Finite Group Databasebeta

← #28743616

Subgroups — #28743616

Order 15104.

610 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434677C211yesmaximal, characteristic, complemented
2C2×D8×Dic59C211yesmaximal, characteristic
3#21426294C211yesmaximal, characteristic, complemented
4#21426291C211yesmaximal, characteristic, complemented
5#21426241C212yesmaximal, complemented
6#21426139C212yesmaximal, complemented
7#21426025C211yesmaximal, characteristic
8#21425951C212yesmaximal, complemented
9SD32×Dic59C212yesmaximal, complemented
10#21425892C211yesmaximal, characteristic, complemented
11C59×#2187C211yesmaximal, characteristic
12#16837954—21no—
13C2×D4×Dic59—21no—
14C2×#14273671C2212yes—
15#16837806C2211yescharacteristic, complemented
16#16837780C2211yescharacteristic, complemented
17#16837751—21no—
18#16837643C2211yescharacteristic
19#16837641—21no—
20Q16×Dic59C2211yescharacteristic, complemented
21#16837496C2211yescharacteristic, complemented
22#16837494—21no—
23D8×Dic59—21no—
24D8×Dic59—21no—
25D8×Dic59C2212yes—
26D8×Dic59C2212yes—
27C2×#14273661C2211yescharacteristic
28C2×#14273648—21no—
29C2×C8×Dic59C2211yescharacteristic
30#16837230C2212yescomplemented
31#16837229C2212yescomplemented
32Dic59⋊C16C2212yescomplemented
33#16837207C2212yescomplemented
34#16837206C2212yescomplemented
35#16837203C2212yes—
36C22⋊(C59⋊C16)C2211yescharacteristic
37C16×Dic59C2212yescomplemented
38C22×D8×C59C2211yescharacteristic
39C2×SD32×C59C2212yes—
40C59×#467C2211yescharacteristic
41C59×#357C2212yes—
42C59×C22⋊C16C2211yescharacteristic
43C23×Dic59—41no—
44C2×Dic236C2311yescharacteristic
45C2×#11745672—21no—
46C22×C59⋊C8D411yescharacteristic
47#14273726—21no—
48#14273725—21no—
49#14273717—41no—
50#14273706—21no—
51#14273704—41no—
52Q8×Dic59—21no—
53#14273692—21no—
54#14273692—41no—
55D4×Dic59—22no—
56D4×Dic59—22no—
57D4×Dic59—41no—
58D4×Dic59—41no—
59#14273683—41no—
60C2×C4⋊Dic59—21no—
61C2×C22⋊Dic59—41no—
62C2×C4×Dic59D411yescharacteristic
63#14273671—24no—
64C2×C59⋊C16C2312yes—
65#14273661—21no—
66#14273661C2311yescharacteristic
67#14273661C2312yes—
68#14273660—21no—
69#14273652—21no—
70#14273651—21no—
71#14273650—21no—
72#14273648—22no—
73#14273648—22no—
74C8×Dic59D411yescharacteristic
75C8×Dic59D411yescharacteristic
76C8×Dic59C2311yescharacteristic
77C8×Dic59C2311yescharacteristic
78C22×D4×C59—21no—
79C2×Q16×C59C2311yescharacteristic
80C2×D8×C59—24no—
81C2×D8×C59—21no—
82C2×D8×C59C2312yes—
83C22×C8×C59C2311yescharacteristic
84C22⋊Q8×C59—21no—
85SD32×C59—24no—
86C2×C16×C59C2312yes—
87#107×C59C2311yescharacteristic
88#103×C59—21no—
89C24×C59—41no—
90C2×Q8×C59—21no—
91C2×D4×C59—24no—
92C2×D4×C59—22no—
93C2×D4×C59—42no—
94C2×D4×C59—41no—
95C22×C4×C59C2×D411yescharacteristic
96C22×Dic59—41no—
97C22×Dic59—42no—
98C22×Dic59—42no—
99C22×Dic59—84no—
100C22×Dic59D811yescharacteristic
101C22×Dic59D811yescharacteristic
102C2×Dic118—21no—
103C2×Dic118—41no—
104Q16×C59—22no—
105D8×C59—24no—
106D8×C59—22no—
107D8×C59#4114yes—
108C2×C8×C59—22no—
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59C2411yescharacteristic
111C4⋊C4×C59—21no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊Dic59—41no—
115C22⋊Dic59—42no—
116C22⋊Dic59—42no—
117#11745672—42no—
118C4⋊Dic59—22no—
119C4⋊Dic59—21no—
120C4⋊Dic59—41no—
121C4⋊Dic59—41no—
122Dic59⋊C4—41no—
123Dic59⋊C4—41no—
124Dic59⋊C4—41no—
125Dic59⋊C4—41no—
126Dic59⋊C4—41no—
127C4×Dic59—41no—
128C4×Dic59—41no—
129C4×Dic59D811yescharacteristic
130C4×Dic59D811yescharacteristic
131C4×Dic59C2×D411yescharacteristic
132C4×Dic59C2×D411yescharacteristic
133C2×C59⋊C8—41no—
134C2×C59⋊C8C2×D411yescharacteristic
135C2×C59⋊C8C2×D411yescharacteristic
136Dic236—22no—
137C16×C59—22no—
138C59⋊C16—22no—
139C23×C59—44no—
140C23×C59—42no—
141C23×C59—82no—
142C23×C59—82no—
143C23×C59C2×D811yescharacteristic
144Q8×C59—41no—
145Q8×C59—41no—
146D4×C59—24no—
147D4×C59—44no—
148D4×C59—42no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59C2×D811yescharacteristic
154C2×C4×C59C22×D411yescharacteristic
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—42no—
159C2×Dic59—82no—
160C2×Dic59—84no—
161C2×Dic59—84no—
162C2×Dic59SD3214yescomplemented
163C2×Dic59C2×D811yescharacteristic
164C2×Dic59C2×D811yescharacteristic
165C2×Dic59C2×D811yescharacteristic
166C2×Dic59C2×D811yescharacteristic
167Dic118—41no—
168Dic118—41no—
169Dic118—82no—
170C8×C59—21no—
171C8×C59C2×#4112yes—
172C59⋊C8—21no—
173C59⋊C8—21no—
174#33661—591nomaximal, Sylow
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—44no—
178C22×C59—88no—
179C22×C59—82no—
180C22×C59—82no—
181C22×C59C2×SD3212yes—
182C22×C59C22×D811yescharacteristic
183C4×C59—21no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59#52511yescharacteristic
187C4×C59#57511yescharacteristic
188Dic59—21no—
189Dic59—21no—
190Dic59—81no—
191Dic59—82no—
192Dic59—81no—
193Dic59C2×SD3212yes—
194Dic59C2×SD3212yes—
195#3170—591no—
196C2×C4×D8—591no—
197#2235—591no—
198#2235—591no—
199#2226—592no—
200#2208—592no—
201#2187—591no—
202#2187D5911yescharacteristic
203C4×SD32—592no—
204C4×SD32—592no—
205#2162—591no—
206C2×C59—22no—
207C2×C59—84no—
208C2×C59#310011yescharacteristic
209C2×C59C22×SD3211yescharacteristic
210C2×C59#341411yescharacteristic
211C22×D8C2×D5911yescharacteristic
212#533—1181no—
213C2×C4×D4—1181no—
214C2×SD32—592no—
215C2×SD32C2×D5912yes—
216#500—591no—
217#494—591no—
218#487—1181no—
219#467—591no—
220#467—1181no—
221#467C2×D5911yescharacteristic
222C4×Q16—591no—
223C4×Q16—591no—
224C4×Q16—1181no—
225C4×D8—592no—
226C4×D8—592no—
227C4×D8—1181no—
228C4×D8—1181no—
229C2×#107—591no—
230C2×#102—1181no—
231C2×C4×C8—591no—
232#367—592no—
233#367—592no—
234C4⋊C16—592no—
235#358—592no—
236#358—592no—
237#357—592no—
238#357C2×D5912yes—
239C22⋊C16—591no—
240C22⋊C16C2×D5911yescharacteristic
241C4×C16—592no—
242C59#3366111yescharacteristic, complemented, Sylow
243C22×D4—21no—
244C23×C4—2361no—
245C2×Q16—591no—
246C2×Q16—1181no—
247C2×Q16C22×D5911yescharacteristic
248C2×D8—24no—
249C2×D8—21no—
250C2×D8C22×D5912yes—
251C22×C8—591no—
252C22×C8C22×D5911yescharacteristic
253C4⋊Q8—1181no—
254C4⋊Q8—1181no—
255#125—2361no—
256C22⋊Q8—21no—
257C22⋊Q8—1181no—
258C22⋊Q8—2361no—
259C4×Q8—1181no—
260C4×Q8—1181no—
261C4×Q8—2361no—
262C4×D4—1182no—
263C4×D4—1182no—
264C4×D4—2361no—
265C4×D4—2361no—
266#117—2361no—
267C2×C4⋊C4—1181no—
268C2×C22⋊C4—2361no—
269C2×C42—591no—
270SD32—24no—
271SD32—1184no—
272C2×C16—592no—
273C2×C16C22×D5912yes—
274#107—591no—
275#107—592no—
276#107—1181no—
277#107—1181no—
278#107C22×D5911yescharacteristic
279#103—21no—
280#103—1181no—
281#103—1181no—
282#103—1181no—
283#102—1182no—
284#102—1182no—
285C4×C8—591no—
286C4×C8—591no—
287C4×C8—591no—
288C4×C8—591no—
289C24—41no—
290C2×Q8—21no—
291C2×Q8—1181no—
292C2×Q8—2361no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—41no—
297C22×C4—591no—
298C22×C4—591no—
299C22×C4—4724no—
300C22×C4—2362no—
301C22×C4—2362no—
302C22×C4—2361no—
303C22×C4D4×D5911yescharacteristic
304Q16—22no—
305Q16—1182no—
306Q16—2362no—
307D8—24no—
308D8—22no—
309D8#1174568714yes—
310C2×C8—22no—
311C2×C8—591no—
312C2×C8—591no—
313C2×C8—2361no—
314C2×C8D4×D5911yescharacteristic
315C2×C8C23×D5911yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—1182no—
319C4⋊C4—1181no—
320C4⋊C4—2361no—
321C4⋊C4—2361no—
322C4⋊C4—2361no—
323C4⋊C4—2361no—
324C4⋊C4—2361no—
325C4⋊C4—2361no—
326C4⋊C4—2361no—
327C22⋊C4—41no—
328C22⋊C4—2362no—
329C22⋊C4—2362no—
330C22⋊C4—2361no—
331C42—591no—
332C42—591no—
333C42—591no—
334C42—591no—
335C42—2361no—
336C42—2361no—
337C16—22no—
338C16—1182no—
339C23—44no—
340C23—42no—
341C23—82no—
342C23—82no—
343C23D8×D5911yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—4722no—
347Q8—2361no—
348Q8—2361no—
349D4—24no—
350D4—44no—
351D4—42no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—591no—
357C2×C4—591no—
358C2×C4—594no—
359C2×C4—591no—
360C2×C4—591no—
361C2×C4—4724no—
362C2×C4—4724no—
363C2×C4—4722no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2362no—
368C2×C4D8×D5911yescharacteristic
369C2×C4C2×D4×D5911yescharacteristic
370C8—21no—
371C8—1181no—
372C8—1181no—
373C8C2×#1174568712yes—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22SD32×D5912yes—
381C22C2×D8×D5911yescharacteristic
382C4—21no—
383C4—81no—
384C4—81no—
385C4—592no—
386C4—592no—
387C4—1181no—
388C4—1181no—
389C4—4722no—
390C4—4721no—
391C4—4721no—
392C4#1683790811yescharacteristic
393C4#1683819811yescharacteristic
394C2—22no—
395C2—84no—
396C2#2143409111yescharacteristic
397C2C2×SD32×D5911yescharacteristic
398C2#2143722811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2187C2
socleC22×C59C22×D8
prime radicalC22×D8×C59C22
hypercenter#2187D59
hypocenterC59#33661
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21434677C2semidirect
3. #21426294C2semidirect
4. #21426291C2semidirect
5. #21426241C2semidirect
6. #21426139C2semidirect
8. #21425951C2semidirect
9. SD32×Dic59C2semidirect
10. #21425892C2semidirect
15. #16837806C22semidirect
16. #16837780C22semidirect
20. Q16×Dic59C22semidirect
21. #16837496C22semidirect
30. #16837230C22semidirect
31. #16837229C22semidirect
32. Dic59⋊C16C22semidirect
33. #16837207C22semidirect
34. #16837206C22semidirect
37. C16×Dic59C22semidirect
162. C2×Dic59SD32semidirect
242. C59#33661semidirect

Maximal subgroups

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

subgroupindex
#214346772
C2×D8×Dic592
#214262942
#214262912
#214262412
#214261392
#214260252
#214259512
SD32×Dic592
#214258922
C59×#21872
#3366159