Finite Group Databasebeta

← #31793380

Subgroups — #31793380

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578116C211yesmaximal, characteristic, complemented
2#22578054C211yesmaximal, characteristic, complemented
3#22577800C211yesmaximal, characteristic, complemented
4#22577730C211yesmaximal, characteristic, complemented
5#22577727C211yesmaximal, characteristic, complemented
6#22577723C211yesmaximal, characteristic, complemented
7#22577721C211yesmaximal, characteristic, complemented
8#22577692C211yesmaximal, characteristic, complemented
9#22577682C211yesmaximal, characteristic, complemented
10#22577657C211yesmaximal, characteristic, complemented
11#22577656C211yesmaximal, characteristic, complemented
12#22577485C211yesmaximal, characteristic, complemented
13#22577479C211yesmaximal, characteristic, complemented
14#22577444C211yesmaximal, characteristic, complemented
15#428×C137C211yesmaximal, characteristic, complemented
16C2×#14555510—21no—
17C2×D1096C2211yescharacteristic
18C2×#14555493C2211yescharacteristic
19#17984087C2211yescharacteristic
20#17984087C2211yescharacteristic
21#17984076—21no—
22#17984074—21no—
23#17984073—21no—
24#17984071—21no—
25#17984066—21no—
26#17984063C2211yescharacteristic
27#17984063C2211yescharacteristic
28#17984062C2211yescharacteristic
29#17984062C2211yescharacteristic
30#17984061—21no—
31#17984050C2211yescharacteristic
32#17984050C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34#17984046C2211yescharacteristic
35#17984045—21no—
36C4⋊C4×D137C2211yescharacteristic
37#17984021C2211yescharacteristic
38#17984020C2211yescharacteristic
39#17984019C2211yescharacteristic
40#17984019C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984010C2211yescharacteristic
44#17984010C2211yescharacteristic
45#17984008C2211yescharacteristic
46#17984008C2211yescharacteristic
47#17984008C2211yescharacteristic
48#17984008C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×M16×C137C2211yescharacteristic
53#117×C137C2211yescharacteristic
54C2×C4⋊C4×C137C2211yescharacteristic
55#107×C137C2211yescharacteristic
56#107×C137C2211yescharacteristic
57#106×C137C2211yescharacteristic
58#106×C137C2211yescharacteristic
59#14555510—44no—
60C2×Dic274—21no—
61C2×Dic274—21no—
62C2×D548—21no—
63C2×D548—21no—
64C2×D548C2311yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137D411yescharacteristic
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137C2311yescharacteristic
73D1096—22no—
74#14555493—22no—
75C2×#11764040C2311yescharacteristic
76C2×#11764040C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489D411yescharacteristic
83C22⋊Dic137D411yescharacteristic
84#14555487—21no—
85#14555487—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487—21no—
89#14555487—21no—
90#14555487—21no—
91#14555487D411yescharacteristic
92C4×Dic137—21no—
93C4×Dic137C2311yescharacteristic
94C22×C4×C137—21no—
95C22×C4×C137C2311yescharacteristic
96M16×C137—22no—
97C2×C8×C137C2311yescharacteristic
98C2×C8×C137C2311yescharacteristic
99C4⋊C4×C137—21no—
100C4⋊C4×C137C2311yescharacteristic
101C4⋊C4×C137C2311yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C4⋊C4×C137C2311yescharacteristic
104C22⋊C4×C137—21no—
105C42×C137—21no—
106C22×D137—21no—
107C22×D137—21no—
108C22×D137C2×D411yescharacteristic
109Dic274—42no—
110Dic274—81no—
111D548—21no—
112D548—21no—
113D548—21no—
114D548—21no—
115D548—42no—
116D548—81no—
117C22⋊D137—22no—
118C22⋊D137—42no—
119C22⋊D137—81no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137C2×D411yescharacteristic
123C2×Dic137C2×D411yescharacteristic
124C2×Dic137C2×D411yescharacteristic
125C4×D137—21no—
126C4×D137—21no—
127C4×D137—42no—
128C4×D137—42no—
129C4×D137—42no—
130#11764040—21no—
131#11764040—21no—
132C23×C137C2×D411yescharacteristic
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—42no—
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2411yescharacteristic
142C8×C137—21no—
143C8×C137#4112yescomplemented
144C22×C137—21no—
145C22×C137—21no—
146C22×C137—21no—
147C22×C137C22×D411yescharacteristic
148C2×D137—21no—
149C2×D137—21no—
150C2×D137—21no—
151C2×D137—41no—
152C2×D137—41no—
153C2×D137—41no—
154C2×D137—41no—
155C2×D137—41no—
156C2×D137—41no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137C2×#4111yescharacteristic
163C4×C1372−1+411yescharacteristic
164Dic137—21no—
165Dic137—21no—
166Dic137—41no—
167Dic137—41no—
168Dic137AGL(1,ℤ/8)12yes—
169C2×C137—41no—
170C2×C137#54711yescharacteristic
171C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C137#57711yescharacteristic
173D137—41no—
174D137—81no—
175D137—81no—
176C137#335411yescharacteristic, complemented, Sylow
177#3354—1371nomaximal, Sylow
178#555—1371no—
179#547—1371no—
180#498—1371no—
181#482—1371no—
182#482—1371no—
183#481—1371no—
184#481—1371no—
185#476—1371no—
186#474—1371no—
187#469—1371no—
188#469—1371no—
189C4×D8—1371no—
190#436—1371no—
191#428—1371no—
192#428D13711yescharacteristic, complemented
193C2×#41—2741no—
194C2×D8—1371no—
195C2×M16—1371no—
196C2×M16C2×D13711yescharacteristic
197C4⋊Q8—1371no—
198C4⋊Q8—1371no—
199#124—2741no—
200#124—2741no—
201#123—2741no—
202#123—2741no—
203C22⋊Q8—2741no—
204#121—2741no—
205#121—1371no—
206#121—1371no—
207#121—1371no—
208#121—1371no—
209C4×D4—1371no—
210C4×D4—1371no—
211C4×D4—1371no—
212#117—2741no—
213#117—1371no—
214#117C2×D13711yescharacteristic
215C2×C4⋊C4—1371no—
216C2×C4⋊C4C2×D13711yescharacteristic
217#107—1371no—
218#107C2×D13711yescharacteristic
219#107C2×D13711yescharacteristic
220#106—1371no—
221#106—1371no—
222#106—1371no—
223#106—1371no—
224#106C2×D13711yescharacteristic
225#106C2×D13711yescharacteristic
226C4⋊C8—1371no—
227#102—1371no—
228#102—1371no—
229#102—1371no—
230#102—1371no—
231#102—1371no—
232#102—1371no—
233C22⋊C8—1371no—
234C22⋊C8—1371no—
235C4×C8—1371no—
236#41—5484no—
237C2×Q8—2741no—
238C2×Q8—2741no—
239C2×D4—2741no—
240C2×D4—2741no—
241C2×D4—2741no—
242C2×D4—2741no—
243C2×D4—1371no—
244C2×D4—1371no—
245C2×D4—1371no—
246C22×C4—21no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—1371no—
251C22×C4C22×D13711yescharacteristic
252D8—2742no—
253M16—22no—
254M16—2742no—
255C2×C8—1371no—
256C2×C8—1371no—
257C2×C8C22×D13711yescharacteristic
258C2×C8C22×D13711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—2741no—
261C4⋊C4—2741no—
262C4⋊C4—2741no—
263C4⋊C4—1371no—
264C4⋊C4—1371no—
265C4⋊C4—1371no—
266C4⋊C4C22×D13711yescharacteristic
267C4⋊C4C22×D13711yescharacteristic
268C4⋊C4C22×D13711yescharacteristic
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—2741no—
278C22⋊C4—1371no—
279C22⋊C4—1371no—
280C42—21no—
281C42—2741no—
282C42—1371no—
283C23—2741no—
284C23—2741no—
285C23—1371no—
286C23D4×D13711yescharacteristic
287Q8—5482no—
288Q8—10961no—
289D4—2742no—
290D4—2741no—
291D4—2741no—
292D4—2741no—
293D4—2741no—
294D4—5482no—
295D4—5482no—
296D4—10961no—
297D4—10961no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—2741no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—5482no—
310C2×C4—5482no—
311C2×C4—5482no—
312C2×C4—1371no—
313C2×C4—1371no—
314C2×C4—1371no—
315C2×C4D4×D13711yescharacteristic
316C2×C4C23×D13711yescharacteristic
317C8—21no—
318C8—2741no—
319C8—2741no—
320C8#1455551012yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—2741no—
325C22—2741no—
326C22—2741no—
327C22—5481no—
328C22—5481no—
329C22—5481no—
330C22—5481no—
331C22—5481no—
332C22—5481no—
333C22C2×D4×D13711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2741no—
340C4—2741no—
341C4—5481no—
342C4—5481no—
343C4—1372no—
344C4C2×#1455551011yescharacteristic
345C4#1798413611yescharacteristic
346C2—41no—
347C2—5481no—
348C2—10961no—
349C2—10961no—
350C2#2257805411yescharacteristic
351C2AGL(1,ℤ/8)×D13711yescharacteristic
352C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
hypercenter#428D137
hypocenterC137#3354
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #22578116C2semidirect
2. #22578054C2semidirect
3. #22577800C2semidirect
4. #22577730C2semidirect
5. #22577727C2semidirect
6. #22577723C2semidirect
7. #22577721C2semidirect
8. #22577692C2semidirect
9. #22577682C2semidirect
10. #22577657C2semidirect
11. #22577656C2semidirect
12. #22577485C2semidirect
13. #22577479C2semidirect
14. #22577444C2semidirect
15. #428×C137C2semidirect
143. C8×C137#41semidirect
176. C137#3354semidirect
192. #428D137semidirect
320. C8#14555510semidirect

Maximal subgroups

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

subgroupindex
#225781162
#225780542
#225778002
#225777302
#225777272
#225777232
#225777212
#225776922
#225776822
#225776572
#225776562
#225774852
#225774792
#225774442
#428×C1372
#3354137