Finite Group Databasebeta

← #31793356

Subgroups — #31793356

Order 17536.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578103C211yesmaximal, characteristic
2#22578091C211yesmaximal, characteristic, complemented
3#22577802C211yesmaximal, characteristic, complemented
4#22577746C211yesmaximal, characteristic, complemented
5#22577735C211yesmaximal, characteristic, complemented
6#22577725C211yesmaximal, characteristic, complemented
7#22577720C211yesmaximal, characteristic, complemented
8#22577711C211yesmaximal, characteristic, complemented
9#22577648C211yesmaximal, characteristic, complemented
10#22577648C211yesmaximal, characteristic, complemented
11C4×#14555498C211yesmaximal, characteristic, complemented
12#22577480C211yesmaximal, characteristic, complemented
13#22577444C211yesmaximal, characteristic, complemented
14#22577442C211yesmaximal, characteristic, complemented
15#478×C137C211yesmaximal, characteristic, complemented
16C2×#14555498C2211yescharacteristic
17C2×#14555493C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19#17984083—21no—
20#17984081—21no—
21#17984080C2211yescharacteristic
22#17984079—21no—
23#17984073—21no—
24#17984071—21no—
25#17984068C2211yescharacteristic
26#17984068C2211yescharacteristic
27#17984061—21no—
28#17984059C2211yescharacteristic
29#17984059C2211yescharacteristic
30C4×Dic274C2211yescharacteristic
31C4×D548C2211yescharacteristic
32C4×C22⋊D137—21no—
33#17984046—21no—
34#17984046C2211yescharacteristic
35#17984046C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984019C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984019C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984013C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984009C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17984000C2211yescharacteristic
51C4×#11764040C2211yescharacteristic
52#125×C137C2211yescharacteristic
53C4×D4×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56C4⋊C8×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C2×Dic274C2311yescharacteristic
60C2×D548D411yescharacteristic
61C2×C22⋊D137—21no—
62C2×C22⋊D137—21no—
63C2×C4×D137—21no—
64C2×C4×D137—21no—
65C2×C4×D137C2311yescharacteristic
66C2×C4×D137C2311yescharacteristic
67#14555498—22no—
68#14555493—22no—
69#14555493—22no—
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C4⋊Dic137—21no—
73C4⋊Dic137—21no—
74C4⋊Dic137D411yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487D411yescharacteristic
90#14555487D411yescharacteristic
91C4×Dic137—21no—
92C4×Dic137—21no—
93C4×Dic137—21no—
94C2×D4×C137C2311yescharacteristic
95C22×C4×C137—21no—
96C2×C8×C137C2311yescharacteristic
97C2×C8×C137C2311yescharacteristic
98C4⋊C4×C137—21no—
99C4⋊C4×C137—21no—
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C42×C137C2311yescharacteristic
105C22×D137C2×D411yescharacteristic
106C22×D137C2×D411yescharacteristic
107Dic274—21no—
108Dic274—21no—
109D548—41no—
110C22⋊D137—42no—
111C22⋊D137—81no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137C2×D411yescharacteristic
117C2×Dic137C2×D411yescharacteristic
118C4×D137—21no—
119C4×D137—21no—
120C4×D137—21no—
121C4×D137—21no—
122C4×D137—42no—
123C4×D137—81no—
124#11764040—21no—
125#11764040#4112yes—
126C23×C137—21no—
127D4×C137—21no—
128D4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—81no—
133C2×C4×C137C2×D411yescharacteristic
134C2×C4×C137C2×D411yescharacteristic
135C2×C4×C137C2411yescharacteristic
136C8×C137—21no—
137C8×C137—21no—
138C22×C137—41no—
139C22×C137—41no—
140C22×C137—41no—
141C22×C137C22×D411yescharacteristic
142C2×D137—21no—
143C2×D137—21no—
144C2×D137—21no—
145C2×D137—21no—
146C2×D137—21no—
147C2×D137—21no—
148C4×C137—21no—
149C4×C137—21no—
150C4×C137—41no—
151C4×C137—41no—
152C4×C137—41no—
153C4×C137C2×#4111yescharacteristic
154C4×C1372−1+411yescharacteristic
155Dic137—21no—
156Dic137—21no—
157Dic137—41no—
158Dic137—41no—
159Dic137—41no—
160Dic137—41no—
161C2×C137—81no—
162C2×C137#54711yescharacteristic
163C2×C137#57511yescharacteristic
164C2×C137#57711yescharacteristic
165D137—41no—
166D137—41no—
167C137#335111yescharacteristic, complemented, Sylow
168#3351—1371nomaximal, Sylow
169#553—1371no—
170#552—1371no—
171#499—1371no—
172#485—1371no—
173#483—1371no—
174#482—1371no—
175#481—1371no—
176#479—1371no—
177#478D13711yescharacteristic, complemented
178#468—1371no—
179#468—1371no—
180C4×SD16—1371no—
181#436—1371no—
182#428—1371no—
183#428—1371no—
184C2×SD16—1371no—
185C2×M16—1371no—
186C2×M16—1371no—
187#126—2741no—
188#126—2741no—
189#125—2741no—
190#125—1371no—
191#125C2×D13711yescharacteristic
192#123—2741no—
193#123—2741no—
194C22⋊Q8—1371no—
195C22⋊Q8—1371no—
196#121—2741no—
197#121—1371no—
198#121—1371no—
199C4×Q8—1371no—
200C4×D4—2741no—
201C4×D4—1371no—
202C4×D4C2×D13711yescharacteristic
203#117—2741no—
204#117—1371no—
205#117—1371no—
206C2×C4⋊C4—1371no—
207C2×C4⋊C4—1371no—
208#107—1371no—
209#107—1371no—
210#107—1371no—
211#107C2×D13711yescharacteristic
212#106—1371no—
213#106—1371no—
214#106—1371no—
215#106—1371no—
216#106C2×D13711yescharacteristic
217C4⋊C8C2×D13711yescharacteristic
218#103—1371no—
219#103—1371no—
220#103—1371no—
221#102—1371no—
222#102C2×D13711yescharacteristic
223#102C2×D13711yescharacteristic
224C22⋊C8—1371no—
225C22⋊C8—1371no—
226C4×C8—1371no—
227C2×Q8—1371no—
228C2×D4—2741no—
229C2×D4—2741no—
230C2×D4—1371no—
231C2×D4C22×D13711yescharacteristic
232C22×C4—21no—
233C22×C4—2741no—
234C22×C4—2741no—
235C22×C4—1371no—
236C22×C4—1371no—
237SD16—2742no—
238M16—2742no—
239M16—2742no—
240C2×C8—1371no—
241C2×C8—1371no—
242C2×C8C22×D13711yescharacteristic
243C2×C8C22×D13711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—2741no—
247C4⋊C4—2741no—
248C4⋊C4—2741no—
249C4⋊C4—2741no—
250C4⋊C4—2741no—
251C4⋊C4—2741no—
252C4⋊C4—2741no—
253C4⋊C4—1371no—
254C4⋊C4—1371no—
255C4⋊C4—1371no—
256C4⋊C4—1371no—
257C4⋊C4C22×D13711yescharacteristic
258C4⋊C4C22×D13711yescharacteristic
259C4⋊C4C22×D13711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—2741no—
262C22⋊C4—2741no—
263C22⋊C4—2741no—
264C22⋊C4—2741no—
265C22⋊C4—2741no—
266C22⋊C4—2741no—
267C22⋊C4—1371no—
268C22⋊C4—1371no—
269C42—2741no—
270C42—2741no—
271C42—2741no—
272C42C22×D13711yescharacteristic
273C23—21no—
274C23—1371no—
275C23—1371no—
276Q8—2741no—
277Q8—2741no—
278D4—21no—
279D4—21no—
280D4—5481no—
281D4—5482no—
282D4—10961no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—2741no—
288C2×C4—2741no—
289C2×C4—2741no—
290C2×C4—2741no—
291C2×C4—2741no—
292C2×C4—2741no—
293C2×C4—2741no—
294C2×C4—2741no—
295C2×C4—5482no—
296C2×C4—10961no—
297C2×C4—1371no—
298C2×C4—1371no—
299C2×C4D4×D13711yescharacteristic
300C2×C4D4×D13711yescharacteristic
301C2×C4C23×D13711yescharacteristic
302C8—21no—
303C8—21no—
304C8—2741no—
305C8—1372no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—2741no—
310C22—2741no—
311C22—2741no—
312C22—2741no—
313C22—2741no—
314C22—2741no—
315C22C2×D4×D13711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—2741no—
322C4—2741no—
323C4—5481no—
324C4—5481no—
325C4—5481no—
326C4—5481no—
327C4#41×D13711yescharacteristic
328C4#1798413611yescharacteristic
329C2—81no—
330C2—5481no—
331C2—5481no—
332C2#2257805511yescharacteristic
333C2#2257820011yescharacteristic
334C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#478×C137C2
socleC22×C137C22×D4
prime radical#22578103C2
hypercenter#478D137
hypocenterC137#3351
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
2. #22578091C2semidirect
3. #22577802C2semidirect
4. #22577746C2semidirect
5. #22577735C2semidirect
6. #22577725C2semidirect
7. #22577720C2semidirect
8. #22577711C2semidirect
9. #22577648C2semidirect
10. #22577648C2semidirect
11. C4×#14555498C2semidirect
12. #22577480C2semidirect
13. #22577444C2semidirect
14. #22577442C2semidirect
15. #478×C137C2semidirect
167. C137#3351semidirect
177. #478D137semidirect

Maximal subgroups

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

subgroupindex
#225781032
#225780912
#225778022
#225777462
#225777352
#225777252
#225777202
#225777112
#225776482
C4×#145554982
#225774802
#225774442
#225774422
#478×C1372
#3351137