Finite Group Databasebeta

← #31793400

Subgroups — #31793400

Order 17536.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578121C211yesmaximal, characteristic
2#22578108C211yesmaximal, characteristic
3#22577800C211yesmaximal, characteristic, complemented
4#22577746C211yesmaximal, characteristic, complemented
5#22577744C211yesmaximal, characteristic, complemented
6#22577735C211yesmaximal, characteristic, complemented
7#22577733C211yesmaximal, characteristic, complemented
8#22577697C211yesmaximal, characteristic, complemented
9#22577687C211yesmaximal, characteristic, complemented
10#22577662C211yesmaximal, characteristic, complemented
11#22577661C211yesmaximal, characteristic, complemented
12#22577497C211yesmaximal, characteristic, complemented
13#22577479C211yesmaximal, characteristic
14#22577444C211yesmaximal, characteristic, complemented
15#428×C137C211yesmaximal, characteristic, complemented
16C2×Dic548C2211yescharacteristic
17C2×#14555493C2211yescharacteristic
18#17984087C2211yescharacteristic
19#17984087C2211yescharacteristic
20#17984082—21no—
21#17984081—21no—
22#17984078—21no—
23#17984074—21no—
24#17984068C2211yescharacteristic
25#17984068C2211yescharacteristic
26#17984067—21no—
27#17984067C2211yescharacteristic
28#17984067C2211yescharacteristic
29#17984066—21no—
30C4×Dic274—21no—
31#17984053C2211yescharacteristic
32#17984053C2211yescharacteristic
33C4×C22⋊D137—21no—
34C4×C22⋊D137C2211yescharacteristic
35#17984046C2211yescharacteristic
36C4⋊C4×D137C2211yescharacteristic
37#17984021C2211yescharacteristic
38#17984020C2211yescharacteristic
39#17984019C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
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×Dic274—21no—
60C2×Dic274—21no—
61C2×Dic274C2311yescharacteristic
62C2×Dic274C2311yescharacteristic
63C2×C22⋊D137D411yescharacteristic
64C2×C4×D137—21no—
65C2×C4×D137C2311yescharacteristic
66Dic548—22no—
67#14555493—22no—
68C2×#11764040C2311yescharacteristic
69C2×#11764040C2311yescharacteristic
70C4⋊Dic137—21no—
71C4⋊Dic137—21no—
72C4⋊Dic137C2311yescharacteristic
73C4⋊Dic137C2311yescharacteristic
74#14555489—21no—
75#14555489—21no—
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489D411yescharacteristic
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137D411yescharacteristic
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487D411yescharacteristic
89C4×Dic137—21no—
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×D137C2×D411yescharacteristic
106Dic274—21no—
107Dic274—21no—
108Dic274—21no—
109Dic274—21no—
110Dic274—42no—
111Dic274—81no—
112C22⋊D137—22no—
113C2×Dic137—21no—
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—81no—
123#11764040—21no—
124#11764040—21no—
125C23×C137C2×D411yescharacteristic
126C2×C4×C137—21no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—81no—
133C2×C4×C137C2×D411yescharacteristic
134C2×C4×C137C2411yescharacteristic
135C8×C137—21no—
136C8×C137#4112yes—
137C22×C137—21no—
138C22×C137—21no—
139C22×C137—21no—
140C22×C137C22×D411yescharacteristic
141C2×D137—21no—
142C2×D137—21no—
143C2×D137—21no—
144C4×C137—21no—
145C4×C137—41no—
146C4×C137—41no—
147C4×C137—41no—
148C4×C137—41no—
149C4×C137C2×#4111yescharacteristic
150C4×C1372−1+411yescharacteristic
151Dic137—21no—
152Dic137—21no—
153Dic137—41no—
154Dic137—41no—
155Dic137—41no—
156Dic137—41no—
157Dic137#13712yes—
158C2×C137—41no—
159C2×C137#54711yescharacteristic
160C2×C137C2×#13711yescharacteristic
161C2×C137#57711yescharacteristic
162D137—41no—
163C137#335611yescharacteristic, complemented, Sylow
164#3356—1371nomaximal, Sylow
165#556—1371no—
166#554—1371no—
167#498—1371no—
168#485—1371no—
169#485—1371no—
170#483—1371no—
171#483—1371no—
172#477—1371no—
173#475—1371no—
174#470—1371no—
175#470—1371no—
176C4×Q16—1371no—
177#436—1371no—
178#428—1371no—
179#428D13711yescharacteristic, complemented
180C2×Q16—1371no—
181C2×M16—1371no—
182C2×M16C2×D13711yescharacteristic
183C4⋊Q8—1371no—
184C4⋊Q8—1371no—
185#126—2741no—
186#126—2741no—
187#125—2741no—
188#124—2741no—
189C22⋊Q8—2741no—
190C22⋊Q8—2741no—
191C22⋊Q8—1371no—
192C22⋊Q8—1371no—
193C22⋊Q8—1371no—
194C22⋊Q8—1371no—
195C4×Q8—2741no—
196C4×Q8—1371no—
197C4×Q8—1371no—
198C4×D4—2741no—
199C4×D4—1371no—
200#117—1371no—
201#117C2×D13711yescharacteristic
202C2×C4⋊C4—1371no—
203C2×C4⋊C4C2×D13711yescharacteristic
204#107—1371no—
205#107C2×D13711yescharacteristic
206#107C2×D13711yescharacteristic
207#106—1371no—
208#106—1371no—
209#106—1371no—
210#106—1371no—
211#106C2×D13711yescharacteristic
212#106C2×D13711yescharacteristic
213C4⋊C8—1371no—
214#103—1371no—
215#103—1371no—
216#103—1371no—
217#103—1371no—
218#103—1371no—
219#103—1371no—
220C22⋊C8—1371no—
221C22⋊C8—1371no—
222C4×C8—1371no—
223C2×Q8—2741no—
224C2×Q8—2741no—
225C2×Q8—1371no—
226C2×Q8—1371no—
227C2×D4—1371no—
228C22×C4—21no—
229C22×C4—2741no—
230C22×C4—1371no—
231C22×C4C22×D13711yescharacteristic
232Q16—2742no—
233M16—22no—
234M16—2742no—
235C2×C8—1371no—
236C2×C8—1371no—
237C2×C8C22×D13711yescharacteristic
238C2×C8C22×D13711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—2741no—
241C4⋊C4—2741no—
242C4⋊C4—2741no—
243C4⋊C4—2741no—
244C4⋊C4—2741no—
245C4⋊C4—2741no—
246C4⋊C4—2741no—
247C4⋊C4—2741no—
248C4⋊C4—2741no—
249C4⋊C4—1371no—
250C4⋊C4—1371no—
251C4⋊C4—1371no—
252C4⋊C4C22×D13711yescharacteristic
253C4⋊C4C22×D13711yescharacteristic
254C4⋊C4C22×D13711yescharacteristic
255C4⋊C4C22×D13711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—2741no—
258C22⋊C4—2741no—
259C22⋊C4—2741no—
260C22⋊C4—2741no—
261C22⋊C4—2741no—
262C22⋊C4—1371no—
263C22⋊C4—1371no—
264C42—21no—
265C42—2741no—
266C42—2741no—
267C42—2741no—
268C42—1371no—
269C23—1371no—
270C23D4×D13711yescharacteristic
271Q8—2741no—
272Q8—2741no—
273Q8—2741no—
274Q8—2741no—
275Q8—5482no—
276Q8—10961no—
277D4—2742no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—2741no—
286C2×C4—2741no—
287C2×C4—2741no—
288C2×C4—2741no—
289C2×C4—2741no—
290C2×C4—2741no—
291C2×C4—10961no—
292C2×C4—1371no—
293C2×C4—1371no—
294C2×C4—1371no—
295C2×C4D4×D13711yescharacteristic
296C2×C4C23×D13711yescharacteristic
297C8—21no—
298C8—2741no—
299C8—2741no—
300C8#1455551012yes—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—2741no—
305C22—2741no—
306C22—2741no—
307C22C2×D4×D13711yescharacteristic
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—2741no—
314C4—2741no—
315C4—5481no—
316C4—5481no—
317C4—5481no—
318C4—5481no—
319C4—1372no—
320C4C2×#1455551011yescharacteristic
321C4#1798413611yescharacteristic
322C2—41no—
323C2—5481no—
324C2#2257805411yescharacteristic
325C2#137×D13711yescharacteristic
326C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radicalC2×C22⋊D137D4
hypercenter#428D137
hypocenterC137#3356
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
3. #22577800C2semidirect
4. #22577746C2semidirect
5. #22577744C2semidirect
6. #22577735C2semidirect
7. #22577733C2semidirect
8. #22577697C2semidirect
9. #22577687C2semidirect
10. #22577662C2semidirect
11. #22577661C2semidirect
12. #22577497C2semidirect
14. #22577444C2semidirect
15. #428×C137C2semidirect
163. C137#3356semidirect
179. #428D137semidirect

Maximal subgroups

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

subgroupindex
#225781212
#225781082
#225778002
#225777462
#225777442
#225777352
#225777332
#225776972
#225776872
#225776622
#225776612
#225774972
#225774792
#225774442
#428×C1372
#3356137