214. #486C2×D73semidirect
369. C8#14556616semidirect

Maximal subgroups

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

subgroupindex
#226193912
C2×C8×D2922
#226093652
#226093172
#226092832
#226086902
#226086082
C73×#31692
#1925573