Finite Group Databasebeta

← #31792597

Subgroups — #31792597

Order 17536.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578053C211yesmaximal, characteristic, complemented
2#22578037C211yesmaximal, characteristic, complemented
3#22577748C211yesmaximal, characteristic, complemented
4#22577718C211yesmaximal, characteristic, complemented
5#22577714C211yesmaximal, characteristic, complemented
6#22577682C211yesmaximal, characteristic, complemented
7#22577638C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577536C211yesmaximal, characteristic, complemented
10#22577535C211yesmaximal, characteristic, complemented
11#22577483C211yesmaximal, characteristic, complemented
12C8×C22⋊D137C211yesmaximal, characteristic, complemented
13C2×#17984010C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#466×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C22×D548C2211yescharacteristic
18C2×D4×D137—21no—
19C2×D4×D137C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C2×#14555495C2211yescharacteristic, complemented
22C2×C8×D137C2211yescharacteristic, complemented
23#17984087C2211yescharacteristic, complemented
24#17984084C2211yescharacteristic
25#17984070—21no—
26#17984069—21no—
27#17984065—21no—
28#17984063C2211yescharacteristic
29#17984061—21no—
30#17984059C2211yescharacteristic, complemented
31#17984058—21no—
32#17984056—21no—
33D4×Dic137C2211yescharacteristic, complemented
34#17984050C2211yescharacteristic
35C4×C22⋊D137C2211yescharacteristic
36C2×C4⋊Dic137C2211yescharacteristic, complemented
37C4⋊C4×D137C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984021C2211yescharacteristic, complemented
40#17984017C2211yescharacteristic, complemented
41#17984010C2211yescharacteristic
42#17984010C2211yescharacteristic
43#17984010C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic, complemented
45#17984009C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic, complemented
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic, complemented
49C22⋊#11764040C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C2×D8×C137C2211yescharacteristic, complemented
53C22×C8×C137C2211yescharacteristic, complemented
54#121×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic, complemented
56#107×C137C2211yescharacteristic, complemented
57#102×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C23×D137—21no—
61#14555509—44no—
62C2×Dic274—21no—
63C2×D548—22no—
64C2×D548—21no—
65C2×D548—21no—
66C2×D548—41no—
67C2×D548C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137D411yescharacteristic
74D4×D137—22no—
75D4×D137—41no—
76D4×D137—44no—
77C22×Dic137—21no—
78C22×Dic137—21no—
79C2×C4×D137—21no—
80C2×C4×D137—21no—
81C2×C4×D137C2311yescharacteristic
82#14555495—22no—
83#14555495—41no—
84C2×#11764040C2311yescharacteristic
85C2×#11764040C2311yescharacteristic
86C8×D137—41no—
87C4⋊Dic137—21no—
88C4⋊Dic137C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95C22⋊Dic137D411yescharacteristic
96#14555487—21no—
97#14555487—21no—
98#14555487D411yescharacteristic
99C4×Dic137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137—21no—
102C2×D4×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137C2311yescharacteristic
105D8×C137—41no—
106C2×C8×C137—41no—
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22⋊C4×C137—21no—
113C22×D137—21no—
114C22×D137—21no—
115C22×D137—42no—
116C22×D137—42no—
117C22×D137—42no—
118C22×D137—42no—
119C22×D137—81no—
120C22×D137—81no—
121C22×D137C2×D411yescharacteristic
122Dic274—42no—
123D548—21no—
124D548—21no—
125D548—21no—
126D548—42no—
127D548—42no—
128D548#4112yescomplemented
129C22⋊D137—42no—
130C22⋊D137—42no—
131C22⋊D137—42no—
132C22⋊D137—81no—
133C22⋊D137D814yescomplemented
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—42no—
137C2×Dic137—42no—
138C2×Dic137C2×D411yescharacteristic
139C2×Dic137C2×D411yescharacteristic
140C2×Dic137C2×D411yescharacteristic
141C4×D137—21no—
142C4×D137—21no—
143C4×D137—42no—
144C4×D137—42no—
145#11764040—21no—
146#11764040—21no—
147C23×C137—21no—
148C23×C137—21no—
149C23×C137C2×D411yescharacteristic
150D4×C137—21no—
151D4×C137—21no—
152D4×C137—21no—
153D4×C137—21no—
154D4×C137—42no—
155D4×C137—42no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137C2×D411yescharacteristic
161C2×C4×C137C2411yescharacteristic
162C8×C137—21no—
163C8×C137—21no—
164C22×C137—21no—
165C22×C137—21no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137C2×D812yes—
173C22×C137C22×D411yescharacteristic
174C2×D137—21no—
175C2×D137—21no—
176C2×D137—22no—
177C2×D137—41no—
178C2×D137—41no—
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—84no—
183C2×D137—84no—
184C2×D137C2×D812yes—
185C4×C137—21no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137C2×#4111yescharacteristic
189C4×C137H(5,𝔽2)11yescharacteristic
190Dic137—21no—
191Dic137—21no—
192Dic137—41no—
193Dic137—41no—
194Dic137C2×D812yes—
195C2×C137—22no—
196C2×C137—81no—
197C2×C137—81no—
198C2×C137#54611yescharacteristic
199C2×C137C22×D811yescharacteristic
200C2×C137#57711yescharacteristic
201D137—22no—
202D137—42no—
203D137—81no—
204C137#329411yescharacteristic, complemented, Sylow
205#3294—1371nomaximal, Sylow
206#547—1371no—
207D42—1371no—
208#486—1371no—
209#480—1371no—
210#480—1371no—
211#474—1371no—
212#466—1371no—
213#466D13711yescharacteristic, complemented
214#459—1371no—
215#447—1371no—
216#447—1371no—
217C4×D8—1371no—
218C8×D4—1371no—
219C2×#102—1371no—
220C2×#102—1371no—
221C2×#41—2741no—
222C22×D4—2741no—
223C22×D4—1371no—
224C22×D4—1371no—
225C2×D8—1371no—
226C2×D8—1371no—
227C2×D8C2×D13711yescharacteristic, complemented
228C22×C8—1371no—
229C22×C8C2×D13711yescharacteristic, complemented
230C4⋊Q8—1371no—
231#127—1371no—
232#123—2741no—
233#123—2741no—
234C22⋊Q8—2741no—
235#121—2741no—
236#121—1371no—
237#121—1371no—
238#121C2×D13711yescharacteristic
239#121C2×D13711yescharacteristic, complemented
240#120—2741no—
241#120—2741no—
242C4×D4—1371no—
243C4×D4—1371no—
244C4×D4—1371no—
245C2×C4⋊C4—1371no—
246C2×C4⋊C4—1371no—
247#107—1371no—
248#107—1371no—
249#107C2×D13711yescharacteristic, complemented
250C4⋊C8—1371no—
251#102—1371no—
252#102—1371no—
253#102—1371no—
254#102—1371no—
255#102—1371no—
256#102—1371no—
257#102—1371no—
258#102—1371no—
259#102C2×D13711yescharacteristic, complemented
260#102C2×D13711yescharacteristic, complemented
261C22⋊C8—1371no—
262C22⋊C8—1371no—
263C4×C8—1371no—
264C24—2741no—
265C24—2741no—
266#41—5484no—
267C2×Q8—2741no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—2742no—
271C2×D4—2741no—
272C2×D4—2742no—
273C2×D4—2741no—
274C2×D4—2741no—
275C2×D4—2741no—
276C2×D4—2741no—
277C2×D4—2741no—
278C2×D4—5481no—
279C2×D4—5481no—
280C2×D4—5484no—
281C2×D4—1371no—
282C2×D4—1371no—
283C2×D4—1371no—
284C2×D4C22×D13711yescharacteristic
285C2×D4C22×D13711yescharacteristic
286C22×C4—2741no—
287C22×C4—2741no—
288C22×C4—2741no—
289C22×C4—2741no—
290C22×C4—1371no—
291C22×C4C22×D13711yescharacteristic
292D8—41no—
293D8—2742no—
294D8—5481no—
295C2×C8—41no—
296C2×C8—5481no—
297C2×C8—1371no—
298C2×C8—1371no—
299C2×C8C22×D13711yescharacteristic
300C2×C8C22×D13711yescharacteristic
301C4⋊C4—2741no—
302C4⋊C4—2741no—
303C4⋊C4—2741no—
304C4⋊C4—1371no—
305C4⋊C4—1371no—
306C4⋊C4—1371no—
307C4⋊C4C22×D13711yescharacteristic
308C4⋊C4C22×D13711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—2741no—
312C22⋊C4—2741no—
313C22⋊C4—2741no—
314C22⋊C4—2741no—
315C22⋊C4—1371no—
316C22⋊C4—1371no—
317C42—1371no—
318C23—21no—
319C23—21no—
320C23—2741no—
321C23—2741no—
322C23—5482no—
323C23—5482no—
324C23—5482no—
325C23—5482no—
326C23—10961no—
327C23—10961no—
328C23—1371no—
329C23D4×D13711yescharacteristic
330Q8—5482no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—42no—
336D4—42no—
337D4—2741no—
338D4—2741no—
339D4—2741no—
340D4—5482no—
341D4—5482no—
342D4—5482no—
343D4—5482no—
344D4—5482no—
345D4—10961no—
346D4—1374no—
347D4—1372no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—2741no—
353C2×C4—2741no—
354C2×C4—2741no—
355C2×C4—2741no—
356C2×C4—5482no—
357C2×C4—5482no—
358C2×C4—5482no—
359C2×C4—5482no—
360C2×C4—1371no—
361C2×C4—1371no—
362C2×C4—1371no—
363C2×C4D4×D13711yescharacteristic
364C2×C4C23×D13711yescharacteristic
365C8—21no—
366C8—21no—
367C8—2741no—
368C8—2741no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2741no—
378C22—2741no—
379C22—2742no—
380C22—5481no—
381C22—5481no—
382C22—5481no—
383C22—5481no—
384C22—5481no—
385C22—10964no—
386C22—10964no—
387C22—1372no—
388C22D8×D13712yes—
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—2741no—
394C4—2741no—
395C4—5481no—
396C4—5481no—
397C4—1372no—
398C4#41×D13711yescharacteristic
399C4#1798413311yescharacteristic
400C2—22no—
401C2—81no—
402C2—81no—
403C2—2742no—
404C2—5482no—
405C2—10961no—
406C2#2257804211yescharacteristic
407C2C2×D8×D13711yescharacteristic
408C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#466×C137C2
socleC22×C137C22×D4
hypercenter#466D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22578053C2semidirect
2. #22578037C2semidirect
3. #22577748C2semidirect
4. #22577718C2semidirect
5. #22577714C2semidirect
6. #22577682C2semidirect
7. #22577638C2semidirect
8. #22577596C2semidirect
9. #22577536C2semidirect
10. #22577535C2semidirect
11. #22577483C2semidirect
12. C8×C22⋊D137C2semidirect
13. C2×#17984010C2semidirect
14. #102×D137C2semidirect
15. #466×C137C2semidirect
21. C2×#14555495C22semidirect
22. C2×C8×D137C22semidirect
23. #17984087C22semidirect
30. #17984059C22semidirect
33. D4×Dic137C22semidirect
36. C2×C4⋊Dic137C22semidirect
37. C4⋊C4×D137C22semidirect
38. #17984021C22semidirect
39. #17984021C22semidirect
40. #17984017C22semidirect
43. #17984010C22semidirect
44. #17984010C22semidirect
45. #17984009C22semidirect
46. #17984009C22semidirect
48. #17984008C22semidirect
49. C22⋊#11764040C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
52. C2×D8×C137C22semidirect
53. C22×C8×C137C22semidirect
55. #121×C137C22semidirect
56. #107×C137C22semidirect
57. #102×C137C22semidirect
58. #102×C137C22semidirect
128. D548#41semidirect
133. C22⋊D137D8semidirect
204. C137#3294semidirect
213. #466D137semidirect
227. C2×D8C2×D137semidirect
229. C22×C8C2×D137semidirect
239. #121C2×D137semidirect
249. #107C2×D137semidirect
259. #102C2×D137semidirect
260. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780532
#225780372
#225777482
#225777182
#225777142
#225776822
#225776382
#225775962
#225775362
#225775352
#225774832
C8×C22⋊D1372
C2×#179840102
#102×D1372
#466×C1372
#3294137