Finite Group Databasebeta

← #31789625

Subgroups — #31789625

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×#17984117C211yesmaximal, characteristic
2#22577882C211yesmaximal, characteristic
3#22577879C211yesmaximal, characteristic, complemented
4#22577509C211yesmaximal, characteristic, complemented
5#22577508C211yesmaximal, characteristic, complemented
6#22577504C211yesmaximal, characteristic, complemented
7#22577503C211yesmaximal, characteristic, complemented
8C4×Dic548C211yesmaximal, characteristic, complemented
9#22577495C211yesmaximal, characteristic, complemented
10#22577494C211yesmaximal, characteristic, complemented
11#22577442C211yesmaximal, characteristic, complemented
12#22577396C211yesmaximal, characteristic
13#103×D137C211yesmaximal, characteristic, complemented
14#22577357C211yesmaximal, characteristic, complemented
15C4×SD16×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18#17984117C418yescomplemented
19C2×Dic548C2211yescharacteristic
20C2×#14555500C2211yescharacteristic
21C2×#14555498C2211yescharacteristic
22C2×#14555493C2211yescharacteristic
23C2×C8×D137C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25C4×Dic274—21no—
26C4×Dic274C2211yescharacteristic
27C4×Dic274C2211yescharacteristic
28#17984053—21no—
29#17984053C2211yescharacteristic
30#17984053C2211yescharacteristic
31D4×Dic137C2211yescharacteristic
32C4×C22⋊D137—21no—
33#17984048—21no—
34#17984046—21no—
35#17984046C2211yescharacteristic
36#17984045—21no—
37#17984045C2211yescharacteristic
38#17984044—21no—
39C2×#14555489—21no—
40C4⋊C4×D137C2211yescharacteristic
41#17984022C2211yescharacteristic
42#17984021C2211yescharacteristic
43#17984019C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984013C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49#17984009C2211yescharacteristic
50#17983999C2211yescharacteristic
51#17983998C2211yescharacteristic
52C8×Dic137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56#106×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C4×C8×C137C2211yescharacteristic
60#14555509—22no—
61#14555509C2×C414yes—
62C2×Dic274—21no—
63C2×Dic274C2311yescharacteristic
64C2×Dic274C2311yescharacteristic
65Q8×D137—22no—
66Q8×D137C2×C414yescomplemented
67C2×C22⋊D137—21no—
68C22×Dic137—21no—
69C22×Dic137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137C2311yescharacteristic
73Dic548C2×C414yescomplemented
74#14555500C2×C414yescomplemented
75#14555498C2×C414yescomplemented
76#14555493C2×C414yescomplemented
77C2×#11764040C2311yescharacteristic
78C2×#11764040C2311yescharacteristic
79C8×D137—22no—
80C4⋊Dic137—21no—
81C4⋊Dic137C2311yescharacteristic
82C4⋊Dic137C2311yescharacteristic
83#14555489—21no—
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489D411yescharacteristic
89#14555489D411yescharacteristic
90C22⋊Dic137—21no—
91#14555487—21no—
92#14555487D411yescharacteristic
93#14555487D411yescharacteristic
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137—21no—
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137—21no—
100C4×Dic137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×D4×C137C2311yescharacteristic
103C22×C4×C137—21no—
104SD16×C137C2×C414yescomplemented
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137—21no—
108C4⋊C4×C137C2311yescharacteristic
109C4⋊C4×C137C2311yescharacteristic
110C22⋊C4×C137—21no—
111C42×C137—21no—
112C42×C137C2311yescharacteristic
113C22×D137C2×D411yescharacteristic
114Dic274—21no—
115Dic274—21no—
116Dic274—24no—
117Dic274C22×C412yes—
118Dic274C22×C412yes—
119C22⋊D137—24no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—24no—
123C2×Dic137—21no—
124C2×Dic137—21no—
125C2×Dic137—81no—
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—24no—
130C4×D137—81no—
131C4×D137C22×C412yes—
132C4×D137#4112yes—
133#11764040—21no—
134#11764040C22×C412yes—
135C23×C137—21no—
136Q8×C137—21no—
137Q8×C137C22×C412yes—
138D4×C137—21no—
139D4×C137C22×C412yes—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—81no—
144C2×C4×C137C2×D411yescharacteristic
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137C22×C412yes—
149C22×C137—22no—
150C22×C137—41no—
151C22×C137—41no—
152C22×C137C22×D411yescharacteristic
153C2×D137—21no—
154C2×D137—21no—
155C2×D137C4×D412yes—
156C4×C137—22no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137C23×C411yescharacteristic
162C4×C137C2×#4111yescharacteristic
163Dic137—22no—
164Dic137—22no—
165Dic137—21no—
166Dic137—21no—
167Dic137—41no—
168Dic137—41no—
169Dic137C4×D412yes—
170C2×C137—42no—
171C2×C137C2×C4×D411yescharacteristic
172C2×C137#57711yescharacteristic
173C2×C137#57811yescharacteristic
174D137—22no—
175C137#294711yescharacteristic, complemented, Sylow
176#2947—1371nomaximal, Sylow
177C2×#137—1371no—
178#520—1371no—
179#519—1371no—
180#441—1371no—
181#441—1371no—
182#440—1371no—
183#440—1371no—
184C4×Q16—1371no—
185C4×Q16—1371no—
186C4×SD16—1371no—
187C4×SD16D13711yescharacteristic, complemented
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×Q16—1371no—
197C2×SD16—1371no—
198C2×SD16C2×D13711yescharacteristic
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×Q8—1371no—
208C4×Q8C2×D13711yescharacteristic
209C4×D4—2741no—
210C4×D4—2741no—
211C4×D4—1371no—
212C4×D4C2×D13711yescharacteristic
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#103—1371no—
229#103C2×D13711yescharacteristic
230#102—1371no—
231#102C2×D13711yescharacteristic
232#97—1371no—
233#97—1371no—
234C4×C8—1371no—
235C4×C8C2×D13711yescharacteristic
236#41—2742no—
237#41—1374no—
238C2×Q8—2741no—
239C2×Q8—2742no—
240C2×Q8—1374no—
241C2×Q8—1371no—
242C2×Q8—1371no—
243C2×Q8C22×D13711yescharacteristic
244C2×D4—2741no—
245C2×D4C22×D13711yescharacteristic
246C22×C4—21no—
247C22×C4—2741no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—1371no—
252Q16—1374no—
253Q16—1374no—
254SD16—1374no—
255SD16C4×D13714yescomplemented
256M16—1374no—
257C2×C8—2742no—
258C2×C8—1371no—
259C2×C8—1371no—
260C2×C8C22×D13711yescharacteristic
261C2×C8C22×D13711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—2741no—
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—21no—
276C22⋊C4—2741no—
277C22⋊C4—2741no—
278C22⋊C4—1371no—
279C22⋊C4—1371no—
280C42—21no—
281C42—2741no—
282C42—2741no—
283C42—2741no—
284C42—2741no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C42C22×D13711yescharacteristic
289C23—21no—
290C23—1371no—
291Q8—21no—
292Q8—2744no—
293Q8—2741no—
294Q8—2741no—
295Q8—1372no—
296Q8—1372no—
297Q8C2×C4×D13712yes—
298D4—21no—
299D4—2744no—
300D4C2×C4×D13712yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—2744no—
306C2×C4—2741no—
307C2×C4—2741no—
308C2×C4—2744no—
309C2×C4—2741no—
310C2×C4—2741no—
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—22no—
326C22—41no—
327C22—41no—
328C22—2741no—
329C22—2741no—
330C22—1372no—
331C22C2×D4×D13711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—2741no—
338C4—2741no—
339C4—2742no—
340C4—2742no—
341C4—5481no—
342C4—5481no—
343C4—1372no—
344C4C22×C4×D13711yescharacteristic
345C4#41×D13711yescharacteristic
346C2—42no—
347C2—2742no—
348C2C4×D4×D13711yescharacteristic
349C2#2257820911yescharacteristic
350C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
FittingC4×SD16×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555509C22
cosocleC2×C4×C137C24
hypercenterC4×SD16D137
hypocenterC137#2947
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
3. #22577879C2semidirect
4. #22577509C2semidirect
5. #22577508C2semidirect
6. #22577504C2semidirect
7. #22577503C2semidirect
8. C4×Dic548C2semidirect
9. #22577495C2semidirect
10. #22577494C2semidirect
11. #22577442C2semidirect
13. #103×D137C2semidirect
14. #22577357C2semidirect
15. C4×SD16×C137C2semidirect
18. #17984117C4semidirect
66. Q8×D137C2×C4semidirect
73. Dic548C2×C4semidirect
74. #14555500C2×C4semidirect
75. #14555498C2×C4semidirect
76. #14555493C2×C4semidirect
104. SD16×C137C2×C4semidirect
175. C137#2947semidirect
187. C4×SD16D137semidirect
255. SD16C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841172
#225778822
#225778792
#225775092
#225775082
#225775042
#225775032
C4×Dic5482
#225774952
#225774942
#225774422
#225773962
#103×D1372
#225773572
C4×SD16×C1372
#2947137