Finite Group Databasebeta

← #28592957

Subgroups — #28592957

Order 15104.

766 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
1C2×C4×#11745672C211yesmaximal, characteristic
2#21426748C211yesmaximal, characteristic
3#21424956C212yesmaximal
4#21424826C212yesmaximal
5#21424789C212yesmaximal
6#21424785C212yesmaximal
7C4×#14273660C211yesmaximal, characteristic
8#21423929C211yesmaximal, characteristic
9#21423926C211yesmaximal, characteristic
10#21423783C211yesmaximal, characteristic
11C59×#2320C211yesmaximal, characteristic
12C22×#11745672C2211yescharacteristic
13C2×C4×Dic118C2211yescharacteristic
14C4×#11745672C418yescomplemented
15C2×#14273660C2212yes—
16C2×#14273660C2211yescharacteristic
17C2×#14273651C2212yes—
18C2×#14273651C2211yescharacteristic
19C2×#14273650C2212yes—
20C2×#14273650C2211yescharacteristic
21C2×#14273638C2211yescharacteristic
22C2×#14273638C2212yes—
23C2×C4×C59⋊C8C2212yes—
24C2×C4×C59⋊C8C2211yescharacteristic
25#16837308C2212yes—
26#16837300C2212yes—
27#16837284—22no—
28#16837269—21no—
29#16837266—21no—
30C4×C4⋊Dic59C2211yescharacteristic
31#16837164C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34C2×C4×Q8×C59C2211yescharacteristic
35C59×#391C2212yes—
36C59×#389C2212yes—
37C59×#387—22no—
38C59×#383—21no—
39C59×#382—21no—
40C4×C4⋊C4×C59C2211yescharacteristic
41C59×#376C2211yescharacteristic
42C22×Dic118C2311yescharacteristic
43C2×#11745672—24no—
44C2×#11745672C2×C414yes—
45C22×C59⋊C8C2311yescharacteristic
46C22×C59⋊C8C2312yes—
47C22×C59⋊C8C2311yescharacteristic
48C4×Dic118—22no—
49C4×Dic118C2×C414yes—
50C2×C4⋊Dic59C2312yes—
51C2×C4⋊Dic59C2311yescharacteristic
52C2×Dic59⋊C4—21no—
53C2×Dic59⋊C4—22no—
54C2×C4×Dic59—21no—
55C2×C4×Dic59—21no—
56#14273660—24no—
57#14273660C2×C414yes—
58#14273651—24no—
59#14273651C2×C414yes—
60#14273650—24no—
61#14273650C2×C414yes—
62#14273638—22no—
63#14273638—24no—
64C4×C59⋊C8—24no—
65C4×C59⋊C8C2×C414yes—
66C59⋊#95—21no—
67C59⋊#95—22no—
68C59⋊#95—22no—
69C59⋊#95—22no—
70C22×Q8×C59C2311yescharacteristic
71C4×Q8×C59—22no—
72C4×Q8×C59C2×C414yes—
73C2×C4⋊C4×C59—21no—
74C2×C4⋊C4×C59—22no—
75C2×C4⋊C4×C59C2312yes—
76C2×C4⋊C4×C59C2311yescharacteristic
77C2×C42×C59—21no—
78C2×C42×C59—21no—
79C2×C42×C59C2312yes—
80C2×C42×C59C2311yescharacteristic
81#95×C59—21no—
82#95×C59—22no—
83#95×C59—22no—
84#95×C59—22no—
85#95×C59D411yescharacteristic
86#95×C59D411yescharacteristic
87#95×C59D412yes—
88C2×Q8×C59—21no—
89C2×Q8×C59—22no—
90C2×Q8×C59—22no—
91C2×Q8×C59C22×C412yes—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—22no—
95C22×C4×C59C2×D412yes—
96C22×C4×C59C2×D412yes—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2411yescharacteristic
100C22×Dic59—21no—
101C22×Dic59—21no—
102C22×Dic59—22no—
103C2×Dic118—22no—
104C2×Dic118—22no—
105C2×Dic118—21no—
106C2×Dic118C22×C412yes—
107C4⋊C4×C59—24no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—82no—
112C4⋊C4×C59C22×C412yes—
113C42×C59—24no—
114C42×C59—22no—
115C42×C59—22no—
116C42×C59—81no—
117C42×C59C22×C412yes—
118C42×C59#4112yes—
119#11745672—28no—
120C4⋊Dic59—21no—
121C4⋊Dic59—22no—
122C4⋊Dic59—22no—
123C4⋊Dic59C22×C412yes—
124Dic59⋊C4—24no—
125Dic59⋊C4—82no—
126C4×Dic59—24no—
127C4×Dic59—81no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8C22×C412yes—
132C2×C59⋊C8C22×C412yes—
133C2×C59⋊C8#4114yes—
134C2×C59⋊C8#4114yes—
135C23×C59C22×D411yescharacteristic
136Q8×C59—22no—
137Q8×C59—22no—
138Q8×C59—42no—
139C2×C4×C59—21no—
140C2×C4×C59—22no—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
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×C59—22no—
150C2×C4×C59—42no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59C4×D412yes—
157C2×C4×C59C4×D412yes—
158C2×C4×C59C23×C411yescharacteristic
159C2×C4×C59C2×#4112yes—
160C2×C4×C59C2×#4112yes—
161C2×C4×C59C2×#4111yescharacteristic
162C2×Dic59—22no—
163C2×Dic59—22no—
164C2×Dic59—42no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—42no—
168C2×Dic59—42no—
169C2×Dic59—42no—
170Dic118—22no—
171Dic118—22no—
172Dic118—42no—
173C59⋊C8—22no—
174C59⋊C8—24no—
175C59⋊C8#11714yes—
176#21281—591nomaximal, Sylow
177C22×C59C2×C4×D411yescharacteristic
178C22×C59#53212yes—
179C22×C59#57611yescharacteristic
180C22×C59#57812yes—
181C22×C59#57811yescharacteristic
182C4×C59—22no—
183C4×C59—22no—
184C4×C59—42no—
185C4×C59—42no—
186C4×C59—42no—
187C4×C59—42no—
188C4×C59—82no—
189C4×C59C2×#11711yescharacteristic
190C4×C59C4×#4111yescharacteristic
191C4×C59#52912yes—
192Dic59—42no—
193Dic59—42no—
194Dic59—82no—
195C2×C4×Q16—591no—
196#2320—591no—
197#2320D5911yescharacteristic
198#1969—592no—
199#1942—592no—
200#1935—592no—
201#1935—592no—
202C4×#107—591no—
203#1763—591no—
204#1763—591no—
205#1726—591no—
206C2×C59#229311yescharacteristic
207C2×C59#295011yescharacteristic
208C2×C59#295111yescharacteristic
209C2×C59#317512yes—
210C2×C59#317712yes—
211C22×Q16—591no—
212C2×C4×Q8—591no—
213C2×C4×Q8C2×D5911yescharacteristic
214C4×Q16—598no—
215C2×#107—591no—
216C2×#107—592no—
217C2×#103—591no—
218C2×#103—591no—
219C2×#103—592no—
220C2×#103—592no—
221C2×#97—592no—
222C2×#97—591no—
223C2×C4×C8—591no—
224C2×C4×C8—592no—
225#391—592no—
226#391C2×D5912yes—
227#389—592no—
228#389C2×D5912yes—
229#387—22no—
230#387—1182no—
231#383—21no—
232#383—1181no—
233#382—21no—
234#382—1181no—
235C4×C4⋊C4—591no—
236C4×C4⋊C4C2×D5911yescharacteristic
237#376C2×D5911yescharacteristic
238#340—592no—
239#340—592no—
240#340—592no—
241C59#2128111yescharacteristic, complemented, Sylow
242C22×Q8—591no—
243C22×Q8C22×D5911yescharacteristic
244C2×Q16—594no—
245C2×Q16—1184no—
246C22×C8—591no—
247C22×C8—591no—
248C22×C8—592no—
249C4×Q8—22no—
250C4×Q8—594no—
251C4×Q8—1182no—
252C4×Q8C4×D5914yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—591no—
256C2×C4⋊C4—592no—
257C2×C4⋊C4—1181no—
258C2×C4⋊C4—1182no—
259C2×C4⋊C4C22×D5912yes—
260C2×C4⋊C4C22×D5911yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—1181no—
264C2×C42—1181no—
265C2×C42C22×D5912yes—
266C2×C42C22×D5911yescharacteristic
267#107—594no—
268#107—1184no—
269#103—594no—
270#103—594no—
271#103—1184no—
272#103—1184no—
273#97—1184no—
274#97—1182no—
275C4×C8—594no—
276C4×C8—1184no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—1181no—
282#95—1182no—
283#95—1182no—
284#95—1182no—
285#95C22⋊D5911yescharacteristic
286#95C22⋊D5911yescharacteristic
287#95C22⋊D5912yes—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—592no—
292C2×Q8—1181no—
293C2×Q8—1182no—
294C2×Q8—1182no—
295C2×Q8C2×C4×D5912yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—1181no—
300C22×C4—1181no—
301C22×C4—1182no—
302C22×C4C2×C22⋊D5912yes—
303C22×C4C2×C22⋊D5912yes—
304C22×C4C2×C22⋊D5911yescharacteristic
305C22×C4C2×C22⋊D5911yescharacteristic
306C22×C4C23×D5911yescharacteristic
307Q16—1188no—
308C2×C8—592no—
309C2×C8—594no—
310C2×C8—592no—
311C2×C8—594no—
312C2×C8—1182no—
313C2×C8—1182no—
314C2×C8—1182no—
315C4⋊C4—24no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—82no—
320C4⋊C4—592no—
321C4⋊C4—1184no—
322C4⋊C4—1181no—
323C4⋊C4—1182no—
324C4⋊C4—1182no—
325C4⋊C4—4722no—
326C4⋊C4C2×C4×D5912yes—
327C42—24no—
328C42—22no—
329C42—22no—
330C42—81no—
331C42—1184no—
332C42—4721no—
333C42C2×C4×D5912yes—
334C42#1174568512yes—
335C23C22×C22⋊D5911yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—42no—
339Q8—1182no—
340Q8—1182no—
341Q8—2362no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—1182no—
360C2×C4—1182no—
361C2×C4—2362no—
362C2×C4—2362no—
363C2×C4—2362no—
364C2×C4—2362no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4C4×C22⋊D5912yes—
368C2×C4C4×C22⋊D5912yes—
369C2×C4C22×C4×D5911yescharacteristic
370C2×C4#41×D5912yes—
371C2×C4#41×D5912yes—
372C2×C4C2×#1174568511yescharacteristic
373C8—594no—
374C8—1184no—
375C8—1182no—
376C22C2×C4×C22⋊D5911yescharacteristic
377C22#1683794812yes—
378C22#1683820511yescharacteristic
379C22#1683821712yes—
380C22#1683821711yescharacteristic
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—4722no—
389C4—2362no—
390C4—2362no—
391C4#117×D5911yescharacteristic
392C4#1683787011yescharacteristic
393C4#1683793012yes—
394C2#2142655511yescharacteristic
395C2#2143296211yescharacteristic
396C2#2143296911yescharacteristic
397C2#2143472912yes—
398C2#2143474912yes—
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2320C2
socle, prime radicalC23×C59C22×D4
cosocleC22×C4×C59C24
hypercenter#2320D59
hypocenterC59#21281
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
14. C4×#11745672C4semidirect
241. C59#21281semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117456722
#214267482
#214249562
#214248262
#214247892
#214247852
C4×#142736602
#214239292
#214239262
#214237832
C59×#23202
#2128159