Finite Group Databasebeta

← #31792254

Subgroups — #31792254

Order 17536.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578024C211yesmaximal, characteristic, complemented
2#22577693C211yesmaximal, characteristic, complemented
3#22577690C211yesmaximal, characteristic, complemented
4#22577688C211yesmaximal, characteristic, complemented
5#22577685C211yesmaximal, characteristic, complemented
6#22577659C211yesmaximal, characteristic, complemented
7#22577654C211yesmaximal, characteristic
8#22577634C211yesmaximal, characteristic, complemented
9#22577630C211yesmaximal, characteristic, complemented
10#22577625C211yesmaximal, characteristic, complemented
11#22577622C211yesmaximal, characteristic, complemented
12#22577618C211yesmaximal, characteristic, complemented
13#22577615C211yesmaximal, characteristic, complemented
14#22577430C211yesmaximal, characteristic, complemented
15#542×C137C211yesmaximal, characteristic, complemented
16C2×#14555500C2211yescharacteristic
17C2×#14555498C2211yescharacteristic
18C2×#14555497C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20C2×#14555494C2211yescharacteristic
21C22×#11764040C2211yescharacteristic
22#17984088C2211yescharacteristic
23#17984083—21no—
24#17984076C2211yescharacteristic
25#17984074—21no—
26#17984070—21no—
27#17984067C2211yescharacteristic
28#17984066—21no—
29#17984062C2211yescharacteristic
30C4×Dic274C2211yescharacteristic
31C4×D548C2211yescharacteristic
32#17984019C2211yescharacteristic
33#17984019C2211yescharacteristic
34#17984019C2211yescharacteristic
35#17984019C2211yescharacteristic
36C4⋊#11764040C2211yescharacteristic
37C4⋊#11764040C2211yescharacteristic
38C4⋊#11764040C2211yescharacteristic
39C4⋊#11764040C2211yescharacteristic
40#17984012C2211yescharacteristic
41#17984012C2211yescharacteristic
42#17984011C2211yescharacteristic
43#17984011C2211yescharacteristic
44#17984009C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48C4⋊Q8×C137C2211yescharacteristic
49#126×C137—21no—
50#124×C137—21no—
51#124×C137C2211yescharacteristic
52#123×C137—21no—
53C22⋊Q8×C137—21no—
54C22⋊Q8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56C4×Q8×C137C2211yescharacteristic
57C4×D4×C137C2211yescharacteristic
58#117×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C4×D137—21no—
64#14555500—41no—
65#14555498—41no—
66#14555497—41no—
67#14555495—41no—
68#14555494—41no—
69C2×#11764040—41no—
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C4⋊Dic137—21no—
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80#14555487—21no—
81#14555487—21no—
82#14555487—21no—
83#14555487—21no—
84C4×Dic137—21no—
85C2×Q8×C137—21no—
86C2×Q8×C137C2311yescharacteristic
87C2×D4×C137—21no—
88C2×D4×C137C2311yescharacteristic
89C22×C4×C137—21no—
90C22×C4×C137C2311yescharacteristic
91C4⋊C4×C137—21no—
92C4⋊C4×C137—21no—
93C4⋊C4×C137—21no—
94C4⋊C4×C137D411yescharacteristic
95C4⋊C4×C137D411yescharacteristic
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—
103C22⋊C4×C137D411yescharacteristic
104C22⋊C4×C137D411yescharacteristic
105C42×C137—21no—
106C42×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109Dic274—21no—
110Dic274—21no—
111Dic274—81no—
112D548—21no—
113D548—21no—
114C22⋊D137—81no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C4×D137—81no—
119#11764040—21no—
120#11764040—21no—
121#11764040—21no—
122#11764040—21no—
123C23×C137—21no—
124C23×C137C2×D411yescharacteristic
125Q8×C137—21no—
126Q8×C137—21no—
127Q8×C137—81no—
128D4×C137—21no—
129D4×C137—21no—
130D4×C137—81no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—81no—
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2×D411yescharacteristic
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C22×C137—21no—
144C22×C137—21no—
145C22×C137—21no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137—41no—
149C22×C137C22×D411yescharacteristic
150C2×D137—41no—
151C2×D137—41no—
152C2×D137—41no—
153C4×C137—21no—
154C4×C137—21no—
155C4×C137—21no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137H(5,𝔽2)11yescharacteristic
162C4×C1372−1+411yescharacteristic
163Dic137—41no—
164Dic137—41no—
165Dic137—41no—
166C2×C137—41no—
167C2×C137—81no—
168C2×C137#53711yescharacteristic
169C2×C137#57711yescharacteristic
170C2×C137#57711yescharacteristic
171D137—81no—
172C137#326311yescharacteristic, complemented, Sylow
173#3263—1371nomaximal, Sylow
174#542—1371no—
175#542D13711yescharacteristic, complemented
176#476—1371no—
177#476—1371no—
178#475—1371no—
179#475—1371no—
180#470—1371no—
181#469—1371no—
182#465—1371no—
183#465—1371no—
184#464—1371no—
185#464—1371no—
186#463—1371no—
187#463—1371no—
188#424—1371no—
189C2×Q16—1371no—
190C2×SD16—1371no—
191C2×SD16—1371no—
192C2×D8—1371no—
193C2×M16—1371no—
194C22×C8—1371no—
195C4⋊Q8—1371no—
196C4⋊Q8C2×D13711yescharacteristic
197#126—21no—
198#126—2741no—
199#124—21no—
200#124—2741no—
201#124—1371no—
202#124C2×D13711yescharacteristic
203#123—21no—
204#123—2741no—
205C22⋊Q8—21no—
206C22⋊Q8—2741no—
207C22⋊Q8—1371no—
208C22⋊Q8C2×D13711yescharacteristic
209#121—1371no—
210#121C2×D13711yescharacteristic
211C4×Q8—1371no—
212C4×Q8C2×D13711yescharacteristic
213C4×D4—1371no—
214C4×D4C2×D13711yescharacteristic
215#117C2×D13711yescharacteristic
216#106—1371no—
217#106—1371no—
218#106—1371no—
219#106—1371no—
220C4⋊C8—1371no—
221C4⋊C8—1371no—
222C4⋊C8—1371no—
223C4⋊C8—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231#102—1371no—
232C2×Q8—21no—
233C2×Q8—2741no—
234C2×Q8—1371no—
235C2×Q8C22×D13711yescharacteristic
236C2×D4—21no—
237C2×D4—2741no—
238C2×D4—1371no—
239C2×D4C22×D13711yescharacteristic
240C22×C4—21no—
241C22×C4—2741no—
242C22×C4C22×D13711yescharacteristic
243Q16—5481no—
244SD16—5481no—
245SD16—5481no—
246D8—5481no—
247M16—5481no—
248C2×C8—5481no—
249C2×C8—1371no—
250C2×C8—1371no—
251C2×C8—1371no—
252C2×C8—1371no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—2741no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—1371no—
260C4⋊C4—1371no—
261C4⋊C4C22⋊D13711yescharacteristic
262C4⋊C4C22⋊D13711yescharacteristic
263C4⋊C4C22×D13711yescharacteristic
264C4⋊C4C22×D13711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4C22⋊D13711yescharacteristic
276C22⋊C4C22⋊D13711yescharacteristic
277C42—21no—
278C42—2741no—
279C42C22×D13711yescharacteristic
280C42C22×D13711yescharacteristic
281C23—21no—
282C23—2741no—
283C23C2×C22⋊D13711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—2741no—
288Q8—2741no—
289Q8—10961no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—2741no—
294D4—2741no—
295D4—10961no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—10961no—
306C2×C4C2×C22⋊D13711yescharacteristic
307C2×C4C2×C22⋊D13711yescharacteristic
308C2×C4C2×C22⋊D13711yescharacteristic
309C2×C4C2×C22⋊D13711yescharacteristic
310C2×C4C2×C22⋊D13711yescharacteristic
311C2×C4C23×D13711yescharacteristic
312C8—2741no—
313C8—2741no—
314C8—2741no—
315C8—2741no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—5481no—
323C22—5481no—
324C22—5481no—
325C22C22×C22⋊D13711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—5481no—
335C4—5481no—
336C4—5481no—
337C4#1798413411yescharacteristic
338C4#1798413511yescharacteristic
339C2—41no—
340C2—81no—
341C2—10961no—
342C2#2257798111yescharacteristic
343C2#2257821111yescharacteristic
344C2#2257821411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#542×C137C2
socleC22×C137C22×D4
prime radical#22577654C2
hypercenter#542D137
hypocenterC137#3263
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #22578024C2semidirect
2. #22577693C2semidirect
3. #22577690C2semidirect
4. #22577688C2semidirect
5. #22577685C2semidirect
6. #22577659C2semidirect
8. #22577634C2semidirect
9. #22577630C2semidirect
10. #22577625C2semidirect
11. #22577622C2semidirect
12. #22577618C2semidirect
13. #22577615C2semidirect
14. #22577430C2semidirect
15. #542×C137C2semidirect
172. C137#3263semidirect
175. #542D137semidirect

Maximal subgroups

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

subgroupindex
#225780242
#225776932
#225776902
#225776882
#225776852
#225776592
#225776542
#225776342
#225776302
#225776252
#225776222
#225776182
#225776152
#225774302
#542×C1372
#3263137