Finite Group Databasebeta

← #31792595

Subgroups — #31792595

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
2D4×C22⋊D137C211yesmaximal, characteristic, complemented
3#22577748C211yesmaximal, characteristic, complemented
4#22577717C211yesmaximal, characteristic, complemented
5#22577715C211yesmaximal, characteristic, complemented
6#22577681C211yesmaximal, characteristic, complemented
7#22577637C211yesmaximal, characteristic, complemented
8#22577636C211yesmaximal, characteristic, complemented
9#22577596C211yesmaximal, characteristic, complemented
10#22577535C211yesmaximal, characteristic, complemented
11D8×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13C2×#17984009C211yesmaximal, characteristic, complemented
14#102×D137C211yesmaximal, characteristic, complemented
15#447×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C22×C22⋊D137—21no—
18C2×D4×D137C2211yescharacteristic
19C2×#14555495C2211yescharacteristic, complemented
20C22×#11764040C2211yescharacteristic, complemented
21C2×C8×D137C2211yescharacteristic, complemented
22#17984087C2211yescharacteristic, complemented
23#17984084C2211yescharacteristic
24#17984070—21no—
25#17984069—21no—
26#17984065—21no—
27#17984062C2211yescharacteristic
28#17984060—21no—
29#17984059C2211yescharacteristic
30#17984059C2211yescharacteristic, complemented
31#17984057—21no—
32#17984056—21no—
33D4×Dic137C2211yescharacteristic
34D4×Dic137C2211yescharacteristic, complemented
35C4×C22⋊D137C2211yescharacteristic
36C2×C4⋊Dic137C2211yescharacteristic, complemented
37C4⋊C4×D137C2211yescharacteristic, complemented
38#17984022C2211yescharacteristic, complemented
39#17984022C2211yescharacteristic, complemented
40#17984021C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984010C2211yescharacteristic, complemented
43#17984010C2211yescharacteristic, complemented
44#17984009C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984009C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984000C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×D8×C137C2211yescharacteristic
54C2×D8×C137C2211yescharacteristic, complemented
55#121×C137C2211yescharacteristic, complemented
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60#14555509—44no—
61C2×Dic274—21no—
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—44no—
70C2×C22⋊D137D411yescharacteristic
71D4×D137—22no—
72D4×D137—41no—
73C22×Dic137—21no—
74C22×Dic137—21no—
75C22×Dic137—21no—
76C2×C4×D137—21no—
77C2×C4×D137C2311yescharacteristic
78#14555495—41no—
79C2×#11764040—41no—
80C2×#11764040C2311yescharacteristic
81C2×#11764040C2311yescharacteristic
82C8×D137—41no—
83C4⋊Dic137—21no—
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489D411yescharacteristic
90C22⋊Dic137—21no—
91C22⋊Dic137—21no—
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137C2311yescharacteristic
98C24×C137—21no—
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—21no—
102C2×D4×C137—41no—
103C2×D4×C137C2311yescharacteristic
104C2×D4×C137C2311yescharacteristic
105C2×D4×C137C2311yescharacteristic
106C22×C4×C137C2311yescharacteristic
107D8×C137—22no—
108D8×C137—41no—
109C2×C8×C137C2311yescharacteristic
110C2×C8×C137C2311yescharacteristic
111C4⋊C4×C137C2311yescharacteristic
112C22⋊C4×C137—21no—
113C22×D137—21no—
114C22×D137—42no—
115C22×D137—42no—
116C22×D137—81no—
117C22×D137C2×D411yescharacteristic
118Dic274—42no—
119D548—21no—
120D548—21no—
121C22⋊D137—42no—
122C22⋊D137—42no—
123C22⋊D137—42no—
124C22⋊D137—42no—
125C22⋊D137—42no—
126C22⋊D137—81no—
127C22⋊D137D814yescomplemented
128C2×Dic137—21no—
129C2×Dic137—21no—
130C2×Dic137—21no—
131C2×Dic137—42no—
132C2×Dic137—42no—
133C2×Dic137—42no—
134C2×Dic137C2×D411yescharacteristic
135C2×Dic137C2×D411yescharacteristic
136C2×Dic137C2×D411yescharacteristic
137C4×D137—21no—
138C4×D137—21no—
139C4×D137—42no—
140#11764040—21no—
141#11764040—21no—
142C23×C137—21no—
143C23×C137—21no—
144C23×C137—21no—
145C23×C137—42no—
146C23×C137—42no—
147C23×C137—81no—
148C23×C137C2×D411yescharacteristic
149D4×C137—21no—
150D4×C137—21no—
151D4×C137—21no—
152D4×C137—21no—
153D4×C137—21no—
154D4×C137—42no—
155D4×C137—42no—
156D4×C137#4112yescomplemented
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—22no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—84no—
176C22×C137C2×D812yes—
177C22×C137C22×D411yescharacteristic
178C2×D137—21no—
179C2×D137—21no—
180C2×D137—41no—
181C2×D137—41no—
182C2×D137—41no—
183C2×D137—84no—
184C2×D137C2×D812yes—
185C4×C137—21no—
186C4×C137—41no—
187C4×C137C2×#4111yescharacteristic
188C4×C137H(5,𝔽2)11yescharacteristic
189Dic137—21no—
190Dic137—21no—
191Dic137—41no—
192Dic137—41no—
193Dic137—41no—
194Dic137C2×D812yes—
195C2×C137—22no—
196C2×C137—42no—
197C2×C137—81no—
198C2×C137—81no—
199C2×C137#54611yescharacteristic
200C2×C137C22×D811yescharacteristic
201C2×C137#57711yescharacteristic
202D137—22no—
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#466—1371no—
214#459—1371no—
215#447—1371no—
216#447D13711yescharacteristic, complemented
217C4×D8—1371no—
218C8×D4—1371no—
219C2×#102—1371no—
220C2×#102—1371no—
221C2×#41—2741no—
222C22×D4—2741no—
223C22×D4—1371no—
224C22×D4C2×D13711yescharacteristic
225C2×D8—1371no—
226C2×D8C2×D13711yescharacteristic
227C2×D8C2×D13711yescharacteristic, complemented
228C22×C8—1371no—
229C22×C8—1371no—
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#121—1371no—
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#107—1371no—
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
260#102C2×D13711yescharacteristic, complemented
261C22⋊C8—1371no—
262C22⋊C8C2×D13711yescharacteristic, complemented
263C4×C8—1371no—
264C24—21no—
265C24—2741no—
266#41—5484no—
267C2×Q8—2741no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—2742no—
273C2×D4—2741no—
274C2×D4—2741no—
275C2×D4—2741no—
276C2×D4—2741no—
277C2×D4—2741no—
278C2×D4—2741no—
279C2×D4—5481no—
280C2×D4—5484no—
281C2×D4—1371no—
282C2×D4—1371no—
283C2×D4C22×D13711yescharacteristic
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—22no—
293D8—41no—
294D8—5481no—
295C2×C8—5481no—
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⋊C4—1371no—
308C4⋊C4C22×D13711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—2741no—
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—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—2741no—
325C23—5482no—
326C23—5482no—
327C23—10961no—
328C23—1371no—
329C23D4×D13711yescharacteristic
330Q8—5482no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—2741no—
339D4—2741no—
340D4—5482no—
341D4—5482no—
342D4—5482no—
343D4—5482no—
344D4—5482no—
345D4—10961no—
346D4—1374no—
347D4#1455550912yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—2741no—
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—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—2741no—
382C22—2741no—
383C22—5481no—
384C22—5481no—
385C22—5481no—
386C22—10964no—
387C22—1372no—
388C22D8×D13712yes—
389C22C2×D4×D13711yescharacteristic
390C4—21no—
391C4—41no—
392C4—2741no—
393C4—2741no—
394C4—5481no—
395C4—5481no—
396C4—5481no—
397C4—1372no—
398C4C2×#1455550911yescharacteristic
399C4#1798413311yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—2742no—
405C2—10961no—
406C2#2257804011yescharacteristic
407C2C2×D8×D13711yescharacteristic
408C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#447×C137C2
socleC22×C137C22×D4
hypercenter#447D137
hypocenterC137#3294
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22578053C2semidirect
2. D4×C22⋊D137C2semidirect
3. #22577748C2semidirect
4. #22577717C2semidirect
5. #22577715C2semidirect
6. #22577681C2semidirect
7. #22577637C2semidirect
8. #22577636C2semidirect
9. #22577596C2semidirect
10. #22577535C2semidirect
11. D8×Dic137C2semidirect
12. #22577466C2semidirect
13. C2×#17984009C2semidirect
14. #102×D137C2semidirect
15. #447×C137C2semidirect
19. C2×#14555495C22semidirect
20. C22×#11764040C22semidirect
21. C2×C8×D137C22semidirect
22. #17984087C22semidirect
30. #17984059C22semidirect
34. D4×Dic137C22semidirect
36. C2×C4⋊Dic137C22semidirect
37. C4⋊C4×D137C22semidirect
38. #17984022C22semidirect
39. #17984022C22semidirect
40. #17984021C22semidirect
41. #17984017C22semidirect
42. #17984010C22semidirect
43. #17984010C22semidirect
46. #17984009C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
49. #17984008C22semidirect
50. #17984000C22semidirect
51. C8×Dic137C22semidirect
54. C2×D8×C137C22semidirect
55. #121×C137C22semidirect
57. #102×C137C22semidirect
58. C22⋊C8×C137C22semidirect
127. C22⋊D137D8semidirect
156. D4×C137#41semidirect
204. C137#3294semidirect
216. #447D137semidirect
227. C2×D8C2×D137semidirect
239. #121C2×D137semidirect
260. #102C2×D137semidirect
262. C22⋊C8C2×D137semidirect
347. D4#14555509semidirect

Maximal subgroups

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

subgroupindex
#225780532
D4×C22⋊D1372
#225777482
#225777172
#225777152
#225776812
#225776372
#225776362
#225775962
#225775352
D8×Dic1372
#225774662
C2×#179840092
#102×D1372
#447×C1372
#3294137