Finite Group Databasebeta

← #31790801

Subgroups — #31790801

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14555510C211yesmaximal, characteristic, complemented
2#22577741C211yesmaximal, characteristic, complemented
3#22577724C211yesmaximal, characteristic, complemented
4#22577707C211yesmaximal, characteristic, complemented
5#22577700C211yesmaximal, characteristic, complemented
6#22577622C211yesmaximal, characteristic, complemented
7#22577615C211yesmaximal, characteristic, complemented
8#22577563C211yesmaximal, characteristic, complemented
9#22577546C211yesmaximal, characteristic
10C4×#14555500C211yesmaximal, characteristic, complemented
11C4×#14555497C211yesmaximal, characteristic, complemented
12C4×#14555498C211yesmaximal, characteristic, complemented
13C4×#14555495C211yesmaximal, characteristic, complemented
14#22577456C211yesmaximal, characteristic, complemented
15#529×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555500C2211yescharacteristic
18C2×#14555498C2211yescharacteristic
19C2×#14555497C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21C4×Dic274C2211yescharacteristic
22C4×Dic274C2211yescharacteristic
23C4×D548C2211yescharacteristic
24C4×D548C2211yescharacteristic
25C4×C22⋊D137—21no—
26C4×C22⋊D137—21no—
27#17984047C2211yescharacteristic
28#17984045—21no—
29C42×D137—21no—
30#17984021C2211yescharacteristic
31#17984021C2211yescharacteristic
32#17984019C2211yescharacteristic
33#17984019C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic
35C4⋊#11764040C2211yescharacteristic
36#17984012C2211yescharacteristic
37#17984012C2211yescharacteristic
38#17984011C2211yescharacteristic
39#17984011C2211yescharacteristic
40#17984009C2211yescharacteristic
41#17984009C2211yescharacteristic
42#17984008C2211yescharacteristic
43#17984008C2211yescharacteristic
44C22⋊#11764040C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C4×#11764040C2211yescharacteristic
47C4×#11764040C2211yescharacteristic
48#126×C137—21no—
49#125×C137C2211yescharacteristic
50#124×C137C2211yescharacteristic
51#123×C137—21no—
52C22⋊Q8×C137C2211yescharacteristic
53#121×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×D4×C137—21no—
56C4×D4×C137C2211yescharacteristic
57#117×C137—21no—
58C2×C42×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61C2×Dic274C2311yescharacteristic
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C4×D137—21no—
65C2×C4×D137—21no—
66C2×C4×D137—21no—
67#14555500—22no—
68#14555498—22no—
69#14555497—22no—
70#14555495—22no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80#14555487—21no—
81#14555487—21no—
82C4×Dic137—21no—
83C4×Dic137—21no—
84C4×Dic137—21no—
85C2×Q8×C137C2311yescharacteristic
86C2×D4×C137—21no—
87C2×D4×C137C2311yescharacteristic
88C22×C4×C137—21no—
89C22×C4×C137D411yescharacteristic
90C22×C4×C137D411yescharacteristic
91C22×C4×C137C2311yescharacteristic
92C4⋊C4×C137—21no—
93C4⋊C4×C137—21no—
94C4⋊C4×C137—21no—
95C4⋊C4×C137—21no—
96C4⋊C4×C137C2311yescharacteristic
97C4⋊C4×C137C2311yescharacteristic
98C22⋊C4×C137—21no—
99C22⋊C4×C137—21no—
100C22⋊C4×C137—21no—
101C22⋊C4×C137—21no—
102C22⋊C4×C137—21no—
103C42×C137—21no—
104C42×C137D411yescharacteristic
105C42×C137D411yescharacteristic
106C42×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109Dic274—21no—
110Dic274#4112yes—
111D548—21no—
112D548#4112yes—
113C22⋊D137—24no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C4×D137—24no—
118C4×D137—42no—
119C4×D137—42no—
120#11764040—21no—
121#11764040—21no—
122#11764040—21no—
123#11764040—21no—
124C23×C137—21no—
125C23×C137C2×D411yescharacteristic
126Q8×C137—21no—
127Q8×C137—21no—
128D4×C137—21no—
129D4×C137—21no—
130D4×C137—42no—
131C2×C4×C137—22no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—41no—
138C2×C4×C137—42no—
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2×D411yescharacteristic
143C2×C4×C137C2×D411yescharacteristic
144C2×C4×C137C2411yescharacteristic
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—21no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137C22×D411yescharacteristic
152C2×D137—22no—
153C2×D137—41no—
154C2×D137—41no—
155C4×C137—21no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137#13512yes—
163C4×C137C2×#4111yescharacteristic
164C4×C137C2×#4111yescharacteristic
165Dic137—22no—
166Dic137—41no—
167Dic137—41no—
168C2×C137—41no—
169C2×C137—81no—
170C2×C137#52511yescharacteristic
171C2×C137C2×#13511yescharacteristic
172C2×C137#57511yescharacteristic
173D137—42no—
174C137#310211yescharacteristic, complemented, Sylow
175#3102—1371nomaximal, Sylow
176#529D13711yescharacteristic, complemented
177C4×#41—1371no—
178#485—1371no—
179#482—1371no—
180#479—1371no—
181#478—1371no—
182#464—1371no—
183#463—1371no—
184#452—1371no—
185#449—1371no—
186C4×Q16—1371no—
187C4×SD16—1371no—
188C4×SD16—1371no—
189C4×D8—1371no—
190#431—1371no—
191C2×#41—1371no—
192C2×Q16—1371no—
193C2×SD16—1371no—
194C2×SD16—1371no—
195C2×D8—1371no—
196#126—21no—
197#125C2×D13711yescharacteristic
198#124C2×D13711yescharacteristic
199#123—21no—
200C22⋊Q8C2×D13711yescharacteristic
201#121C2×D13711yescharacteristic
202C4×Q8—1371no—
203C4×Q8—1371no—
204C4×Q8C2×D13711yescharacteristic
205C4×D4—21no—
206C4×D4—2741no—
207C4×D4—2741no—
208C4×D4—1371no—
209C4×D4—1371no—
210C4×D4C2×D13711yescharacteristic
211#117—21no—
212#117—2741no—
213#117—1371no—
214C2×C42—2741no—
215C2×C42C2×D13711yescharacteristic
216#107—1371no—
217#107—1371no—
218#106—1371no—
219#106—1371no—
220C4⋊C8—1371no—
221C4⋊C8—1371no—
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#102—1371no—
227#102—1371no—
228#102—1371no—
229#102—1371no—
230C22⋊C8—1371no—
231C22⋊C8—1371no—
232C4×C8—1371no—
233C4×C8—1371no—
234#41—2742no—
235#41—2742no—
236C2×Q8—1371no—
237C2×Q8C22×D13711yescharacteristic
238C2×D4—21no—
239C2×D4—2741no—
240C2×D4—1371no—
241C2×D4C22×D13711yescharacteristic
242C22×C4—21no—
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4C22⋊D13711yescharacteristic
247C22×C4C22⋊D13711yescharacteristic
248C22×C4C22×D13711yescharacteristic
249Q16—2742no—
250SD16—2742no—
251SD16—2742no—
252D8—2742no—
253C2×C8—1371no—
254C2×C8—1371no—
255C2×C8—1371no—
256C2×C8—1371no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4C22×D13711yescharacteristic
266C4⋊C4C22×D13711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C42—21no—
276C42—2741no—
277C42—2741no—
278C42—2741no—
279C42C22⋊D13711yescharacteristic
280C42C22⋊D13711yescharacteristic
281C42C22×D13711yescharacteristic
282C42C22×D13711yescharacteristic
283C23—21no—
284C23—2741no—
285C23C2×C22⋊D13711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—2741no—
289Q8—1372no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—2744no—
294D4—2741no—
295D4—1372no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—2744no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—5482no—
309C2×C4—5482no—
310C2×C4C2×C22⋊D13711yescharacteristic
311C2×C4C2×C22⋊D13711yescharacteristic
312C2×C4C2×C22⋊D13711yescharacteristic
313C2×C4C2×C22⋊D13711yescharacteristic
314C2×C4C2×C22⋊D13711yescharacteristic
315C2×C4C23×D13711yescharacteristic
316C8—2741no—
317C8—2741no—
318C8—2741no—
319C8—2741no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2742no—
327C22—5481no—
328C22—5481no—
329C22C22×C22⋊D13711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2742no—
338C4—5481no—
339C4—5481no—
340C4#1798410812yes—
341C4#41×D13711yescharacteristic
342C4#41×D13711yescharacteristic
343C2—41no—
344C2—81no—
345C2—5482no—
346C2#2257791011yescharacteristic
347C2C2×#1798410811yescharacteristic
348C2#2257820111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#529×C137C2
socleC22×C137C22×D4
prime radical#22577546C2
hypercenter#529D137
hypocenterC137#3102
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C4×#14555510C2semidirect
2. #22577741C2semidirect
3. #22577724C2semidirect
4. #22577707C2semidirect
5. #22577700C2semidirect
6. #22577622C2semidirect
7. #22577615C2semidirect
8. #22577563C2semidirect
10. C4×#14555500C2semidirect
11. C4×#14555497C2semidirect
12. C4×#14555498C2semidirect
13. C4×#14555495C2semidirect
14. #22577456C2semidirect
15. #529×C137C2semidirect
174. C137#3102semidirect
176. #529D137semidirect

Maximal subgroups

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

subgroupindex
C4×#145555102
#225777412
#225777242
#225777072
#225777002
#225776222
#225776152
#225775632
#225775462
C4×#145555002
C4×#145554972
C4×#145554982
C4×#145554952
#225774562
#529×C1372
#3102137