Finite Group Databasebeta

← #31793350

Subgroups — #31793350

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#22578101C211yesmaximal, characteristic
2#22578092C211yesmaximal, characteristic, complemented
3#22577804C211yesmaximal, characteristic, complemented
4#22577746C211yesmaximal, characteristic, complemented
5#22577733C211yesmaximal, characteristic, complemented
6#22577727C211yesmaximal, characteristic, complemented
7#22577723C211yesmaximal, characteristic, complemented
8#22577713C211yesmaximal, characteristic, complemented
9#22577706C211yesmaximal, characteristic, complemented
10#22577650C211yesmaximal, characteristic, complemented
11#22577649C211yesmaximal, characteristic, complemented
12#22577490C211yesmaximal, characteristic, complemented
13#22577479C211yesmaximal, characteristic, complemented
14#22577442C211yesmaximal, characteristic, complemented
15#428×C137C211yesmaximal, characteristic, complemented
16C2×#14555499C2211yescharacteristic
17C2×#14555493C2211yescharacteristic
18#17984083—21no—
19#17984081—21no—
20#17984080—21no—
21#17984079C2211yescharacteristic
22#17984079C2211yescharacteristic
23#17984073—21no—
24#17984070—21no—
25#17984068C2211yescharacteristic
26#17984067C2211yescharacteristic
27#17984063C2211yescharacteristic
28#17984062—21no—
29#17984062C2211yescharacteristic
30#17984053C2211yescharacteristic
31#17984050C2211yescharacteristic
32C4×C22⋊D137—21no—
33C4×C22⋊D137C2211yescharacteristic
34#17984046C2211yescharacteristic
35#17984045—21no—
36C4⋊C4×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984019C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984011C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984010C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×M16×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#107×C137C2211yescharacteristic
56#107×C137C2211yescharacteristic
57#106×C137C2211yescharacteristic
58#106×C137C2211yescharacteristic
59C2×Dic274C2311yescharacteristic
60C2×D548—21no—
61C2×D548C2311yescharacteristic
62C2×C22⋊D137—21no—
63C2×C22⋊D137D411yescharacteristic
64C2×C4×D137—21no—
65C2×C4×D137—21no—
66C2×C4×D137C2311yescharacteristic
67#14555499—22no—
68#14555493—22no—
69C2×#11764040C2311yescharacteristic
70C2×#11764040C2311yescharacteristic
71C4⋊Dic137—21no—
72C4⋊Dic137—21no—
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489D411yescharacteristic
82C22⋊Dic137—21no—
83C22⋊Dic137D411yescharacteristic
84#14555487—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487D411yescharacteristic
90C4×Dic137—21no—
91C4×Dic137—21no—
92C4×Dic137C2311yescharacteristic
93C22×C4×C137—21no—
94C22×C4×C137C2311yescharacteristic
95M16×C137—22no—
96C2×C8×C137C2311yescharacteristic
97C2×C8×C137C2311yescharacteristic
98C4⋊C4×C137—21no—
99C4⋊C4×C137C2311yescharacteristic
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C42×C137—21no—
105C22×D137—21no—
106C22×D137C2×D411yescharacteristic
107Dic274—21no—
108Dic274—21no—
109D548—21no—
110D548—21no—
111D548—81no—
112C22⋊D137—41no—
113C22⋊D137—42no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137C2×D411yescharacteristic
118C2×Dic137C2×D411yescharacteristic
119C2×Dic137C2×D411yescharacteristic
120C4×D137—21no—
121C4×D137—21no—
122C4×D137—42no—
123C4×D137—81no—
124#11764040—21no—
125#11764040—21no—
126C23×C137C2×D411yescharacteristic
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—81no—
134C2×C4×C137C2×D411yescharacteristic
135C2×C4×C137C2411yescharacteristic
136C8×C137—21no—
137C8×C137#4112yes—
138C22×C137—21no—
139C22×C137—21no—
140C22×C137—21no—
141C22×C137C22×D411yescharacteristic
142C2×D137—21no—
143C2×D137—21no—
144C2×D137—21no—
145C2×D137—41no—
146C2×D137—41no—
147C2×D137—41no—
148C4×C137—21no—
149C4×C137—41no—
150C4×C137—41no—
151C4×C137—41no—
152C4×C137—41no—
153C4×C137C2×#4111yescharacteristic
154C4×C1372−1+411yescharacteristic
155Dic137—21no—
156Dic137—21no—
157Dic137—21no—
158Dic137—41no—
159Dic137—41no—
160Dic137—41no—
161C2×C137—41no—
162C2×C137#54711yescharacteristic
163C2×C137#57511yescharacteristic
164C2×C137#57711yescharacteristic
165D137—41no—
166D137—81no—
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#478—1371no—
178#468—1371no—
179#468—1371no—
180C4×SD16—1371no—
181#436—1371no—
182#428—1371no—
183#428D13711yescharacteristic, complemented
184C2×SD16—1371no—
185C2×M16—1371no—
186C2×M16C2×D13711yescharacteristic
187#126—2741no—
188#126—2741no—
189#125—2741no—
190#125—1371no—
191#125—1371no—
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×D4—1371no—
203#117—2741no—
204#117—1371no—
205#117C2×D13711yescharacteristic
206C2×C4⋊C4—1371no—
207C2×C4⋊C4C2×D13711yescharacteristic
208#107—1371no—
209#107—1371no—
210#107C2×D13711yescharacteristic
211#107C2×D13711yescharacteristic
212#106—1371no—
213#106—1371no—
214#106—1371no—
215#106C2×D13711yescharacteristic
216#106C2×D13711yescharacteristic
217C4⋊C8—1371no—
218#103—1371no—
219#103—1371no—
220#103—1371no—
221#102—1371no—
222#102—1371no—
223#102—1371no—
224C22⋊C8—1371no—
225C22⋊C8—1371no—
226C4×C8—1371no—
227C2×Q8—1371no—
228C2×D4—2741no—
229C2×D4—2741no—
230C2×D4—1371no—
231C2×D4—1371no—
232C22×C4—21no—
233C22×C4—2741no—
234C22×C4—2741no—
235C22×C4—1371no—
236C22×C4C22×D13711yescharacteristic
237SD16—2742no—
238M16—22no—
239M16—2742no—
240C2×C8—1371no—
241C2×C8—1371no—
242C2×C8C22×D13711yescharacteristic
243C2×C8C22×D13711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—2741no—
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⋊C4C22×D13711yescharacteristic
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—21no—
270C42—2741no—
271C42—2741no—
272C42—1371no—
273C23—2741no—
274C23—1371no—
275C23D4×D13711yescharacteristic
276Q8—2741no—
277Q8—2741no—
278D4—2741no—
279D4—2741no—
280D4—5481no—
281D4—5482no—
282D4—10961no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—81no—
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×C4—1371no—
300C2×C4D4×D13711yescharacteristic
301C2×C4C23×D13711yescharacteristic
302C8—21no—
303C8—2741no—
304C8—2741no—
305C8#1455551012yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—2741no—
310C22—2741no—
311C22—2741no—
312C22—5481no—
313C22—5481no—
314C22—5481no—
315C22C2×D4×D13711yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—2741no—
322C4—2741no—
323C4—2741no—
324C4—5481no—
325C4—5481no—
326C4—5481no—
327C4C2×#1455551011yescharacteristic
328C4#1798413611yescharacteristic
329C2—41no—
330C2—5481no—
331C2—10961no—
332C2#2257805411yescharacteristic
333C2#2257820211yescharacteristic
334C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radical#22578101C2
hypercenter#428D137
hypocenterC137#3351
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #22578092C2semidirect
3. #22577804C2semidirect
4. #22577746C2semidirect
5. #22577733C2semidirect
6. #22577727C2semidirect
7. #22577723C2semidirect
8. #22577713C2semidirect
9. #22577706C2semidirect
10. #22577650C2semidirect
11. #22577649C2semidirect
12. #22577490C2semidirect
13. #22577479C2semidirect
14. #22577442C2semidirect
15. #428×C137C2semidirect
167. C137#3351semidirect
183. #428D137semidirect

Maximal subgroups

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

subgroupindex
#225781012
#225780922
#225778042
#225777462
#225777332
#225777272
#225777232
#225777132
#225777062
#225776502
#225776492
#225774902
#225774792
#225774422
#428×C1372
#3351137