Finite Group Databasebeta

← #28477278

Subgroups — #28477278

Order 15104.

570 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432397C211yesmaximal, characteristic
2#21424076C211yesmaximal, characteristic, complemented
3#21421921C212yesmaximal, complemented
4#21421916C212yesmaximal, complemented
5C59×#1728C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×#14273706C411yescharacteristic
8C2×#14273701C411yescharacteristic
9C2×C4⋊(C59⋊C8)—21no—
10C2×C22⋊(C59⋊C8)—21no—
11#16837308C2211yescharacteristic
12#16837304C2211yescharacteristic
13#16837293—21no—
14#16837293—21no—
15#16837281—21no—
16#16837281—21no—
17#16837268C411yescharacteristic
18#16837157C2212yes—
19C22×M16×C59—21no—
20C22×C42×C59C2211yescharacteristic
21C59×#336—21no—
22C59×#336C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C22×C59⋊C8—22no—
27#14273715—42no—
28#14273706—42no—
29#14273701—42no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C4⋊(C59⋊C8)—42no—
40C22⋊(C59⋊C8)—42no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C2×M16×C59—24no—
49C2×M16×C59—42no—
50C2×M16×C59—42no—
51C22×C8×C59—22no—
52C2×C42×C59—42no—
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59D411yescharacteristic
56C2×C42×C59C2311yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C23×D59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59—41no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C2×D236—22no—
83C2×D236—41no—
84C2×D236—41no—
85C2×Dic118—22no—
86C2×Dic118—41no—
87C2×Dic118—41no—
88M16×C59—48no—
89C2×C8×C59—24no—
90C2×C8×C59—42no—
91C2×C8×C59—42no—
92C42×C59—22no—
93C42×C59—22no—
94C42×C59—41no—
95C42×C59—41no—
96C22⋊Dic59—42no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—42no—
99(C2×D59)⋊C4—42no—
100(C2×D59)⋊C4—41no—
101(C2×D59)⋊C4—41no—
102(C2×D59)⋊C4—41no—
103(C2×D59)⋊C4—41no—
104C4⋊Dic59—41no—
105C4⋊Dic59—41no—
106Dic59⋊C4—42no—
107C2×C59⋊C8—42no—
108C2×C59⋊C8—42no—
109C2×C59⋊C8—42no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59#10412yes—
117C23×C59C2×C22⋊C411yescharacteristic
118C22×D59—41no—
119C22×D59—41no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C2×C4×C59—24no—
126C2×C4×C59—24no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—44no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59#9912yes—
144C2×C4×C59#10012yes—
145C2×C4×C59#13112yes—
146C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C2×C4×C59#13711yescharacteristic
150C22⋊D59—84no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160D236—82no—
161Dic118—82no—
162C8×C59—44no—
163C59⋊C8—82no—
164#7710—591nomaximal, Sylow
165C22×C59—22no—
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59#40811yescharacteristic
173C22×C59C2×#9911yescharacteristic
174C22×C59C2×#10011yescharacteristic
175C22×C59#41711yescharacteristic
176C22×C59#41711yescharacteristic
177C22×C59C2×#10411yescharacteristic
178C22×C59#42111yescharacteristic
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—81no—
184C2×D59—81no—
185C2×D59—81no—
186C4×C59—22no—
187C4×C59—22no—
188C4×C59—21no—
189C4×C59—22no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—41no—
194Dic59—81no—
195Dic59—81no—
196Dic59—81no—
197#2850—591no—
198#1797—591no—
199#1728D5911yescharacteristic, complemented
200#1272—592no—
201#1270—592no—
202C2×C59—42no—
203C2×C59#147111yescharacteristic
204C2×C59#148311yescharacteristic
205C2×C59#149511yescharacteristic
206C2×C59#150411yescharacteristic
207C2×C59#151311yescharacteristic
208C2×C59#152711yescharacteristic
209C2×C59#156011yescharacteristic
210D59—161no—
211C22×M16—21no—
212C2×#124—591no—
213C2×C22⋊Q8—591no—
214C2×#121—591no—
215C22×C42C2×D5911yescharacteristic
216C2×C4⋊C8—1181no—
217C2×C22⋊C8—1181no—
218#391—591no—
219#390—591no—
220#388—1181no—
221#388—1181no—
222#386—1181no—
223#386—1181no—
224#382—591no—
225#336—21no—
226#336—592no—
227#336C2×D5912yes—
228C59#771011yescharacteristic, complemented, Sylow
229C22×Q8—591no—
230C22×D4—591no—
231C22×D4—1181no—
232C23×C4C4×D5911yescharacteristic
233C23×C4D23611yescharacteristic
234C23×C4C22⋊D5911yescharacteristic
235C2×M16—24no—
236C2×M16—42no—
237C2×M16—42no—
238C22×C8—22no—
239C22×C8—1182no—
240#124—2362no—
241C22⋊Q8—2362no—
242#121—2362no—
243C2×C4⋊C4—591no—
244C2×C4⋊C4—591no—
245C2×C4⋊C4—1181no—
246C2×C22⋊C4—1181no—
247C2×C22⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1181no—
251C2×C22⋊C4—1181no—
252C2×C42—42no—
253C2×C42C4×D5911yescharacteristic
254C2×C42D23611yescharacteristic
255C2×C42C22⋊D5911yescharacteristic
256C2×C42C22×D5911yescharacteristic
257C4⋊C8—2362no—
258C22⋊C8—2362no—
259#95—1181no—
260#95—1181no—
261#95—1181no—
262#95—1181no—
263C24—1181no—
264C24(C2×D59)⋊C411yescharacteristic
265C2×Q8—1182no—
266C2×Q8—2361no—
267C2×Q8—2361no—
268C2×D4—1182no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2362no—
272C2×D4—2362no—
273C2×D4—2362no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—1181no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4(C2×D59)⋊C411yescharacteristic
289C22×C4(C2×D59)⋊C411yescharacteristic
290C22×C4(C2×D59)⋊C411yescharacteristic
291C22×C4C2×C4×D5911yescharacteristic
292C22×C4C2×D23611yescharacteristic
293C22×C4C2×C22⋊D5911yescharacteristic
294M16—48no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—2362no—
299C2×C8—2362no—
300C2×C8—2362no—
301C4⋊C4—2361no—
302C4⋊C4—2361no—
303C4⋊C4—2362no—
304C22⋊C4—2362no—
305C22⋊C4—2362no—
306C22⋊C4—2362no—
307C22⋊C4—2362no—
308C22⋊C4—2361no—
309C22⋊C4—2361no—
310C22⋊C4—2361no—
311C22⋊C4—2361no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—2361no—
323C23—2361no—
324C23—2361no—
325C23—2361no—
326C23—2361no—
327C23—2361no—
328C23—2361no—
329C23#1427365512yes—
330C23C2×(C2×D59)⋊C411yescharacteristic
331Q8—4722no—
332D4—4722no—
333D4—4724no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4#1427364112yes—
362C2×C4#1427364312yes—
363C2×C4#1427373512yes—
364C2×C4#1427375311yescharacteristic
365C2×C4#1427375511yescharacteristic
366C2×C4#1427375811yescharacteristic
367C2×C4#1427376011yescharacteristic
368C8—44no—
369C8—4722no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—4721no—
378C22—4721no—
379C22—4721no—
380C22—4721no—
381C22—4721no—
382C22—4721no—
383C22—4721no—
384C22#1683737011yescharacteristic
385C22C2×#1427364111yescharacteristic
386C22C2×#1427364311yescharacteristic
387C22#1683740111yescharacteristic
388C22#1683740411yescharacteristic
389C22C2×#1427365511yescharacteristic
390C22#1683741811yescharacteristic
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—41no—
399C4—4721no—
400C4—4721no—
401C4—4721no—
402C2—42no—
403C2—9441no—
404C2#2142245711yescharacteristic
405C2#2142251011yescharacteristic
406C2#2142255911yescharacteristic
407C2#2142259811yescharacteristic
408C2#2142264011yescharacteristic
409C2#2142269311yescharacteristic
410C2#2142282911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1728C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C4
cosocleC2×C42×C59C23
hypercenter#1728D59
hypocenterC59#7710
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
2. #21424076C2semidirect
3. #21421921C2semidirect
4. #21421916C2semidirect
5. C59×#1728C2semidirect
199. #1728D59semidirect
228. C59#7710semidirect

Maximal subgroups

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

subgroupindex
#214323972
#214240762
#214219212
#214219162
C59×#17282
#771059