Finite Group Databasebeta

← #28606374

Subgroups — #28606374

Order 15104.

766 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×#16837440C211yesmaximal, characteristic
2#21426987C211yesmaximal, characteristic
3#21426979C211yesmaximal, characteristic
4#21424839C212yesmaximal, complemented
5#21424689C212yesmaximal, complemented
6#21424668C212yesmaximal, complemented
7#21424192C211yesmaximal, characteristic
8#21424192C211yesmaximal, characteristic
9#21424125C211yesmaximal, characteristic
10#21423993C212yesmaximal, complemented
11C59×#1752C211yesmaximal, characteristic
12C2×#14273686C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14#16837440C418yescomplemented
15C2×#14273661C2212yes—
16C2×#14273661C2211yescharacteristic
17C2×#14273661C2211yescharacteristic
18C2×#14273659C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20#16837300C2212yes—
21#16837300C2212yes—
22#16837296—22no—
23#16837290—22no—
24#16837269—21no—
25#16837266—21no—
26#16837262C2211yescharacteristic
27#16837262C2211yescharacteristic
28C4×C4⋊Dic59C2212yes—
29C4×C4⋊Dic59C2212yes—
30#16837253—21no—
31#16837247—21no—
32#16837165C2212yes—
33#16837165C2212yes—
34#16837165C2212yes—
35#16837165C2212yes—
36#16837165C2212yes—
37C23×C8×C59C2211yescharacteristic
38C2×C22⋊C8×C59C2211yescharacteristic
39C2×#97×C59C2212yes—
40C4×C22⋊C4×C59C2211yescharacteristic
41C59×#336C2212yes—
42#14273686—22no—
43#14273686—22no—
44#14273686C2×C414yes—
45#14273686C2×C414yes—
46C2×C4⋊Dic59C2312yes—
47C2×C4⋊Dic59C2312yes—
48C2×C4⋊Dic59C2311yescharacteristic
49C2×C4⋊Dic59C2311yescharacteristic
50C2×C4⋊Dic59C2311yescharacteristic
51C2×C4⋊Dic59C2311yescharacteristic
52C2×Dic59⋊C4—22no—
53C2×Dic59⋊C4—22no—
54C2×C22⋊Dic59—21no—
55C2×C22⋊Dic59—21no—
56C2×C22⋊Dic59—21no—
57C2×C22⋊Dic59—21no—
58C2×C4×Dic59—21no—
59C2×C4×Dic59—21no—
60C2×C4×Dic59—22no—
61C2×C4×Dic59—22no—
62#14273661—24no—
63#14273661C2×C414yes—
64#14273661C2×C414yes—
65#14273659C2×C414yes—
66#14273659C2×C414yes—
67C59⋊#95—21no—
68C59⋊#95—22no—
69C59⋊#95—21no—
70C59⋊#95—22no—
71C59⋊#95—22no—
72C59⋊#95—22no—
73C23×C4×C59C2311yescharacteristic
74C22×C8×C59—24no—
75C22×C8×C59C2×C414yes—
76C22×C8×C59C2312yes—
77C22×C8×C59C2311yescharacteristic
78C22×C8×C59C2311yescharacteristic
79C2×C22⋊C4×C59D412yes—
80C2×C42×C59C2312yes—
81C22⋊C8×C59—22no—
82#97×C59—24no—
83#95×C59D411yescharacteristic
84#95×C59D411yescharacteristic
85C24×C59C2×D411yescharacteristic
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59C22×C412yes—
89C22×C4×C59C2×D412yes—
90C22×C4×C59C2×D412yes—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4112yes—
93C22×C4×C59C2411yescharacteristic
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—22no—
99C22×Dic59—22no—
100C2×C8×C59—22no—
101C2×C8×C59—28no—
102C2×C8×C59—22no—
103C2×C8×C59—22no—
104C2×C8×C59—22no—
105C2×C8×C59C22×C412yes—
106C2×C8×C59C22×C412yes—
107C2×C8×C59#4114yes—
108C22⋊C4×C59—42no—
109C42×C59—22no—
110C42×C59—22no—
111C22⋊Dic59—24no—
112C22⋊Dic59—24no—
113C22⋊Dic59—81no—
114C22⋊Dic59—81no—
115C4⋊Dic59—21no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—22no—
120C4⋊Dic59—22no—
121C4⋊Dic59—22no—
122C4⋊Dic59—22no—
123C4⋊Dic59C22×C412yes—
124C4⋊Dic59C22×C412yes—
125C4⋊Dic59C22×C412yes—
126C4⋊Dic59C22×C412yes—
127Dic59⋊C4—44no—
128Dic59⋊C4—82no—
129C4×Dic59—24no—
130C4×Dic59—24no—
131C4×Dic59—44no—
132C4×Dic59—82no—
133C23×C59—22no—
134C23×C59—22no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59C4×D412yes—
138C23×C59C22×D411yescharacteristic
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—22no—
142C2×C4×C59—24no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—22no—
148C2×C4×C59—22no—
149C2×C4×C59C4×D412yes—
150C2×C4×C59C23×C411yescharacteristic
151C2×C4×C59C2×#4112yes—
152C2×C4×C59C2×#4111yescharacteristic
153C2×C4×C592−1+412yes—
154C2×Dic59—22no—
155C2×Dic59—22no—
156C2×Dic59—22no—
157C2×Dic59—22no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—42no—
165C2×Dic59—42no—
166C2×Dic59—42no—
167C2×Dic59—42no—
168C2×Dic59—42no—
169C2×Dic59—42no—
170C8×C59—22no—
171C8×C59—42no—
172C8×C59#11714yes—
173#22616—591nomaximal, Sylow
174C22×C59—22no—
175C22×C59—22no—
176C22×C59—22no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59C2×C4×D411yescharacteristic
180C22×C59#54712yes—
181C22×C59#57512yes—
182C22×C59#57611yescharacteristic
183C22×C59#57811yescharacteristic
184C4×C59—22no—
185C4×C59—42no—
186C4×C59—42no—
187C4×C59C2×#11711yescharacteristic
188C4×C59#51911yescharacteristic
189C4×C59#55212yes—
190Dic59—42no—
191Dic59—42no—
192Dic59—42no—
193Dic59—42no—
194Dic59—82no—
195Dic59—82no—
196C2×#427—591no—
197#2347—591no—
198#2346—591no—
199#1944—592no—
200#1921—592no—
201#1918—592no—
202#1825—591no—
203#1825—591no—
204#1808—591no—
205#1776—592no—
206#1752D5911yescharacteristic
207C2×C59—42no—
208C2×C59#234411yescharacteristic
209C2×C59#294311yescharacteristic
210C2×C59#295111yescharacteristic
211C2×C59#334712yes—
212C2×C59#334912yes—
213C23×C8C2×D5911yescharacteristic
214C2×#117—591no—
215C2×#117—591no—
216#427—598no—
217C2×#107—591no—
218C2×#107—591no—
219C2×#107—592no—
220C2×#106—591no—
221C2×#106—591no—
222C2×C22⋊C8C2×D5911yescharacteristic
223C2×#97C2×D5912yes—
224#389—592no—
225#389—592no—
226#388—1182no—
227#388—1182no—
228#383—1181no—
229#382—1181no—
230#380—591no—
231#380—591no—
232C4×C4⋊C4—592no—
233C4×C4⋊C4—592no—
234C4×C4⋊C4—1181no—
235C4×C22⋊C4C2×D5911yescharacteristic
236#376—1181no—
237#340—592no—
238#340—592no—
239#340—592no—
240#340—592no—
241#340—592no—
242#336C2×D5912yes—
243C59#2261611yescharacteristic, complemented, Sylow
244C23×C4C22×D5911yescharacteristic
245C22×C8—24no—
246C22×C8C4×D5914yes—
247C22×C8C22×D5912yes—
248C22×C8C22×D5911yescharacteristic
249C22×C8C22×D5911yescharacteristic
250#117—594no—
251#117—594no—
252#117—1182no—
253#117—1182no—
254C2×C4⋊C4—591no—
255C2×C4⋊C4—591no—
256C2×C4⋊C4—591no—
257C2×C4⋊C4—591no—
258C2×C4⋊C4—592no—
259C2×C4⋊C4—592no—
260C2×C4⋊C4—1182no—
261C2×C4⋊C4—1182no—
262C2×C22⋊C4—1181no—
263C2×C22⋊C4—1181no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4—1181no—
266C2×C22⋊C4D23612yes—
267C2×C42—1181no—
268C2×C42—1181no—
269C2×C42—1182no—
270C2×C42—1182no—
271C2×C42C22×D5912yes—
272#107—594no—
273#107—594no—
274#107—1184no—
275#106—594no—
276#106—594no—
277C22⋊C8—22no—
278#97—24no—
279#95—1181no—
280#95—1182no—
281#95—1182no—
282#95—1182no—
283#95—1181no—
284#95—1182no—
285#95D23611yescharacteristic
286#95D23611yescharacteristic
287C24C2×D23611yescharacteristic
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—1181no—
291C22×C4—1181no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—1182no—
295C22×C4—1182no—
296C22×C4C2×C4×D5912yes—
297C22×C4C2×D23612yes—
298C22×C4C2×D23612yes—
299C22×C4C2×D23611yescharacteristic
300C22×C4#1174568512yes—
301C22×C4C23×D5911yescharacteristic
302C2×C8—22no—
303C2×C8—28no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8C2×C4×D5912yes—
308C2×C8C2×C4×D5912yes—
309C2×C8#1174568714yes—
310C4⋊C4—592no—
311C4⋊C4—592no—
312C4⋊C4—592no—
313C4⋊C4—592no—
314C4⋊C4—1181no—
315C4⋊C4—1181no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1182no—
319C4⋊C4—1182no—
320C4⋊C4—1182no—
321C4⋊C4—1182no—
322C4⋊C4—4722no—
323C4⋊C4—2364no—
324C22⋊C4—42no—
325C22⋊C4—1184no—
326C22⋊C4—1184no—
327C22⋊C4—4721no—
328C22⋊C4—4721no—
329C42—22no—
330C42—22no—
331C42—1184no—
332C42—1184no—
333C42—4722no—
334C42—2364no—
335C23—22no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23C4×D23612yes—
340C23C22×D23611yescharacteristic
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—1182no—
352C2×C4—1182no—
353C2×C4—1182no—
354C2×C4—1182no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4—2362no—
361C2×C4—2362no—
362C2×C4—2362no—
363C2×C4—2361no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4C4×D23612yes—
368C2×C4C22×C4×D5911yescharacteristic
369C2×C4C2×#1174568712yes—
370C2×C4C2×#1174568511yescharacteristic
371C2×C4#1427377412yes—
372C8—22no—
373C8—42no—
374C8#1427368314yes—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22C2×C4×D23611yescharacteristic
381C22#1683805612yes—
382C22#1683820212yes—
383C22#1683820711yescharacteristic
384C22#1683821911yescharacteristic
385C4—22no—
386C4—42no—
387C4—42no—
388C4—4722no—
389C4—4722no—
390C4—2362no—
391C4—2362no—
392C4—2362no—
393C4—2362no—
394C4C2×#1427368311yescharacteristic
395C4#1683787911yescharacteristic
396C4#1683809212yes—
397C2—42no—
398C2#2142696311yescharacteristic
399C2#2143289811yescharacteristic
400C2#2143297311yescharacteristic
401C2#2143659812yes—
402C2#2143662812yes—
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1752C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1752D59
hypocenterC59#22616
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
4. #21424839C2semidirect
5. #21424689C2semidirect
6. #21424668C2semidirect
10. #21423993C2semidirect
14. #16837440C4semidirect
243. C59#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#168374402
#214269872
#214269792
#214248392
#214246892
#214246682
#214241922
#214241252
#214239932
C59×#17522
#2261659