Finite Group Databasebeta

← #31793382

Subgroups — #31793382

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578116C211yesmaximal, characteristic, complemented
2#22578053C211yesmaximal, characteristic, complemented
3#22577800C211yesmaximal, characteristic, complemented
4#22577726C211yesmaximal, characteristic, complemented
5#22577723C211yesmaximal, characteristic, complemented
6#22577722C211yesmaximal, characteristic, complemented
7#22577691C211yesmaximal, characteristic, complemented
8#22577682C211yesmaximal, characteristic, complemented
9#22577655C211yesmaximal, characteristic, complemented
10#22577654C211yesmaximal, characteristic, complemented
11#22577483C211yesmaximal, characteristic, complemented
12#22577478C211yesmaximal, characteristic, complemented
13#22577446C211yesmaximal, characteristic, complemented
14#22577444C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C2×#14555495C2211yescharacteristic
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20#17984087C2211yescharacteristic
21#17984087C2211yescharacteristic
22#17984076—21no—
23#17984074—21no—
24#17984070—21no—
25#17984069—21no—
26#17984065—21no—
27#17984063C2211yescharacteristic
28#17984062C2211yescharacteristic
29#17984061—21no—
30#17984059C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32#17984050C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34#17984046C2211yescharacteristic
35#17984045—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984020C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984019C2211yescharacteristic
45#17984017C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555509—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×D548—21no—
63C2×D548C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137—21no—
66C2×C22⋊D137D411yescharacteristic
67C22×Dic137—21no—
68C22×Dic137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137C2311yescharacteristic
72#14555495—22no—
73#14555494—22no—
74#14555493—22no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C4⋊Dic137—21no—
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80C4⋊Dic137C2311yescharacteristic
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489D411yescharacteristic
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137D411yescharacteristic
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93C4×Dic137—21no—
94C4×Dic137C2311yescharacteristic
95C2×D4×C137—21no—
96C2×D4×C137C2311yescharacteristic
97C22×C4×C137C2311yescharacteristic
98C2×C8×C137C2311yescharacteristic
99C2×C8×C137C2311yescharacteristic
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C42×C137—21no—
106C22×D137—21no—
107C22×D137C2×D411yescharacteristic
108Dic274—42no—
109Dic274—81no—
110D548—21no—
111D548—21no—
112D548—81no—
113C22⋊D137—22no—
114C22⋊D137—42no—
115C22⋊D137—81no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—42no—
120C2×Dic137—42no—
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C4×D137—21no—
125C4×D137—21no—
126C4×D137—42no—
127C4×D137—42no—
128#11764040—21no—
129#11764040#4112yescomplemented
130C23×C137—21no—
131C23×C137C2×D411yescharacteristic
132D4×C137—21no—
133D4×C137—21no—
134D4×C137—42no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2411yescharacteristic
142C8×C137—21no—
143C8×C137—21no—
144C22×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—41no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137C22×D411yescharacteristic
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—21no—
154C2×D137—41no—
155C2×D137—41no—
156C2×D137—41no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137C2×#4111yescharacteristic
162C4×C1372−1+411yescharacteristic
163Dic137—21no—
164Dic137—21no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168Dic137AGL(1,ℤ/8)12yes—
169C2×C137—41no—
170C2×C137—81no—
171C2×C137#54711yescharacteristic
172C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C137#57711yescharacteristic
174D137—41no—
175D137—81no—
176C137#335411yescharacteristic, complemented, Sylow
177#3354—1371nomaximal, Sylow
178#555—1371no—
179#547—1371no—
180#498—1371no—
181#482—1371no—
182#482D13711yescharacteristic, complemented
183#481—1371no—
184#481—1371no—
185#476—1371no—
186#474—1371no—
187#469—1371no—
188#469—1371no—
189C4×D8—1371no—
190#436—1371no—
191#428—1371no—
192#428—1371no—
193C2×#41—2741no—
194C2×D8—1371no—
195C2×M16—1371no—
196C2×M16—1371no—
197C4⋊Q8—1371no—
198C4⋊Q8—1371no—
199#124—2741no—
200#124—2741no—
201#123—2741no—
202#123—2741no—
203C22⋊Q8—2741no—
204#121—2741no—
205#121—1371no—
206#121—1371no—
207#121—1371no—
208#121C2×D13711yescharacteristic
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212#117—2741no—
213#117—1371no—
214#117C2×D13711yescharacteristic
215C2×C4⋊C4—1371no—
216C2×C4⋊C4—1371no—
217#107—1371no—
218#107—1371no—
219#107C2×D13711yescharacteristic
220#106—1371no—
221#106—1371no—
222#106—1371no—
223#106—1371no—
224#106—1371no—
225#106C2×D13711yescharacteristic
226C4⋊C8—1371no—
227#102—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231#102C2×D13711yescharacteristic
232#102C2×D13711yescharacteristic
233C22⋊C8—1371no—
234C22⋊C8C2×D13711yescharacteristic
235C4×C8—1371no—
236#41—5484no—
237C2×Q8—2741no—
238C2×Q8—2741no—
239C2×D4—21no—
240C2×D4—2741no—
241C2×D4—2741no—
242C2×D4—2741no—
243C2×D4—1371no—
244C2×D4—1371no—
245C2×D4C22×D13711yescharacteristic
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—1371no—
251C22×C4C22×D13711yescharacteristic
252D8—2742no—
253M16—2742no—
254M16—2742no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8C22×D13711yescharacteristic
258C2×C8C22×D13711yescharacteristic
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C4⋊C4C22×D13711yescharacteristic
268C4⋊C4C22×D13711yescharacteristic
269C4⋊C4C22×D13711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—1371no—
279C22⋊C4—1371no—
280C42—21no—
281C42—2741no—
282C42—1371no—
283C23—21no—
284C23—2741no—
285C23—1371no—
286C23D4×D13711yescharacteristic
287Q8—5482no—
288Q8—10961no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—2742no—
293D4—2741no—
294D4—2741no—
295D4—5482no—
296D4—10961no—
297D4—10961no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—5482no—
309C2×C4—5482no—
310C2×C4—5482no—
311C2×C4—5482no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1371no—
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2741no—
320C8—1372no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—2741no—
328C22—2741no—
329C22—2741no—
330C22—5481no—
331C22—5481no—
332C22—5481no—
333C22C2×D4×D13711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2741no—
339C4—2741no—
340C4—5481no—
341C4—5481no—
342C4—5481no—
343C4—1372no—
344C4#41×D13711yescharacteristic
345C4#1798413511yescharacteristic
346C2—41no—
347C2—81no—
348C2—5481no—
349C2—10961no—
350C2#2257805111yescharacteristic
351C2AGL(1,ℤ/8)×D13711yescharacteristic
352C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
hypercenter#482D137
hypocenterC137#3354
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22578116C2semidirect
2. #22578053C2semidirect
3. #22577800C2semidirect
4. #22577726C2semidirect
5. #22577723C2semidirect
6. #22577722C2semidirect
7. #22577691C2semidirect
8. #22577682C2semidirect
9. #22577655C2semidirect
10. #22577654C2semidirect
11. #22577483C2semidirect
12. #22577478C2semidirect
13. #22577446C2semidirect
14. #22577444C2semidirect
15. #482×C137C2semidirect
129. #11764040#41semidirect
176. C137#3354semidirect
182. #482D137semidirect

Maximal subgroups

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

subgroupindex
#225781162
#225780532
#225778002
#225777262
#225777232
#225777222
#225776912
#225776822
#225776552
#225776542
#225774832
#225774782
#225774462
#225774442
#482×C1372
#3354137