Finite Group Databasebeta

← #28743775

Subgroups — #28743775

Order 15104.

548 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#21434682C211yesmaximal, characteristic, complemented
2#21426304C212yesmaximal, complemented
3#21426248C211yesmaximal, characteristic, complemented
4#21426243C211yesmaximal, characteristic, complemented
5#21426146C211yesmaximal, characteristic, complemented
6#21426141C211yesmaximal, characteristic, complemented
7#21426036C212yesmaximal
8#21425965C212yesmaximal, complemented
9#21425960C212yesmaximal, complemented
10#21425899C211yesmaximal, characteristic, complemented
11C4×#135×C59C211yesmaximal, characteristic
12#16837959—21no—
13C2×#14273673C2212yes—
14C2×#14273671C2212yes—
15#16837807C2211yescharacteristic
16#16837782C2211yescharacteristic
17#16837753—21no—
18#16837643—21no—
19#16837643C2212yes—
20C4×Dic236—21no—
21C4×Dic236C2212yes—
22#16837229C2212yes—
23#16837227C2212yes—
24C4⋊(C59⋊C16)C2211yescharacteristic
25C4⋊(C59⋊C16)C2211yescharacteristic
26#16837206C2212yes—
27#16837205C2212yes—
28#16837205C2212yes—
29#16837203C2212yes—
30C22⋊(C59⋊C16)C2212yes—
31#16837178C2212yes—
32C2×#135×C59C2211yescharacteristic
33C4×#41×C59—21no—
34C4×Q16×C59C2211yescharacteristic
35C4×Q16×C59C2211yescharacteristic
36C4×SD16×C59—21no—
37C4×SD16×C59—21no—
38C4×D8×C59C2211yescharacteristic
39C4×D8×C59C2211yescharacteristic
40C59×#427C2211yescharacteristic
41C59×#416—21no—
42C2×C4×C8×C59C2211yescharacteristic
43C2×Dic236—21no—
44C2×Dic236C2312yes—
45#14273727—21no—
46#14273727—21no—
47#14273719—41no—
48#14273706—22no—
49#14273706—41no—
50C4×Dic118—22no—
51C4×Dic118—41no—
52#14273686—41no—
53#14273673—24no—
54#14273671—24no—
55C2×C59⋊C16C2312yes—
56C2×C59⋊C16C2312yes—
57#14273661—21no—
58#14273661C2312yes—
59#14273652—22no—
60#14273652—22no—
61C2×#41×C59—21no—
62#135×C59—22no—
63#135×C59—22no—
64C2×Q16×C59C2311yescharacteristic
65C2×SD16×C59—21no—
66C2×D8×C59C2311yescharacteristic
67C22×C8×C59D411yescharacteristic
68C22×C8×C59C2311yescharacteristic
69C4×Q8×C59—21no—
70C4×Q8×C59—21no—
71C4×D4×C59—21no—
72C4×D4×C59—21no—
73C4×D4×C59—41no—
74C4×D4×C59—41no—
75#117×C59—21no—
76#117×C59—41no—
77C2×C42×C59—41no—
78C2×C42×C59D411yescharacteristic
79#107×C59C2311yescharacteristic
80#107×C59C2311yescharacteristic
81#106×C59—21no—
82#103×C59—21no—
83#103×C59—21no—
84#102×C59—21no—
85#102×C59—21no—
86C4×C8×C59D411yescharacteristic
87C4×C8×C59D411yescharacteristic
88C4×C8×C59C2311yescharacteristic
89C4×C8×C59C2311yescharacteristic
90#41×C59—24no—
91#41×C59—42no—
92C2×Q8×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59D811yescharacteristic
99C22×C4×C59D811yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C2×Dic118—22no—
102C2×Dic118—41no—
103Q16×C59—21no—
104Q16×C59#4112yes—
105SD16×C59—24no—
106D8×C59—21no—
107D8×C59#4112yes—
108C2×C8×C59—21no—
109C2×C8×C59—21no—
110C2×C8×C59—41no—
111C2×C8×C59C2×D411yescharacteristic
112C2×C8×C59C2×D411yescharacteristic
113C2×C8×C59C2×D411yescharacteristic
114C2×C8×C59C2411yescharacteristic
115C4⋊C4×C59—21no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—41no—
118C4⋊C4×C59—41no—
119C22⋊C4×C59—41no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C42×C59—41no—
123C42×C59—41no—
124C42×C59—41no—
125C42×C59D811yescharacteristic
126C42×C59D811yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59C2×D411yescharacteristic
129C22⋊Dic59—42no—
130C4⋊Dic59—22no—
131C4⋊Dic59—41no—
132C4⋊Dic59—41no—
133C4⋊Dic59—41no—
134Dic59⋊C4—42no—
135Dic59⋊C4—42no—
136C4×Dic59—42no—
137Dic236—22no—
138Dic236—22no—
139Dic236—42no—
140C59⋊C16—22no—
141C59⋊C16—22no—
142C23×C59—41no—
143C23×C59C2×D811yescharacteristic
144Q8×C59—22no—
145Q8×C59—41no—
146D4×C59—22no—
147D4×C59—44no—
148D4×C59—41no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—22no—
152C2×C4×C59—41no—
153C2×C4×C59—44no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—82no—
158C2×C4×C59—82no—
159C2×C4×C59C2×D811yescharacteristic
160C2×C4×C59C2×D811yescharacteristic
161C2×C4×C59C2×D811yescharacteristic
162C2×C4×C59C2×D811yescharacteristic
163C2×C4×C59C2×D811yescharacteristic
164C2×C4×C59C22×D411yescharacteristic
165C2×Dic59—42no—
166C2×Dic59—42no—
167Dic118—42no—
168Dic118—42no—
169Dic118—82no—
170C8×C59—21no—
171C8×C59—21no—
172C8×C59—21no—
173C8×C59C2×#4111yescharacteristic
174C8×C59C2×#4111yescharacteristic
175#33676—591nomaximal, Sylow
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—42no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59C22×D811yescharacteristic
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—42no—
188C4×C59—81no—
189C4×C59—81no—
190C4×C59#51012yes—
191C4×C59#52511yescharacteristic
192C4×C59#57511yescharacteristic
193Dic59—82no—
194Dic59—82no—
195#3170—591no—
196C4×#135D5911yescharacteristic
197#2237—592no—
198#2227—591no—
199#2226—591no—
200#2209—591no—
201#2208—591no—
202#2189—592no—
203#2176—592no—
204#2175—592no—
205#2163—591no—
206C2×C59—41no—
207C2×C59—82no—
208C2×C59#310011yescharacteristic
209C2×C59C2×#51011yescharacteristic
210C2×C59#341411yescharacteristic
211C2×#135C2×D5911yescharacteristic
212#533—1181no—
213C4×#41—21no—
214C2×Q32—592no—
215C2×SD32—592no—
216#500—591no—
217#494—591no—
218#487—1181no—
219#467—592no—
220#467—1181no—
221C4×Q16—592no—
222C4×Q16—1181no—
223C4×Q16C2×D5911yescharacteristic
224C4×Q16C2×D5911yescharacteristic
225C4×SD16—21no—
226C4×SD16—21no—
227C4×D8C2×D5911yescharacteristic
228C4×D8C2×D5911yescharacteristic
229#427C2×D5911yescharacteristic
230#416—21no—
231C2×C4×C8C2×D5911yescharacteristic
232#367—592no—
233#366—592no—
234C4⋊C16—591no—
235C4⋊C16—591no—
236#358—592no—
237#358—592no—
238#358—592no—
239#357—592no—
240C22⋊C16—592no—
241#346—592no—
242C59#3367611yescharacteristic, complemented, Sylow
243C2×#41—21no—
244#135—22no—
245#135—22no—
246C2×Q16—592no—
247C2×Q16—1181no—
248C2×Q16C22×D5911yescharacteristic
249C2×SD16—21no—
250C2×D8C22×D5911yescharacteristic
251C22×C8C22⋊D5911yescharacteristic
252C22×C8C22×D5911yescharacteristic
253C4⋊Q8—1181no—
254C4⋊Q8—1181no—
255#125—2361no—
256C22⋊Q8—1182no—
257C22⋊Q8—2361no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—1182no—
261C4×Q8—2361no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266#117—21no—
267#117—41no—
268#117—2361no—
269C2×C42—41no—
270C2×C42C22⋊D5911yescharacteristic
271Q32—1184no—
272SD32—1184no—
273C2×C16—592no—
274C2×C16—592no—
275#107—592no—
276#107—1181no—
277#107C22×D5911yescharacteristic
278#107C22×D5911yescharacteristic
279#106—21no—
280#103—21no—
281#103—21no—
282#103—1182no—
283#103—1182no—
284#102—21no—
285#102—21no—
286C4×C8C22⋊D5911yescharacteristic
287C4×C8C22⋊D5911yescharacteristic
288C4×C8C22×D5911yescharacteristic
289C4×C8C22×D5911yescharacteristic
290#41—24no—
291#41—42no—
292C2×Q8—21no—
293C2×Q8—1182no—
294C2×Q8—2361no—
295C2×D4—21no—
296C2×D4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4#1174566911yescharacteristic
301C22×C4#1174566911yescharacteristic
302C22×C4C2×C22⋊D5911yescharacteristic
303Q16—21no—
304Q16—1182no—
305Q16—1182no—
306Q16—2362no—
307Q16#1174568512yes—
308SD16—24no—
309D8—21no—
310D8#1174568512yes—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8C2×C22⋊D5911yescharacteristic
315C2×C8C2×C22⋊D5911yescharacteristic
316C2×C8C2×C22⋊D5911yescharacteristic
317C2×C8C23×D5911yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—1182no—
323C4⋊C4—2361no—
324C4⋊C4—2361no—
325C4⋊C4—2362no—
326C4⋊C4—2362no—
327C4⋊C4—2361no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—2362no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—2362no—
336C42#1174566911yescharacteristic
337C42#1174566911yescharacteristic
338C42C2×C22⋊D5911yescharacteristic
339C42C2×C22⋊D5911yescharacteristic
340C16—1182no—
341C16—1182no—
342C23—41no—
343C23C2×#1174566911yescharacteristic
344Q8—22no—
345Q8—41no—
346Q8—4722no—
347Q8—2362no—
348Q8—2362no—
349D4—22no—
350D4—44no—
351D4—41no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—2362no—
363C2×C4—2362no—
364C2×C4C2×#1174566911yescharacteristic
365C2×C4C2×#1174566911yescharacteristic
366C2×C4C2×#1174566911yescharacteristic
367C2×C4C2×#1174566911yescharacteristic
368C2×C4C2×#1174566911yescharacteristic
369C2×C4C22×C22⋊D5911yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8C2×#1174568511yescharacteristic
374C8C2×#1174568511yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22C22×#1174566911yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—4722no—
390C4—4722no—
391C4#1683784212yes—
392C4#1683791311yescharacteristic
393C4#1683820011yescharacteristic
394C2—41no—
395C2—82no—
396C2#2143409611yescharacteristic
397C2C2×#1683784211yescharacteristic
398C2#2143723011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC4×#135×C59C2
socleC22×C59C22×D8
prime radicalC2×#135×C59C22
hypercenterC4×#135D59
hypocenterC59#33676
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #21434682C2semidirect
2. #21426304C2semidirect
3. #21426248C2semidirect
4. #21426243C2semidirect
5. #21426146C2semidirect
6. #21426141C2semidirect
8. #21425965C2semidirect
9. #21425960C2semidirect
10. #21425899C2semidirect
242. C59#33676semidirect

Maximal subgroups

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

subgroupindex
#214346822
#214263042
#214262482
#214262432
#214261462
#214261412
#214260362
#214259652
#214259602
#214258992
C4×#135×C592
#3367659