Finite Group Databasebeta

← #31789857

Subgroups — #31789857

Order 17536.

394 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577950C211yesmaximal, characteristic
2#22577477C211yesmaximal, characteristic, complemented
3#22577476C211yesmaximal, characteristic, complemented
4#22577475C211yesmaximal, characteristic, complemented
5#22577474C211yesmaximal, characteristic, complemented
6#22577473C211yesmaximal, characteristic, complemented
7#22577472C211yesmaximal, characteristic, complemented
8#22577471C211yesmaximal, characteristic, complemented
9#22577459C211yesmaximal, characteristic, complemented
10#22577458C211yesmaximal, characteristic, complemented
11#22577371C211yesmaximal, characteristic, complemented
12#22577369C211yesmaximal, characteristic, complemented
13#22577368C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555510C2211yescharacteristic
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20C2×#14555493C2211yescharacteristic
21C22×#11764040C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24#17984087C411yescharacteristic
25#17984084C411yescharacteristic
26#17984074C411yescharacteristic
27#17984074C411yescharacteristic
28#17984063C411yescharacteristic
29#17984061C411yescharacteristic
30#17984061C411yescharacteristic
31#17984059C411yescharacteristic
32C4×D548C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35C42×D137C2211yescharacteristic
36C4⋊#11764040C2211yescharacteristic
37#17984017C2211yescharacteristic
38#17984017C2211yescharacteristic
39C22⋊#11764040C2211yescharacteristic
40C22⋊#11764040C2211yescharacteristic
41#17984000C2211yescharacteristic
42#17984000C2211yescharacteristic
43#17984000C2211yescharacteristic
44#17984000C2211yescharacteristic
45#17984000C2211yescharacteristic
46#17984000C2211yescharacteristic
47#17984000C2211yescharacteristic
48#17984000C2211yescharacteristic
49#17983999C2211yescharacteristic
50#17983998C2211yescharacteristic
51#17983998C2211yescharacteristic
52C2×M16×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55C4⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic
58#97×C137C2211yescharacteristic
59#14555510—41no—
60#14555510—41no—
61#14555510—41no—
62#14555510—41no—
63C2×Dic274C2×C411yescharacteristic
64C2×Dic274C2×C411yescharacteristic
65C2×D548C2×C411yescharacteristic
66C2×D548C2×C411yescharacteristic
67C2×D548C2×C411yescharacteristic
68C2×C22⋊D137C2×C411yescharacteristic
69C2×C22⋊D137C2×C411yescharacteristic
70C2×C22⋊D137C2×C411yescharacteristic
71C2×C22⋊D137C2×C411yescharacteristic
72C2×C22⋊D137C2×C411yescharacteristic
73C2×C22⋊D137C2×C411yescharacteristic
74C2×C4×D137C2×C411yescharacteristic
75C2×C4×D137C2×C411yescharacteristic
76C2×C4×D137C2311yescharacteristic
77C2×C4×D137C2311yescharacteristic
78C2×C4×D137C2311yescharacteristic
79#14555494—41no—
80#14555493—41no—
81#14555493—41no—
82C2×#11764040—41no—
83C2×#11764040C2311yescharacteristic
84C2×#11764040C2311yescharacteristic
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C8×D137—41no—
88C8×D137—41no—
89C4⋊Dic137C2×C411yescharacteristic
90#14555489C2×C411yescharacteristic
91#14555489C2×C411yescharacteristic
92C22⋊Dic137C2×C411yescharacteristic
93C22⋊Dic137C2×C411yescharacteristic
94#14555487C2×C411yescharacteristic
95#14555487C2×C411yescharacteristic
96#14555487C2×C411yescharacteristic
97#14555487C2×C411yescharacteristic
98C4×Dic137C2×C411yescharacteristic
99C4×Dic137C2×C411yescharacteristic
100C4×Dic137C2311yescharacteristic
101C2×D4×C137C2×C411yescharacteristic
102C22×C4×C137C2311yescharacteristic
103C22×C4×C137C2311yescharacteristic
104M16×C137—41no—
105C2×C8×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C2×C8×C137C2311yescharacteristic
109C2×C8×C137C2311yescharacteristic
110C4⋊C4×C137C2×C411yescharacteristic
111C22⋊C4×C137C2×C411yescharacteristic
112C22⋊C4×C137C2×C411yescharacteristic
113C42×C137C2311yescharacteristic
114C22×D137C22×C411yescharacteristic
115C22×D137C22×C411yescharacteristic
116C22×D137C22×C411yescharacteristic
117Dic274—41no—
118Dic274—41no—
119D548—41no—
120D548—41no—
121D548—41no—
122C22⋊D137—41no—
123C22⋊D137—41no—
124C22⋊D137—41no—
125C22⋊D137—41no—
126C22⋊D137—41no—
127C22⋊D137—41no—
128C2×Dic137C22×C411yescharacteristic
129C2×Dic137C22×C411yescharacteristic
130C2×Dic137C22×C411yescharacteristic
131C2×Dic137C22×C411yescharacteristic
132C2×Dic137C22×C411yescharacteristic
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—21no—
136C4×D137—21no—
137C4×D137—21no—
138C4×D137—21no—
139C4×D137—41no—
140C4×D137—41no—
141#11764040—21no—
142#11764040—21no—
143#11764040—21no—
144#11764040—21no—
145C23×C137C22×C411yescharacteristic
146C23×C137C22×C411yescharacteristic
147D4×C137—41no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137C22×C411yescharacteristic
153C2×C4×C137C22×C411yescharacteristic
154C2×C4×C137C22×C411yescharacteristic
155C2×C4×C137C22×C411yescharacteristic
156C2×C4×C137C2411yescharacteristic
157C8×C137—21no—
158C8×C137—21no—
159C8×C137—21no—
160C8×C137—21no—
161C22×C137—21no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—21no—
165C22×C137—21no—
166C22×C137—21no—
167C22×C137C23×C411yescharacteristic
168C2×D137—21no—
169C2×D137—21no—
170C2×D137—21no—
171C2×D137—21no—
172C2×D137—21no—
173C2×D137—21no—
174C2×D137—21no—
175C2×D137—21no—
176C2×D137—21no—
177C4×C137—21no—
178C4×C137—21no—
179C4×C137—21no—
180C4×C137—21no—
181C4×C137H(5,𝔽2)11yescharacteristic
182C4×C137H(5,𝔽2)11yescharacteristic
183Dic137—21no—
184Dic137—21no—
185Dic137—21no—
186Dic137—21no—
187Dic137—21no—
188C2×C137—41no—
189C2×C137—41no—
190C2×C137#51811yescharacteristic
191C2×C137#56811yescharacteristic
192C2×C137#56811yescharacteristic
193D137—41no—
194D137—41no—
195D137—41no—
196C137#298011yescharacteristic, complemented, Sylow
197#2980—1371nomaximal, Sylow
198#532—1371no—
199#435—1371no—
200#435—1371no—
201#435—1371no—
202#435—1371no—
203#435—1371no—
204#435—1371no—
205#435—1371no—
206#435D13711yescharacteristic, complemented
207#432—1371no—
208#432—1371no—
209#408—1371no—
210#408—1371no—
211#408—1371no—
212#408—1371no—
213C2×#41—1371no—
214C2×#41—1371no—
215C2×M16—1371no—
216C2×M16—1371no—
217C2×M16—1371no—
218C2×M16C2×D13711yescharacteristic
219C22×C8—1371no—
220C22×C8—1371no—
221C22×C8—1371no—
222C22×C8C2×D13711yescharacteristic
223C4⋊Q8—1371no—
224#127—1371no—
225#124—1371no—
226#124—1371no—
227#121—1371no—
228#121—1371no—
229#121—1371no—
230#121—1371no—
231C4×D4—1371no—
232C4×D4—1371no—
233C4×D4—1371no—
234C4×D4C2×D13711yescharacteristic
235C2×C42—1371no—
236C4⋊C8—1371no—
237C4⋊C8—1371no—
238C4⋊C8—1371no—
239C4⋊C8C2×D13711yescharacteristic
240C22⋊C8—1371no—
241C22⋊C8—1371no—
242C22⋊C8—1371no—
243C22⋊C8—1371no—
244C22⋊C8—1371no—
245C22⋊C8—1371no—
246C22⋊C8—1371no—
247C22⋊C8—1371no—
248C22⋊C8—1371no—
249C22⋊C8—1371no—
250C22⋊C8C2×D13711yescharacteristic
251C22⋊C8C2×D13711yescharacteristic
252#97—1371no—
253#97—1371no—
254#97—1371no—
255#97C2×D13711yescharacteristic
256#41—5481no—
257#41—5481no—
258#41—5481no—
259#41—5481no—
260C2×Q8—1371no—
261C2×Q8—1371no—
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4—1371no—
266C2×D4—1371no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4—1371no—
270C2×D4—1371no—
271C2×D4C4×D13711yescharacteristic
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1371no—
275C22×C4—1371no—
276C22×C4—1371no—
277C22×C4C22×D13711yescharacteristic
278C22×C4C22×D13711yescharacteristic
279M16—41no—
280M16—5481no—
281M16—5481no—
282M16—5481no—
283C2×C8—41no—
284C2×C8—5481no—
285C2×C8—5481no—
286C2×C8—5481no—
287C2×C8—1371no—
288C2×C8—1371no—
289C2×C8—1371no—
290C2×C8—1371no—
291C2×C8C22×D13711yescharacteristic
292C2×C8C22×D13711yescharacteristic
293C2×C8C22×D13711yescharacteristic
294C2×C8C22×D13711yescharacteristic
295C4⋊C4—1371no—
296C4⋊C4—1371no—
297C4⋊C4—1371no—
298C4⋊C4C4×D13711yescharacteristic
299C22⋊C4—1371no—
300C22⋊C4—1371no—
301C22⋊C4—1371no—
302C22⋊C4—1371no—
303C22⋊C4—1371no—
304C22⋊C4—1371no—
305C22⋊C4C4×D13711yescharacteristic
306C22⋊C4C4×D13711yescharacteristic
307C42—1371no—
308C42—1371no—
309C42—1371no—
310C42C22×D13711yescharacteristic
311C23—1371no—
312C23—1371no—
313C23—1371no—
314C23C2×C4×D13711yescharacteristic
315C23C2×C4×D13711yescharacteristic
316Q8—5481no—
317Q8—5481no—
318D4—41no—
319D4—5481no—
320D4—5481no—
321D4—5481no—
322D4—5481no—
323D4—5481no—
324D4—5481no—
325D4—5481no—
326D4—5481no—
327D4—5481no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2741no—
333C2×C4—2741no—
334C2×C4—2741no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—5481no—
339C2×C4—5481no—
340C2×C4—1371no—
341C2×C4—1371no—
342C2×C4—1371no—
343C2×C4—1371no—
344C2×C4—1371no—
345C2×C4C2×C4×D13711yescharacteristic
346C2×C4C2×C4×D13711yescharacteristic
347C2×C4C2×C4×D13711yescharacteristic
348C2×C4C2×C4×D13711yescharacteristic
349C2×C4C23×D13711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—2741no—
355C8—2741no—
356C8—2741no—
357C8—2741no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—2741no—
365C22—2741no—
366C22—2741no—
367C22—2741no—
368C22—2741no—
369C22—2741no—
370C22—2741no—
371C22—2741no—
372C22—2741no—
373C22C22×C4×D13711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—2741no—
379C4—2741no—
380C4—2741no—
381C4—2741no—
382C4—2741no—
383C4#1798413311yescharacteristic
384C4#1798413411yescharacteristic
385C2—41no—
386C2—41no—
387C2—5481no—
388C2—5481no—
389C2—5481no—
390C2#2257787511yescharacteristic
391C2#2257816811yescharacteristic
392C2#2257816911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D137
commutator, socleC22×C137C23×C4
Fitting#435×C137C2
prime radical#22577950C2
cosocleC2×C4×C137C24
hypercenter#435D137
hypocenterC137#2980
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #22577477C2semidirect
3. #22577476C2semidirect
4. #22577475C2semidirect
5. #22577474C2semidirect
6. #22577473C2semidirect
7. #22577472C2semidirect
8. #22577471C2semidirect
9. #22577459C2semidirect
10. #22577458C2semidirect
11. #22577371C2semidirect
12. #22577369C2semidirect
13. #22577368C2semidirect
14. #22577368C2semidirect
15. #435×C137C2semidirect
196. C137#2980semidirect
206. #435D137semidirect

Maximal subgroups

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

subgroupindex
#225779502
#225774772
#225774762
#225774752
#225774742
#225774732
#225774722
#225774712
#225774592
#225774582
#225773712
#225773692
#225773682
#435×C1372
#2980137