Finite Group Databasebeta

← #22620406

Subgroups — #22620406

Order 9344.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556623C211yesmaximal, characteristic, complemented
2C2×#14556618C211yesmaximal, characteristic, complemented
3C2×#14556614C211yesmaximal, characteristic, complemented
4C2×SD16×D73C211yesmaximal, characteristic, complemented
5#17989162C211yesmaximal, characteristic, complemented
6#17989155C211yesmaximal, characteristic, complemented
7#17988980C211yesmaximal, characteristic, complemented
8#17988976C211yesmaximal, characteristic, complemented
9#17988973C211yesmaximal, characteristic, complemented
10#17988972C211yesmaximal, characteristic, complemented
11#17988969C211yesmaximal, characteristic, complemented
12#17988961C211yesmaximal, characteristic, complemented
13#17988958C211yesmaximal, characteristic, complemented
14#17988544C211yesmaximal, characteristic, complemented
15#497×C73C211yesmaximal, characteristic, complemented
16C2×#11924029C2211yescharacteristic
17C2×#11924029C2211yescharacteristic
18C2×Q8×D73C2211yescharacteristic
19C2×D4×D73C2211yescharacteristic
20#14556623—24no—
21#14556618—24no—
22#14556614—24no—
23C2×#11924018C2211yescharacteristic, complemented
24C2×#11924017C2211yescharacteristic
25C2×#11924017C2211yescharacteristic, complemented
26C2×#11924016C2211yescharacteristic
27C2×#11924015C2211yescharacteristic
28C2×#11924015C2211yescharacteristic
29C2×#11924015C2211yescharacteristic, complemented
30C2×#11924015C2211yescharacteristic, complemented
31C2×#11924015C2211yescharacteristic, complemented
32SD16×D73—24no—
33C2×D584C2211yescharacteristic
34C2×D584C2211yescharacteristic, complemented
35C2×#11924013C2211yescharacteristic
36C2×#11924011C2211yescharacteristic
37C2×C8×D73C2211yescharacteristic
38#14556589C2211yescharacteristic, complemented
39#14556588C2211yescharacteristic
40#14556587C2211yescharacteristic
41#14556580C2211yescharacteristic
42#14556580C2211yescharacteristic, complemented
43#14556579C2211yescharacteristic, complemented
44#14556576—22no—
45#14556569—22no—
46#14556564—22no—
47#14556561—22no—
48#14556548C2211yescharacteristic
49#14556501C2211yescharacteristic, complemented
50C4×#10596831C2211yescharacteristic, complemented
51C8×Dic73C2211yescharacteristic, complemented
52C2×Q16×C73C2211yescharacteristic, complemented
53C2×Q16×C73C2211yescharacteristic, complemented
54C2×SD16×C73C2211yescharacteristic
55C2×SD16×C73C2211yescharacteristic
56C4⋊Q8×C73C2211yescharacteristic, complemented
57#124×C73C2211yescharacteristic
58#97×C73C2211yescharacteristic, complemented
59C23×D73—21no—
60#11924029—22no—
61#11924029—22no—
62#11924029—22no—
63#11924029—22no—
64C2×Dic146—21no—
65C2×Dic146C2311yescharacteristic
66Q8×D73—22no—
67Q8×D73—22no—
68C2×D292—21no—
69C2×D292—21no—
70C2×D292—22no—
71C2×D292C2311yescharacteristic
72C2×D292C2311yescharacteristic
73C2×D292C2311yescharacteristic
74C2×C22⋊D73—21no—
75C2×C22⋊D73—22no—
76D4×D73—22no—
77D4×D73—22no—
78C2×C4×D73—21no—
79C2×C4×D73—21no—
80C2×C4×D73—21no—
81C2×C4×D73C2311yescharacteristic
82#11924018—22no—
83#11924017—22no—
84#11924017—22no—
85#11924016—22no—
86#11924015—22no—
87#11924015—22no—
88#11924015—22no—
89#11924015—22no—
90#11924015—22no—
91D584—22no—
92D584—22no—
93#11924013—22no—
94#11924011D414yescomplemented
95C2×#10596831C2311yescharacteristic
96C2×#10596831C2311yescharacteristic
97C8×D73D414yescomplemented
98#11924007—22no—
99#11924007D412yes—
100#11924005—22no—
101#11924005—22no—
102#11924005—22no—
103#11924005D412yes—
104C4×Dic73C2311yescharacteristic
105C2×Q8×C73C2311yescharacteristic
106C2×Q8×C73C2311yescharacteristic
107C2×Q8×C73C2311yescharacteristic
108C2×D4×C73C2311yescharacteristic
109Q16×C73—22no—
110Q16×C73—22no—
111SD16×C73—22no—
112SD16×C73—22no—
113C2×C8×C73C2311yescharacteristic
114C2×C8×C73C2311yescharacteristic
115C4⋊C4×C73—22no—
116C22⋊C4×C73—22no—
117C42×C73C2311yescharacteristic
118C23×C73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—42no—
123C22×D73—82no—
124C22×D73C2×D411yescharacteristic
125Q8×C73—21no—
126Q8×C73—21no—
127Q8×C73—21no—
128Q8×C73—21no—
129Q8×C73—21no—
130Q8×C73—21no—
131D4×C73—21no—
132D4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73C2×D412yes—
137C2×C4×C73C2411yescharacteristic
138C22⋊D73—42no—
139C22⋊D73—82no—
140C2×Dic73—21no—
141C2×Dic73C2×D412yes—
142C2×Dic73C2×D411yescharacteristic
143D292—21no—
144D292—21no—
145D292—21no—
146D292—21no—
147D292—21no—
148D292—21no—
149D292—42no—
150D292—42no—
151D292—82no—
152C4×D73—42no—
153C4×D73—42no—
154C4×D73—42no—
155C4×D73C2×D412yes—
156C4×D73C2×D412yes—
157Dic146—21no—
158Dic146—21no—
159Dic146—42no—
160C8×C73C2×D412yes—
161C8×C73C2×D412yes—
162#10596831C2×D412yes—
163#10596831C2×D412yes—
164C22×C73—41no—
165C22×C73—42no—
166C22×C73C22×D411yescharacteristic
167C2×D73—22no—
168C2×D73—41no—
169C2×D73—41no—
170C2×D73—41no—
171C2×D73—42no—
172C2×D73—42no—
173C2×D73—42no—
174C2×D73—84no—
175C2×D73#12712yes—
176C4×C73—22no—
177C4×C73—41no—
178C4×C73—41no—
179C4×C73—41no—
180C4×C73C22×D411yescharacteristic
181C4×C73C22×D411yescharacteristic
182Dic73—22no—
183Dic73—41no—
184Dic73#12712yes—
185C2×C73—81no—
186C2×C73C2×#12711yescharacteristic
187C2×C73#57712yes—
188D73—22no—
189D73—81no—
190D73—81no—
191D73—81no—
192#3154—731nomaximal, Sylow
193C73#315411yescharacteristic, complemented, Sylow
194C2×#137—731no—
195C2×AGL(1,ℤ/8)—731no—
196C2×#135—731no—
197C22×SD16—731no—
198#536—731no—
199#535—731no—
200#497—731no—
201#497D7311yescharacteristic, complemented
202#496—731no—
203#496—731no—
204#495—731no—
205#495—731no—
206#492—731no—
207#492—731no—
208#405—731no—
209C2×#41—731no—
210C2×#41—731no—
211C22×Q8—731no—
212C22×D4—731no—
213#137—1464no—
214AGL(1,ℤ/8)—1464no—
215#135—1464no—
216C2×Q16—731no—
217C2×Q16C2×D7311yescharacteristic, complemented
218C2×Q16C2×D7311yescharacteristic, complemented
219C2×SD16—731no—
220C2×SD16—731no—
221C2×SD16—731no—
222C2×SD16—731no—
223C2×SD16—731no—
224C2×SD16—731no—
225C2×SD16—731no—
226C2×SD16—731no—
227C2×SD16—1464no—
228C2×SD16C2×D7311yescharacteristic
229C2×SD16C2×D7311yescharacteristic
230C2×D8—731no—
231C2×D8—731no—
232C2×D8—731no—
233C2×M16—731no—
234C22×C8—731no—
235C4⋊Q8—731no—
236C4⋊Q8C2×D7311yescharacteristic, complemented
237#127—731no—
238#127—731no—
239#124—731no—
240#124—731no—
241#124—731no—
242#124C2×D7311yescharacteristic
243#123—1462no—
244C22⋊Q8—1462no—
245#121—1462no—
246#120—1462no—
247#117—731no—
248#97—731no—
249#97C2×D7311yescharacteristic, complemented
250C4×C8—731no—
251C4×C8—731no—
252C24—1461no—
253#41—1462no—
254#41—1462no—
255#41—1462no—
256#41—1462no—
257C2×Q8—731no—
258C2×Q8—1461no—
259C2×Q8—1462no—
260C2×Q8—1462no—
261C2×Q8C22×D7311yescharacteristic
262C2×Q8C22×D7311yescharacteristic
263C2×Q8C22×D7311yescharacteristic
264C2×D4—731no—
265C2×D4—731no—
266C2×D4—731no—
267C2×D4—1461no—
268C2×D4—1461no—
269C2×D4—1462no—
270C2×D4—1462no—
271C2×D4—1461no—
272C2×D4—1462no—
273C2×D4—1462no—
274C2×D4C22×D7311yescharacteristic
275C22×C4—731no—
276C22×C4—1461no—
277C22×C4—1461no—
278C22×C4—1461no—
279Q16—22no—
280Q16—22no—
281Q16—1462no—
282SD16—22no—
283SD16—22no—
284SD16—1462no—
285SD16—1462no—
286SD16—1462no—
287SD16—1462no—
288SD16—1462no—
289SD16—1462no—
290SD16—1462no—
291SD16—1462no—
292D8—1462no—
293D8—1462no—
294D8—1462no—
295M16—734no—
296C2×C8—731no—
297C2×C8—731no—
298C2×C8—734no—
299C2×C8C22×D7311yescharacteristic
300C2×C8C22×D7311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—732no—
303C4⋊C4—1462no—
304C22⋊C4—22no—
305C22⋊C4—732no—
306C22⋊C4—1462no—
307C22⋊C4—1462no—
308C22⋊C4—1462no—
309C42—731no—
310C42C22×D7311yescharacteristic
311C23—21no—
312C23—2922no—
313C23—5842no—
314C23—731no—
315C23—1461no—
316C23—1461no—
317C23—1461no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—2922no—
325Q8—1461no—
326Q8—1461no—
327D4—21no—
328D4—21no—
329D4—2922no—
330D4—2922no—
331D4—2922no—
332D4—5842no—
333D4—5842no—
334D4—1461no—
335D4—1461no—
336D4—1461no—
337D4—1461no—
338D4—1461no—
339D4—1461no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—731no—
347C2×C4—732no—
348C2×C4—732no—
349C2×C4—732no—
350C2×C4—1461no—
351C2×C4D4×D7312yes—
352C2×C4C23×D7311yescharacteristic
353C8—732no—
354C8—732no—
355C8D4×D7312yes—
356C8D4×D7312yes—
357C22—41no—
358C22—42no—
359C22—2921no—
360C22—2921no—
361C22—2921no—
362C22—2922no—
363C22—2922no—
364C22—2922no—
365C22—5844no—
366C22—732no—
367C22—1462no—
368C22C2×D4×D7311yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—2921no—
374C4—732no—
375C4—1462no—
376C4C2×D4×D7311yescharacteristic
377C4C2×D4×D7311yescharacteristic
378C2—81no—
379C2—5841no—
380C2—5841no—
381C2—5841no—
382C2—1462no—
383C2#127×D7311yescharacteristic
384C2#1798939612yes—
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#497×C73C2
socleC22×C73C22×D4
hypercenter#497D73
hypocenterC73#3154
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14556623C2semidirect
2. C2×#14556618C2semidirect
3. C2×#14556614C2semidirect
4. C2×SD16×D73C2semidirect
5. #17989162C2semidirect
6. #17989155C2semidirect
7. #17988980C2semidirect
8. #17988976C2semidirect
9. #17988973C2semidirect
10. #17988972C2semidirect
11. #17988969C2semidirect
12. #17988961C2semidirect
13. #17988958C2semidirect
14. #17988544C2semidirect
15. #497×C73C2semidirect
23. C2×#11924018C22semidirect
25. C2×#11924017C22semidirect
29. C2×#11924015C22semidirect
30. C2×#11924015C22semidirect
31. C2×#11924015C22semidirect
34. C2×D584C22semidirect
38. #14556589C22semidirect
42. #14556580C22semidirect
43. #14556579C22semidirect
49. #14556501C22semidirect
50. C4×#10596831C22semidirect
51. C8×Dic73C22semidirect
52. C2×Q16×C73C22semidirect
53. C2×Q16×C73C22semidirect
56. C4⋊Q8×C73C22semidirect
58. #97×C73C22semidirect
94. #11924011D4semidirect
97. C8×D73D4semidirect
193. C73#3154semidirect
201. #497D73semidirect
217. C2×Q16C2×D73semidirect
218. C2×Q16C2×D73semidirect
236. C4⋊Q8C2×D73semidirect
249. #97C2×D73semidirect