Finite Group Databasebeta

← #31789631

Subgroups — #31789631

Order 17536.

470 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
1C2×#17984120C211yesmaximal, characteristic
2#22577885C211yesmaximal, characteristic
3#22577879C211yesmaximal, characteristic, complemented
4#22577507C211yesmaximal, characteristic, complemented
5#22577506C211yesmaximal, characteristic, complemented
6#22577505C211yesmaximal, characteristic, complemented
7Q16×Dic137C211yesmaximal, characteristic, complemented
8C4×#14555500C211yesmaximal, characteristic, complemented
9C4×#14555497C211yesmaximal, characteristic, complemented
10#22577490C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577398C211yesmaximal, characteristic
13#103×D137C211yesmaximal, characteristic, complemented
14#22577360C211yesmaximal, characteristic, complemented
15#441×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18#17984120C418yescomplemented
19C2×#14555500C2211yescharacteristic
20C2×#14555499C2211yescharacteristic
21C2×#14555497C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic
26C4×Dic274—21no—
27C4×Dic274C2211yescharacteristic
28#17984053—21no—
29#17984053C2211yescharacteristic
30C4×D548—21no—
31C4×D548C2211yescharacteristic
32#17984050—21no—
33#17984050C2211yescharacteristic
34#17984046—21no—
35#17984046C2211yescharacteristic
36#17984045—21no—
37#17984045—21no—
38#17984045C2211yescharacteristic
39C4⋊C4×D137—21no—
40C4⋊C4×D137C2211yescharacteristic
41#17984022C2211yescharacteristic
42#17984021C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984010C2211yescharacteristic
49#17984008C2211yescharacteristic
50#17983998C2211yescharacteristic
51C4×#11764040C2211yescharacteristic
52C8×Dic137C2211yescharacteristic
53C2×Q16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×Q8×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59#97×C137C2211yescharacteristic
60#14555511—22no—
61#14555511C2×C414yes—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64Q8×D137—22no—
65Q8×D137C2×C414yescomplemented
66C2×D548—21no—
67C2×D548C2311yescharacteristic
68C2×C4×D137—21no—
69C2×C4×D137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74#14555500C2×C414yescomplemented
75#14555499C2×C414yescomplemented
76#14555497C2×C414yescomplemented
77#14555493C2×C414yescomplemented
78C2×#11764040C2311yescharacteristic
79C2×#11764040C2311yescharacteristic
80C8×D137—22no—
81C4⋊Dic137—21no—
82C4⋊Dic137—21no—
83C4⋊Dic137C2311yescharacteristic
84C4⋊Dic137C2311yescharacteristic
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89#14555489D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×Q8×C137C2311yescharacteristic
103Q16×C137C2×C414yescomplemented
104C2×C8×C137C2311yescharacteristic
105C2×C8×C137C2311yescharacteristic
106C4⋊C4×C137—21no—
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C42×C137—21no—
111C42×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137—21no—
114C22×D137C2×D411yescharacteristic
115Dic274—21no—
116Dic274—24no—
117Dic274C22×C412yes—
118D548—24no—
119D548—21no—
120D548C22×C412yes—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137C2×D411yescharacteristic
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C4×D137—24no—
128C4×D137—24no—
129C4×D137—81no—
130C4×D137—81no—
131C4×D137—81no—
132C4×D137C22×C412yes—
133C4×D137#4112yes—
134#11764040—21no—
135#11764040C22×C412yes—
136Q8×C137—21no—
137Q8×C137—21no—
138Q8×C137C22×C412yes—
139Q8×C137C22×C412yes—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137C22×C412yes—
149C22×C137C22×D411yescharacteristic
150C2×D137—21no—
151C2×D137—21no—
152C2×D137—22no—
153C2×D137—41no—
154C2×D137—41no—
155C2×D137C4×D412yes—
156C4×C137—22no—
157C4×C137—22no—
158C4×C137—21no—
159C4×C137—21no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137C23×C411yescharacteristic
163C4×C137C2×#4111yescharacteristic
164Dic137—22no—
165Dic137—21no—
166Dic137—21no—
167Dic137—41no—
168Dic137—41no—
169Dic137C4×D412yes—
170C2×C137C2×C4×D411yescharacteristic
171C2×C137#57711yescharacteristic
172C2×C137#57811yescharacteristic
173D137—22no—
174D137—42no—
175C137#294711yescharacteristic, complemented, Sylow
176#2947—1371nomaximal, Sylow
177C2×#137—1371no—
178#520—1371no—
179#519—1371no—
180#441—1371no—
181#441D13711yescharacteristic, complemented
182#440—1371no—
183#440—1371no—
184C4×Q16—1371no—
185C4×Q16—1371no—
186C4×SD16—1371no—
187C4×SD16—1371no—
188#428—1371no—
189#416—1371no—
190C2×#103—1371no—
191#405—1371no—
192C2×#41—1371no—
193C22×Q8—1371no—
194#137—1378no—
195C2×Q16—1371no—
196C2×Q16C2×D13711yescharacteristic
197C2×SD16—1371no—
198C2×SD16—1371no—
199C2×M16—1371no—
200C22×C8—1371no—
201C4×Q8—2741no—
202C4×Q8—2741no—
203C4×Q8—1371no—
204C4×Q8—1371no—
205C4×Q8—1371no—
206C4×Q8—1371no—
207C4×Q8C2×D13711yescharacteristic
208C4×Q8C2×D13711yescharacteristic
209C4×D4—2741no—
210C4×D4—2741no—
211C4×D4—1371no—
212C4×D4—1371no—
213#117—2741no—
214#117—2741no—
215#117—2741no—
216#117—1371no—
217#117—1371no—
218C2×C4⋊C4—2741no—
219C2×C4⋊C4—1371no—
220#107—1371no—
221#107—1371no—
222#106—1371no—
223#106C2×D13711yescharacteristic
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103C2×D13711yescharacteristic
229#103C2×D13711yescharacteristic
230#102—1371no—
231#102—1371no—
232#97—1371no—
233#97C2×D13711yescharacteristic
234C4×C8—1371no—
235C4×C8—1371no—
236#41—2742no—
237#41—1374no—
238C2×Q8—2741no—
239C2×Q8—2742no—
240C2×Q8—1374no—
241C2×Q8—1371no—
242C2×Q8C22×D13711yescharacteristic
243C2×Q8C22×D13711yescharacteristic
244C2×D4—2741no—
245C2×D4—1371no—
246C22×C4—2741no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—1371no—
252Q16—1374no—
253Q16C4×D13714yescomplemented
254SD16—1374no—
255SD16—1374no—
256M16—1374no—
257C2×C8—2742no—
258C2×C8—1371no—
259C2×C8—1371no—
260C2×C8C22×D13711yescharacteristic
261C2×C8C22×D13711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—1371no—
270C4⋊C4—1371no—
271C4⋊C4—1371no—
272C4⋊C4—1371no—
273C4⋊C4C22×D13711yescharacteristic
274C4⋊C4C22×D13711yescharacteristic
275C22⋊C4—2741no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—1371no—
279C22⋊C4—1371no—
280C42—21no—
281C42—21no—
282C42—2741no—
283C42—2741no—
284C42—2741no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C42C22×D13711yescharacteristic
289C23—2741no—
290C23—1371no—
291Q8—21no—
292Q8—21no—
293Q8—2744no—
294Q8—2741no—
295Q8—1372no—
296Q8C2×C4×D13712yes—
297Q8C2×C4×D13712yes—
298D4—2744no—
299D4—2741no—
300D4—1372no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2744no—
306C2×C4—2744no—
307C2×C4—2741no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—10961no—
311C2×C4—10961no—
312C2×C4—10961no—
313C2×C4—1371no—
314C2×C4—1371no—
315C2×C4—1371no—
316C2×C4—1372no—
317C2×C4—1372no—
318C2×C4D4×D13711yescharacteristic
319C2×C4D4×D13711yescharacteristic
320C2×C4C23×D13711yescharacteristic
321C8—21no—
322C8—2741no—
323C8—1372no—
324C8C2×C4×D13712yes—
325C22—2741no—
326C22—2741no—
327C22—2742no—
328C22—5481no—
329C22—5481no—
330C22—1372no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—2741no—
339C4—2741no—
340C4—2742no—
341C4—5481no—
342C4—5481no—
343C4—1372no—
344C4C22×C4×D13711yescharacteristic
345C4#41×D13711yescharacteristic
346C2—2742no—
347C2—5482no—
348C2C4×D4×D13711yescharacteristic
349C2#2257821211yescharacteristic
350C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#441×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555511C22
cosocleC2×C4×C137C24
hypercenter#441D137
hypocenterC137#2947
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #22577879C2semidirect
4. #22577507C2semidirect
5. #22577506C2semidirect
6. #22577505C2semidirect
7. Q16×Dic137C2semidirect
8. C4×#14555500C2semidirect
9. C4×#14555497C2semidirect
10. #22577490C2semidirect
11. #22577442C2semidirect
13. #103×D137C2semidirect
14. #22577360C2semidirect
15. #441×C137C2semidirect
18. #17984120C4semidirect
65. Q8×D137C2×C4semidirect
74. #14555500C2×C4semidirect
75. #14555499C2×C4semidirect
76. #14555497C2×C4semidirect
77. #14555493C2×C4semidirect
103. Q16×C137C2×C4semidirect
175. C137#2947semidirect
181. #441D137semidirect
253. Q16C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841202
#225778852
#225778792
#225775072
#225775062
#225775052
Q16×Dic1372
C4×#145555002
C4×#145554972
#225774902
#225774422
#225773982
#103×D1372
#225773602
#441×C1372
#2947137