Finite Group Databasebeta

← #16833145

Subgroups — #16833145

Order 3712.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745365C211yesmaximal, characteristic, complemented
2C2×#41×D29C211yesmaximal, characteristic, complemented
3#14271833C213yesmaximal, complemented
4#14271832C211yesmaximal, characteristic, complemented
5#14271831C213yesmaximal, complemented
6#14271825C211yesmaximal, characteristic, complemented
7#14271822C213yesmaximal, complemented
8#14271820C211yesmaximal, characteristic, complemented
9#14271819C213yesmaximal, complemented
10#14271729C213yesmaximal, complemented
11#14271726C213yesmaximal, complemented
12#14271673C213yesmaximal, complemented
13#14271672C213yesmaximal, complemented
14#14271639C211yesmaximal, characteristic, complemented
15C29×#550C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C2×#41×C29C2211yescharacteristic, complemented
18#11745365—28no—
19#41×D29—28no—
20C2×#90809C2211yescharacteristic
21C2×Q8×D29C2211yescharacteristic
22C2×Q8×D29C2211yescharacteristic, complemented
23C2×#90807C2213yes—
24C2×#90807C2213yes—
25C2×#90807C2213yescomplemented
26C2×D4×D29C2213yes—
27C2×#90805C2213yes—
28C2×#90805C2213yes—
29C22×C4×D29C2213yes—
30C22×Dic58C2213yescomplemented
31C29×#127C2213yescomplemented
32C29×#121C2213yescomplemented
33C29×#121C2213yescomplemented
34C4×Q8×C29C2211yescharacteristic, complemented
35C4×D4×C29C2213yescomplemented
36Q8×Dic29C2211yescharacteristic, complemented
37#11745302C2213yescomplemented
38#11745302C2216yescomplemented
39#11745301C2216yescomplemented
40#11745301C2216yescomplemented
41#11745301C2213yescomplemented
42#11745300C2213yescomplemented
43#11745300C2213yescomplemented
44#11745299C2213yes—
45#11745299C2216yes—
46#11745298C2213yescomplemented
47D4×Dic29C2216yescomplemented
48#11745292C2213yescomplemented
49C4×C22⋊D29C2213yescomplemented
50C4×C22⋊D29C2213yescomplemented
51#11745286C2213yescomplemented
52#11745259C2213yescomplemented
53#11745258C2213yescomplemented
54#11745257C2213yescomplemented
55#11745256C2211yescharacteristic
56#11745254C2213yes—
57#11745249C2213yescomplemented
58#11745247C2216yescomplemented
59#11745246C2213yescomplemented
60#11745245C2213yes—
61#11745244C2213yes—
62#11745243C2216yescomplemented
63#11745240C2213yescomplemented
64#11745237C2213yescomplemented
65C4×D116C2213yescomplemented
66#11745234C2213yes—
67C4×Dic58C2213yescomplemented
68C23×D29C2313yes—
69#41×C29—24no—
70#41×C29—24no—
71C2×Q8×C29C2311yescharacteristic
72C2×D4×C29C2313yes—
73C2×D4×C29C2313yes—
74C2×D4×C29C2316yes—
75C2×D4×C29C2313yes—
76C22×C4×C29C2313yes—
77C22×C4×C29C2313yes—
78C2×C22⋊D29C2313yes—
79C2×C22⋊D29C2316yes—
80C2×C22⋊D29C2313yes—
81C2×C22⋊D29C2313yes—
82C2×C22⋊D29C2313yes—
83C2×C22⋊D29C2316yes—
84C22×Dic29C2313yes—
85C22×Dic29C2316yes—
86#90809—24no—
87Q8×D29—24no—
88Q8×D29D418yescomplemented
89#90807—212no—
90#90807—212no—
91#90807—212no—
92D4×D29—212no—
93#90805—212no—
94#90805—212no—
95C2×D116C2313yes—
96C2×D116C2311yescharacteristic
97C2×D116C2313yes—
98C2×C4×D29—212no—
99C2×C4×D29C2313yes—
100C2×C4×D29C2313yes—
101C2×C4×D29C2311yescharacteristic
102C2×C4×D29C2313yes—
103C2×C4×D29C2313yes—
104C2×Dic58—212no—
105C2×Dic58C2313yes—
106C2×Dic58C2313yes—
107C2×Dic58C2313yes—
108C4⋊C4×C29C2311yescharacteristic
109C4⋊C4×C29C2313yes—
110C22⋊C4×C29C2313yes—
111C22⋊C4×C29C2313yes—
112C42×C29C2313yes—
113C22⋊Dic29C2316yes—
114C22⋊Dic29C2313yes—
115C22⋊Dic29C2313yes—
116#90781C2311yescharacteristic
117#90781C2313yes—
118#90781C2311yescharacteristic
119#90781C2313yes—
120#90781C2313yes—
121#90781C2316yes—
122#90781C2311yescharacteristic
123C4⋊Dic29C2313yes—
124C4⋊Dic29C2313yes—
125#90779C2313yes—
126#90779C2311yescharacteristic
127#90779C2313yes—
128#90779C2313yes—
129C4×Dic29C2313yes—
130C4×Dic29C2316yes—
131C23×C29C2413yes—
132C23×C29C2413yes—
133C22×D29—26no—
134C22×D29—43no—
135C22×D29—43no—
136C22×D29C2413yes—
137C22×D29C2411yescharacteristic
138C22×D29C2411yescharacteristic
139Q8×C29C2×D414yes—
140D4×C29—26no—
141D4×C29—26no—
142D4×C29—46no—
143D4×C29—43no—
144C2×C4×C29—26no—
145C2×C4×C29—26no—
146C2×C4×C29C2413yes—
147C2×C4×C29C2411yescharacteristic
148C2×C4×C29C2411yescharacteristic
149C2×C4×C29C2411yescharacteristic
150C2×C4×C29C2413yes—
151C22⋊D29—26no—
152C22⋊D29—212no—
153C22⋊D29—26no—
154C22⋊D29—43no—
155C22⋊D29—43no—
156C22⋊D29—46no—
157C2×Dic29—212no—
158C2×Dic29—26no—
159C2×Dic29C2411yescharacteristic
160C2×Dic29C2413yes—
161C2×Dic29C2411yescharacteristic
162C2×Dic29C2413yes—
163C2×Dic29C2413yes—
164D116—22no—
165D116—26no—
166D116—43no—
167C4×D29—22no—
168C4×D29—26no—
169C4×D29—26no—
170C4×D29—26no—
171C4×D29C2×D4112yes—
172Dic58—26no—
173Dic58—26no—
174Dic58C2×D4112yes—
175#3487—291nomaximal, Sylow
176C22×C29—23no—
177C22×C29—23no—
178C22×C29—23no—
179C22×C29—23no—
180C22×C29—23no—
181C22×C29—23no—
182C22×C29C2511yescharacteristic
183C2×D29—21no—
184C2×D29—21no—
185C2×D29—21no—
186C2×D29—23no—
187C2×D29—21no—
188C2×D29—23no—
189C2×D29—21no—
190C2×D29—23no—
191C2×D29—412no—
192C2×D29C22×D412yes—
193C4×C29—21no—
194C4×C29—21no—
195C4×C29—23no—
196C4×C29—21no—
197C4×C29C22×D416yes—
198Dic29—23no—
199Dic29—21no—
200Dic29—21no—
201Dic29—23no—
202Dic29C22×D416yes—
203C2×2−1+4—291no—
204C22×#41—291no—
205#550—291no—
206#550—293no—
207#550—293no—
208#550D2911yescharacteristic, complemented
209#548—291no—
210#548—293no—
211#548—291no—
212#548—293no—
213#535—293no—
214#535—293no—
215#525—293no—
216#525—293no—
217#519—291no—
218C2×C29—43no—
219C2×C29—43no—
220C2×C29C23×D411yescharacteristic
221C2×C29#58511yescharacteristic
222C2×C29#58511yescharacteristic
223D29—22no—
224D29—41no—
225D29—43no—
2262−1+4—588no—
227C2×#41—291no—
228C2×#41—293no—
229C2×#41—293no—
230C2×#41—293no—
231C2×#41—293no—
232C2×#41—293no—
233C2×#41—588no—
234C2×#41C2×D2911yescharacteristic
235C2×#41C2×D2911yescharacteristic, complemented
236C22×Q8—291no—
237C22×Q8—291no—
238C22×Q8—293no—
239C22×D4—293no—
240C23×C4—293no—
241#127—296no—
242#127—293no—
243#127C2×D2913yescomplemented
244#124—293no—
245#124—296no—
246#124—293no—
247#123—291no—
248#123—293no—
249C22⋊Q8—293no—
250C22⋊Q8—293no—
251C22⋊Q8—293no—
252C22⋊Q8—293no—
253#121—293no—
254#121—293no—
255#121—293no—
256#121—296no—
257#121—296no—
258#121—293no—
259#121—293no—
260#121—293no—
261#121C2×D2913yescomplemented
262#121C2×D2913yescomplemented
263#120—296no—
264#120—293no—
265#120—293no—
266C4×Q8—291no—
267C4×Q8—293no—
268C4×Q8—293no—
269C4×Q8C2×D2911yescharacteristic, complemented
270C4×D4—296no—
271C4×D4—293no—
272C4×D4—293no—
273C4×D4—296no—
274C4×D4—293no—
275C4×D4C2×D2913yescomplemented
276#117—293no—
277#117—293no—
278C29#348711yescharacteristic, complemented, Sylow
279C24—293no—
280#41—24no—
281#41—24no—
282#41—5812no—
283#41—5812no—
284#41—5812no—
285#41—5812no—
286#41—5812no—
287#41—584no—
288C2×Q8—298no—
289C2×Q8—293no—
290C2×Q8—293no—
291C2×Q8—293no—
292C2×Q8—5812no—
293C2×Q8—584no—
294C2×Q8C22×D2911yescharacteristic
295C2×D4—293no—
296C2×D4—296no—
297C2×D4—293no—
298C2×D4—293no—
299C2×D4—296no—
300C2×D4—293no—
301C2×D4—291no—
302C2×D4—293no—
303C2×D4—293no—
304C2×D4—5812no—
305C2×D4C22×D2913yes—
306C2×D4C22×D2913yes—
307C2×D4C22×D2916yes—
308C2×D4C22×D2913yes—
309C22×C4—293no—
310C22×C4—296no—
311C22×C4—293no—
312C22×C4—293no—
313C22×C4—293no—
314C22×C4—293no—
315C22×C4—291no—
316C22×C4—5812no—
317C22×C4C22×D2913yes—
318C22×C4C22×D2913yes—
319C4⋊C4—293no—
320C4⋊C4—293no—
321C4⋊C4—293no—
322C4⋊C4—293no—
323C4⋊C4—291no—
324C4⋊C4—293no—
325C4⋊C4C22×D2913yes—
326C4⋊C4C22×D2911yescharacteristic
327C22⋊C4—293no—
328C22⋊C4—296no—
329C22⋊C4—293no—
330C22⋊C4—293no—
331C22⋊C4—291no—
332C22⋊C4—291no—
333C22⋊C4—291no—
334C22⋊C4—293no—
335C22⋊C4—296no—
336C22⋊C4—293no—
337C22⋊C4C22×D2913yes—
338C22⋊C4C22×D2913yes—
339C42—296no—
340C42—293no—
341C42C22×D2913yes—
342C23—291no—
343C23—293no—
344C23—291no—
345C23—586no—
346C23—1163no—
347C23—1163no—
348C23C23×D2913yes—
349C23C23×D2913yes—
350Q8—2912no—
351Q8—586no—
352Q8—586no—
353Q8D4×D2914yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—582no—
359D4—5812no—
360D4—586no—
361D4—586no—
362D4—586no—
363D4—1163no—
364D4—1163no—
365D4—1166no—
366D4—1163no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—293no—
370C2×C4—2912no—
371C2×C4—291no—
372C2×C4—293no—
373C2×C4—293no—
374C2×C4—291no—
375C2×C4—586no—
376C2×C4—582no—
377C2×C4—586no—
378C2×C4—5812no—
379C2×C4—586no—
380C2×C4—586no—
381C2×C4C23×D2913yes—
382C2×C4C23×D2911yescharacteristic
383C2×C4C23×D2911yescharacteristic
384C2×C4C23×D2911yescharacteristic
385C2×C4C23×D2913yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—292no—
393C22—581no—
394C22—581no—
395C22—581no—
396C22—583no—
397C22—581no—
398C22—583no—
399C22—581no—
400C22—583no—
401C22—11612no—
402C22C24×D2911yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—296no—
408C4—581no—
409C4—583no—
410C4—581no—
411C4—583no—
412C4C2×D4×D2916yes—
413C2—43no—
414C2—43no—
415C2—582no—
416C2—1163no—
417C2—1161no—
418C2C22×D4×D2911yescharacteristic
419C2#1427199811yescharacteristic
420C2#1427199911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingC29×#550C2
hypercenter#550D29
hypocenterC29#3487
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#11745365C2semidirect
2. C2×#41×D29C2semidirect
3. #14271833C2semidirect
4. #14271832C2semidirect
5. #14271831C2semidirect
6. #14271825C2semidirect
7. #14271822C2semidirect
8. #14271820C2semidirect
9. #14271819C2semidirect
10. #14271729C2semidirect
11. #14271726C2semidirect
12. #14271673C2semidirect
13. #14271672C2semidirect
14. #14271639C2semidirect
15. C29×#550C2semidirect
17. C2×#41×C29C22semidirect
22. C2×Q8×D29C22semidirect
25. C2×#90807C22semidirect
30. C22×Dic58C22semidirect
31. C29×#127C22semidirect
32. C29×#121C22semidirect
33. C29×#121C22semidirect
34. C4×Q8×C29C22semidirect
35. C4×D4×C29C22semidirect
36. Q8×Dic29C22semidirect
37. #11745302C22semidirect
38. #11745302C22semidirect
39. #11745301C22semidirect
40. #11745301C22semidirect
41. #11745301C22semidirect
42. #11745300C22semidirect
43. #11745300C22semidirect
46. #11745298C22semidirect
47. D4×Dic29C22semidirect
48. #11745292C22semidirect
49. C4×C22⋊D29C22semidirect
50. C4×C22⋊D29C22semidirect
51. #11745286C22semidirect
52. #11745259C22semidirect
53. #11745258C22semidirect
54. #11745257C22semidirect
57. #11745249C22semidirect
58. #11745247C22semidirect
59. #11745246C22semidirect
62. #11745243C22semidirect
63. #11745240C22semidirect
64. #11745237C22semidirect
65. C4×D116C22semidirect
67. C4×Dic58C22semidirect
88. Q8×D29D4semidirect
208. #550D29semidirect
235. C2×#41C2×D29semidirect
243. #127C2×D29semidirect
261. #121C2×D29semidirect
262. #121C2×D29semidirect
269. C4×Q8C2×D29semidirect
275. C4×D4C2×D29semidirect
278. C29#3487semidirect

Maximal subgroups

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

subgroupindex
C2×#117453652
C2×#41×D292
#142718332
#142718322
#142718312
#142718252
#142718222
#142718202
#142718192
#142717292
#142717262
#142716732
#142716722
#142716392
C29×#5502
#348729