Finite Group Databasebeta

← #28745512

Subgroups — #28745512

Order 15104.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436444C211yesmaximal, characteristic, complemented
2#21426319C211yesmaximal, characteristic, complemented
3#21426274C212yesmaximal, complemented
4#21426233C211yesmaximal, characteristic, complemented
5#21426180C212yesmaximal, complemented
6#21426128C211yesmaximal, characteristic, complemented
7#21426008C212yesmaximal, complemented
8C4×#14273668C211yesmaximal, characteristic, complemented
9D4×C59⋊C16C211yesmaximal, characteristic, complemented
10C2×#16837203C212yesmaximal, complemented
11D4×D8×C59C211yesmaximal, characteristic, complemented
12#16838056—21no—
13C2×#14273668C2211yescharacteristic
14C2×#14273668C2212yescomplemented
15C22×C59⋊C16C2212yescomplemented
16#16837807C2211yescharacteristic, complemented
17#16837716—22no—
18#16837683—21no—
19#16837638C2212yescomplemented
20C4×D472C2211yescharacteristic, complemented
21C2×#14273661C2212yescomplemented
22#16837227C2212yescomplemented
23#16837227C2211yescharacteristic, complemented
24C4⋊(C59⋊C16)C2211yescharacteristic, complemented
25#16837203C2211yescharacteristic
26#16837203C2212yes—
27#16837203C2212yescomplemented
28#16837203C2212yescomplemented
29#16837203C2212yescomplemented
30#16837202C2211yescharacteristic, complemented
31C22⋊(C59⋊C16)C2212yescomplemented
32C4×C59⋊C16C2211yescharacteristic, complemented
33C22×D8×C59C2212yes—
34D42×C59—21no—
35C59×#493C2211yescharacteristic
36C59×#466C2212yes—
37C59×#459—21no—
38C59×#447—22no—
39C4×D8×C59C2211yescharacteristic
40C8×D4×C59C2211yescharacteristic
41C2×#11745687—41no—
42C2×D472C2311yescharacteristic
43#14273727—21no—
44#14273711—42no—
45#14273707—41no—
46#14273701—22no—
47C4×D236—21no—
48C2×C4⋊Dic59—22no—
49#14273668—22no—
50#14273668—42no—
51C2×C59⋊C16—42no—
52C2×C59⋊C16C2311yescharacteristic
53C2×C59⋊C16C2311yescharacteristic
54C2×C59⋊C16C2312yes—
55#14273661—22no—
56#14273661—21no—
57#14273661C2312yes—
58#14273661C2311yescharacteristic
59#14273649—22no—
60#14273649—21no—
61C22×D4×C59—22no—
62C22×D4×C59—41no—
63C2×D8×C59—24no—
64C2×D8×C59—22no—
65C2×D8×C59—21no—
66C2×D8×C59—42no—
67C2×D8×C59C2312yes—
68C2×D8×C59C2311yescharacteristic
69C22×C8×C59C2312yes—
70#127×C59—21no—
71#121×C59—22no—
72#121×C59—41no—
73#120×C59—42no—
74C4×D4×C59—21no—
75C4×D4×C59D411yescharacteristic
76#107×C59C2311yescharacteristic
77C4⋊C8×C59D411yescharacteristic
78#102×C59—21no—
79#102×C59—22no—
80C22⋊C8×C59D412yes—
81C4×C8×C59C2311yescharacteristic
82C24×C59—42no—
83C2×D4×C59—21no—
84C2×D4×C59—22no—
85C2×D4×C59—44no—
86C2×D4×C59—42no—
87C2×D4×C59—42no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—82no—
91C2×D4×C59—84no—
92C2×D4×C59D811yescharacteristic
93C22×C4×C59—41no—
94C22×C4×C59C2×D412yes—
95C2×C22⋊D59—42no—
96C22×Dic59—42no—
97#11745687—84no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×Dic118—41no—
101D8×C59—22no—
102D8×C59—21no—
103D8×C59—22no—
104D8×C59—44no—
105D8×C59—81no—
106D8×C59#4112yescomplemented
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2×D412yes—
111C2×C8×C59C2411yescharacteristic
112C4⋊C4×C59—21no—
113C4⋊C4×C59D811yescharacteristic
114C22⋊C4×C59—41no—
115C22⋊C4×C59—42no—
116C22⋊C4×C59D812yes—
117C42×C59C2×D411yescharacteristic
118(C2×D59)⋊C4—41no—
119(C2×D59)⋊C4—42no—
120C4⋊Dic59—21no—
121C4⋊Dic59—22no—
122C4⋊Dic59—42no—
123C4⋊Dic59—41no—
124D472—21no—
125D472—21no—
126C59⋊C16—21no—
127C59⋊C16—21no—
128C59⋊C16—22no—
129C23×C59—41no—
130C23×C59—42no—
131C23×C59—84no—
132C23×C59—84no—
133C23×C59—162no—
134C23×C59C2×D812yes—
135C22×D59—41no—
136D4×C59—22no—
137D4×C59—42no—
138D4×C59—41no—
139D4×C59—42no—
140D4×C59—84no—
141D4×C59—84no—
142D4×C59—82no—
143D4×C59—161no—
144D4×C59D1614yescomplemented
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—82no—
149C2×C4×C59C2×D811yescharacteristic
150C2×C4×C59C2×D812yes—
151C2×C4×C59C2×D811yescharacteristic
152C2×C4×C59C22×D411yescharacteristic
153C22⋊D59—84no—
154C2×Dic59—42no—
155C2×Dic59—41no—
156C2×Dic59—84no—
157D236—41no—
158D236—41no—
159C4×D59—82no—
160Dic118—82no—
161C8×C59—22no—
162C8×C59—21no—
163C8×C59C2×#4111yescharacteristic
164C8×C59H(5,𝔽2)11yescharacteristic
165#33861—591nomaximal, Sylow
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—42no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—82no—
172C22×C59—82no—
173C22×C59—82no—
174C22×C59—168no—
175C22×C59C2×D1614yes—
176C22×C59C22×D811yescharacteristic
177C2×D59—81no—
178C2×D59—81no—
179C2×D59—81no—
180C4×C59—21no—
181C4×C59—22no—
182C4×C59—81no—
183C4×C59C2×D1612yes—
184C4×C59#54611yescharacteristic
185C4×C59#57711yescharacteristic
186Dic59—82no—
187Dic59—81no—
188#3334—591no—
189D4×D8D5911yescharacteristic, complemented
190#2240—591no—
191#2232—592no—
192#2224—591no—
193#2215—592no—
194#2206—591no—
195#2184—592no—
196C4×D16—591no—
197D4×C16—591no—
198C2×#357—592no—
199C2×C59—24no—
200C2×C59—82no—
201C2×C59—162no—
202C2×C59#329411yescharacteristic
203C2×C59C22×D1611yescharacteristic
204C2×C59#341611yescharacteristic
205D59—161no—
206C22×D8C2×D5912yes—
207#547—1181no—
208D42—21no—
209C2×D16—591no—
210C2×D16—592no—
211C22×C16—592no—
212#500—591no—
213#493C2×D5911yescharacteristic
214#480—1182no—
215#474—1181no—
216#466—592no—
217#466C2×D5912yes—
218#459—21no—
219#447—22no—
220C4×D8—591no—
221C4×D8C2×D5911yescharacteristic
222C8×D4C2×D5911yescharacteristic
223C2×#107—592no—
224#366—591no—
225#366—592no—
226C4⋊C16—591no—
227#357—591no—
228#357—592no—
229#357—592no—
230#357—592no—
231#357—592no—
232#357—591no—
233C22⋊C16—592no—
234C4×C16—591no—
235C59#3386111yescharacteristic, complemented, Sylow
236C2×#41—2361no—
237C22×D4—22no—
238C22×D4—41no—
239C2×D8—24no—
240C2×D8—22no—
241C2×D8—21no—
242C2×D8—42no—
243C2×D8—591no—
244C2×D8C22×D5912yes—
245C2×D8C22×D5911yescharacteristic
246C22×C8C22×D5912yes—
247C4⋊Q8—1181no—
248#127—21no—
249#123—2362no—
250C22⋊Q8—2361no—
251#121—22no—
252#121—41no—
253#121—1182no—
254#120—42no—
255C4×D4—21no—
256C4×D4—1181no—
257C4×D4C22⋊D5911yescharacteristic
258C2×C4⋊C4—1182no—
259D16—1182no—
260D16—2362no—
261C2×C16—591no—
262C2×C16—591no—
263C2×C16—592no—
264C2×C16—2362no—
265#107—591no—
266#107—592no—
267#107—1182no—
268#107—1181no—
269#107C22×D5911yescharacteristic
270C4⋊C8C22⋊D5911yescharacteristic
271#102—21no—
272#102—22no—
273#102—1181no—
274#102—1182no—
275C22⋊C8C22⋊D5912yes—
276C4×C8C22×D5911yescharacteristic
277C24—42no—
278#41—4724no—
279C2×Q8—2361no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4—1181no—
290C2×D4—2362no—
291C2×D4#1174566911yescharacteristic
292C22×C4—41no—
293C22×C4—2362no—
294C22×C4—2361no—
295C22×C4C2×C22⋊D5912yes—
296D8—22no—
297D8—21no—
298D8—22no—
299D8—44no—
300D8—81no—
301D8—1181no—
302D8—1181no—
303D8#1174568712yescomplemented
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×C22⋊D5911yescharacteristic
307C2×C8C2×C22⋊D5912yes—
308C2×C8C23×D5911yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—1181no—
311C4⋊C4—1182no—
312C4⋊C4—2362no—
313C4⋊C4—2361no—
314C4⋊C4#1174566911yescharacteristic
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—2361no—
318C22⋊C4—2362no—
319C22⋊C4#1174566912yes—
320C42C2×C22⋊D5911yescharacteristic
321C16—1181no—
322C16—1181no—
323C16—1182no—
324C23—41no—
325C23—42no—
326C23—84no—
327C23—84no—
328C23—162no—
329C23—2361no—
330C23C2×#1174566912yes—
331Q8—4722no—
332D4—22no—
333D4—41no—
334D4—42no—
335D4—42no—
336D4—84no—
337D4—84no—
338D4—82no—
339D4—161no—
340D4—4724no—
341D4—2361no—
342D4—2361no—
343D4#1427366814yescomplemented
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—4724no—
349C2×C4—4722no—
350C2×C4—2361no—
351C2×C4—2362no—
352C2×C4C2×#1174566911yescharacteristic
353C2×C4C2×#1174566912yes—
354C2×C4C2×#1174566911yescharacteristic
355C2×C4C22×C22⋊D5911yescharacteristic
356C8—21no—
357C8—22no—
358C8C2×#1174568711yescharacteristic
359C8#1427377211yescharacteristic
360C22—22no—
361C22—22no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—168no—
369C22—4721no—
370C22—4721no—
371C22—4721no—
372C22C2×#1427366814yes—
373C22C22×#1174566911yescharacteristic
374C4—22no—
375C4—21no—
376C4—81no—
377C4—4721no—
378C4—4722no—
379C4C2×#1427366812yes—
380C4#1683804611yescharacteristic
381C4#1683821311yescharacteristic
382C2—24no—
383C2—82no—
384C2—162no—
385C2—9441no—
386C2#2143590911yescharacteristic
387C2C22×#1427366811yescharacteristic
388C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×D8×C59C2
socleC22×C59C22×D8
hypercenterD4×D8D59
hypocenterC59#33861
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #21436444C2semidirect
2. #21426319C2semidirect
3. #21426274C2semidirect
4. #21426233C2semidirect
5. #21426180C2semidirect
6. #21426128C2semidirect
7. #21426008C2semidirect
8. C4×#14273668C2semidirect
9. D4×C59⋊C16C2semidirect
10. C2×#16837203C2semidirect
11. D4×D8×C59C2semidirect
14. C2×#14273668C22semidirect
15. C22×C59⋊C16C22semidirect
16. #16837807C22semidirect
19. #16837638C22semidirect
20. C4×D472C22semidirect
21. C2×#14273661C22semidirect
22. #16837227C22semidirect
23. #16837227C22semidirect
24. C4⋊(C59⋊C16)C22semidirect
27. #16837203C22semidirect
28. #16837203C22semidirect
29. #16837203C22semidirect
30. #16837202C22semidirect
31. C22⋊(C59⋊C16)C22semidirect
32. C4×C59⋊C16C22semidirect
106. D8×C59#41semidirect
144. D4×C59D16semidirect
189. D4×D8D59semidirect
235. C59#33861semidirect
303. D8#11745687semidirect
343. D4#14273668semidirect

Maximal subgroups

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

subgroupindex
#214364442
#214263192
#214262742
#214262332
#214261802
#214261282
#214260082
C4×#142736682
D4×C59⋊C162
C2×#168372032
D4×D8×C592
#3386159