Finite Group Databasebeta

← #28743488

Subgroups — #28743488

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433677C211yesmaximal, characteristic
2#21426304C212yesmaximal, complemented
3#21426298C211yesmaximal, characteristic, complemented
4#21426293C211yesmaximal, characteristic, complemented
5#21426287C212yesmaximal, complemented
6#21426281C211yesmaximal, characteristic, complemented
7#21426276C211yesmaximal, characteristic, complemented
8#21425878C212yesmaximal, complemented
9#21425801C212yesmaximal, complemented
10C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
11C2×C59×#427C211yesmaximal, characteristic, complemented
12C2×#14273749C2211yescharacteristic
13#16837913—21no—
14C22×C59⋊C16C2212yes—
15#16837643C2211yescharacteristic
16#16837643C2211yescharacteristic
17#16837638C2211yescharacteristic
18#16837638C2211yescharacteristic
19#16837631—21no—
20#16837631—21no—
21#16837440C2211yescharacteristic
22#16837399—21no—
23#16837229C2212yescomplemented
24#16837229C2212yescomplemented
25#16837227C2212yescomplemented
26#16837227C2212yescomplemented
27#16837205C2212yescomplemented
28#16837205C2212yescomplemented
29#16837202C2212yescomplemented
30#16837202C2212yescomplemented
31C22⋊(C59⋊C16)C2211yescharacteristic
32C22⋊(C59⋊C16)C2211yescharacteristic
33C22⋊(C59⋊C16)C2212yes—
34C23×C8×C59C2211yescharacteristic
35C2×#117×C59—21no—
36C59×#427—21no—
37C59×#427—21no—
38C59×#427C2212yes—
39C59×#427C2212yes—
40C2×#107×C59C2211yescharacteristic
41C2×#107×C59C2211yescharacteristic
42C2×#106×C59—21no—
43C2×#11745685—21no—
44#14273749—24no—
45C2×Dic236C2311yescharacteristic
46C2×#11745660—21no—
47C2×D472C2311yescharacteristic
48#14273710—41no—
49#14273706—21no—
50#14273706—21no—
51#14273701—21no—
52#14273701—21no—
53#14273696—41no—
54C4×C22⋊D59—41no—
55C4×C22⋊D59—41no—
56#14273686—21no—
57C2×C59⋊C16—24no—
58C2×C59⋊C16C2312yes—
59C2×C59⋊C16C2312yes—
60#14273661C2311yescharacteristic
61#14273661C2311yescharacteristic
62#14273659—21no—
63#14273652—21no—
64#14273652—21no—
65#14273649—21no—
66#14273649—21no—
67C23×C4×C59D411yescharacteristic
68C22×C8×C59—24no—
69C22×C8×C59D411yescharacteristic
70C22×C8×C59D411yescharacteristic
71C22×C8×C59D411yescharacteristic
72C22×C8×C59C2311yescharacteristic
73C22×C8×C59C2311yescharacteristic
74C22×C8×C59C2311yescharacteristic
75#117×C59—22no—
76#117×C59—22no—
77#117×C59—41no—
78#117×C59—41no—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59—21no—
81C2×C22⋊C4×C59—41no—
82C2×C42×C59—41no—
83#107×C59—21no—
84#107×C59—21no—
85#107×C59C2312yes—
86#107×C59C2312yes—
87#106×C59—22no—
88#106×C59—22no—
89C24×C59D811yescharacteristic
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—41no—
93C22×C4×C59—41no—
94C22×C4×C59D811yescharacteristic
95C22×C4×C59D811yescharacteristic
96C22×C4×C59D811yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C2×C22⋊D59—41no—
101C2×C22⋊D59—41no—
102C2×C22⋊D59—41no—
103#11745685—42no—
104#11745685—42no—
105C2×D236—21no—
106C2×C4×D59—41no—
107C2×Dic118—21no—
108C2×C8×C59—22no—
109C2×C8×C59—22no—
110C2×C8×C59—22no—
111C2×C8×C59—22no—
112C2×C8×C59—22no—
113C2×C8×C59C2×D411yescharacteristic
114C2×C8×C59C2×D411yescharacteristic
115C2×C8×C59C2×D411yescharacteristic
116C2×C8×C59#4114yes—
117C2×C8×C59C2411yescharacteristic
118C4⋊C4×C59—22no—
119C4⋊C4×C59—22no—
120C4⋊C4×C59—41no—
121C4⋊C4×C59—41no—
122C22⋊C4×C59—42no—
123C22⋊C4×C59—42no—
124C42×C59—42no—
125C42×C59—42no—
126C22⋊Dic59—41no—
127C22⋊Dic59—41no—
128C22⋊Dic59—41no—
129(C2×D59)⋊C4—41no—
130(C2×D59)⋊C4—41no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133Dic59⋊C4—41no—
134Dic59⋊C4—41no—
135C4×Dic59—41no—
136Dic236—22no—
137D472—22no—
138#11745660—42no—
139C59⋊C16—22no—
140C59⋊C16—22no—
141C23×C59—22no—
142C23×C59—22no—
143C23×C59—41no—
144C23×C59C2×D811yescharacteristic
145C23×C59C2×D811yescharacteristic
146C23×C59C2×D811yescharacteristic
147C22×D59—41no—
148C2×C4×C59—22no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—22no—
154C2×C4×C59—22no—
155C2×C4×C59—42no—
156C2×C4×C59—42no—
157C2×C4×C59—82no—
158C2×C4×C59—82no—
159C2×C4×C59#12312yes—
160C2×C4×C59C2×D811yescharacteristic
161C2×C4×C59C2×D811yescharacteristic
162C2×C4×C59C2×D811yescharacteristic
163C2×C4×C59#13712yes—
164C2×C4×C59C22×D411yescharacteristic
165C22⋊D59—82no—
166C22⋊D59—82no—
167C22⋊D59—82no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171D236—41no—
172D236—41no—
173C4×D59—82no—
174Dic118—41no—
175Dic118—41no—
176C8×C59—21no—
177C8×C59—21no—
178C8×C59#12312yes—
179C8×C59C2×#4112yes—
180#33645—591nomaximal, Sylow
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—44no—
189C22×C59#48012yes—
190C22×C59#50812yes—
191C22×C59C22×D811yescharacteristic
192C2×D59—81no—
193C2×D59—81no—
194C2×D59—81no—
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—82no—
198C4×C59—82no—
199C4×C59#48012yes—
200C4×C59C2×#12311yescharacteristic
201C4×C59C2×#13711yescharacteristic
202Dic59—81no—
203Dic59—81no—
204Dic59—81no—
205#3050—591no—
206C2×#427D5911yescharacteristic, complemented
207#2237—592no—
208#2236—591no—
209#2235—591no—
210#2234—592no—
211#2233—591no—
212#2232—591no—
213#2158—592no—
214#2138—592no—
215C2×C22⋊C16—591no—
216C2×C59—22no—
217C2×C59—22no—
218C2×C59—41no—
219C2×C59C2×#48011yescharacteristic
220C2×C59C2×#50811yescharacteristic
221C2×C59#341411yescharacteristic
222D59—161no—
223C2×#135—591no—
224C23×C8C2×D5911yescharacteristic
225#525—1181no—
226C2×#117—21no—
227C22×C16—592no—
228#467—591no—
229#467—591no—
230#466—591no—
231#466—591no—
232#465—1181no—
233#465—1181no—
234#427—21no—
235#427—21no—
236#427—591no—
237#427C2×D5912yes—
238#427C2×D5912yes—
239C2×#107C2×D5911yescharacteristic
240C2×#107C2×D5911yescharacteristic
241C2×#106—21no—
242#416—1181no—
243#367—592no—
244#367—592no—
245#366—592no—
246#366—592no—
247#358—592no—
248#358—592no—
249#357—592no—
250#357—592no—
251C22⋊C16—591no—
252C22⋊C16—591no—
253C22⋊C16—592no—
254C59#3364511yescharacteristic, complemented, Sylow
255C2×#41—1181no—
256C23×C4C22⋊D5911yescharacteristic
257#135—1184no—
258C2×Q16—591no—
259C2×SD16—1181no—
260C2×D8—591no—
261C22×C8—24no—
262C22×C8C22⋊D5911yescharacteristic
263C22×C8C22⋊D5911yescharacteristic
264C22×C8C22⋊D5911yescharacteristic
265C22×C8C22×D5911yescharacteristic
266C22×C8C22×D5911yescharacteristic
267C22×C8C22×D5911yescharacteristic
268#123—2361no—
269C22⋊Q8—1181no—
270C22⋊Q8—1181no—
271#121—1181no—
272#121—1181no—
273#120—2361no—
274C4×D4—2361no—
275C4×D4—2361no—
276#117—22no—
277#117—22no—
278#117—41no—
279#117—41no—
280#117—1181no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C42—41no—
285C2×C16—592no—
286C2×C16—592no—
287C2×C16—1184no—
288#107—21no—
289#107—21no—
290#107—591no—
291#107—591no—
292#107C22×D5912yes—
293#107C22×D5912yes—
294#106—22no—
295#106—22no—
296#106—1181no—
297#103—1181no—
298#103—1181no—
299#102—1181no—
300#102—1181no—
301C24#1174566911yescharacteristic
302#41—2362no—
303#41—2362no—
304C2×Q8—1181no—
305C2×D4—1181no—
306C2×D4—2361no—
307C2×D4—2361no—
308C2×D4—2361no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—2361no—
314C22×C4#1174566911yescharacteristic
315C22×C4#1174566911yescharacteristic
316C22×C4#1174566911yescharacteristic
317C22×C4C2×C22⋊D5911yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319C22×C4C2×C22⋊D5911yescharacteristic
320Q16—1182no—
321SD16—2362no—
322D8—1182no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8#1174568514yes—
329C2×C8C2×C22⋊D5911yescharacteristic
330C2×C8C2×C22⋊D5911yescharacteristic
331C2×C8C2×C22⋊D5911yescharacteristic
332C2×C8C23×D5911yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1181no—
338C4⋊C4—1181no—
339C4⋊C4—2361no—
340C4⋊C4—2361no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—2361no—
344C22⋊C4—2361no—
345C22⋊C4—2361no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C42—42no—
349C42—42no—
350C42—2361no—
351C16—1182no—
352C16—1182no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—2361no—
357C23C2×#1174566911yescharacteristic
358C23C2×#1174566911yescharacteristic
359C23C2×#1174566911yescharacteristic
360Q8—2361no—
361Q8—2361no—
362D4—4722no—
363D4—4722no—
364D4—4722no—
365D4—2361no—
366D4—2361no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—4722no—
379C2×C4—2361no—
380C2×C4—2361no—
381C2×C4—2361no—
382C2×C4#1427371012yes—
383C2×C4C2×#1174566911yescharacteristic
384C2×C4C2×#1174566911yescharacteristic
385C2×C4C2×#1174566911yescharacteristic
386C2×C4#1427375812yes—
387C2×C4C22×C22⋊D5911yescharacteristic
388C8—21no—
389C8—21no—
390C8#1427371012yes—
391C8C2×#1174568512yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—4721no—
401C22—4721no—
402C22—4721no—
403C22#1683771512yes—
404C22#1683783212yes—
405C22C22×#1174566911yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—4721no—
411C4—4721no—
412C4—4721no—
413C4#1683771512yes—
414C4C2×#1427371011yescharacteristic
415C4C2×#1427375811yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—9441no—
420C2C2×#1683771511yescharacteristic
421C2C2×#1683783211yescharacteristic
422C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC2×C59×#427C2
socleC22×C59C22×D8
prime radical#21433677C2
hypercenterC2×#427D59
hypocenterC59#33645
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #21426304C2semidirect
3. #21426298C2semidirect
4. #21426293C2semidirect
5. #21426287C2semidirect
6. #21426281C2semidirect
7. #21426276C2semidirect
8. #21425878C2semidirect
9. #21425801C2semidirect
10. C2×C22⋊(C59⋊C16)C2semidirect
11. C2×C59×#427C2semidirect
23. #16837229C22semidirect
24. #16837229C22semidirect
25. #16837227C22semidirect
26. #16837227C22semidirect
27. #16837205C22semidirect
28. #16837205C22semidirect
29. #16837202C22semidirect
30. #16837202C22semidirect
206. C2×#427D59semidirect
254. C59#33645semidirect

Maximal subgroups

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

subgroupindex
#214336772
#214263042
#214262982
#214262932
#214262872
#214262812
#214262762
#214258782
#214258012
C2×C22⋊(C59⋊C16)2
C2×C59×#4272
#3364559