Finite Group Databasebeta

← #28744194

Subgroups — #28744194

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434633C211yesmaximal, characteristic
2#21434629C211yesmaximal, characteristic
3#21426382C211yesmaximal, characteristic, complemented
4#21426381C211yesmaximal, characteristic, complemented
5#21426380C211yesmaximal, characteristic, complemented
6#21426374C211yesmaximal, characteristic, complemented
7#21426372C211yesmaximal, characteristic, complemented
8#21426370C211yesmaximal, characteristic, complemented
9#21426369C211yesmaximal, characteristic, complemented
10#21425883C211yesmaximal, characteristic
11#21425881C211yesmaximal, characteristic
12#21425879C211yesmaximal, characteristic
13#366×D59C211yesmaximal, characteristic
14#21425694C211yesmaximal, characteristic
15C59×#2255C211yesmaximal, characteristic, complemented
16#16838032—21no—
17#16838029—21no—
18C2×#14273666C2211yescharacteristic
19C2×C16×D59C2211yescharacteristic
20#16837812—21no—
21#16837810—21no—
22#16837809—21no—
23#16837808—21no—
24#16837807C2211yescharacteristic
25#16837806C2211yescharacteristic
26#16837806C2211yescharacteristic
27#16837804C2211yescharacteristic
28#16837803C2211yescharacteristic
29#16837803C2211yescharacteristic
30#16837443—21no—
31#16837441—21no—
32#16837439C2211yescharacteristic
33#16837439C2211yescharacteristic
34#107×D59C2211yescharacteristic
35#107×D59C2211yescharacteristic
36#16837356C2211yescharacteristic
37#16837230C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837229C2211yescharacteristic
40#16837229C2211yescharacteristic
41#16837228C2211yescharacteristic
42#16837228C2211yescharacteristic
43#16837227C2211yescharacteristic
44#16837227C2211yescharacteristic
45#16837227C2211yescharacteristic
46#16837227C2211yescharacteristic
47#16837227C2211yescharacteristic
48#16837227C2211yescharacteristic
49#16837177C2211yescharacteristic
50C4×C59⋊C16C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C59×#500C2211yescharacteristic
53C59×#499C2211yescharacteristic
54C59×#367C2211yescharacteristic
55C59×#367C2211yescharacteristic
56C59×#366C2211yescharacteristic
57C59×#366C2211yescharacteristic
58C59×#346C2211yescharacteristic
59C2×#11745659D411yescharacteristic
60C2×C8×D59C2311yescharacteristic
61#14273727—21no—
62#14273726—21no—
63#14273726—21no—
64#14273726—41no—
65#14273725—41no—
66#14273719—21no—
67#14273718—21no—
68#14273718—21no—
69#14273717—41no—
70#14273717—41no—
71#14273707—41no—
72#14273705—41no—
73#14273705—41no—
74#14273703—41no—
75#14273685—21no—
76#14273685—21no—
77#14273684D411yescharacteristic
78C4⋊C4×D59—21no—
79C4⋊C4×D59—21no—
80#14273666Q814yescomplemented
81C2×C59⋊C16C2311yescharacteristic
82C2×C59⋊C16C2311yescharacteristic
83C16×D59Q814yescomplemented
84#14273661—21no—
85#14273661C2311yescharacteristic
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273660—21no—
90#14273660—21no—
91#14273660—21no—
92#14273660—21no—
93#14273659—21no—
94#14273658—21no—
95#14273658—21no—
96#14273658—21no—
97#14273658—21no—
98#14273638D411yescharacteristic
99#14273637D411yescharacteristic
100C8×Dic59C2311yescharacteristic
101C4⋊Q8×C59—21no—
102#125×C59—21no—
103C2×C16×C59C2311yescharacteristic
104C2×C16×C59C2311yescharacteristic
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111C4×C8×C59C2311yescharacteristic
112C2×Q8×C59—41no—
113C2×C4×D59—41no—
114C2×C4×D59—41no—
115C2×C4×D59C2×D411yescharacteristic
116C2×Dic118—41no—
117C2×Dic118—41no—
118C2×Dic118—41no—
119C2×C8×C59C2×D411yescharacteristic
120C2×C8×C59C2411yescharacteristic
121C4⋊C4×C59—21no—
122C4⋊C4×C59—21no—
123C4⋊C4×C59—21no—
124C4⋊C4×C59—21no—
125C4⋊C4×C59—41no—
126C4⋊C4×C59—41no—
127C4⋊C4×C59—41no—
128C42×C59C2×D411yescharacteristic
129(C2×D59)⋊C4—41no—
130(C2×D59)⋊C4—41no—
131(C2×D59)⋊C4D811yescharacteristic
132(C2×D59)⋊C4D811yescharacteristic
133C4⋊Dic59—21no—
134C4⋊Dic59—21no—
135C4⋊Dic59—21no—
136C4⋊Dic59—21no—
137C4⋊Dic59—41no—
138C4⋊Dic59—41no—
139C4⋊Dic59—41no—
140Dic59⋊C4—41no—
141Dic59⋊C4—41no—
142Dic59⋊C4—41no—
143Dic59⋊C4—41no—
144Dic59⋊C4—41no—
145Dic59⋊C4—41no—
146Dic59⋊C4—41no—
147Dic59⋊C4—41no—
148Dic59⋊C4D811yescharacteristic
149Dic59⋊C4D811yescharacteristic
150C4×Dic59—41no—
151C4×Dic59—41no—
152C4×Dic59C2×D411yescharacteristic
153C2×C59⋊C8C2×D411yescharacteristic
154C2×C59⋊C8C2×D411yescharacteristic
155#11745659Q1614yes—
156C8×D59C2×Q812yes—
157C8×D59C2×Q812yes—
158C16×C59C2×Q812yes—
159C16×C59C2×Q812yes—
160C59⋊C16C2×Q812yes—
161C59⋊C16C2×Q812yes—
162C22×D59C2×D811yescharacteristic
163Q8×C59—161no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59C2×D811yescharacteristic
169C2×C4×C59C2×D811yescharacteristic
170C2×C4×C59C22×D411yescharacteristic
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59C2×D811yescharacteristic
176C2×Dic59C2×D811yescharacteristic
177C2×Dic59C2×D811yescharacteristic
178C4×D59—161no—
179C4×D59—161no—
180C4×D59C4⋊Q812yes—
181C4×D59C2×Q1612yes—
182Dic118—161no—
183Dic118—161no—
184Dic118—161no—
185C8×C59C2×Q1612yes—
186C8×C59C22×Q811yescharacteristic
187C8×C59C22×Q811yescharacteristic
188C59⋊C8C4⋊Q812yes—
189C59⋊C8C2×Q1612yes—
190#33729—591nomaximal, Sylow
191C22×C59C22×D811yescharacteristic
192C2×D59—21no—
193C2×D59—21no—
194C2×D59#50012yes—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—81no—
198C4×C59—81no—
199C4×C59—81no—
200C4×C59—81no—
201C4×C59C2×C4⋊Q811yescharacteristic
202C4×C59C22×Q1611yescharacteristic
203Dic59—21no—
204Dic59—21no—
205Dic59—81no—
206Dic59—81no—
207Dic59—81no—
208Dic59—81no—
209Dic59#50012yes—
210#3164—591no—
211#3164—591no—
212#2255—591no—
213#2255—591no—
214#2255—591no—
215#2255D5911yescharacteristic, complemented
216#2253—591no—
217#2253—591no—
218#2252—591no—
219#2252—591no—
220#2159—591no—
221#2159—591no—
222#2158—591no—
223C2×#366—591no—
224#2108—591no—
225C2×C59C2×#50011yescharacteristic
226C2×C59#341511yescharacteristic
227C2×C59#341711yescharacteristic
228D59—22no—
229#544—1181no—
230#544—1181no—
231C2×M32—591no—
232C22×C16—591no—
233#501—1181no—
234#501—1181no—
235#501—1181no—
236#501—1181no—
237#500—591no—
238#500—591no—
239#500—591no—
240#500C2×D5911yescharacteristic
241#499—591no—
242#499—591no—
243#499—591no—
244#499C2×D5911yescharacteristic
245#428—1181no—
246#428—1181no—
247#427—591no—
248#427—591no—
249C2×#107—591no—
250C2×#107—591no—
251#405—591no—
252#367—591no—
253#367—591no—
254#367—591no—
255#367—591no—
256#367C2×D5911yescharacteristic
257#367C2×D5911yescharacteristic
258#366—591no—
259#366—591no—
260#366—591no—
261#366—591no—
262#366—591no—
263#366—591no—
264#366—591no—
265#366—591no—
266#366C2×D5911yescharacteristic
267#366C2×D5911yescharacteristic
268#346—591no—
269#346C2×D5911yescharacteristic
270C4×C16—591no—
271C4×C16—591no—
272C59#3372911yescharacteristic, complemented, Sylow
273C2×M16—591no—
274C22×C8—591no—
275C4⋊Q8—21no—
276C4⋊Q8—1181no—
277C4⋊Q8—1181no—
278C4⋊Q8—1181no—
279C4⋊Q8—2361no—
280C4⋊Q8—2361no—
281#125—21no—
282#125—1181no—
283#125—1181no—
284#125—1181no—
285#125—2361no—
286#125—2361no—
287C22⋊Q8—2361no—
288C22⋊Q8—2361no—
289C22⋊Q8—2361no—
290C22⋊Q8—2361no—
291#117—591no—
292#117—1181no—
293#117—1181no—
294C2×C4⋊C4—1181no—
295C2×C4⋊C4—1181no—
296M32—594no—
297C2×C16—591no—
298C2×C16—591no—
299C2×C16—594no—
300C2×C16C22×D5911yescharacteristic
301C2×C16C22×D5911yescharacteristic
302#107—21no—
303#107—591no—
304#107—591no—
305#107—591no—
306#107—591no—
307#107—1181no—
308#107—1181no—
309#107—1181no—
310#107—1181no—
311#107—1181no—
312#107C22×D5911yescharacteristic
313#107C22×D5911yescharacteristic
314#107C22×D5911yescharacteristic
315#107C22×D5911yescharacteristic
316#106—21no—
317#106—1181no—
318#106—1181no—
319#106—1181no—
320#106—1181no—
321#106—1181no—
322#97—591no—
323#97—591no—
324C4×C8—591no—
325C4×C8C22×D5911yescharacteristic
326C2×Q8—41no—
327C2×Q8—2361no—
328C2×Q8—2361no—
329C2×Q8—2361no—
330C22×C4—591no—
331C22×C4—2361no—
332C22×C4—2361no—
333M16—594no—
334C2×C8—591no—
335C2×C8—591no—
336C2×C8—592no—
337C2×C8—592no—
338C2×C8D4×D5911yescharacteristic
339C2×C8C23×D5911yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—591no—
348C4⋊C4—591no—
349C4⋊C4—1181no—
350C4⋊C4—1181no—
351C4⋊C4—1181no—
352C4⋊C4—1181no—
353C4⋊C4—2361no—
354C4⋊C4—2361no—
355C4⋊C4—2361no—
356C4⋊C4—2361no—
357C4⋊C4—2361no—
358C4⋊C4—2361no—
359C4⋊C4—2361no—
360C4⋊C4—2361no—
361C4⋊C4—2361no—
362C4⋊C4—2361no—
363C4⋊C4—2361no—
364C22⋊C4—591no—
365C22⋊C4—591no—
366C22⋊C4—2361no—
367C22⋊C4—2361no—
368C42—591no—
369C42—2361no—
370C42—2361no—
371C42D4×D5911yescharacteristic
372C16—592no—
373C16—592no—
374C16Q8×D5912yes—
375C16Q8×D5912yes—
376C23—591no—
377Q8—161no—
378Q8—9441no—
379Q8—9441no—
380Q8—9441no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—591no—
386C2×C4—592no—
387C2×C4—592no—
388C2×C4—591no—
389C2×C4—591no—
390C2×C4—9441no—
391C2×C4—9441no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4D8×D5911yescharacteristic
397C2×C4D8×D5911yescharacteristic
398C2×C4C2×D4×D5911yescharacteristic
399C8—592no—
400C8—592no—
401C8Q16×D5912yes—
402C8C2×Q8×D5911yescharacteristic
403C8C2×Q8×D5911yescharacteristic
404C22—592no—
405C22—1181no—
406C22—1181no—
407C22C2×D8×D5911yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—592no—
415C4—1181no—
416C4—1181no—
417C4—4721no—
418C4—4721no—
419C4—4721no—
420C4—4721no—
421C4C4⋊Q8×D5911yescharacteristic
422C4C2×Q16×D5911yescharacteristic
423C2—1182no—
424C2#500×D5911yescharacteristic
425C2#2143723611yescharacteristic
426C2#2143724811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2255C2
socleC22×C59C22×D8
prime radicalC22×D59C2×D8
hypercenter#2255D59
hypocenterC59#33729
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #21426382C2semidirect
4. #21426381C2semidirect
5. #21426380C2semidirect
6. #21426374C2semidirect
7. #21426372C2semidirect
8. #21426370C2semidirect
9. #21426369C2semidirect
15. C59×#2255C2semidirect
80. #14273666Q8semidirect
83. C16×D59Q8semidirect
215. #2255D59semidirect
272. C59#33729semidirect

Maximal subgroups

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

subgroupindex
#214346332
#214346292
#214263822
#214263812
#214263802
#214263742
#214263722
#214263702
#214263692
#214258832
#214258812
#214258792
#366×D592
#214256942
C59×#22552
#3372959