Finite Group Databasebeta

← #31792680

Subgroups — #31792680

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578095C211yesmaximal, characteristic, complemented
2#22578064C211yesmaximal, characteristic
3#22577759C211yesmaximal, characteristic, complemented
4#22577743C211yesmaximal, characteristic, complemented
5#22577717C211yesmaximal, characteristic, complemented
6#22577712C211yesmaximal, characteristic, complemented
7#22577641C211yesmaximal, characteristic, complemented
8#22577634C211yesmaximal, characteristic, complemented
9#22577627C211yesmaximal, characteristic, complemented
10#22577564C211yesmaximal, characteristic, complemented
11Q16×Dic137C211yesmaximal, characteristic, complemented
12#22577466C211yesmaximal, characteristic, complemented
13#22577398C211yesmaximal, characteristic, complemented
14C2×#17984012C211yesmaximal, characteristic, complemented
15#451×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×#14555497C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19C2×C8×D137C2211yescharacteristic
20#17984083—21no—
21#17984079—21no—
22#17984079C2211yescharacteristic
23#17984077—21no—
24#17984077C2211yescharacteristic
25#17984072—21no—
26#17984064—21no—
27#17984064C2211yescharacteristic
28#17984064C2211yescharacteristic
29#17984062—21no—
30#17984062C2211yescharacteristic
31Q8×Dic137C2211yescharacteristic
32Q8×Dic137C2211yescharacteristic
33C4×C22⋊D137—21no—
34C4×C22⋊D137C2211yescharacteristic
35#17984046C2211yescharacteristic
36#17984044—21no—
37C2×C4⋊Dic137C2211yescharacteristic
38#17984022C2211yescharacteristic
39#17984022C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984017C2211yescharacteristic
42#17984012C2211yescharacteristic
43#17984012C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984010C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C22×Q8×C137C2211yescharacteristic
53C2×Q16×C137C2211yescharacteristic
54C2×Q16×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58C22⋊C8×C137C2211yescharacteristic
59#14555511—22no—
60#14555511—41no—
61C2×D548—21no—
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137D411yescharacteristic
65C22×Dic137—21no—
66C2×C4×D137—21no—
67C2×C4×D137C2311yescharacteristic
68#14555497—41no—
69C2×#11764040—41no—
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C8×D137—41no—
73C4⋊Dic137—21no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137—21no—
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79C4⋊Dic137C2311yescharacteristic
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84C22⋊Dic137D411yescharacteristic
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487D411yescharacteristic
92C4×Dic137—21no—
93C4×Dic137—21no—
94C4×Dic137—21no—
95C4×Dic137C2311yescharacteristic
96C2×Q8×C137—21no—
97C2×Q8×C137—22no—
98C2×Q8×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×Q8×C137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104Q16×C137—22no—
105Q16×C137—41no—
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C22×D137—21no—
112C22×D137C2×D411yescharacteristic
113D548—21no—
114D548—21no—
115D548—42no—
116C22⋊D137—22no—
117C22⋊D137—42no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—42no—
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—21no—
126C4×D137—21no—
127C4×D137—42no—
128#11764040—21no—
129#11764040—21no—
130C23×C137C2×D411yescharacteristic
131Q8×C137—21no—
132Q8×C137—21no—
133Q8×C137—21no—
134Q8×C137—21no—
135Q8×C137—21no—
136Q8×C137—42no—
137Q8×C137#4112yes—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—42no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137#13512yes—
152C22×C137C22×D411yescharacteristic
153C2×D137—21no—
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—41no—
157C2×D137—41no—
158C2×D137—41no—
159C4×C137—21no—
160C4×C137—22no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137—41no—
164C4×C137C2×#4111yescharacteristic
165C4×C137H(5,𝔽2)11yescharacteristic
166Dic137—21no—
167Dic137—21no—
168Dic137—21no—
169Dic137—41no—
170Dic137—41no—
171Dic137—41no—
172C2×C137—22no—
173C2×C137#54611yescharacteristic
174C2×C137C2×#13511yescharacteristic
175C2×C137#57811yescharacteristic
176D137—41no—
177D137—81no—
178C137#330011yescharacteristic, complemented, Sylow
179#3300—1371nomaximal, Sylow
180#552—1371no—
181#548—1371no—
182#488—1371no—
183#485—1371no—
184#480—1371no—
185#479—1371no—
186#467—1371no—
187#465—1371no—
188#464—1371no—
189#452—1371no—
190#451D13711yescharacteristic, complemented
191C4×Q16—1371no—
192C8×D4—1371no—
193#416—1371no—
194C2×#103—1371no—
195C2×#41—1371no—
196C22×Q8C2×D13711yescharacteristic
197C2×Q16C2×D13711yescharacteristic
198C2×Q16C2×D13711yescharacteristic
199C2×SD16—1371no—
200C22×C8—1371no—
201C22×C8—1371no—
202#126—2741no—
203#125—2741no—
204#125—1371no—
205#124—2741no—
206#124—1371no—
207#123—2741no—
208C22⋊Q8—2741no—
209C22⋊Q8—1371no—
210C22⋊Q8—1371no—
211C22⋊Q8C2×D13711yescharacteristic
212#121—2741no—
213#121—1371no—
214C4×Q8—1371no—
215C4×Q8—1371no—
216C4×D4—2741no—
217C4×D4—1371no—
218#117—2741no—
219#117—1371no—
220C2×C4⋊C4—1371no—
221#107—1371no—
222#107—1371no—
223#106—1371no—
224C4⋊C8—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103C2×D13711yescharacteristic
231#103C2×D13711yescharacteristic
232#102—1371no—
233#102—1371no—
234#102—1371no—
235C22⋊C8—1371no—
236C22⋊C8C2×D13711yescharacteristic
237C4×C8—1371no—
238#41—2742no—
239#41—5481no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D13711yescharacteristic
244C2×Q8C22×D13711yescharacteristic
245C2×Q8C22×D13711yescharacteristic
246C2×D4—2741no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4—1371no—
250C22×C4—21no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—1371no—
254C22×C4C22×D13711yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—5481no—
258C2×C8—5481no—
259C2×C8—5481no—
260C2×C8—1371no—
261C2×C8—1371no—
262C2×C8C22×D13711yescharacteristic
263C2×C8C22×D13711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—1371no—
285C22⋊C4—1371no—
286C42—2741no—
287C42—2741no—
288C42—2741no—
289C42—1371no—
290C23—2741no—
291C23—1371no—
292C23D4×D13711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1455550912yes—
300D4—2742no—
301D4—2741no—
302D4—2741no—
303D4—5482no—
304D4—5482no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—5482no—
318C2×C4—5482no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4D4×D13711yescharacteristic
323C2×C4C23×D13711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2741no—
327C8—2741no—
328C22—21no—
329C22—21no—
330C22—2741no—
331C22—2741no—
332C22—2741no—
333C22—5481no—
334C22—5481no—
335C22—5481no—
336C22#1798411112yes—
337C22C2×D4×D13711yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2741no—
344C4—2741no—
345C4—2741no—
346C4—5481no—
347C4—5481no—
348C4—5481no—
349C4C2×#1455550911yescharacteristic
350C4#1798413311yescharacteristic
351C2—22no—
352C2—5481no—
353C2—10961no—
354C2#2257804011yescharacteristic
355C2C2×#1798411111yescharacteristic
356C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#451×C137C2
socleC22×C137C22×D4
prime radical#22578064C2
hypercenter#451D137
hypocenterC137#3300
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #22578095C2semidirect
3. #22577759C2semidirect
4. #22577743C2semidirect
5. #22577717C2semidirect
6. #22577712C2semidirect
7. #22577641C2semidirect
8. #22577634C2semidirect
9. #22577627C2semidirect
10. #22577564C2semidirect
11. Q16×Dic137C2semidirect
12. #22577466C2semidirect
13. #22577398C2semidirect
14. C2×#17984012C2semidirect
15. #451×C137C2semidirect
178. C137#3300semidirect
190. #451D137semidirect

Maximal subgroups

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

subgroupindex
#225780952
#225780642
#225777592
#225777432
#225777172
#225777122
#225776412
#225776342
#225776272
#225775642
Q16×Dic1372
#225774662
#225773982
C2×#179840122
#451×C1372
#3300137