Finite Group Databasebeta

← #31793359

Subgroups — #31793359

Order 17536.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578095C211yesmaximal, characteristic
2#22578087C211yesmaximal, characteristic
3#22577803C211yesmaximal, characteristic, complemented
4#22577742C211yesmaximal, characteristic, complemented
5#22577741C211yesmaximal, characteristic, complemented
6#22577737C211yesmaximal, characteristic, complemented
7#22577736C211yesmaximal, characteristic, complemented
8#22577712C211yesmaximal, characteristic, complemented
9#22577708C211yesmaximal, characteristic, complemented
10#22577658C211yesmaximal, characteristic, complemented
11Q16×Dic137C211yesmaximal, characteristic, complemented
12#22577479C211yesmaximal, characteristic
13#22577445C211yesmaximal, characteristic, complemented
14#22577442C211yesmaximal, characteristic, complemented
15#470×C137C211yesmaximal, characteristic, complemented
16C2×#14555493C2211yescharacteristic
17#17984083—21no—
18#17984081—21no—
19#17984079—21no—
20#17984079C2211yescharacteristic
21#17984079C2211yescharacteristic
22#17984078—21no—
23#17984072—21no—
24#17984070—21no—
25#17984064C2211yescharacteristic
26#17984064C2211yescharacteristic
27Q8×Dic137C2211yescharacteristic
28Q8×Dic137C2211yescharacteristic
29C4×C22⋊D137C2211yescharacteristic
30#17984047C2211yescharacteristic
31#17984046—21no—
32#17984046C2211yescharacteristic
33#17984044—21no—
34C2×C4⋊Dic137C2211yescharacteristic
35C4⋊C4×D137C2211yescharacteristic
36#17984022C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984020C2211yescharacteristic
42#17984020C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984017C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984012C2211yescharacteristic
48#17984012C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×Q16×C137C2211yescharacteristic
53C2×M16×C137C2211yescharacteristic
54C22⋊Q8×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59C2×C22⋊D137D411yescharacteristic
60C22×Dic137—21no—
61C2×C4×D137—21no—
62C2×C4×D137C2311yescharacteristic
63#14555493—22no—
64C2×#11764040C2311yescharacteristic
65C2×#11764040C2311yescharacteristic
66C4⋊Dic137—21no—
67C4⋊Dic137—21no—
68C4⋊Dic137—21no—
69C4⋊Dic137—21no—
70C4⋊Dic137—21no—
71C4⋊Dic137C2311yescharacteristic
72C4⋊Dic137C2311yescharacteristic
73C4⋊Dic137C2311yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75#14555489—21no—
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489D411yescharacteristic
81C22⋊Dic137—21no—
82C22⋊Dic137D411yescharacteristic
83#14555487—21no—
84#14555487—21no—
85#14555487—21no—
86#14555487D411yescharacteristic
87C4×Dic137—21no—
88C4×Dic137—21no—
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C2×Q8×C137C2311yescharacteristic
93C2×Q8×C137C2311yescharacteristic
94C22×C4×C137C2311yescharacteristic
95Q16×C137—22no—
96M16×C137—22no—
97C2×C8×C137C2311yescharacteristic
98C2×C8×C137C2311yescharacteristic
99C4⋊C4×C137—21no—
100C4⋊C4×C137—21no—
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22×D137C2×D411yescharacteristic
106C22⋊D137—41no—
107C2×Dic137—21no—
108C2×Dic137—21no—
109C2×Dic137—21no—
110C2×Dic137—21no—
111C2×Dic137—42no—
112C2×Dic137C2×D411yescharacteristic
113C2×Dic137C2×D411yescharacteristic
114C2×Dic137C2×D411yescharacteristic
115C4×D137—21no—
116C4×D137—21no—
117C4×D137—42no—
118#11764040—21no—
119#11764040—21no—
120C23×C137C2×D411yescharacteristic
121Q8×C137—21no—
122Q8×C137—21no—
123Q8×C137—21no—
124Q8×C137—21no—
125C2×C4×C137—21no—
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137C2×D411yescharacteristic
132C2×C4×C137C2411yescharacteristic
133C8×C137—21no—
134C8×C137#4112yes—
135C22×C137—21no—
136C22×C137—21no—
137C22×C137—21no—
138C22×C137C22×D411yescharacteristic
139C2×D137—21no—
140C2×D137—21no—
141C2×D137—21no—
142C4×C137—21no—
143C4×C137—41no—
144C4×C137—41no—
145C4×C137—41no—
146C4×C137—41no—
147C4×C137C2×#4111yescharacteristic
148C4×C1372−1+411yescharacteristic
149Dic137—21no—
150Dic137—21no—
151Dic137—21no—
152Dic137—41no—
153Dic137—41no—
154Dic137—41no—
155Dic137—41no—
156C2×C137—41no—
157C2×C137#54711yescharacteristic
158C2×C137#57511yescharacteristic
159C2×C137#57811yescharacteristic
160D137—41no—
161C137#335211yescharacteristic, complemented, Sylow
162#3352—1371nomaximal, Sylow
163#552—1371no—
164#552—1371no—
165#499—1371no—
166#485—1371no—
167#485—1371no—
168#484—1371no—
169#484—1371no—
170#479—1371no—
171#479—1371no—
172#470—1371no—
173#470D13711yescharacteristic, complemented
174C4×Q16—1371no—
175#436—1371no—
176#428—1371no—
177#428—1371no—
178C2×Q16C2×D13711yescharacteristic
179C2×M16—1371no—
180C2×M16C2×D13711yescharacteristic
181#126—2741no—
182#126—2741no—
183#125—2741no—
184#125—2741no—
185#125—1371no—
186#125—1371no—
187#123—2741no—
188#123—2741no—
189C22⋊Q8—1371no—
190C22⋊Q8—1371no—
191C22⋊Q8C2×D13711yescharacteristic
192C22⋊Q8C2×D13711yescharacteristic
193C4×Q8—1371no—
194C4×Q8—1371no—
195C4×D4—1371no—
196#117—2741no—
197#117—2741no—
198#117—1371no—
199#117—1371no—
200C2×C4⋊C4—1371no—
201C2×C4⋊C4—1371no—
202#107—1371no—
203#107—1371no—
204#107—1371no—
205#107—1371no—
206#107—1371no—
207#106—1371no—
208#106—1371no—
209#106—1371no—
210#106C2×D13711yescharacteristic
211C4⋊C8—1371no—
212#103—1371no—
213#103—1371no—
214#103—1371no—
215#103—1371no—
216#103C2×D13711yescharacteristic
217#103C2×D13711yescharacteristic
218C22⋊C8—1371no—
219C22⋊C8—1371no—
220C4×C8—1371no—
221C2×Q8C22×D13711yescharacteristic
222C2×Q8C22×D13711yescharacteristic
223C2×D4—1371no—
224C22×C4—2741no—
225C22×C4—2741no—
226C22×C4—1371no—
227C22×C4C22×D13711yescharacteristic
228Q16—22no—
229M16—22no—
230M16—2742no—
231C2×C8—1371no—
232C2×C8—1371no—
233C2×C8C22×D13711yescharacteristic
234C2×C8C22×D13711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—2741no—
238C4⋊C4—2741no—
239C4⋊C4—2741no—
240C4⋊C4—2741no—
241C4⋊C4—2741no—
242C4⋊C4—2741no—
243C4⋊C4—2741no—
244C4⋊C4—2741no—
245C4⋊C4—2741no—
246C4⋊C4—2741no—
247C4⋊C4—1371no—
248C4⋊C4—1371no—
249C4⋊C4—1371no—
250C4⋊C4—1371no—
251C4⋊C4—1371no—
252C4⋊C4C22×D13711yescharacteristic
253C4⋊C4C22×D13711yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—2741no—
257C22⋊C4—2741no—
258C22⋊C4—2741no—
259C22⋊C4—2741no—
260C22⋊C4—1371no—
261C22⋊C4—1371no—
262C42—2741no—
263C42—2741no—
264C42—2741no—
265C42—2741no—
266C42—1371no—
267C23—1371no—
268C23D4×D13711yescharacteristic
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273D4—5481no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—2741no—
281C2×C4—2741no—
282C2×C4—2741no—
283C2×C4—2741no—
284C2×C4—2741no—
285C2×C4—2741no—
286C2×C4—5482no—
287C2×C4—5482no—
288C2×C4—1371no—
289C2×C4—1371no—
290C2×C4—1371no—
291C2×C4D4×D13711yescharacteristic
292C2×C4C23×D13711yescharacteristic
293C8—21no—
294C8—2741no—
295C8—2741no—
296C8#1455550912yes—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—2741no—
301C22—2741no—
302C22—2741no—
303C22C2×D4×D13711yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—2741no—
310C4—2741no—
311C4—2741no—
312C4—5481no—
313C4—5481no—
314C4—5481no—
315C4—5481no—
316C4C2×#1455550911yescharacteristic
317C4#1798413511yescharacteristic
318C2—41no—
319C2—5481no—
320C2#2257805311yescharacteristic
321C2#2257820211yescharacteristic
322C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#470×C137C2
socleC22×C137C22×D4
prime radicalC2×C22⋊D137D4
hypercenter#470D137
hypocenterC137#3352
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #22577803C2semidirect
4. #22577742C2semidirect
5. #22577741C2semidirect
6. #22577737C2semidirect
7. #22577736C2semidirect
8. #22577712C2semidirect
9. #22577708C2semidirect
10. #22577658C2semidirect
11. Q16×Dic137C2semidirect
13. #22577445C2semidirect
14. #22577442C2semidirect
15. #470×C137C2semidirect
161. C137#3352semidirect
173. #470D137semidirect

Maximal subgroups

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

subgroupindex
#225780952
#225780872
#225778032
#225777422
#225777412
#225777372
#225777362
#225777122
#225777082
#225776582
Q16×Dic1372
#225774792
#225774452
#225774422
#470×C1372
#3352137