Finite Group Databasebeta

← #31791149

Subgroups — #31791149

Order 17536.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578030C211yesmaximal, characteristic, complemented
2#22577978C211yesmaximal, characteristic
3#22577771C211yesmaximal, characteristic, complemented
4#22577770C211yesmaximal, characteristic, complemented
5#22577765C211yesmaximal, characteristic, complemented
6#22577764C211yesmaximal, characteristic, complemented
7#22577759C211yesmaximal, characteristic, complemented
8#22577758C211yesmaximal, characteristic, complemented
9#22577752C211yesmaximal, characteristic, complemented
10#22577410C211yesmaximal, characteristic, complemented
11#22577403C211yesmaximal, characteristic
12#22577398C211yesmaximal, characteristic
13#103×D137C211yesmaximal, characteristic, complemented
14#22577357C211yesmaximal, characteristic, complemented
15#487×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984087C2211yescharacteristic
21#17984086—21no—
22#17984086C2211yescharacteristic
23#17984080C2211yescharacteristic
24#17984079C2211yescharacteristic
25#17984078—21no—
26#17984077C2211yescharacteristic
27#17984076C2211yescharacteristic
28#17984073—21no—
29#17984073—21no—
30#17984066—21no—
31#17984066—21no—
32#17984066—21no—
33#17984064—21no—
34#17984046C2211yescharacteristic
35#17984045C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984013C2211yescharacteristic
38#17984013C2211yescharacteristic
39#17984012C2211yescharacteristic
40#17984012C2211yescharacteristic
41#17984012C2211yescharacteristic
42#17984012C2211yescharacteristic
43#17984011C2211yescharacteristic
44#17984011C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17983999C2211yescharacteristic
50#17983998C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C4⋊Q8×C137C2211yescharacteristic
53C4⋊Q8×C137C2211yescharacteristic
54#103×C137C2211yescharacteristic
55#103×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic
59#14555511—22no—
60#14555511—22no—
61C2×Dic274—21no—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64Q8×D137—22no—
65Q8×D137—22no—
66C2×D548—21no—
67C2×D548C2311yescharacteristic
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137C2311yescharacteristic
72#14555493—22no—
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C8×D137—22no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489D411yescharacteristic
87#14555489D411yescharacteristic
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97C2×Q8×C137—21no—
98C2×Q8×C137C2311yescharacteristic
99C2×Q8×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137—21no—
103C4⋊C4×C137—21no—
104C4⋊C4×C137—21no—
105C4⋊C4×C137C2311yescharacteristic
106C4⋊C4×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137C2×D411yescharacteristic
110Dic274—21no—
111Dic274—21no—
112Dic274—42no—
113Dic274—81no—
114D548—21no—
115D548—21no—
116D548—42no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C4×D137—42no—
124C4×D137—42no—
125C4×D137—81no—
126C4×D137#4112yes—
127C4×D137#4112yes—
128#11764040—21no—
129#11764040—21no—
130Q8×C137—21no—
131Q8×C137—21no—
132Q8×C137—21no—
133Q8×C137—21no—
134Q8×C137—81no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2411yescharacteristic
142C8×C137—21no—
143C8×C137—21no—
144C22×C137C22×D411yescharacteristic
145C2×D137—21no—
146C2×D137—21no—
147C2×D137—41no—
148C2×D137—41no—
149C2×D137—41no—
150C2×D137#12412yes—
151C4×C137—21no—
152C4×C137—21no—
153C4×C137—41no—
154C4×C137—41no—
155C4×C137—41no—
156C4×C137—41no—
157C4×C137C2×#4111yescharacteristic
158C4×C137C2×#4111yescharacteristic
159Dic137—21no—
160Dic137—21no—
161Dic137—41no—
162Dic137—41no—
163Dic137—41no—
164Dic137#12412yes—
165C2×C137C2×#12411yescharacteristic
166C2×C137#57711yescharacteristic
167C2×C137#57811yescharacteristic
168D137—22no—
169D137—81no—
170C137#313711yescharacteristic, complemented, Sylow
171#3137—1371nomaximal, Sylow
172#544—1371no—
173#536—1371no—
174#491—1371no—
175#491—1371no—
176#489—1371no—
177#489—1371no—
178#488—1371no—
179#488—1371no—
180#487—1371no—
181#487D13711yescharacteristic, complemented
182#419—1371no—
183#417—1371no—
184#416—1371no—
185C2×#103—1371no—
186#405—1371no—
187C2×#41—1371no—
188C22×Q8—1371no—
189C2×M16—1371no—
190C22×C8—1371no—
191C4⋊Q8—2741no—
192C4⋊Q8—1371no—
193C4⋊Q8—1371no—
194C4⋊Q8C2×D13711yescharacteristic
195C4⋊Q8C2×D13711yescharacteristic
196#125—2741no—
197#125—1371no—
198#125—1371no—
199#124—1371no—
200#124—1371no—
201#123—2741no—
202#123—2741no—
203C22⋊Q8—2741no—
204C22⋊Q8—2741no—
205C22⋊Q8—2741no—
206C22⋊Q8—2741no—
207#117—1371no—
208#117—1371no—
209C2×C4⋊C4—1371no—
210#103—1371no—
211#103—1371no—
212#103—1371no—
213#103—1371no—
214#103—1371no—
215#103—1371no—
216#103—1371no—
217#103—1371no—
218#103C2×D13711yescharacteristic
219#103C2×D13711yescharacteristic
220#103C2×D13711yescharacteristic
221#103C2×D13711yescharacteristic
222#102—1371no—
223#102—1371no—
224#102—1371no—
225#102—1371no—
226#97—1371no—
227#97—1371no—
228C4×C8—1371no—
229C4×C8C2×D13711yescharacteristic
230#41—2742no—
231#41—2742no—
232C2×Q8—21no—
233C2×Q8—2741no—
234C2×Q8—2741no—
235C2×Q8—2742no—
236C2×Q8—2742no—
237C2×Q8—1371no—
238C2×Q8C22×D13711yescharacteristic
239C2×Q8C22×D13711yescharacteristic
240C2×D4—2741no—
241C2×D4—1371no—
242C22×C4—2741no—
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—1371no—
246M16—2742no—
247C2×C8—2742no—
248C2×C8—1371no—
249C2×C8—1371no—
250C2×C8C22×D13711yescharacteristic
251C2×C8C22×D13711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—2741no—
256C4⋊C4—2741no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
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
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—1371no—
275C22⋊C4—1371no—
276C42—2741no—
277C42—1371no—
278C42C22×D13711yescharacteristic
279C23—2741no—
280C23—1371no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—81no—
286Q8—2741no—
287Q8—2741no—
288Q8—5482no—
289Q8—10961no—
290D4—2741no—
291D4—2741no—
292D4—5482no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—5482no—
301C2×C4—5482no—
302C2×C4—10961no—
303C2×C4—1371no—
304C2×C4—1371no—
305C2×C4—1371no—
306C2×C4—1372no—
307C2×C4—1372no—
308C2×C4D4×D13711yescharacteristic
309C2×C4D4×D13711yescharacteristic
310C2×C4C23×D13711yescharacteristic
311C8—21no—
312C8—21no—
313C8—2741no—
314C8—2741no—
315C22—2741no—
316C22—2741no—
317C22—5481no—
318C22—5481no—
319C22—5481no—
320C22—1372no—
321C22C2×D4×D13711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—2741no—
329C4—2741no—
330C4—5481no—
331C4—5481no—
332C4—5481no—
333C4—1372no—
334C4#41×D13711yescharacteristic
335C4#41×D13711yescharacteristic
336C2—2742no—
337C2—10961no—
338C2#124×D13711yescharacteristic
339C2#2257820911yescharacteristic
340C2#2257821611yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555511C22
hypercenter#487D137
hypocenterC137#3137
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22578030C2semidirect
3. #22577771C2semidirect
4. #22577770C2semidirect
5. #22577765C2semidirect
6. #22577764C2semidirect
7. #22577759C2semidirect
8. #22577758C2semidirect
9. #22577752C2semidirect
10. #22577410C2semidirect
13. #103×D137C2semidirect
14. #22577357C2semidirect
15. #487×C137C2semidirect
170. C137#3137semidirect
181. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225780302
#225779782
#225777712
#225777702
#225777652
#225777642
#225777592
#225777582
#225777522
#225774102
#225774032
#225773982
#103×D1372
#225773572
#487×C1372
#3137137