Finite Group Databasebeta

← #31792652

Subgroups — #31792652

Order 17536.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578066C211yesmaximal, characteristic, complemented
2#22578054C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577687C211yesmaximal, characteristic, complemented
6#22577634C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577607C211yesmaximal, characteristic, complemented
9#22577544C211yesmaximal, characteristic, complemented
10#22577543C211yesmaximal, characteristic, complemented
11#22577490C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13C2×#17984011C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#483×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C22×Dic274C2211yescharacteristic
18C2×Q8×D137C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C2×#14555497C2211yescharacteristic
22C22×#11764040C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24#17984087—21no—
25#17984087C2211yescharacteristic
26#17984086C2211yescharacteristic
27#17984073—21no—
28#17984071—21no—
29#17984068C2211yescharacteristic
30#17984066—21no—
31#17984066—21no—
32#17984065—21no—
33#17984063C2211yescharacteristic
34#17984062C2211yescharacteristic
35#17984053C2211yescharacteristic
36#17984050C2211yescharacteristic
37C4×C22⋊D137C2211yescharacteristic
38#17984048—21no—
39C4⋊C4×D137C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C22⋊Q8×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#106×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555510—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×Dic274—22no—
63C2×Dic274—21no—
64C2×Dic274—41no—
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67Q8×D137—22no—
68Q8×D137—41no—
69C2×D548—21no—
70C2×D548C2311yescharacteristic
71C2×C22⋊D137—21no—
72C2×C22⋊D137D411yescharacteristic
73C22×Dic137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—21no—
77C2×C4×D137C2311yescharacteristic
78#14555499—22no—
79#14555499—41no—
80#14555497—41no—
81C2×#11764040—41no—
82C2×#11764040C2311yescharacteristic
83C2×#11764040C2311yescharacteristic
84C8×D137—41no—
85C4⋊Dic137—21no—
86C4⋊Dic137C2311yescharacteristic
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137D411yescharacteristic
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97#14555487—21no—
98#14555487D411yescharacteristic
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C22×D137—21no—
113C22×D137C2×D411yescharacteristic
114Dic274—21no—
115Dic274—21no—
116Dic274—21no—
117Dic274—42no—
118Dic274—42no—
119Dic274—42no—
120Dic274#4112yescomplemented
121D548—21no—
122D548—21no—
123D548—42no—
124C22⋊D137—42no—
125C22⋊D137SD1614yescomplemented
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—21no—
129C2×Dic137—42no—
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C2×Dic137C2×D411yescharacteristic
133C4×D137—21no—
134C4×D137—21no—
135C4×D137—42no—
136C4×D137—42no—
137C4×D137—42no—
138#11764040—21no—
139#11764040—21no—
140C23×C137C2×D411yescharacteristic
141Q8×C137—21no—
142Q8×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—42no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2411yescharacteristic
152C8×C137—21no—
153C8×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137C2×SD1612yes—
157C22×C137C22×D411yescharacteristic
158C2×D137—21no—
159C2×D137—21no—
160C2×D137—41no—
161C2×D137—41no—
162C2×D137—41no—
163C2×D137C2×SD1612yes—
164C4×C137—21no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137H(5,𝔽2)11yescharacteristic
171Dic137—21no—
172Dic137—22no—
173Dic137—21no—
174Dic137—41no—
175Dic137—41no—
176Dic137C2×SD1612yes—
177C2×C137—22no—
178C2×C137#54611yescharacteristic
179C2×C137C22×SD1611yescharacteristic
180C2×C137#57811yescharacteristic
181D137—22no—
182D137—81no—
183C137#329811yescharacteristic, complemented, Sylow
184#3298—1371nomaximal, Sylow
185D4×Q8—1371no—
186#547—1371no—
187#487—1371no—
188#483—1371no—
189#483D13711yescharacteristic, complemented
190#475—1371no—
191#465—1371no—
192#465—1371no—
193#461—1371no—
194#448—1371no—
195#448—1371no—
196C4×SD16—1371no—
197C8×D4—1371no—
198C2×#103—1371no—
199C2×#103—1371no—
200C2×#41—2741no—
201C22×Q8—1371no—
202C22×Q8—1371no—
203C2×SD16—1371no—
204C2×SD16—1371no—
205C2×SD16—1371no—
206C22×C8—1371no—
207C22×C8—1371no—
208C4⋊Q8—2741no—
209C4⋊Q8—1371no—
210C4⋊Q8—1371no—
211#123—2741no—
212#123—2741no—
213C22⋊Q8—2741no—
214C22⋊Q8—2741no—
215C22⋊Q8—2741no—
216C22⋊Q8—1371no—
217C22⋊Q8C2×D13711yescharacteristic
218#121—1371no—
219#121—1371no—
220C4×Q8—1371no—
221C4×D4—2741no—
222C4×D4—1371no—
223C4×D4—1371no—
224C2×C4⋊C4—1371no—
225C2×C4⋊C4C2×D13711yescharacteristic
226#106—1371no—
227#106C2×D13711yescharacteristic
228#106C2×D13711yescharacteristic
229C4⋊C8—1371no—
230#103—1371no—
231#103—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#103—1371no—
237#103C2×D13711yescharacteristic
238#103C2×D13711yescharacteristic
239#102—1371no—
240C22⋊C8—1371no—
241C22⋊C8C2×D13711yescharacteristic
242C4×C8—1371no—
243#41—5484no—
244C2×Q8—2741no—
245C2×Q8—2741no—
246C2×Q8—2742no—
247C2×Q8—2741no—
248C2×Q8—2742no—
249C2×Q8—5481no—
250C2×Q8—5481no—
251C2×Q8—1371no—
252C2×Q8—1371no—
253C2×Q8C22×D13711yescharacteristic
254C2×D4—2741no—
255C2×D4—2741no—
256C2×D4—1371no—
257C2×D4—1371no—
258C22×C4—21no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—2741no—
263C22×C4—1371no—
264C22×C4C22×D13711yescharacteristic
265SD16—2742no—
266SD16—5481no—
267SD16—5481no—
268C2×C8—5481no—
269C2×C8—5481no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8C22×D13711yescharacteristic
273C2×C8C22×D13711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—2741no—
279C4⋊C4—2741no—
280C4⋊C4—2741no—
281C4⋊C4—2741no—
282C4⋊C4—1371no—
283C4⋊C4—1371no—
284C4⋊C4C22×D13711yescharacteristic
285C4⋊C4C22×D13711yescharacteristic
286C4⋊C4C22×D13711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—2741no—
295C42—1371no—
296C23—2741no—
297C23—1371no—
298C23D4×D13711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2741no—
302Q8—2741no—
303Q8—2741no—
304Q8—5482no—
305Q8—5482no—
306Q8—5482no—
307Q8—1372no—
308D4—2741no—
309D4—2741no—
310D4—5482no—
311D4—5482no—
312D4—1374no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—5482no—
326C2×C4—5482no—
327C2×C4—5482no—
328C2×C4—5482no—
329C2×C4—1371no—
330C2×C4—1371no—
331C2×C4—1371no—
332C2×C4D4×D13711yescharacteristic
333C2×C4C23×D13711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2741no—
337C8—2741no—
338C22—21no—
339C22—21no—
340C22—2741no—
341C22—2741no—
342C22—5481no—
343C22—5481no—
344C22—5481no—
345C22—1372no—
346C22SD16×D13712yes—
347C22C2×D4×D13711yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2741no—
354C4—2741no—
355C4—2742no—
356C4—5481no—
357C4—5481no—
358C4—1372no—
359C4#41×D13711yescharacteristic
360C4#1798413411yescharacteristic
361C2—22no—
362C2—2742no—
363C2—10961no—
364C2#2257804511yescharacteristic
365C2C2×SD16×D13711yescharacteristic
366C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#483×C137C2
socleC22×C137C22×D4
prime radical#22578054C2
hypercenter#483D137
hypocenterC137#3298
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22578066C2semidirect
3. #22577752C2semidirect
4. #22577735C2semidirect
5. #22577687C2semidirect
6. #22577634C2semidirect
7. #22577633C2semidirect
8. #22577607C2semidirect
9. #22577544C2semidirect
10. #22577543C2semidirect
11. #22577490C2semidirect
12. #22577466C2semidirect
13. C2×#17984011C2semidirect
14. #103×D137C2semidirect
15. #483×C137C2semidirect
120. Dic274#41semidirect
125. C22⋊D137SD16semidirect
183. C137#3298semidirect
189. #483D137semidirect

Maximal subgroups

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

subgroupindex
#225780662
#225780542
#225777522
#225777352
#225776872
#225776342
#225776332
#225776072
#225775442
#225775432
#225774902
#225774662
C2×#179840112
#103×D1372
#483×C1372
#3298137