Finite Group Databasebeta

← #28484517

Subgroups — #28484517

Order 15104.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434218C211yesmaximal, characteristic
2#21433185C211yesmaximal, characteristic, complemented
3#21422706C211yesmaximal, characteristic
4#21422590C211yesmaximal, characteristic
5#21421995C211yesmaximal, characteristic, complemented
6#21421952C211yesmaximal, characteristic, complemented
7C59×#1763C211yesmaximal, characteristic, complemented
8#16837952C2211yescharacteristic
9#16837715—21no—
10#16837697C411yescharacteristic, complemented
11#16837697C411yescharacteristic, complemented
12#16837602C411yescharacteristic, complemented
13#16837602C411yescharacteristic, complemented
14#16837541—21no—
15#16837533—22no—
16#16837533—21no—
17Q8×C59⋊C8—21no—
18#16837500—21no—
19#16837500—21no—
20#16837459—21no—
21#16837459C2211yescharacteristic
22#16837459C2211yescharacteristic
23#16837455—21no—
24#16837455C2211yescharacteristic
25#16837141—22no—
26#16837135—21no—
27#16837135—21no—
28C2×C4×Q8×C59C2211yescharacteristic
29C2×#103×C59—21no—
30C2×#97×C59C2211yescharacteristic
31C4×C4⋊C4×C59C2211yescharacteristic
32C59×#340—21no—
33C2×#11745685—21no—
34C2×#11745671—21no—
35C2×#11745671—21no—
36#14273727C2×C411yescharacteristic
37#14273724C2×C411yescharacteristic
38#14273715—21no—
39#14273701—21no—
40#14273701—21no—
41C4×D236—21no—
42C4×D236—21no—
43#14273660—21no—
44#14273660—21no—
45C4⋊(C59⋊C8)—22no—
46C4⋊(C59⋊C8)—22no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50C4⋊(C59⋊C8)C2×C411yescharacteristic
51#14273651—21no—
52#14273651—21no—
53#14273647—21no—
54#14273647—21no—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57C22⋊(C59⋊C8)D411yescharacteristic
58C22⋊(C59⋊C8)D411yescharacteristic
59#14273638—22no—
60#14273638—21no—
61#14273638D412yes—
62C4×C59⋊C8—22no—
63C4×C59⋊C8—21no—
64C22×Q8×C59D411yescharacteristic
65C22×C8×C59—21no—
66C4×Q8×C59—21no—
67C4×Q8×C59—21no—
68C4×Q8×C59C2×C411yescharacteristic
69C4×Q8×C59C2×C411yescharacteristic
70C4×Q8×C59D411yescharacteristic
71C4×Q8×C59D411yescharacteristic
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59—21no—
74C2×C4⋊C4×C59D411yescharacteristic
75C2×C42×C59—21no—
76C2×C42×C59—21no—
77C2×C42×C59C2311yescharacteristic
78#103×C59—24no—
79#97×C59—21no—
80#97×C59—21no—
81#95×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—22no—
84C2×Q8×C59—41no—
85C2×Q8×C59C22⋊C411yescharacteristic
86C2×Q8×C59C22⋊C411yescharacteristic
87C22×C4×C59—21no—
88C22×C4×C59—21no—
89C22×C4×C59—41no—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×C22⋊D59—41no—
94#11745685—81no—
95#11745685—81no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×Dic118—21no—
101C2×C8×C59—22no—
102C2×C8×C59—41no—
103C2×C8×C59—41no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59C22⋊C411yescharacteristic
112C4⋊C4×C59C22⋊C411yescharacteristic
113C42×C59—21no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—21no—
117C42×C59—81no—
118C42×C59C22×C411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59#4111yescharacteristic
122#11745671—42no—
123#11745671—42no—
124(C2×D59)⋊C4—41no—
125(C2×D59)⋊C4—41no—
126C4⋊Dic59—21no—
127C4⋊Dic59—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—22no—
133C23×C59#12111yescharacteristic
134C22×D59—41no—
135Q8×C59—21no—
136Q8×C59—21no—
137Q8×C59—21no—
138Q8×C59—42no—
139Q8×C59#10412yes—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59#12011yescharacteristic
159C2×C4×C59#12111yescharacteristic
160C2×C4×C59#12311yescharacteristic
161C22⋊D59—81no—
162C22⋊D59—81no—
163C2×Dic59—41no—
164D236—41no—
165D236—41no—
166D236—81no—
167D236—81no—
168C4×D59—82no—
169Dic118—81no—
170C8×C59—42no—
171C59⋊C8—41no—
172C59⋊C8—41no—
173C59⋊C8—42no—
174C59⋊C8—41no—
175C59⋊C8—41no—
176#9164—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—21no—
179C22×C59#38611yescharacteristic
180C22×C59#47111yescharacteristic
181C22×C59#47211yescharacteristic
182C2×D59—81no—
183C2×D59—81no—
184C2×D59—81no—
185C4×C59—21no—
186C4×C59—22no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—41no—
194C4×C59—82no—
195C4×C59C2×#10411yescharacteristic
196C4×C59#42111yescharacteristic
197Dic59—81no—
198#3112—591no—
199#2984—591no—
200#1763D5911yescharacteristic, complemented
201#1530—591no—
202#1502—591no—
203#1302—591no—
204#1285—591no—
205C2×C59—21no—
206C2×C59—21no—
207C2×C59#187111yescharacteristic
208C2×C59#190511yescharacteristic
209C2×C59#190711yescharacteristic
210D59—161no—
211#532—591no—
212C2×C4×Q8C2×D5911yescharacteristic
213#480—1181no—
214#477—591no—
215#477—591no—
216#460—591no—
217#460—591no—
218#448—1181no—
219#446—1182no—
220#446—1181no—
221C8×Q8—1181no—
222#440—1181no—
223#440—1181no—
224#432—591no—
225#432—591no—
226#432—1181no—
227#431—591no—
228#431—1181no—
229C2×#103—21no—
230C2×#97C2×D5911yescharacteristic
231C4×C4⋊C4C2×D5911yescharacteristic
232#340—21no—
233#329—1182no—
234#327—1181no—
235#327—1181no—
236C59#916411yescharacteristic, complemented, Sylow
237C2×#41—1181no—
238C22×Q8C22⋊D5911yescharacteristic
239C2×SD16—1181no—
240C2×SD16—1181no—
241C22×C8—21no—
242C4⋊Q8—591no—
243#127—591no—
244#124—1181no—
245#121—1181no—
246#121—1181no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C4×D5911yescharacteristic
250C4×Q8C4×D5911yescharacteristic
251C4×Q8D23611yescharacteristic
252C4×Q8D23611yescharacteristic
253C4×D4—1181no—
254C4×D4—1181no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D5911yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42C22×D5911yescharacteristic
261#107—1181no—
262#107—1181no—
263C4⋊C8—591no—
264C4⋊C8—591no—
265C4⋊C8—1182no—
266C4⋊C8—1182no—
267C4⋊C8—1181no—
268C4⋊C8—1181no—
269#103—24no—
270#103—1181no—
271#103—1181no—
272#102—1181no—
273#102—1181no—
274C22⋊C8—591no—
275C22⋊C8—591no—
276C22⋊C8—1181no—
277C22⋊C8—1181no—
278#97—21no—
279#97—21no—
280#97—592no—
281#97—1182no—
282#97—1181no—
283C4×C8—1182no—
284C4×C8—1181no—
285#95—21no—
286#41—4721no—
287#41—4721no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—1181no—
292C2×Q8(C2×D59)⋊C411yescharacteristic
293C2×Q8(C2×D59)⋊C411yescharacteristic
294C2×D4—1181no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—2361no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2361no—
302C22×C4D4×D5911yescharacteristic
303C22×C4#1174568711yescharacteristic
304C22×C4C2×C22⋊D5911yescharacteristic
305SD16—2362no—
306SD16—2362no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—1181no—
311C2×C8—1181no—
312C2×C8—1181no—
313C2×C8—1182no—
314C2×C8—1181no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—1181no—
323C4⋊C4—1181no—
324C4⋊C4(C2×D59)⋊C411yescharacteristic
325C4⋊C4(C2×D59)⋊C411yescharacteristic
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—81no—
333C42C2×C4×D5911yescharacteristic
334C42C2×D23611yescharacteristic
335C42D4×D5911yescharacteristic
336C42#1174568711yescharacteristic
337C23—2361no—
338C23#1427369911yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—4721no—
344Q8#1427365512yes—
345D4—4721no—
346D4—4721no—
347D4—4721no—
348D4—4721no—
349D4—2361no—
350D4—2361no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—4722no—
367C2×C4—2361no—
368C2×C4C2×(C2×D59)⋊C411yescharacteristic
369C2×C4(C22⋊D59)⋊C411yescharacteristic
370C2×C4(C22⋊D59)⋊C411yescharacteristic
371C2×C4#1427369711yescharacteristic
372C2×C4#1427369911yescharacteristic
373C2×C4#1427371111yescharacteristic
374C8—42no—
375C8—2361no—
376C8—2362no—
377C8—2361no—
378C8—2361no—
379C8—2361no—
380C22—21no—
381C22—21no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22#1683727911yescharacteristic
386C22#1683766711yescharacteristic
387C22#1683767011yescharacteristic
388C4—21no—
389C4—22no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—4721no—
399C4C2×#1427365511yescharacteristic
400C4#1683742011yescharacteristic
401C2—21no—
402C2—21no—
403C2—9441no—
404C2#2142439611yescharacteristic
405C2#2142458811yescharacteristic
406C2#2142460211yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1763C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1763D59
hypocenterC59#9164
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. #21433185C2semidirect
5. #21421995C2semidirect
6. #21421952C2semidirect
7. C59×#1763C2semidirect
10. #16837697C4semidirect
11. #16837697C4semidirect
12. #16837602C4semidirect
13. #16837602C4semidirect
200. #1763D59semidirect
236. C59#9164semidirect

Maximal subgroups

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

subgroupindex
#214342182
#214331852
#214227062
#214225902
#214219952
#214219522
C59×#17632
#916459