Finite Group Databasebeta

← #28591887

Subgroups — #28591887

Order 15104.

814 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11745672C211yesmaximal, characteristic
2#21426742C211yesmaximal, characteristic
3#21426461C211yesmaximal, characteristic
4#21424735C212yesmaximal
5#21424665C212yesmaximal
6#21424435C212yesmaximal
7#21424348C212yesmaximal
8C4×#14273660C211yesmaximal, characteristic
9#21423927C211yesmaximal, characteristic
10#21423849C211yesmaximal, characteristic
11C59×#1763C211yesmaximal, characteristic
12C22×#11745672C2211yescharacteristic
13C2×Q8×Dic59C2212yes—
14C2×C4×Dic118C2211yescharacteristic
15C4×#11745672C418yescomplemented
16C2×#14273660C2212yes—
17C2×#14273660C2211yescharacteristic
18C2×#14273652C2212yes—
19C2×#14273651C2211yescharacteristic
20C2×#14273650C2211yescharacteristic
21C2×C4×C59⋊C8C2211yescharacteristic
22#16837307C2212yes—
23#16837298C2212yes—
24#16837286—22no—
25#16837269—21no—
26#16837266—21no—
27C4⋊C4×Dic59—22no—
28C4⋊C4×Dic59C2212yes—
29C4×C4⋊Dic59—21no—
30C4×C4⋊Dic59C2211yescharacteristic
31#16837248—21no—
32#16837248C2211yescharacteristic
33#16837165C2212yes—
34#16837164C2212yes—
35#16837156C2212yes—
36#16837156C2212yes—
37C2×C4×Q8×C59C2211yescharacteristic
38C2×#103×C59C2212yes—
39C2×#97×C59C2211yescharacteristic
40C4×C4⋊C4×C59C2211yescharacteristic
41C59×#340C2212yes—
42C22×Dic118C2311yescharacteristic
43C2×#11745672—24no—
44C2×#11745672C2×C414yes—
45C22×C59⋊C8C2311yescharacteristic
46C22×C59⋊C8C2311yescharacteristic
47Q8×Dic59—24no—
48Q8×Dic59—24no—
49C4×Dic118—22no—
50C4×Dic118C2×C414yes—
51C2×C4⋊Dic59—22no—
52C2×C4⋊Dic59C2311yescharacteristic
53C2×C4⋊Dic59C2312yes—
54C2×Dic59⋊C4—21no—
55C2×Dic59⋊C4—22no—
56C2×C4×Dic59—21no—
57C2×C4×Dic59—22no—
58C2×C4×Dic59—21no—
59C2×C4×Dic59—22no—
60C2×C4×Dic59C2312yes—
61#14273660—24no—
62#14273660C2×C414yes—
63#14273652—24no—
64#14273651C2×C414yes—
65#14273650C2×C414yes—
66C4×C59⋊C8C2×C414yes—
67C59⋊#95—21no—
68C59⋊#95—22no—
69C59⋊#95—22no—
70C59⋊#95—22no—
71C59⋊#95—22no—
72C59⋊#95D411yescharacteristic
73C59⋊#95D411yescharacteristic
74C59⋊#95D412yes—
75C22×Q8×C59C2311yescharacteristic
76C22×C8×C59C2312yes—
77C4×Q8×C59—22no—
78C4×Q8×C59C2×C414yes—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59C2312yes—
81C2×C4⋊C4×C59C2311yescharacteristic
82C2×C42×C59—21no—
83C2×C42×C59—21no—
84C2×C42×C59C2311yescharacteristic
85#103×C59—24no—
86#97×C59—22no—
87#95×C59—21no—
88C2×Q8×C59—21no—
89C2×Q8×C59—22no—
90C2×Q8×C59C22×C412yes—
91C2×Q8×C59#4114yes—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—22no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2411yescharacteristic
98C22×Dic59—21no—
99C22×Dic59—22no—
100C22×Dic59—21no—
101C22×Dic59C2×D412yes—
102C22×Dic59C2×D412yes—
103C2×Dic118—22no—
104C2×Dic118—22no—
105C2×Dic118—21no—
106C2×Dic118C22×C412yes—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—22no—
110C4⋊C4×C59—24no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—22no—
113C4⋊C4×C59—22no—
114C4⋊C4×C59C22×C412yes—
115C42×C59—24no—
116C42×C59—81no—
117C42×C59C22×C412yes—
118C42×C59#4112yes—
119#11745672—28no—
120C4⋊Dic59—28no—
121C4⋊Dic59—21no—
122C4⋊Dic59—22no—
123C4⋊Dic59C22×C412yes—
124C4⋊Dic59#4114yes—
125Dic59⋊C4—24no—
126Dic59⋊C4—82no—
127C4×Dic59—28no—
128C4×Dic59—24no—
129C4×Dic59—22no—
130C4×Dic59—22no—
131C4×Dic59—82no—
132C4×Dic59—81no—
133C2×C59⋊C8—22no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8C22×C412yes—
136C2×C59⋊C8C22×C412yes—
137C23×C59C22×D411yescharacteristic
138Q8×C59—22no—
139Q8×C59—24no—
140Q8×C59#11714yes—
141C2×C4×C59—22no—
142C2×C4×C59—24no—
143C2×C4×C59—21no—
144C2×C4×C59—22no—
145C2×C4×C59—21no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—42no—
152C2×C4×C59C4×D412yes—
153C2×C4×C59C4×D412yes—
154C2×C4×C59C23×C411yescharacteristic
155C2×C4×C59C2×#4112yes—
156C2×C4×C59C2×#4111yescharacteristic
157C2×C4×C59C2×#4112yes—
158C2×Dic59—22no—
159C2×Dic59—22no—
160C2×Dic59—22no—
161C2×Dic59—22no—
162C2×Dic59—24no—
163C2×Dic59—22no—
164C2×Dic59—22no—
165C2×Dic59—22no—
166C2×Dic59—22no—
167C2×Dic59—42no—
168C2×Dic59—42no—
169C2×Dic59—41no—
170C2×Dic59—42no—
171C2×Dic59—41no—
172Dic118—22no—
173Dic118—22no—
174Dic118—42no—
175C8×C59—42no—
176C59⋊C8—22no—
177C59⋊C8—22no—
178#21135—591nomaximal, Sylow
179C22×C59C2×C4×D411yescharacteristic
180C22×C59#52512yes—
181C22×C59#57611yescharacteristic
182C22×C59#57812yes—
183C22×C59#57811yescharacteristic
184C4×C59—22no—
185C4×C59—22no—
186C4×C59—24no—
187C4×C59—42no—
188C4×C59—82no—
189C4×C59C2×#11711yescharacteristic
190C4×C59C4×#4111yescharacteristic
191C4×C59#52912yes—
192Dic59—42no—
193Dic59—44no—
194Dic59—42no—
195Dic59—42no—
196Dic59—42no—
197C2×C4×Q16—591no—
198#2320—591no—
199#2276—591no—
200#1928—592no—
201#1918—592no—
202#1878—592no—
203#1863—592no—
204C4×#107—591no—
205#1763—591no—
206#1763D5911yescharacteristic
207#1746—591no—
208C2×C59#229011yescharacteristic
209C2×C59#295011yescharacteristic
210C2×C59#295111yescharacteristic
211C2×C59#312412yes—
212C2×C59#312712yes—
213C22×Q16—591no—
214C2×C4×Q8—592no—
215C2×C4×Q8—591no—
216C2×C4×Q8C2×D5911yescharacteristic
217C4×Q16—598no—
218C2×#107—591no—
219C2×#107—592no—
220C2×#103—591no—
221C2×#103—591no—
222C2×#103—592no—
223C2×#103C2×D5912yes—
224C2×#97C2×D5911yescharacteristic
225C2×C4×C8—591no—
226#391—592no—
227#389—592no—
228#387—1182no—
229#383—1181no—
230#382—1181no—
231C4×C4⋊C4—592no—
232C4×C4⋊C4—591no—
233C4×C4⋊C4—1181no—
234C4×C4⋊C4—1182no—
235C4×C4⋊C4C2×D5911yescharacteristic
236#376—591no—
237#376—1181no—
238#340—592no—
239#340—592no—
240#340C2×D5912yes—
241#336—592no—
242#336—592no—
243C59#2113511yescharacteristic, complemented, Sylow
244C22×Q8—591no—
245C22×Q8C22×D5911yescharacteristic
246C2×Q16—594no—
247C2×Q16—1184no—
248C22×C8—591no—
249C22×C8—591no—
250C22×C8C22×D5912yes—
251C4×Q8—22no—
252C4×Q8—594no—
253C4×Q8—1184no—
254C4×Q8—1184no—
255C4×Q8—1182no—
256C4×Q8C4×D5914yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—592no—
259C2×C4⋊C4—591no—
260C2×C4⋊C4—1182no—
261C2×C4⋊C4—1181no—
262C2×C4⋊C4—1182no—
263C2×C4⋊C4C22×D5912yes—
264C2×C4⋊C4C22×D5911yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—592no—
268C2×C42—1182no—
269C2×C42—1181no—
270C2×C42—1182no—
271C2×C42—1181no—
272C2×C42C22×D5911yescharacteristic
273#107—594no—
274#107—1184no—
275#103—24no—
276#103—594no—
277#103—594no—
278#103—1184no—
279#97—22no—
280C4×C8—594no—
281#95—21no—
282#95—591no—
283#95—591no—
284#95—592no—
285#95—1182no—
286#95—1182no—
287#95—1182no—
288#95—1182no—
289#95—1181no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—592no—
293C2×Q8—1181no—
294C2×Q8—1182no—
295C2×Q8—1182no—
296C2×Q8C2×C4×D5912yes—
297C2×Q8#1174568514yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—592no—
302C22×C4—592no—
303C22×C4—1181no—
304C22×C4—1182no—
305C22×C4—1181no—
306C22×C4D4×D5911yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4C23×D5911yescharacteristic
309Q16—1188no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—592no—
314C2×C8—592no—
315C2×C8—1182no—
316C2×C8—1182no—
317C4⋊C4—24no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—592no—
322C4⋊C4—594no—
323C4⋊C4—1188no—
324C4⋊C4—1184no—
325C4⋊C4—1181no—
326C4⋊C4—1182no—
327C4⋊C4—4722no—
328C4⋊C4C2×C4×D5912yes—
329C42—24no—
330C42—81no—
331C42—1188no—
332C42—1184no—
333C42—1182no—
334C42—1182no—
335C42—4721no—
336C42—4722no—
337C42C2×C4×D5912yes—
338C42#1174568712yes—
339C23C2×D4×D5911yescharacteristic
340Q8—24no—
341Q8—22no—
342Q8—1182no—
343Q8—1182no—
344Q8—2362no—
345Q8#1427368414yes—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—1182no—
358C2×C4—1184no—
359C2×C4—1182no—
360C2×C4—1182no—
361C2×C4—1182no—
362C2×C4—1182no—
363C2×C4—1182no—
364C2×C4—1182no—
365C2×C4—1182no—
366C2×C4—2362no—
367C2×C4—2362no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2362no—
371C2×C4(C22⋊D59)⋊C412yes—
372C2×C4(C22⋊D59)⋊C412yes—
373C2×C4C22×C4×D5911yescharacteristic
374C2×C4#41×D5912yes—
375C2×C4C2×#1174568711yescharacteristic
376C2×C4C2×#1174568512yes—
377C8—42no—
378C8—1182no—
379C8—1182no—
380C22C2×(C22⋊D59)⋊C411yescharacteristic
381C22#1683791112yes—
382C22#1683820611yescharacteristic
383C22#1683821811yescharacteristic
384C22#1683821812yes—
385C4—24no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—82no—
390C4—2362no—
391C4—2362no—
392C4—2362no—
393C4—2362no—
394C4—2364no—
395C4C2×#1427368411yescharacteristic
396C4#1683787011yescharacteristic
397C4#1683793212yes—
398C2#2142653511yescharacteristic
399C2#2143296011yescharacteristic
400C2#2143296711yescharacteristic
401C2#2143435612yes—
402C2#2143439512yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1763C2
socle, prime radicalC23×C59C22×D4
cosocleC22×C4×C59C24
hypercenter#1763D59
hypocenterC59#21135
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
15. C4×#11745672C4semidirect
243. C59#21135semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117456722
#214267422
#214264612
#214247352
#214246652
#214244352
#214243482
C4×#142736602
#214239272
#214238492
C59×#17632
#2113559