Finite Group Databasebeta

← #31790802

Subgroups — #31790802

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#22577934C211yesmaximal, characteristic, complemented
2#22577871C211yesmaximal, characteristic, complemented
3#22577744C211yesmaximal, characteristic, complemented
4#22577713C211yesmaximal, characteristic, complemented
5#22577706C211yesmaximal, characteristic, complemented
6#22577624C211yesmaximal, characteristic, complemented
7#22577617C211yesmaximal, characteristic, complemented
8#22577567C211yesmaximal, characteristic, complemented
9#22577546C211yesmaximal, characteristic
10#22577497C211yesmaximal, characteristic, complemented
11#22577494C211yesmaximal, characteristic, complemented
12#22577490C211yesmaximal, characteristic, complemented
13#22577483C211yesmaximal, characteristic, complemented
14#22577453C211yesmaximal, characteristic, complemented
15#482×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C2×Dic548C2211yescharacteristic
18C2×#14555499C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×#14555495C2211yescharacteristic
21#17984081—21no—
22#17984079C2211yescharacteristic
23#17984075C2211yescharacteristic
24#17984069—21no—
25#17984067C2211yescharacteristic
26#17984053C2211yescharacteristic
27#17984053C2211yescharacteristic
28#17984053C2211yescharacteristic
29D4×Dic137—21no—
30D4×Dic137C2211yescharacteristic
31#17984050C2211yescharacteristic
32#17984050C2211yescharacteristic
33#17984048—21no—
34#17984048—21no—
35#17984045—21no—
36#17984044—21no—
37C2×C4×Dic137C2211yescharacteristic
38C42×D137—21no—
39#17984021C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984011C2211yescharacteristic
45#17984011C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50C8×Dic137C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52#121×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54#107×C137C2211yescharacteristic
55#106×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555510—22no—
60#14555510—22no—
61C2×Dic274C2311yescharacteristic
62C2×Dic274C2311yescharacteristic
63C2×D548C2311yescharacteristic
64C2×C22⋊D137—21no—
65C22×Dic137—21no—
66C22×Dic137D411yescharacteristic
67C22×Dic137D411yescharacteristic
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71Dic548—22no—
72#14555499—22no—
73#14555498—22no—
74#14555495—22no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C4⋊Dic137—21no—
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88#14555487—21no—
89#14555487—21no—
90C4×Dic137—21no—
91C4×Dic137—21no—
92C4×Dic137—21no—
93C4×Dic137D411yescharacteristic
94C4×Dic137D411yescharacteristic
95C4×Dic137C2311yescharacteristic
96C4×Dic137C2311yescharacteristic
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C22×C4×C137C2311yescharacteristic
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C4⋊C4×C137C2311yescharacteristic
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C42×C137—21no—
108C22×D137—21no—
109Dic274—21no—
110Dic274—21no—
111Dic274—21no—
112Dic274#4112yes—
113D548—21no—
114D548#4112yes—
115C22⋊D137—24no—
116C2×Dic137—21no—
117C2×Dic137—22no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—41no—
121C2×Dic137—42no—
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C4×D137—24no—
127C4×D137—42no—
128C4×D137—42no—
129#11764040—21no—
130#11764040—21no—
131C23×C137—21no—
132C23×C137C2×D411yescharacteristic
133D4×C137—21no—
134D4×C137—21no—
135D4×C137—42no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137—21no—
148C22×C137—41no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137C22×D411yescharacteristic
152C2×D137—22no—
153C2×D137—41no—
154C2×D137—41no—
155C4×C137—21no—
156C4×C137—41no—
157C4×C137—41no—
158C4×C137—41no—
159C4×C137C2×#4111yescharacteristic
160C4×C137C2×#4111yescharacteristic
161Dic137—22no—
162Dic137—21no—
163Dic137—21no—
164Dic137—21no—
165Dic137—41no—
166Dic137—41no—
167Dic137#13512yes—
168C2×C137—41no—
169C2×C137—81no—
170C2×C137#52511yescharacteristic
171C2×C137C2×#13511yescharacteristic
172C2×C137#57511yescharacteristic
173D137—42no—
174C137#310211yescharacteristic, complemented, Sylow
175#3102—1371nomaximal, Sylow
176#529—1371no—
177C4×#41—1371no—
178#485—1371no—
179#482D13711yescharacteristic, complemented
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#431—1371no—
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#121C2×D13711yescharacteristic
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#117C2×D13711yescharacteristic
214C2×C42—2741no—
215C2×C42—1371no—
216#107—1371no—
217#107C2×D13711yescharacteristic
218#106—1371no—
219#106C2×D13711yescharacteristic
220C4⋊C8—1371no—
221C4⋊C8—1371no—
222#103—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#102—1371no—
227#102—1371no—
228#102C2×D13711yescharacteristic
229#102C2×D13711yescharacteristic
230C22⋊C8—1371no—
231C22⋊C8C2×D13711yescharacteristic
232C4×C8—1371no—
233C4×C8—1371no—
234#41—2742no—
235#41—2742no—
236C2×Q8—1371no—
237C2×Q8—1371no—
238C2×D4—21no—
239C2×D4—2741no—
240C2×D4—1371no—
241C2×D4C22×D13711yescharacteristic
242C22×C4—2741no—
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—1371no—
247C22×C4—1371no—
248C22×C4C22×D13711yescharacteristic
249Q16—2742no—
250SD16—2742no—
251SD16—2742no—
252D8—2742no—
253C2×C8—1371no—
254C2×C8—1371no—
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⋊C4C22×D13711yescharacteristic
265C4⋊C4C22×D13711yescharacteristic
266C4⋊C4C22×D13711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—2741no—
270C22⋊C4—2741no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C42—21no—
276C42—2741no—
277C42—2741no—
278C42—2741no—
279C42—1371no—
280C42—1371no—
281C42—1371no—
282C42—1371no—
283C23—21no—
284C23—2741no—
285C23D4×D13711yescharacteristic
286Q8—2741no—
287Q8—2741no—
288Q8—2741no—
289Q8—1372no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—2744no—
294D4—2741no—
295D4—1372no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2744no—
302C2×C4—2741no—
303C2×C4—2742no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—5482no—
307C2×C4—5482no—
308C2×C4—5481no—
309C2×C4—5482no—
310C2×C4—1371no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4D4×D13711yescharacteristic
315C2×C4C23×D13711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2741no—
319C8—2741no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2742no—
327C22—5481no—
328C22—5481no—
329C22C2×D4×D13711yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—2741no—
335C4—2741no—
336C4—2741no—
337C4—2742no—
338C4—5481no—
339C4—5481no—
340C4—1372no—
341C4#41×D13711yescharacteristic
342C4#41×D13711yescharacteristic
343C2—41no—
344C2—81no—
345C2—5482no—
346C2#2257791111yescharacteristic
347C2#135×D13711yescharacteristic
348C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#482×C137C2
socleC22×C137C22×D4
prime radical#22577546C2
hypercenter#482D137
hypocenterC137#3102
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22577934C2semidirect
2. #22577871C2semidirect
3. #22577744C2semidirect
4. #22577713C2semidirect
5. #22577706C2semidirect
6. #22577624C2semidirect
7. #22577617C2semidirect
8. #22577567C2semidirect
10. #22577497C2semidirect
11. #22577494C2semidirect
12. #22577490C2semidirect
13. #22577483C2semidirect
14. #22577453C2semidirect
15. #482×C137C2semidirect
174. C137#3102semidirect
179. #482D137semidirect

Maximal subgroups

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

subgroupindex
#225779342
#225778712
#225777442
#225777132
#225777062
#225776242
#225776172
#225775672
#225775462
#225774972
#225774942
#225774902
#225774832
#225774532
#482×C1372
#3102137