Finite Group Databasebeta

← #31790988

Subgroups — #31790988

Order 17536.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578008C211yesmaximal, characteristic, complemented
2#22577721C211yesmaximal, characteristic, complemented
3#22577716C211yesmaximal, characteristic, complemented
4#22577710C211yesmaximal, characteristic, complemented
5#22577703C211yesmaximal, characteristic, complemented
6#22577603C211yesmaximal, characteristic, complemented
7#22577598C211yesmaximal, characteristic, complemented
8#22577548C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577513C211yesmaximal, characteristic, complemented
11#22577501C211yesmaximal, characteristic, complemented
12C4×#14555497C211yesmaximal, characteristic, complemented
13C4×#14555495C211yesmaximal, characteristic, complemented
14#22577464C211yesmaximal, characteristic, complemented
15#520×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic
17C2×#14555497C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19C2×#14555495C2211yescharacteristic
20#17984085C2211yescharacteristic
21#17984080C2211yescharacteristic
22#17984073—21no—
23#17984073—21no—
24#17984062C2211yescharacteristic
25#17984062C2211yescharacteristic
26#17984061—21no—
27#17984061—21no—
28C4×D548C2211yescharacteristic
29C4×D548C2211yescharacteristic
30#17984021C2211yescharacteristic
31#17984021C2211yescharacteristic
32#17984019C2211yescharacteristic
33#17984019C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic
35C4⋊#11764040C2211yescharacteristic
36#17984012C2211yescharacteristic
37#17984012C2211yescharacteristic
38#17984009C2211yescharacteristic
39#17984009C2211yescharacteristic
40#17984008C2211yescharacteristic
41#17984008C2211yescharacteristic
42#17984008C2211yescharacteristic
43#17984008C2211yescharacteristic
44C22⋊#11764040C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46#17983999C2211yescharacteristic
47C4×#11764040C2211yescharacteristic
48C2×#41×C137C2211yescharacteristic
49C4×Q8×C137C2211yescharacteristic
50C4×Q8×C137C2211yescharacteristic
51C4×D4×C137—21no—
52C4×D4×C137—21no—
53C4×D4×C137C2211yescharacteristic
54C4×D4×C137C2211yescharacteristic
55#117×C137—21no—
56#117×C137C2211yescharacteristic
57C2×C4⋊C4×C137—21no—
58C2×C4⋊C4×C137C2211yescharacteristic
59C2×D548—21no—
60C2×D548—21no—
61C2×D548C2311yescharacteristic
62C2×D548C2311yescharacteristic
63C2×C22⋊D137—21no—
64C2×C22⋊D137—21no—
65C2×C4×D137—21no—
66C2×C4×D137—21no—
67#14555497—22no—
68#14555497—22no—
69#14555495—22no—
70#14555495—22no—
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C2×#11764040C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79#14555487—21no—
80#14555487—21no—
81#14555487—21no—
82#14555487—21no—
83#41×C137—22no—
84#41×C137—22no—
85C2×Q8×C137C2311yescharacteristic
86C2×D4×C137—21no—
87C2×D4×C137C2311yescharacteristic
88C22×C4×C137—21no—
89C22×C4×C137—21no—
90C22×C4×C137—21no—
91C22×C4×C137—21no—
92C22×C4×C137C2311yescharacteristic
93C4⋊C4×C137—21no—
94C4⋊C4×C137—21no—
95C4⋊C4×C137—21no—
96C4⋊C4×C137D411yescharacteristic
97C4⋊C4×C137D411yescharacteristic
98C4⋊C4×C137C2311yescharacteristic
99C4⋊C4×C137C2311yescharacteristic
100C22⋊C4×C137—21no—
101C22⋊C4×C137—21no—
102C22⋊C4×C137D411yescharacteristic
103C22⋊C4×C137D411yescharacteristic
104C42×C137—21no—
105C42×C137—21no—
106C42×C137C2311yescharacteristic
107C42×C137C2311yescharacteristic
108C22×D137—21no—
109C22×D137—21no—
110D548—21no—
111D548—21no—
112D548—21no—
113D548—21no—
114D548—81no—
115D548—81no—
116C22⋊D137—81no—
117C22⋊D137—81no—
118C2×Dic137—21no—
119C2×Dic137—21no—
120C4×D137—81no—
121C4×D137—81no—
122#11764040—21no—
123#11764040—21no—
124#11764040—21no—
125#11764040—21no—
126C23×C137—21no—
127C23×C137C2×D411yescharacteristic
128Q8×C137—21no—
129Q8×C137#4112yes—
130D4×C137—24no—
131D4×C137—21no—
132D4×C137#4112yes—
133C2×C4×C137—24no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—81no—
140C2×C4×C137—81no—
141C2×C4×C137—81no—
142C2×C4×C137C2×D411yescharacteristic
143C2×C4×C137C2×D411yescharacteristic
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C22×C137—21no—
149C22×C137—22no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—41no—
153C22×C137—41no—
154C22×C137C22×D411yescharacteristic
155C2×D137—41no—
156C2×D137—41no—
157C2×D137—41no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—41no—
161C4×C137—21no—
162C4×C137—22no—
163C4×C137—21no—
164C4×C137—21no—
165C4×C137—21no—
166C4×C137—21no—
167C4×C137—41no—
168C4×C137—41no—
169C4×C137C2×#4111yescharacteristic
170C4×C137C2×#4111yescharacteristic
171Dic137—41no—
172Dic137—41no—
173C2×C137—41no—
174C2×C137—42no—
175C2×C137#52511yescharacteristic
176C2×C137#57611yescharacteristic
177C2×C137#57711yescharacteristic
178D137—81no—
179D137—81no—
180C137#311911yescharacteristic, complemented, Sylow
181#3119—1371nomaximal, Sylow
182#540—1371no—
183#520D13711yescharacteristic, complemented
184#481—1371no—
185#480—1371no—
186#479—1371no—
187#478—1371no—
188#460—1371no—
189#459—1371no—
190#449—1371no—
191#449—1371no—
192#442—1371no—
193#440—1371no—
194C4×SD16—1371no—
195C4×D8—1371no—
196#433—1371no—
197C2×#41C2×D13711yescharacteristic
198C2×SD16—1371no—
199C2×SD16—1371no—
200C2×D8—1371no—
201C2×D8—1371no—
202#127—1371no—
203#125—1371no—
204#123—2741no—
205#123—2741no—
206#121—2741no—
207#121—2741no—
208#121—1371no—
209#121—1371no—
210C4×Q8C2×D13711yescharacteristic
211C4×Q8C2×D13711yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1371no—
215C4×D4—1371no—
216C4×D4C2×D13711yescharacteristic
217C4×D4C2×D13711yescharacteristic
218#117—21no—
219#117C2×D13711yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D13711yescharacteristic
222#107—1371no—
223#107—1371no—
224#106—1371no—
225#106—1371no—
226C4⋊C8—1371no—
227C4⋊C8—1371no—
228#103—1371no—
229#103—1371no—
230#102—1371no—
231#102—1371no—
232#102—1371no—
233#102—1371no—
234#102—1371no—
235#102—1371no—
236C22⋊C8—1371no—
237C22⋊C8—1371no—
238#97—1371no—
239C4×C8—1371no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D13711yescharacteristic
243C2×D4—21no—
244C2×D4—2741no—
245C2×D4—2741no—
246C2×D4—2741no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4—1371no—
250C2×D4C22×D13711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—2741no—
256C22×C4—2741no—
257C22×C4C22×D13711yescharacteristic
258SD16—2742no—
259SD16—2742no—
260D8—2742no—
261D8—2742no—
262C2×C8—1371no—
263C2×C8—1371no—
264C2×C8—1371no—
265C2×C8—1371no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4C22⋊D13711yescharacteristic
274C4⋊C4C22⋊D13711yescharacteristic
275C4⋊C4C22×D13711yescharacteristic
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4C22⋊D13711yescharacteristic
284C22⋊C4C22⋊D13711yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D13711yescharacteristic
288C42C22×D13711yescharacteristic
289C23—21no—
290C23—2741no—
291C23—2741no—
292C23C2×C22⋊D13711yescharacteristic
293Q8—21no—
294Q8#1455551012yes—
295D4—24no—
296D4—21no—
297D4—2741no—
298D4—2741no—
299D4—2741no—
300D4—2741no—
301D4—10961no—
302D4—10961no—
303D4—10961no—
304D4—10961no—
305D4#1455551012yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—10961no—
318C2×C4—10961no—
319C2×C4C2×C22⋊D13711yescharacteristic
320C2×C4C2×C22⋊D13711yescharacteristic
321C2×C4C2×C22⋊D13711yescharacteristic
322C2×C4C2×C22⋊D13711yescharacteristic
323C2×C4C2×C22⋊D13711yescharacteristic
324C2×C4C23×D13711yescharacteristic
325C8—2741no—
326C8—2741no—
327C8—2741no—
328C8—2741no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—5481no—
336C22—5481no—
337C22—5481no—
338C22—5481no—
339C22—5481no—
340C22—5481no—
341C22C22×C22⋊D13711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—41no—
349C4—41no—
350C4—5481no—
351C4—5481no—
352C4C2×#1455551011yescharacteristic
353C4C2×#1455551011yescharacteristic
354C2—41no—
355C2—42no—
356C2—10961no—
357C2—10961no—
358C2#2257791411yescharacteristic
359C2#2257820611yescharacteristic
360C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#520×C137C2
socleC22×C137C22×D4
hypercenter#520D137
hypocenterC137#3119
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #22578008C2semidirect
2. #22577721C2semidirect
3. #22577716C2semidirect
4. #22577710C2semidirect
5. #22577703C2semidirect
6. #22577603C2semidirect
7. #22577598C2semidirect
8. #22577548C2semidirect
9. #22577548C2semidirect
10. #22577513C2semidirect
11. #22577501C2semidirect
12. C4×#14555497C2semidirect
13. C4×#14555495C2semidirect
14. #22577464C2semidirect
15. #520×C137C2semidirect
180. C137#3119semidirect
183. #520D137semidirect

Maximal subgroups

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

subgroupindex
#225780082
#225777212
#225777162
#225777102
#225777032
#225776032
#225775982
#225775482
#225775132
#225775012
C4×#145554972
C4×#145554952
#225774642
#520×C1372
#3119137