Finite Group Databasebeta

← #31791039

Subgroups — #31791039

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#22578014C211yesmaximal, characteristic, complemented
2#22577875C211yesmaximal, characteristic
3#22577743C211yesmaximal, characteristic, complemented
4#22577742C211yesmaximal, characteristic, complemented
5#22577704C211yesmaximal, characteristic, complemented
6#22577703C211yesmaximal, characteristic, complemented
7#22577605C211yesmaximal, characteristic, complemented
8#22577552C211yesmaximal, characteristic
9#22577552C211yesmaximal, characteristic
10#22577506C211yesmaximal, characteristic, complemented
11#22577504C211yesmaximal, characteristic, complemented
12SD16×Dic137C211yesmaximal, characteristic, complemented
13#22577488C211yesmaximal, characteristic, complemented
14#22577463C211yesmaximal, characteristic, complemented
15#461×C137C211yesmaximal, characteristic, complemented
16C2×D4×D137C2211yescharacteristic
17C2×#14555497C2211yescharacteristic
18C2×#14555497C2211yescharacteristic
19#17984083—21no—
20#17984083—21no—
21#17984080C2211yescharacteristic, complemented
22#17984078—21no—
23#17984078—21no—
24#17984064C2211yescharacteristic
25#17984064C2211yescharacteristic
26Q8×Dic137C2211yescharacteristic, complemented
27Q8×Dic137C2211yescharacteristic, complemented
28C4×D548C2211yescharacteristic
29D4×Dic137C2211yescharacteristic
30#17984050C2211yescharacteristic
31C4×C22⋊D137—21no—
32#17984048—21no—
33#17984046C2211yescharacteristic
34#17984045C2211yescharacteristic
35C2×#14555487—21no—
36C22⋊C4×D137—21no—
37#17984022C2211yescharacteristic, complemented
38#17984021C2211yescharacteristic, complemented
39#17984020C2211yescharacteristic, complemented
40#17984019C2211yescharacteristic, complemented
41C4⋊#11764040C2211yescharacteristic, complemented
42#17984012C2211yescharacteristic, complemented
43#17984012C2211yescharacteristic, complemented
44#17984010C2211yescharacteristic
45#17984010C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C2×SD16×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C4⋊Q8×C137C2211yescharacteristic, complemented
55C4×D4×C137C2211yescharacteristic
56C4⋊C8×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58#103×C137C2211yescharacteristic, complemented
59C23×D137—21no—
60C2×D548C2311yescharacteristic
61C2×C22⋊D137—21no—
62D4×D137—22no—
63D4×D137—22no—
64C22×Dic137—21no—
65C2×C4×D137—21no—
66C2×C4×D137—21no—
67C2×C4×D137C2311yescharacteristic
68#14555497—22no—
69#14555497—22no—
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C4⋊Dic137—21no—
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489D411yescharacteristic
82#14555489D411yescharacteristic
83C22⋊Dic137—21no—
84#14555487—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487D411yescharacteristic
90#14555487D411yescharacteristic
91C4×Dic137—21no—
92C4×Dic137—21no—
93C4×Dic137—21no—
94C4×Dic137C2311yescharacteristic
95C2×Q8×C137C2311yescharacteristic
96C2×Q8×C137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98C22×C4×C137—21no—
99SD16×C137—22no—
100SD16×C137—22no—
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C4⋊C4×C137—21no—
104C4⋊C4×C137—21no—
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—24no—
110C22×D137—81no—
111C22×D137—81no—
112C22×D137C2×D411yescharacteristic
113D548—21no—
114D548#4112yes—
115C22⋊D137—24no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—81no—
120C2×Dic137C2×D411yescharacteristic
121C2×Dic137C2×D411yescharacteristic
122C2×Dic137C2×D411yescharacteristic
123C4×D137—21no—
124C4×D137—21no—
125C4×D137—81no—
126C4×D137—81no—
127#11764040—21no—
128#11764040—21no—
129C23×C137—21no—
130Q8×C137—21no—
131Q8×C137—21no—
132Q8×C137—21no—
133Q8×C137—21no—
134D4×C137—21no—
135D4×C137#4112yes—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—81no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2×D411yescharacteristic
142C2×C4×C137C2411yescharacteristic
143C8×C137—21no—
144C8×C137—21no—
145C22×C137—22no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137C22×D411yescharacteristic
149C2×D137—21no—
150C2×D137—22no—
151C2×D137—21no—
152C2×D137—21no—
153C2×D137—41no—
154C2×D137—41no—
155C2×D137—84no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137C2×#4111yescharacteristic
162C4×C137C2×#4111yescharacteristic
163Dic137—21no—
164Dic137—21no—
165Dic137—21no—
166Dic137—41no—
167Dic137—41no—
168Dic137—41no—
169C2×C137—42no—
170C2×C137#52511yescharacteristic
171C2×C137#57711yescharacteristic
172C2×C137#57711yescharacteristic
173D137—41no—
174D137—42no—
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#461D13711yescharacteristic, complemented
185#450—1371no—
186#450—1371no—
187#440—1371no—
188#440—1371no—
189C4×SD16—1371no—
190C4×SD16—1371no—
191#433—1371no—
192C22×D4—1371no—
193C2×SD16—1371no—
194C2×SD16—1371no—
195C2×SD16C2×D13711yescharacteristic
196C2×SD16C2×D13711yescharacteristic
197C4⋊Q8C2×D13711yescharacteristic, complemented
198#126—2741no—
199#126—2741no—
200#125—2741no—
201#125—2741no—
202#125—1371no—
203C22⋊Q8—1371no—
204C22⋊Q8—1371no—
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×D4—2741no—
208C4×D4—2741no—
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212C4×D4C2×D13711yescharacteristic
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⋊C8C2×D13711yescharacteristic, complemented
223#103—1371no—
224#103—1371no—
225#103C2×D13711yescharacteristic, complemented
226#103C2×D13711yescharacteristic, complemented
227#102—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231C22⋊C8—1371no—
232C22⋊C8—1371no—
233#97—1371no—
234C4×C8—1371no—
235C24—2741no—
236C2×Q8C22×D13711yescharacteristic
237C2×Q8C22×D13711yescharacteristic
238C2×D4—2742no—
239C2×D4—2742no—
240C2×D4—2741no—
241C2×D4—1371no—
242C2×D4C22×D13711yescharacteristic
243C22×C4—21no—
244C22×C4—2741no—
245C22×C4—2741no—
246C22×C4—2741no—
247C22×C4—1371no—
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—21no—
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—
283C42C22×D13711yescharacteristic
284C23—21no—
285C23—2744no—
286C23—2741no—
287C23—10961no—
288C23—10961no—
289C23—1371no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—2744no—
296D4—2741no—
297D4—1372no—
298D4#1455550912yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—2741no—
304C2×C4—2741no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—10961no—
309C2×C4—10961no—
310C2×C4—10961no—
311C2×C4—1371no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4D4×D13711yescharacteristic
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2741no—
320C8—2741no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—2741no—
325C22—2741no—
326C22—2741no—
327C22—2742no—
328C22—5481no—
329C22—5481no—
330C22—10964no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2741no—
338C4—2741no—
339C4—2741no—
340C4—5481no—
341C4—5481no—
342C4—5481no—
343C4#41×D13711yescharacteristic
344C4C2×#1455550911yescharacteristic
345C2—42no—
346C2—5481no—
347C2—5482no—
348C2#2257791311yescharacteristic
349C2#2257820911yescharacteristic
350C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#461×C137C2
socleC22×C137C22×D4
prime radicalC2×D4×D137C22
hypercenter#461D137
hypocenterC137#3123
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22578014C2semidirect
3. #22577743C2semidirect
4. #22577742C2semidirect
5. #22577704C2semidirect
6. #22577703C2semidirect
7. #22577605C2semidirect
10. #22577506C2semidirect
11. #22577504C2semidirect
12. SD16×Dic137C2semidirect
13. #22577488C2semidirect
14. #22577463C2semidirect
15. #461×C137C2semidirect
21. #17984080C22semidirect
26. Q8×Dic137C22semidirect
27. Q8×Dic137C22semidirect
37. #17984022C22semidirect
38. #17984021C22semidirect
39. #17984020C22semidirect
40. #17984019C22semidirect
41. C4⋊#11764040C22semidirect
42. #17984012C22semidirect
43. #17984012C22semidirect
50. #17983998C22semidirect
51. C8×Dic137C22semidirect
54. C4⋊Q8×C137C22semidirect
56. C4⋊C8×C137C22semidirect
57. #103×C137C22semidirect
58. #103×C137C22semidirect
175. C137#3123semidirect
184. #461D137semidirect
197. C4⋊Q8C2×D137semidirect
222. C4⋊C8C2×D137semidirect
225. #103C2×D137semidirect
226. #103C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225780142
#225778752
#225777432
#225777422
#225777042
#225777032
#225776052
#225775522
#225775062
#225775042
SD16×Dic1372
#225774882
#225774632
#461×C1372
#3123137