Finite Group Databasebeta

← #28482689

Subgroups — #28482689

Order 15104.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426714C211yesmaximal, characteristic
2#21424291C212yesmaximal
3#21422349C211yesmaximal, characteristic
4#21421976C212yesmaximal
5C2×C59×#324C211yesmaximal, characteristic
6C2×#14273686—21no—
7C22×C4⋊Dic59C2211yescharacteristic
8C2×#14273659—22no—
9#16837301—22no—
10#16837301—22no—
11#16837262—21no—
12#16837262C2212yes—
13C2×C59⋊#95—21no—
14C2×C59⋊#95—21no—
15C2×C59⋊#95C2211yescharacteristic
16#16837165—22no—
17C22×C4⋊C4×C59C2211yescharacteristic
18C2×C22⋊C8×C59C2212yes—
19C59×#324C414yescomplemented
20C23×Dic59—21no—
21C23×Dic59—21no—
22#14273686—42no—
23#14273686—42no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—42no—
30C2×C4⋊Dic59D412yes—
31C2×C4⋊Dic59D412yes—
32C2×C22⋊Dic59—22no—
33C2×C4×Dic59—22no—
34#14273659—44no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—42no—
39C59⋊#95—42no—
40C23×C4×C59D411yescharacteristic
41C23×C4×C59D411yescharacteristic
42C23×C4×C59C2311yescharacteristic
43C22×C8×C59—22no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59—22no—
46C2×C4⋊C4×C59C2×C412yes—
47C2×C4⋊C4×C59Q812yes—
48C22⋊C8×C59—22no—
49C22⋊C8×C59C2×C414yes—
50C24×C59C2×D411yescharacteristic
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—41no—
58C22×C4×C59—41no—
59C22×C4×C59—41no—
60C22×C4×C59—41no—
61C22×C4×C59C4⋊C412yes—
62C22×C4×C59C4⋊C412yes—
63C22×C4×C59SD1612yes—
64C22×C4×C59SD1612yes—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D412yes—
67C22×C4×C59C2×Q811yescharacteristic
68C22×C4×C59#4112yes—
69C22×Dic59—22no—
70C22×Dic59—22no—
71C22×Dic59—22no—
72C22×Dic59—22no—
73C22×Dic59—41no—
74C22×Dic59—41no—
75C22×Dic59—41no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C22×Dic59—41no—
81C22×Dic59—41no—
82C22×Dic59—41no—
83C22×Dic59—41no—
84C22×Dic59—41no—
85C2×C8×C59—24no—
86C2×C8×C59—42no—
87C2×C8×C59—42no—
88C4⋊C4×C59—24no—
89C4⋊C4×C59—24no—
90C22⋊Dic59—42no—
91C22⋊Dic59—42no—
92C4⋊Dic59—24no—
93C4⋊Dic59—42no—
94C4⋊Dic59—42no—
95C4⋊Dic59—42no—
96C4⋊Dic59—42no—
97C4⋊Dic59—42no—
98C4⋊Dic59—81no—
99C4⋊Dic59—81no—
100C4×Dic59—44no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59C2×C4⋊C411yescharacteristic
106C23×C59#12011yescharacteristic
107C23×C59#12311yescharacteristic
108C23×C59C2×SD1611yescharacteristic
109C23×C59C2×SD1611yescharacteristic
110C23×C59AGL(1,ℤ/8)11yescharacteristic
111C23×C59#13711yescharacteristic
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—81no—
128C2×C4×C59#10614yescomplemented
129C2×C4×C59C4×D412yes—
130C2×C4×C59C22⋊Q812yes—
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—42no—
143C2×Dic59—81no—
144C2×Dic59—81no—
145C2×Dic59—81no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C2×Dic59—81no—
149C2×Dic59—81no—
150C2×Dic59—81no—
151C8×C59—44no—
152#8857—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59#38511yescharacteristic
166C22×C59C2×#10611yescharacteristic
167C22×C59#42811yescharacteristic
168C22×C59#44811yescharacteristic
169C22×C59#45011yescharacteristic
170C22×C59#45411yescharacteristic
171C22×C59#45611yescharacteristic
172C22×C59B(4,𝔽2)11yescharacteristic
173C22×C59#45811yescharacteristic
174C22×C59#48111yescharacteristic
175C22×C59#48311yescharacteristic
176C4×C59—24no—
177C4×C59—22no—
178C4×C59—22no—
179C4×C59—42no—
180Dic59—82no—
181Dic59—82no—
182Dic59—82no—
183Dic59—82no—
184#2316—591no—
185#1853—592no—
186C2×#324D5911yescharacteristic
187#1441—591no—
188#1294—592no—
189C2×C59—21no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—21no—
193C2×C59#159911yescharacteristic
194C2×C59#160411yescharacteristic
195C2×C59#161111yescharacteristic
196C2×C59#161611yescharacteristic
197C2×C59#185311yescharacteristic
198C2×C59#185711yescharacteristic
199C2×C59#186711yescharacteristic
200C2×#117—1181no—
201C22×C4⋊C4—591no—
202C22×C4⋊C4C2×D5911yescharacteristic
203C2×#106—1182no—
204C2×C22⋊C8C2×D5912yes—
205#389—1182no—
206#389—1182no—
207#380—592no—
208#380—1181no—
209C2×#95—591no—
210C2×#95—1181no—
211C2×#95—1181no—
212#340—1182no—
213#324Dic5914yescomplemented
214C59#885711yescharacteristic, complemented, Sylow
215C23×C4—1181no—
216C23×C4—1181no—
217C23×C4D23611yescharacteristic
218C23×C4D23611yescharacteristic
219C23×C4C22×D5911yescharacteristic
220C22×C8—22no—
221#117—2362no—
222#117—2362no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—592no—
226C2×C4⋊C4—592no—
227C2×C4⋊C4—1182no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4—2362no—
233C2×C4⋊C4Dic11812yes—
234C2×C4⋊C4C2×Dic5912yes—
235C2×C22⋊C4—1182no—
236C2×C42—1182no—
237#106—2364no—
238C22⋊C8—22no—
239C22⋊C8C2×Dic5914yes—
240#95—1182no—
241#95—1182no—
242#95—1182no—
243#95—2362no—
244#95—2362no—
245C24C2×D23611yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—1182no—
257C22×C4—1182no—
258C22×C4—1182no—
259C22×C4—1182no—
260C22×C4—2361no—
261C22×C4—2361no—
262C22×C4—2361no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4—2361no—
268C22×C4—2361no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4—2361no—
272C22×C4#1174566012yes—
273C22×C4#1174566012yes—
274C22×C4C4⋊Dic5912yes—
275C22×C4C4⋊Dic5912yes—
276C22×C4C2×Dic11811yescharacteristic
277C22×C4D4×D5912yes—
278C22×C4#1174568712yes—
279C22×C4C22×Dic5911yescharacteristic
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—1184no—
286C4⋊C4—4721no—
287C4⋊C4—4721no—
288C4⋊C4—2362no—
289C4⋊C4—2362no—
290C4⋊C4—2362no—
291C4⋊C4—2362no—
292C4⋊C4—2362no—
293C22⋊C4—2362no—
294C22⋊C4—2362no—
295C42—2364no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×C4⋊Dic5911yescharacteristic
301C23#1427369711yescharacteristic
302C23#1427371111yescharacteristic
303C23C2×#1174566011yescharacteristic
304C23C2×#1174566011yescharacteristic
305C23#1427375511yescharacteristic
306C23#1427376011yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—4721no—
324C2×C4—4721no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—4721no—
329C2×C4—4721no—
330C2×C4—4721no—
331C2×C4—2362no—
332C2×C4—2362no—
333C2×C4—2362no—
334C2×C4—2362no—
335C2×C4—2362no—
336C2×C4—2362no—
337C2×C4—2362no—
338C2×C4—2362no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4#1427365914yescomplemented
344C2×C4D4×Dic5912yes—
345C2×C4#1427370412yes—
346C8—44no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1683727711yescharacteristic
360C22C2×#1427365911yescharacteristic
361C22#1683744411yescharacteristic
362C22#1683754311yescharacteristic
363C22#1683755511yescharacteristic
364C22#1683757711yescharacteristic
365C22#1683758711yescharacteristic
366C22#1683759011yescharacteristic
367C22#1683759311yescharacteristic
368C22#1683772111yescharacteristic
369C22#1683773311yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—4722no—
375C4—4722no—
376C4—4722no—
377C4—4722no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2142303811yescharacteristic
383C2#2142306711yescharacteristic
384C2#2142310611yescharacteristic
385C2#2142313511yescharacteristic
386C2#2142429111yescharacteristic
387C2#2142431311yescharacteristic
388C2#2142437911yescharacteristic
subgroupG/H
centerC23#14273697
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#324C2
socleC23×C59#120
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC2×#324D59
hypocenterC59#8857
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
19. C59×#324C4semidirect
128. C2×C4×C59#106semidirect
213. #324Dic59semidirect
214. C59#8857semidirect
343. C2×C4#14273659semidirect

Maximal subgroups

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

subgroupindex
#214267142
#214242912
#214223492
#214219762
C2×C59×#3242
#885759