Finite Group Databasebeta

← #31790809

Subgroups — #31790809

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577937C211yesmaximal, characteristic, complemented
2C4×#14555510C211yesmaximal, characteristic, complemented
3#22577745C211yesmaximal, characteristic, complemented
4#22577728C211yesmaximal, characteristic, complemented
5#22577711C211yesmaximal, characteristic, complemented
6#22577704C211yesmaximal, characteristic, complemented
7#22577626C211yesmaximal, characteristic, complemented
8#22577619C211yesmaximal, characteristic, complemented
9#22577567C211yesmaximal, characteristic, complemented
10#22577550C211yesmaximal, characteristic
11C4×Dic548C211yesmaximal, characteristic, complemented
12C4×#14555499C211yesmaximal, characteristic, complemented
13C4×#14555499C211yesmaximal, characteristic, complemented
14C4×D1096C211yesmaximal, characteristic, complemented
15#431×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×Dic548C2211yescharacteristic
18C2×#14555499C2211yescharacteristic
19C2×#14555499C2211yescharacteristic
20C2×D1096C2211yescharacteristic
21#17984082—21no—
22#17984080C2211yescharacteristic
23#17984076C2211yescharacteristic
24#17984071—21no—
25#17984067C2211yescharacteristic
26#17984062C2211yescharacteristic
27C4×Dic274C2211yescharacteristic
28C4×Dic274C2211yescharacteristic
29C4×Dic274C2211yescharacteristic
30C4×D548C2211yescharacteristic
31C4×D548C2211yescharacteristic
32C4×D548C2211yescharacteristic
33C4×C22⋊D137—21no—
34C4×C22⋊D137—21no—
35C4×C22⋊D137—21no—
36#17984047C2211yescharacteristic
37#17984045—21no—
38#17984045—21no—
39C42×D137—21no—
40#17984022C2211yescharacteristic
41#17984022C2211yescharacteristic
42#17984020C2211yescharacteristic
43#17984020C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984013C2211yescharacteristic
47#17984013C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984010C2211yescharacteristic
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic
52C2×C42×C137C2211yescharacteristic
53C4⋊C8×C137C2211yescharacteristic
54C4⋊C8×C137C2211yescharacteristic
55C22⋊C8×C137C2211yescharacteristic
56C22⋊C8×C137C2211yescharacteristic
57C4×C8×C137C2211yescharacteristic
58C4×C8×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61C2×Dic274C2311yescharacteristic
62C2×Dic274C2311yescharacteristic
63C2×D548C2311yescharacteristic
64C2×D548C2311yescharacteristic
65C2×C22⋊D137—21no—
66C2×C22⋊D137—21no—
67C2×C4×D137—21no—
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71Dic548—22no—
72#14555499—22no—
73#14555499—22no—
74D1096—22no—
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C22×C4×C137D411yescharacteristic
98C22×C4×C137D411yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C2×C8×C137C2311yescharacteristic
104C42×C137D411yescharacteristic
105C42×C137D411yescharacteristic
106C42×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—21no—
110Dic274—21no—
111Dic274—21no—
112Dic274—21no—
113Dic274#4112yes—
114D548—21no—
115D548—21no—
116D548—21no—
117D548#4112yes—
118C22⋊D137—24no—
119C22⋊D137—42no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—21no—
126C4×D137—24no—
127C4×D137—42no—
128C4×D137—42no—
129C4×D137—42no—
130C23×C137C2×D411yescharacteristic
131C2×C4×C137—22no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—41no—
135C2×C4×C137C2×D411yescharacteristic
136C2×C4×C137C2×D411yescharacteristic
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2×D411yescharacteristic
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2411yescharacteristic
141C8×C137—21no—
142C8×C137—21no—
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—21no—
148C22×C137C22×D411yescharacteristic
149C2×D137—22no—
150C2×D137—41no—
151C2×D137—41no—
152C2×D137—41no—
153C2×D137—41no—
154C2×D137—41no—
155C4×C137—21no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137#13512yes—
160C4×C137C2×#4111yescharacteristic
161C4×C137C2×#4111yescharacteristic
162Dic137—22no—
163Dic137—41no—
164Dic137—41no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168C2×C137—41no—
169C2×C137#52511yescharacteristic
170C2×C137C2×#13511yescharacteristic
171C2×C137#57511yescharacteristic
172D137—42no—
173D137—81no—
174C137#310211yescharacteristic, complemented, Sylow
175#3102—1371nomaximal, Sylow
176#529—1371no—
177C4×#41—1371no—
178#485—1371no—
179#482—1371no—
180#479—1371no—
181#478—1371no—
182#464—1371no—
183#463—1371no—
184#452—1371no—
185#449—1371no—
186C4×Q16—1371no—
187C4×SD16—1371no—
188C4×SD16—1371no—
189C4×D8—1371no—
190#431D13711yescharacteristic, complemented
191C2×#41—1371no—
192C2×Q16—1371no—
193C2×SD16—1371no—
194C2×SD16—1371no—
195C2×D8—1371no—
196#126—2741no—
197#125—1371no—
198#124—1371no—
199#123—2741no—
200C22⋊Q8—1371no—
201#121—1371no—
202C4×Q8—1371no—
203C4×Q8—1371no—
204C4×Q8—1371no—
205C4×D4—2741no—
206C4×D4—2741no—
207C4×D4—2741no—
208C4×D4—1371no—
209C4×D4—1371no—
210C4×D4—1371no—
211#117—2741no—
212#117—2741no—
213#117—1371no—
214C2×C42—2741no—
215C2×C42C2×D13711yescharacteristic
216#107—1371no—
217#107—1371no—
218#106—1371no—
219#106—1371no—
220C4⋊C8C2×D13711yescharacteristic
221C4⋊C8C2×D13711yescharacteristic
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#102—1371no—
227#102—1371no—
228#102—1371no—
229#102—1371no—
230C22⋊C8C2×D13711yescharacteristic
231C22⋊C8C2×D13711yescharacteristic
232C4×C8C2×D13711yescharacteristic
233C4×C8C2×D13711yescharacteristic
234#41—2742no—
235#41—2742no—
236C2×Q8—1371no—
237C2×Q8—1371no—
238C2×D4—2741no—
239C2×D4—2741no—
240C2×D4—1371no—
241C2×D4—1371no—
242C22×C4—2741no—
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4D54811yescharacteristic
247C22×C4D54811yescharacteristic
248C22×C4C22×D13711yescharacteristic
249Q16—2742no—
250SD16—2742no—
251SD16—2742no—
252D8—2742no—
253C2×C8C22×D13711yescharacteristic
254C2×C8C22×D13711yescharacteristic
255C2×C8C22×D13711yescharacteristic
256C2×C8C22×D13711yescharacteristic
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C22⋊C4—2741no—
268C22⋊C4—2741no—
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C42—2741no—
276C42—2741no—
277C42—2741no—
278C42—2741no—
279C42D54811yescharacteristic
280C42D54811yescharacteristic
281C42C22×D13711yescharacteristic
282C42C22×D13711yescharacteristic
283C23—2741no—
284C23—2741no—
285C23C2×D54811yescharacteristic
286Q8—2741no—
287Q8—2741no—
288Q8—2741no—
289Q8—1372no—
290D4—2744no—
291D4—2741no—
292D4—2741no—
293D4—2741no—
294D4—5482no—
295D4—1372no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—2744no—
301C2×C4—2741no—
302C2×C4—2741no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—5482no—
308C2×C4—5482no—
309C2×C4—5482no—
310C2×C4C2×D54811yescharacteristic
311C2×C4C2×D54811yescharacteristic
312C2×C4C2×D54811yescharacteristic
313C2×C4C2×D54811yescharacteristic
314C2×C4C2×D54811yescharacteristic
315C2×C4C23×D13711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—2742no—
324C22—5481no—
325C22—5481no—
326C22—5481no—
327C22—5481no—
328C22—5481no—
329C22C22×D54811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—2742no—
335C4—5481no—
336C4—5481no—
337C4—5481no—
338C4—5481no—
339C4—5481no—
340C4#1798411012yes—
341C4#41×D13711yescharacteristic
342C4#41×D13711yescharacteristic
343C2—41no—
344C2—5482no—
345C2—10961no—
346C2#2257791511yescharacteristic
347C2C2×#1798411011yescharacteristic
348C2#2257820311yescharacteristic
subgroupG/H
centerC2×C4C2×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#431×C137C2
socleC22×C137C22×D4
prime radical#22577550C2
hypercenter#431D137
hypocenterC137#3102
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #22577937C2semidirect
2. C4×#14555510C2semidirect
3. #22577745C2semidirect
4. #22577728C2semidirect
5. #22577711C2semidirect
6. #22577704C2semidirect
7. #22577626C2semidirect
8. #22577619C2semidirect
9. #22577567C2semidirect
11. C4×Dic548C2semidirect
12. C4×#14555499C2semidirect
13. C4×#14555499C2semidirect
14. C4×D1096C2semidirect
15. #431×C137C2semidirect
174. C137#3102semidirect
190. #431D137semidirect

Maximal subgroups

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

subgroupindex
#225779372
C4×#145555102
#225777452
#225777282
#225777112
#225777042
#225776262
#225776192
#225775672
#225775502
C4×Dic5482
C4×#145554992
C4×D10962
#431×C1372
#3102137