Finite Group Databasebeta

← #31789862

Subgroups — #31789862

Order 17536.

402 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#22577949C211yesmaximal, characteristic
2#22577479C211yesmaximal, characteristic, complemented
3#22577478C211yesmaximal, characteristic, complemented
4#22577477C211yesmaximal, characteristic, complemented
5#22577474C211yesmaximal, characteristic, complemented
6#22577472C211yesmaximal, characteristic, complemented
7#22577471C211yesmaximal, characteristic, complemented
8C8×C22⋊D137C211yesmaximal, characteristic, complemented
9#22577466C211yesmaximal, characteristic, complemented
10#22577457C211yesmaximal, characteristic, complemented
11#22577453C211yesmaximal, characteristic, complemented
12#22577370C211yesmaximal, characteristic, complemented
13#22577368C211yesmaximal, characteristic, complemented
14#22577368C211yesmaximal, characteristic, complemented
15#408×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×#14555494C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19C2×#14555493C2211yescharacteristic
20C22×#11764040C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic
23#17984086C411yescharacteristic
24#17984084C411yescharacteristic
25#17984077C411yescharacteristic
26#17984075C411yescharacteristic
27#17984060C411yescharacteristic
28#17984060C411yescharacteristic
29#17984060C411yescharacteristic
30#17984060C411yescharacteristic
31C4×C22⋊D137C2211yescharacteristic
32C4×C22⋊D137C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35C2×C4×Dic137C2211yescharacteristic
36#17984017C2211yescharacteristic
37#17984017C2211yescharacteristic
38#17984017C2211yescharacteristic
39#17984017C2211yescharacteristic
40C22⋊#11764040C2211yescharacteristic
41C22⋊#11764040C2211yescharacteristic
42C22⋊#11764040C2211yescharacteristic
43C22⋊#11764040C2211yescharacteristic
44#17984000C2211yescharacteristic
45#17984000C2211yescharacteristic
46#17984000C2211yescharacteristic
47#17984000C2211yescharacteristic
48#17983998C2211yescharacteristic
49#17983998C2211yescharacteristic
50C8×Dic137C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic
54C22×C8×C137C2211yescharacteristic
55C22⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57C22⋊C8×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555510—41no—
60#14555510—41no—
61C2×Dic274C2×C411yescharacteristic
62C2×D548C2×C411yescharacteristic
63C2×C22⋊D137C2×C411yescharacteristic
64C2×C22⋊D137C2×C411yescharacteristic
65C2×C22⋊D137C2×C411yescharacteristic
66C2×C22⋊D137C2×C411yescharacteristic
67C2×C22⋊D137C2×C411yescharacteristic
68C2×C22⋊D137C2×C411yescharacteristic
69C22×Dic137C2×C411yescharacteristic
70C22×Dic137C2×C411yescharacteristic
71C2×C4×D137C2311yescharacteristic
72C2×C4×D137C2311yescharacteristic
73#14555494—41no—
74#14555493—41no—
75#14555493—41no—
76C2×#11764040—41no—
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C8×D137—41no—
82C8×D137—41no—
83#14555489C2×C411yescharacteristic
84#14555489C2×C411yescharacteristic
85#14555489C2×C411yescharacteristic
86#14555489C2×C411yescharacteristic
87C22⋊Dic137C2×C411yescharacteristic
88C22⋊Dic137C2×C411yescharacteristic
89C22⋊Dic137C2×C411yescharacteristic
90C22⋊Dic137C2×C411yescharacteristic
91#14555487C2×C411yescharacteristic
92#14555487C2×C411yescharacteristic
93#14555487C2×C411yescharacteristic
94#14555487C2×C411yescharacteristic
95C4×Dic137C2×C411yescharacteristic
96C4×Dic137C2×C411yescharacteristic
97C4×Dic137C2311yescharacteristic
98C4×Dic137C2311yescharacteristic
99#41×C137—41no—
100#41×C137—41no—
101C2×Q8×C137C2×C411yescharacteristic
102C2×D4×C137C2×C411yescharacteristic
103C2×D4×C137C2×C411yescharacteristic
104C2×D4×C137C2×C411yescharacteristic
105C22×C4×C137C2311yescharacteristic
106C22×C4×C137C2311yescharacteristic
107C22×C4×C137C2311yescharacteristic
108M16×C137—41no—
109C2×C8×C137—41no—
110C2×C8×C137C2311yescharacteristic
111C2×C8×C137C2311yescharacteristic
112C2×C8×C137C2311yescharacteristic
113C2×C8×C137C2311yescharacteristic
114C22×D137C22×C411yescharacteristic
115C22×D137C22×C411yescharacteristic
116Dic274—41no—
117D548—41no—
118C22⋊D137—22no—
119C22⋊D137—22no—
120C22⋊D137—41no—
121C22⋊D137—41no—
122C22⋊D137—41no—
123C22⋊D137—41no—
124C2×Dic137—22no—
125C2×Dic137—41no—
126C2×Dic137C22×C411yescharacteristic
127C2×Dic137C22×C411yescharacteristic
128C2×Dic137C22×C411yescharacteristic
129C2×Dic137C22×C411yescharacteristic
130C2×Dic137C22×C411yescharacteristic
131C2×Dic137C22×C411yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—21no—
136#11764040—21no—
137#11764040—21no—
138#11764040—21no—
139#11764040—21no—
140C23×C137C22×C411yescharacteristic
141C23×C137C22×C411yescharacteristic
142C23×C137C22×C411yescharacteristic
143Q8×C137—41no—
144D4×C137—41no—
145D4×C137—41no—
146D4×C137—41no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
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×C137—21no—
168C22×C137—21no—
169C22×C137—21no—
170C22×C137C23×C411yescharacteristic
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×C137H(5,𝔽2)11yescharacteristic
181C4×C137H(5,𝔽2)11yescharacteristic
182Dic137—21no—
183Dic137—21no—
184Dic137—21no—
185Dic137—21no—
186Dic137—21no—
187Dic137#13112yes—
188C2×C137—41no—
189C2×C137—41no—
190C2×C137—41no—
191C2×C137#51811yescharacteristic
192C2×C137C2×#13111yescharacteristic
193C2×C137#56811yescharacteristic
194D137—41no—
195D137—41no—
196C137#298111yescharacteristic, complemented, Sylow
197#2981—1371nomaximal, Sylow
198#532—1371no—
199#436—1371no—
200#436—1371no—
201#435—1371no—
202#435—1371no—
203#435—1371no—
204#435—1371no—
205C8×D4—1371no—
206C8×D4—1371no—
207#432—1371no—
208#431—1371no—
209#408—1371no—
210#408—1371no—
211#408—1371no—
212#408D13711yescharacteristic, complemented
213C2×#41—1371no—
214C2×#41C2×D13711yescharacteristic
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×D4—1371no—
235C2×C42—1371no—
236C4⋊C8—1371no—
237C4⋊C8—1371no—
238C4⋊C8—1371no—
239C4⋊C8—1371no—
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⋊C8C2×D13711yescharacteristic
249C22⋊C8C2×D13711yescharacteristic
250C22⋊C8C2×D13711yescharacteristic
251C22⋊C8C2×D13711yescharacteristic
252#97—1371no—
253#97—1371no—
254C4×C8—1371no—
255C4×C8—1371no—
256#41—41no—
257#41—41no—
258#41—5481no—
259#41—5481no—
260C2×Q8—1371no—
261C2×Q8C4×D13711yescharacteristic
262C2×D4—1371no—
263C2×D4—1371no—
264C2×D4—1371no—
265C2×D4—1371no—
266C2×D4—1371no—
267C2×D4—1371no—
268C2×D4—1371no—
269C2×D4C4×D13711yescharacteristic
270C2×D4C4×D13711yescharacteristic
271C2×D4C4×D13711yescharacteristic
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1371no—
275C22×C4—1371no—
276C22×C4C22×D13711yescharacteristic
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⋊C4—1371no—
299C22⋊C4—1371no—
300C22⋊C4—1371no—
301C22⋊C4—1371no—
302C22⋊C4—1371no—
303C22⋊C4—1371no—
304C22⋊C4—1371no—
305C22⋊C4—1371no—
306C22⋊C4—1371no—
307C42—1371no—
308C42—1371no—
309C42—1371no—
310C42—1371no—
311C23—1371no—
312C23—1371no—
313C23C2×C4×D13711yescharacteristic
314C23C2×C4×D13711yescharacteristic
315C23C2×C4×D13711yescharacteristic
316Q8—41no—
317Q8—5481no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—2742no—
322D4—2742no—
323D4—5481no—
324D4—5481no—
325D4—5481no—
326D4—5481no—
327D4—5481no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2742no—
335C2×C4—2741no—
336C2×C4—2741no—
337C2×C4—2741no—
338C2×C4—2741no—
339C2×C4—5481no—
340C2×C4—1371no—
341C2×C4—1371no—
342C2×C4—1371no—
343C2×C4—1371no—
344C2×C4—1371no—
345C2×C4—1371no—
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—21no—
365C22—21no—
366C22—21no—
367C22—2741no—
368C22—2741no—
369C22—2741no—
370C22—2741no—
371C22—2741no—
372C22—2741no—
373C22C22×C4×D13711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—2741no—
378C4—2741no—
379C4—2741no—
380C4—2741no—
381C4—2741no—
382C4—1372no—
383C4#1798413311yescharacteristic
384C4#1798413311yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—5481no—
389C2—5481no—
390C2#2257787611yescharacteristic
391C2#131×D13711yescharacteristic
392C2#2257816811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D137
commutator, socleC22×C137C23×C4
Fitting#408×C137C2
prime radical#22577949C2
cosocleC2×C4×C137C24
hypercenter#408D137
hypocenterC137#2981
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22577479C2semidirect
3. #22577478C2semidirect
4. #22577477C2semidirect
5. #22577474C2semidirect
6. #22577472C2semidirect
7. #22577471C2semidirect
8. C8×C22⋊D137C2semidirect
9. #22577466C2semidirect
10. #22577457C2semidirect
11. #22577453C2semidirect
12. #22577370C2semidirect
13. #22577368C2semidirect
14. #22577368C2semidirect
15. #408×C137C2semidirect
196. C137#2981semidirect
212. #408D137semidirect

Maximal subgroups

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

subgroupindex
#225779492
#225774792
#225774782
#225774772
#225774742
#225774722
#225774712
C8×C22⋊D1372
#225774662
#225774572
#225774532
#225773702
#225773682
#408×C1372
#2981137