Finite Group Databasebeta

← #31791035

Subgroups — #31791035

Order 17536.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578013C211yesmaximal, characteristic, complemented
2#22577878C211yesmaximal, characteristic
3#22577745C211yesmaximal, characteristic, complemented
4#22577741C211yesmaximal, characteristic, complemented
5#22577705C211yesmaximal, characteristic, complemented
6#22577701C211yesmaximal, characteristic, complemented
7#22577606C211yesmaximal, characteristic, complemented
8#22577606C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic
10#22577504C211yesmaximal, characteristic, complemented
11#22577500C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13SD16×Dic137C211yesmaximal, characteristic, complemented
14#22577461C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic
16C2×#14555498C2211yescharacteristic
17C2×#14555498C2211yescharacteristic
18#17984086C2211yescharacteristic, complemented
19#17984081—21no—
20#17984081—21no—
21#17984079—21no—
22#17984079C2211yescharacteristic, complemented
23#17984078—21no—
24#17984067C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic, complemented
26#17984053C2211yescharacteristic, complemented
27D4×Dic137—21no—
28D4×Dic137—21no—
29D4×Dic137C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32D4×Dic137C2211yescharacteristic
33#17984047C2211yescharacteristic
34#17984047C2211yescharacteristic
35C2×C22⋊Dic137—21no—
36C2×C22⋊Dic137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41#17984017C2211yescharacteristic, complemented
42#17984017C2211yescharacteristic, complemented
43#17984013C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984012C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C22×D4×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C2×SD16×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#102×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59C2×Dic274C2311yescharacteristic
60C22×Dic137—21no—
61C22×Dic137—21no—
62C22×Dic137—21no—
63C22×Dic137—21no—
64#14555498—22no—
65#14555498—22no—
66C2×#11764040C2311yescharacteristic
67C2×#11764040C2311yescharacteristic
68C4⋊Dic137—21no—
69C4⋊Dic137—21no—
70C4⋊Dic137D411yescharacteristic
71C4⋊Dic137D411yescharacteristic
72C4⋊Dic137C2311yescharacteristic
73C4⋊Dic137C2311yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75#14555489—21no—
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80C22⋊Dic137—21no—
81C22⋊Dic137—21no—
82C22⋊Dic137—21no—
83C22⋊Dic137—21no—
84C22⋊Dic137—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137D411yescharacteristic
87C22⋊Dic137D411yescharacteristic
88C4×Dic137—21no—
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137C2311yescharacteristic
92C4×Dic137C2311yescharacteristic
93C24×C137—21no—
94C2×Q8×C137C2311yescharacteristic
95C2×D4×C137—22no—
96C2×D4×C137—22no—
97C2×D4×C137—21no—
98C2×D4×C137C2311yescharacteristic
99C2×D4×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101SD16×C137—22no—
102SD16×C137—22no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137C2311yescharacteristic
107C22⋊C4×C137—21no—
108Dic274—21no—
109Dic274—21no—
110C2×Dic137—21no—
111C2×Dic137—21no—
112C2×Dic137—21no—
113C2×Dic137—21no—
114C2×Dic137—81no—
115C2×Dic137—81no—
116C2×Dic137—81no—
117C2×Dic137—81no—
118C2×Dic137C2×D411yescharacteristic
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122#11764040—21no—
123#11764040—21no—
124C23×C137—24no—
125C23×C137—21no—
126C23×C137—21no—
127C23×C137—81no—
128C23×C137—81no—
129C23×C137C2×D411yescharacteristic
130Q8×C137—21no—
131Q8×C137—21no—
132D4×C137—24no—
133D4×C137—21no—
134D4×C137—21no—
135D4×C137#4112yes—
136D4×C137#4112yes—
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—22no—
147C22×C137—22no—
148C22×C137—21no—
149C22×C137—21no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137—84no—
155C22×C137C22×D411yescharacteristic
156C4×C137—21no—
157C4×C137—41no—
158C4×C137—41no—
159C4×C137C2×#4111yescharacteristic
160C4×C137C2×#4111yescharacteristic
161Dic137—21no—
162Dic137—21no—
163Dic137—21no—
164Dic137—21no—
165Dic137—41no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169C2×C137—41no—
170C2×C137—42no—
171C2×C137—42no—
172C2×C137#52511yescharacteristic
173C2×C137#57711yescharacteristic
174C2×C137#57711yescharacteristic
175C137#312311yescharacteristic, complemented, Sylow
176#3123—1371nomaximal, Sylow
177#541—1371no—
178#518—1371no—
179#485—1371no—
180#485—1371no—
181#478—1371no—
182#478—1371no—
183#461—1371no—
184#461—1371no—
185#450—1371no—
186#450D13711yescharacteristic
187#440—1371no—
188#440—1371no—
189C4×SD16—1371no—
190C4×SD16—1371no—
191#433—1371no—
192C22×D4C2×D13711yescharacteristic
193C2×SD16—1371no—
194C2×SD16—1371no—
195C2×SD16C2×D13711yescharacteristic
196C2×SD16C2×D13711yescharacteristic
197C4⋊Q8—1371no—
198#126—2741no—
199#126—2741no—
200#125—2741no—
201#125—2741no—
202#125—1371no—
203C22⋊Q8—1371no—
204C22⋊Q8C2×D13711yescharacteristic
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×D4—2741no—
208C4×D4—2741no—
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212C4×D4—1371no—
213#117—1371no—
214#117—1371no—
215C2×C22⋊C4—2741no—
216C2×C22⋊C4—2741no—
217#107—1371no—
218#107—1371no—
219#106—1371no—
220#106—1371no—
221C4⋊C8—1371no—
222C4⋊C8—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#102—1371no—
228#102—1371no—
229#102C2×D13711yescharacteristic
230#102C2×D13711yescharacteristic
231C22⋊C8—1371no—
232C22⋊C8C2×D13711yescharacteristic
233#97—1371no—
234C4×C8—1371no—
235C24—21no—
236C2×Q8—1371no—
237C2×Q8C22×D13711yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D13711yescharacteristic
242C2×D4C22×D13711yescharacteristic
243C22×C4—2741no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—2741no—
247C22×C4C22×D13711yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—2742no—
251SD16—2742no—
252C2×C8—1371no—
253C2×C8—1371no—
254C2×C8C22×D13711yescharacteristic
255C2×C8C22×D13711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—2741no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4—1371no—
267C4⋊C4—1371no—
268C4⋊C4—1371no—
269C4⋊C4C22×D13711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2741no—
272C22⋊C4—2741no—
273C22⋊C4—2741no—
274C22⋊C4—2741no—
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—1371no—
278C22⋊C4—1371no—
279C42—2741no—
280C42—2741no—
281C42—2741no—
282C42—1371no—
283C42—1371no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D13711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—2741no—
293Q8—2741no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1455550912yes—
298D4#1455550912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—10961no—
308C2×C4—10961no—
309C2×C4—10961no—
310C2×C4—10961no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1371no—
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2741no—
320C8—2741no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—2741no—
336C4—2741no—
337C4—2741no—
338C4—2741no—
339C4—5481no—
340C4—5481no—
341C4—5481no—
342C4—5481no—
343C4C2×#1455550911yescharacteristic
344C4C2×#1455550911yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2257790911yescharacteristic
349C2#2257820911yescharacteristic
350C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
prime radicalC22×D4×C137C22
hypercenter#450D137
hypocenterC137#3123
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22578013C2semidirect
3. #22577745C2semidirect
4. #22577741C2semidirect
5. #22577705C2semidirect
6. #22577701C2semidirect
7. #22577606C2semidirect
8. #22577606C2semidirect
10. #22577504C2semidirect
11. #22577500C2semidirect
12. #22577494C2semidirect
13. SD16×Dic137C2semidirect
14. #22577461C2semidirect
18. #17984086C22semidirect
22. #17984079C22semidirect
25. Q8×Dic137C22semidirect
26. #17984053C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. #17984017C22semidirect
42. #17984017C22semidirect
43. #17984013C22semidirect
44. #17984013C22semidirect
45. #17984012C22semidirect
46. #17984012C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
175. C137#3123semidirect

Maximal subgroups

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

subgroupindex
#225780132
#225778782
#225777452
#225777412
#225777052
#225777012
#225776062
#225775542
#225775042
#225775002
#225774942
SD16×Dic1372
#225774612
#450×C1372
#3123137