Finite Group Databasebeta

← #31792691

Subgroups — #31792691

Order 17536.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578099C211yesmaximal, characteristic, complemented
2#22577757C211yesmaximal, characteristic, complemented
3#22577741C211yesmaximal, characteristic, complemented
4#22577719C211yesmaximal, characteristic, complemented
5#22577710C211yesmaximal, characteristic, complemented
6#22577637C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577615C211yesmaximal, characteristic, complemented
9#22577548C211yesmaximal, characteristic, complemented
10#22577542C211yesmaximal, characteristic, complemented
11C4×#14555497C211yesmaximal, characteristic, complemented
12D4×#11764040C211yesmaximal, characteristic, complemented
13#22577399C211yesmaximal, characteristic, complemented
14C2×#17984012C211yesmaximal, characteristic, complemented
15#548×C137C211yesmaximal, characteristic, complemented
16C2×#14555497C2211yescharacteristic
17C2×#14555497C2211yescharacteristic
18C2×#14555495C2211yescharacteristic
19C22×#11764040C2211yescharacteristic
20C22×#11764040C2211yescharacteristic
21#17984083—21no—
22#17984080C2211yescharacteristic
23#17984070—21no—
24#17984062C2211yescharacteristic
25#17984062C2211yescharacteristic
26#17984060—21no—
27C4×D548C2211yescharacteristic
28#17984048—21no—
29#17984047C2211yescharacteristic
30C2×C4⋊Dic137C2211yescharacteristic
31#17984021C2211yescharacteristic
32#17984019C2211yescharacteristic
33#17984019C2211yescharacteristic
34C4⋊#11764040C2211yescharacteristic
35#17984012C2211yescharacteristic
36#17984012C2211yescharacteristic
37#17984012C2211yescharacteristic
38#17984012C2211yescharacteristic
39#17984012C2211yescharacteristic
40#17984012C2211yescharacteristic
41#17984009C2211yescharacteristic
42#17984009C2211yescharacteristic
43#17984009C2211yescharacteristic
44#17984008C2211yescharacteristic
45C22⋊#11764040C2211yescharacteristic
46C22⋊#11764040C2211yescharacteristic
47C4×#11764040C2211yescharacteristic
48C2×#41×C137C2211yescharacteristic
49C22×Q8×C137C2211yescharacteristic
50#124×C137—21no—
51#124×C137C2211yescharacteristic
52C22⋊Q8×C137—21no—
53C22⋊Q8×C137C2211yescharacteristic
54#121×C137—21no—
55#121×C137C2211yescharacteristic
56C4×Q8×C137C2211yescharacteristic
57C4×D4×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C2×D548C2311yescharacteristic
60C2×C22⋊D137—21no—
61C2×C22⋊D137—21no—
62C22×Dic137—21no—
63C2×C4×D137—21no—
64#14555497—22no—
65#14555497—41no—
66#14555495—41no—
67C2×#11764040—41no—
68C2×#11764040—41no—
69C2×#11764040C2311yescharacteristic
70C2×#11764040C2311yescharacteristic
71C2×#11764040C2311yescharacteristic
72C2×#11764040C2311yescharacteristic
73C4⋊Dic137—21no—
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76C4⋊Dic137C2311yescharacteristic
77#14555489—21no—
78#14555489—21no—
79C22⋊Dic137—21no—
80#14555487—21no—
81#14555487—21no—
82#14555487—21no—
83C4×Dic137—21no—
84#41×C137—22no—
85#41×C137—41no—
86C2×Q8×C137—21no—
87C2×Q8×C137—22no—
88C2×Q8×C137—41no—
89C2×Q8×C137C2311yescharacteristic
90C2×Q8×C137C2311yescharacteristic
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137D411yescharacteristic
94C2×D4×C137C2311yescharacteristic
95C22×C4×C137—21no—
96C22×C4×C137—21no—
97C22×C4×C137C2311yescharacteristic
98C22×C4×C137C2311yescharacteristic
99C4⋊C4×C137—21no—
100C4⋊C4×C137—21no—
101C4⋊C4×C137D411yescharacteristic
102C4⋊C4×C137C2311yescharacteristic
103C22⋊C4×C137—21no—
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C22⋊C4×C137—21no—
107C22⋊C4×C137D411yescharacteristic
108C22⋊C4×C137D411yescharacteristic
109C42×C137—21no—
110C42×C137C2311yescharacteristic
111C22×D137—21no—
112D548—21no—
113D548—21no—
114C22⋊D137—42no—
115C22⋊D137—81no—
116C2×Dic137—21no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—81no—
120C4×D137—81no—
121#11764040—21no—
122#11764040—21no—
123#11764040—21no—
124#11764040—21no—
125C23×C137—21no—
126C23×C137C2×D411yescharacteristic
127C23×C137C2×D411yescharacteristic
128Q8×C137—21no—
129Q8×C137—21no—
130Q8×C137—21no—
131Q8×C137—42no—
132Q8×C137#4112yes—
133D4×C137—22no—
134D4×C137—21no—
135D4×C137—21no—
136D4×C137—42no—
137D4×C137—42no—
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—21no—
145C2×C4×C137—42no—
146C2×C4×C137—42no—
147C2×C4×C137C2×D411yescharacteristic
148C2×C4×C137C2×D411yescharacteristic
149C2×C4×C137C2×D411yescharacteristic
150C2×C4×C137C2×D411yescharacteristic
151C2×C4×C137C2411yescharacteristic
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137#13512yes—
161C22×C137C22×D411yescharacteristic
162C2×D137—41no—
163C2×D137—41no—
164C2×D137—41no—
165C4×C137—21no—
166C4×C137—21no—
167C4×C137—22no—
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137C2×#4111yescharacteristic
173C4×C137H(5,𝔽2)11yescharacteristic
174Dic137—41no—
175Dic137—41no—
176Dic137—41no—
177C2×C137—22no—
178C2×C137—41no—
179C2×C137—81no—
180C2×C137#54611yescharacteristic
181C2×C137C2×#13511yescharacteristic
182C2×C137#57711yescharacteristic
183D137—81no—
184C137#330111yescharacteristic, complemented, Sylow
185#3301—1371nomaximal, Sylow
186#553—1371no—
187#548D13711yescharacteristic, complemented
188#488—1371no—
189#485—1371no—
190#481—1371no—
191#479—1371no—
192#466—1371no—
193#465—1371no—
194#463—1371no—
195#449—1371no—
196#448—1371no—
197C4×SD16—1371no—
198C8×D4—1371no—
199#416—1371no—
200C2×#103—1371no—
201C2×#41C2×D13711yescharacteristic
202C22×Q8C2×D13711yescharacteristic
203C2×SD16—1371no—
204C2×SD16—1371no—
205C2×D8—1371no—
206C22×C8—1371no—
207C22×C8—1371no—
208#126—2741no—
209#125—1371no—
210#124—21no—
211#124C2×D13711yescharacteristic
212#123—2741no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D13711yescharacteristic
215#121—21no—
216#121—2741no—
217#121—1371no—
218#121—1371no—
219#121C2×D13711yescharacteristic
220C4×Q8C2×D13711yescharacteristic
221C4×D4—21no—
222C4×D4—2741no—
223C4×D4—1371no—
224C4×D4C2×D13711yescharacteristic
225#117—1371no—
226C2×C4⋊C4—1371no—
227#107—1371no—
228#106—1371no—
229#106—1371no—
230C4⋊C8—1371no—
231#103—1371no—
232#103—1371no—
233#103—1371no—
234#103—1371no—
235#103—1371no—
236#103—1371no—
237#102—1371no—
238#102—1371no—
239#102—1371no—
240#102—1371no—
241C22⋊C8—1371no—
242C22⋊C8—1371no—
243C4×C8—1371no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D13711yescharacteristic
250C2×Q8C22×D13711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2741no—
254C2×D4—2741no—
255C2×D4—1371no—
256C2×D4C22⋊D13711yescharacteristic
257C2×D4C22×D13711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2741no—
261C22×C4—2741no—
262C22×C4C22×D13711yescharacteristic
263C22×C4C22×D13711yescharacteristic
264SD16—2742no—
265SD16—5481no—
266D8—5481no—
267C2×C8—5481no—
268C2×C8—5481no—
269C2×C8—1371no—
270C2×C8—1371no—
271C2×C8—1371no—
272C2×C8—1371no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2741no—
276C4⋊C4—2741no—
277C4⋊C4—2741no—
278C4⋊C4—1371no—
279C4⋊C4—1371no—
280C4⋊C4—1371no—
281C4⋊C4C22⋊D13711yescharacteristic
282C4⋊C4C22×D13711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2741no—
288C22⋊C4—2741no—
289C22⋊C4—2741no—
290C22⋊C4—2741no—
291C22⋊C4C22⋊D13711yescharacteristic
292C22⋊C4C22⋊D13711yescharacteristic
293C42—21no—
294C42—2741no—
295C42C22×D13711yescharacteristic
296C23—21no—
297C23—2741no—
298C23C2×C22⋊D13711yescharacteristic
299C23C2×C22⋊D13711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1455550912yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—2741no—
311D4—2741no—
312D4—5482no—
313D4—10961no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—2741no—
324C2×C4—2741no—
325C2×C4—2741no—
326C2×C4—10961no—
327C2×C4—10961no—
328C2×C4C2×C22⋊D13711yescharacteristic
329C2×C4C2×C22⋊D13711yescharacteristic
330C2×C4C2×C22⋊D13711yescharacteristic
331C2×C4C2×C22⋊D13711yescharacteristic
332C2×C4C23×D13711yescharacteristic
333C8—2741no—
334C8—2741no—
335C8—2741no—
336C8—2741no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—5481no—
346C22—5481no—
347C22—5481no—
348C22#1798410812yes—
349C22C22×C22⋊D13711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—5481no—
358C4—5481no—
359C4—5481no—
360C4C2×#1455550911yescharacteristic
361C4#1798413311yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—10961no—
366C2#2257804711yescharacteristic
367C2C2×#1798410811yescharacteristic
368C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#548×C137C2
socleC22×C137C22×D4
hypercenter#548D137
hypocenterC137#3301
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #22578099C2semidirect
2. #22577757C2semidirect
3. #22577741C2semidirect
4. #22577719C2semidirect
5. #22577710C2semidirect
6. #22577637C2semidirect
7. #22577634C2semidirect
8. #22577615C2semidirect
9. #22577548C2semidirect
10. #22577542C2semidirect
11. C4×#14555497C2semidirect
12. D4×#11764040C2semidirect
13. #22577399C2semidirect
14. C2×#17984012C2semidirect
15. #548×C137C2semidirect
184. C137#3301semidirect
187. #548D137semidirect

Maximal subgroups

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

subgroupindex
#225780992
#225777572
#225777412
#225777192
#225777102
#225776372
#225776342
#225776152
#225775482
#225775422
C4×#145554972
D4×#117640402
#225773992
C2×#179840122
#548×C1372
#3301137