Finite Group Databasebeta

← #31792649

Subgroups — #31792649

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#22578053C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577735C211yesmaximal, characteristic, complemented
5#22577731C211yesmaximal, characteristic, complemented
6#22577687C211yesmaximal, characteristic, complemented
7#22577631C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577541C211yesmaximal, characteristic, complemented
10#22577540C211yesmaximal, characteristic, complemented
11#22577494C211yesmaximal, characteristic, complemented
12C8×C22⋊D137C211yesmaximal, characteristic, complemented
13C2×#17984013C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#465×C137C211yesmaximal, characteristic, complemented
16C2×#14555509—21no—
17C22×Dic274C2211yescharacteristic
18C2×Q8×D137C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C2×C8×D137C2211yescharacteristic
22#17984087—21no—
23#17984087C2211yescharacteristic
24#17984086C2211yescharacteristic
25#17984070—21no—
26#17984069—21no—
27#17984068C2211yescharacteristic
28#17984066—21no—
29#17984065—21no—
30#17984065—21no—
31#17984059C2211yescharacteristic
32#17984053C2211yescharacteristic
33D4×Dic137C2211yescharacteristic
34C4×C22⋊D137C2211yescharacteristic
35#17984048—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984019C2211yescharacteristic
39#17984019C2211yescharacteristic
40#17984017C2211yescharacteristic
41#17984013C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×SD16×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C22⋊Q8×C137C2211yescharacteristic
55#121×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—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×C22⋊D137—21no—
70C2×C22⋊D137D411yescharacteristic
71C22×Dic137—21no—
72C22×Dic137—21no—
73C22×Dic137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137C2311yescharacteristic
77#14555498—22no—
78#14555498—41no—
79C2×#11764040C2311yescharacteristic
80C2×#11764040C2311yescharacteristic
81C8×D137—41no—
82C4⋊Dic137—21no—
83C4⋊Dic137—21no—
84C4⋊Dic137C2311yescharacteristic
85C4⋊Dic137C2311yescharacteristic
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489—21no—
91#14555489D411yescharacteristic
92C22⋊Dic137—21no—
93C22⋊Dic137—21no—
94C22⋊Dic137D411yescharacteristic
95#14555487—21no—
96#14555487D411yescharacteristic
97C4×Dic137—21no—
98C4×Dic137C2311yescharacteristic
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137C2311yescharacteristic
103SD16×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22⋊C4×C137—21no—
112C22×D137C2×D411yescharacteristic
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117Dic274—42no—
118Dic274—42no—
119Dic274#4112yescomplemented
120C22⋊D137—42no—
121C22⋊D137SD1614yescomplemented
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C2×Dic137—42no—
127C2×Dic137—42no—
128C2×Dic137—42no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—42no—
135C4×D137—42no—
136#11764040—21no—
137#11764040—21no—
138C23×C137—21no—
139C23×C137C2×D411yescharacteristic
140Q8×C137—21no—
141Q8×C137—21no—
142D4×C137—21no—
143D4×C137—21no—
144D4×C137—42no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2411yescharacteristic
152C8×C137—21no—
153C8×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137C2×SD1612yes—
160C22×C137C22×D411yescharacteristic
161C2×D137—21no—
162C2×D137—21no—
163C2×D137C2×SD1612yes—
164C4×C137—21no—
165C4×C137—41no—
166C4×C137—41no—
167C4×C137—41no—
168C4×C137C2×#4111yescharacteristic
169C4×C137H(5,𝔽2)11yescharacteristic
170Dic137—21no—
171Dic137—22no—
172Dic137—21no—
173Dic137—41no—
174Dic137—41no—
175Dic137—41no—
176Dic137C2×SD1612yes—
177C2×C137—22no—
178C2×C137—81no—
179C2×C137#54611yescharacteristic
180C2×C137C22×SD1611yescharacteristic
181C2×C137#57811yescharacteristic
182D137—22no—
183C137#329811yescharacteristic, complemented, Sylow
184#3298—1371nomaximal, Sylow
185D4×Q8—1371no—
186#547—1371no—
187#487—1371no—
188#483—1371no—
189#483—1371no—
190#475—1371no—
191#465—1371no—
192#465D13711yescharacteristic, complemented
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×SD16C2×D13711yescharacteristic
206C22×C8—1371no—
207C22×C8C2×D13711yescharacteristic
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#121C2×D13711yescharacteristic
220C4×Q8—1371no—
221C4×D4—2741no—
222C4×D4—1371no—
223C4×D4—1371no—
224C2×C4⋊C4—1371no—
225C2×C4⋊C4—1371no—
226#106—1371no—
227#106—1371no—
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#103—1371no—
238#103C2×D13711yescharacteristic
239#102C2×D13711yescharacteristic
240C22⋊C8—1371no—
241C22⋊C8—1371no—
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—21no—
255C2×D4—2741no—
256C2×D4—1371no—
257C2×D4C22×D13711yescharacteristic
258C22×C4—2741no—
259C22×C4—2741no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4—2741no—
263C22×C4—1371no—
264C22×C4C22×D13711yescharacteristic
265SD16—41no—
266SD16—2742no—
267SD16—5481no—
268C2×C8—41no—
269C2×C8—5481no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8C22×D13711yescharacteristic
273C2×C8C22×D13711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—2741no—
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⋊C4—1371no—
285C4⋊C4C22×D13711yescharacteristic
286C4⋊C4C22×D13711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4—2741no—
292C22⋊C4—1371no—
293C22⋊C4—1371no—
294C42—2741no—
295C42—1371no—
296C23—21no—
297C23—1371no—
298C23D4×D13711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2741no—
302Q8—2741no—
303Q8—2741no—
304Q8—5482no—
305Q8—5482no—
306Q8—5482no—
307Q8—1372no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—5482no—
312D4—1374no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2741no—
319C2×C4—2741no—
320C2×C4—2741no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2741no—
324C2×C4—5482no—
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—41no—
341C22—41no—
342C22—41no—
343C22—2741no—
344C22—2741no—
345C22—1372no—
346C22SD16×D13712yes—
347C22C2×D4×D13711yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—2741no—
353C4—2741no—
354C4—2742no—
355C4—5481no—
356C4—5481no—
357C4—5481no—
358C4—1372no—
359C4#41×D13711yescharacteristic
360C4#1798413311yescharacteristic
361C2—22no—
362C2—81no—
363C2—2742no—
364C2#2257804211yescharacteristic
365C2C2×SD16×D13711yescharacteristic
366C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#465×C137C2
socleC22×C137C22×D4
prime radical#22578053C2
hypercenter#465D137
hypocenterC137#3298
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #22578066C2semidirect
3. #22577752C2semidirect
4. #22577735C2semidirect
5. #22577731C2semidirect
6. #22577687C2semidirect
7. #22577631C2semidirect
8. #22577606C2semidirect
9. #22577541C2semidirect
10. #22577540C2semidirect
11. #22577494C2semidirect
12. C8×C22⋊D137C2semidirect
13. C2×#17984013C2semidirect
14. #103×D137C2semidirect
15. #465×C137C2semidirect
119. Dic274#41semidirect
121. C22⋊D137SD16semidirect
183. C137#3298semidirect
192. #465D137semidirect

Maximal subgroups

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

subgroupindex
#225780662
#225780532
#225777522
#225777352
#225777312
#225776872
#225776312
#225776062
#225775412
#225775402
#225774942
C8×C22⋊D1372
C2×#179840132
#103×D1372
#465×C1372
#3298137