Finite Group Databasebeta

← #31792674

Subgroups — #31792674

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578088C211yesmaximal, characteristic, complemented
2#22578059C211yesmaximal, characteristic
3#22577756C211yesmaximal, characteristic, complemented
4#22577746C211yesmaximal, characteristic, complemented
5#22577709C211yesmaximal, characteristic, complemented
6#22577645C211yesmaximal, characteristic, complemented
7#22577630C211yesmaximal, characteristic, complemented
8#22577624C211yesmaximal, characteristic, complemented
9#22577566C211yesmaximal, characteristic, complemented
10#22577561C211yesmaximal, characteristic, complemented
11#22577497C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577396C211yesmaximal, characteristic, complemented
14C2×#17984011C211yesmaximal, characteristic, complemented
15#480×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18C2×Dic548C2211yescharacteristic
19C2×Dic548C2211yescharacteristic
20C2×#14555498C2211yescharacteristic
21C22×#11764040C2211yescharacteristic
22C2×C8×D137C2211yescharacteristic
23#17984083—21no—
24#17984079C2211yescharacteristic
25#17984078—21no—
26#17984075C2211yescharacteristic
27#17984074—21no—
28#17984071—21no—
29#17984068C2211yescharacteristic
30#17984068C2211yescharacteristic
31#17984067C2211yescharacteristic
32#17984065—21no—
33#17984060—21no—
34#17984053C2211yescharacteristic
35#17984053C2211yescharacteristic
36D4×Dic137—21no—
37C4×C22⋊D137C2211yescharacteristic
38#17984047—21no—
39#17984046C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984011C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52#121×C137C2211yescharacteristic
53C2×C4⋊C4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#107×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274—22no—
62C2×Dic274—21no—
63C2×Dic274—41no—
64C2×Dic274C2311yescharacteristic
65C2×Dic274C2311yescharacteristic
66C2×Dic274C2311yescharacteristic
67C2×C22⋊D137—21no—
68C2×C22⋊D137D411yescharacteristic
69C22×Dic137—21no—
70C22×Dic137—21no—
71C2×C4×D137C2311yescharacteristic
72Dic548—22no—
73Dic548—41no—
74#14555498—41no—
75C2×#11764040—41no—
76C2×#11764040C2311yescharacteristic
77C2×#11764040C2311yescharacteristic
78C8×D137—41no—
79C4⋊Dic137—21no—
80C4⋊Dic137—21no—
81C4⋊Dic137—21no—
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489D411yescharacteristic
88C22⋊Dic137—21no—
89C22⋊Dic137—21no—
90C22⋊Dic137—21no—
91C22⋊Dic137D411yescharacteristic
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487D411yescharacteristic
96C4×Dic137—21no—
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—21no—
117Dic274—42no—
118Dic274#4112yes—
119C22⋊D137—22no—
120C22⋊D137—42no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—42no—
126C2×Dic137—42no—
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C2×Dic137C2×D411yescharacteristic
130C4×D137—21no—
131C4×D137—21no—
132#11764040—21no—
133#11764040—21no—
134C23×C137—21no—
135C23×C137C2×D411yescharacteristic
136D4×C137—21no—
137D4×C137—21no—
138D4×C137—42no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—42no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137#13512yes—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C4×C137—21no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137C2×#4111yescharacteristic
164C4×C137H(5,𝔽2)11yescharacteristic
165Dic137—21no—
166Dic137—22no—
167Dic137—21no—
168Dic137—21no—
169Dic137—41no—
170Dic137—41no—
171Dic137—41no—
172C2×C137—22no—
173C2×C137—81no—
174C2×C137#54611yescharacteristic
175C2×C137C2×#13511yescharacteristic
176C2×C137#57811yescharacteristic
177D137—41no—
178C137#330011yescharacteristic, complemented, Sylow
179#3300—1371nomaximal, Sylow
180#552—1371no—
181#548—1371no—
182#488—1371no—
183#485—1371no—
184#480D13711yescharacteristic, complemented
185#479—1371no—
186#467—1371no—
187#465—1371no—
188#464—1371no—
189#452—1371no—
190#451—1371no—
191C4×Q16—1371no—
192C8×D4—1371no—
193#416—1371no—
194C2×#103—1371no—
195C2×#41—1371no—
196C22×Q8—1371no—
197C2×Q16—1371no—
198C2×Q16—1371no—
199C2×SD16—1371no—
200C22×C8—1371no—
201C22×C8—1371no—
202#126—2741no—
203#125—2741no—
204#125—1371no—
205#124—2741no—
206#124—1371no—
207#123—2741no—
208C22⋊Q8—2741no—
209C22⋊Q8—1371no—
210C22⋊Q8—1371no—
211C22⋊Q8—1371no—
212#121—2741no—
213#121C2×D13711yescharacteristic
214C4×Q8—1371no—
215C4×Q8—1371no—
216C4×D4—2741no—
217C4×D4—1371no—
218#117—2741no—
219#117—1371no—
220C2×C4⋊C4C2×D13711yescharacteristic
221#107C2×D13711yescharacteristic
222#107C2×D13711yescharacteristic
223#106—1371no—
224C4⋊C8—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103—1371no—
232#102—1371no—
233#102C2×D13711yescharacteristic
234#102C2×D13711yescharacteristic
235C22⋊C8—1371no—
236C22⋊C8C2×D13711yescharacteristic
237C4×C8—1371no—
238#41—2742no—
239#41—5481no—
240C2×Q8—2742no—
241C2×Q8—2741no—
242C2×Q8—5481no—
243C2×Q8—1371no—
244C2×Q8—1371no—
245C2×Q8—1371no—
246C2×D4—21no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4C22×D13711yescharacteristic
250C22×C4—21no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—1371no—
254C22×C4C22×D13711yescharacteristic
255Q16—2742no—
256Q16—5481no—
257SD16—5481no—
258C2×C8—5481no—
259C2×C8—5481no—
260C2×C8—1371no—
261C2×C8—1371no—
262C2×C8C22×D13711yescharacteristic
263C2×C8C22×D13711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4C22×D13711yescharacteristic
275C4⋊C4C22×D13711yescharacteristic
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—1371no—
285C22⋊C4—1371no—
286C42—2741no—
287C42—2741no—
288C42—2741no—
289C42—1371no—
290C23—21no—
291C23—1371no—
292C23D4×D13711yescharacteristic
293Q8—2741no—
294Q8—2741no—
295Q8—2741no—
296Q8—2741no—
297Q8—2741no—
298Q8—5482no—
299Q8—1372no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—2742no—
304D4—5482no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—5482no—
318C2×C4—5482no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4D4×D13711yescharacteristic
323C2×C4C23×D13711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2741no—
327C8—2741no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—2741no—
334C22—2741no—
335C22—2741no—
336C22#1798410712yes—
337C22C2×D4×D13711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—2741no—
343C4—2741no—
344C4—2741no—
345C4—2742no—
346C4—5481no—
347C4—5481no—
348C4—5481no—
349C4#41×D13711yescharacteristic
350C4#1798413411yescharacteristic
351C2—22no—
352C2—81no—
353C2—5481no—
354C2#2257804511yescharacteristic
355C2C2×#1798410711yescharacteristic
356C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#22578059C2
hypercenter#480D137
hypocenterC137#3300
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #22578088C2semidirect
3. #22577756C2semidirect
4. #22577746C2semidirect
5. #22577709C2semidirect
6. #22577645C2semidirect
7. #22577630C2semidirect
8. #22577624C2semidirect
9. #22577566C2semidirect
10. #22577561C2semidirect
11. #22577497C2semidirect
12. #22577466C2semidirect
13. #22577396C2semidirect
14. C2×#17984011C2semidirect
15. #480×C137C2semidirect
178. C137#3300semidirect
184. #480D137semidirect

Maximal subgroups

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

subgroupindex
#225780882
#225780592
#225777562
#225777462
#225777092
#225776452
#225776302
#225776242
#225775662
#225775612
#225774972
#225774662
#225773962
C2×#179840112
#480×C1372
#3300137