Finite Group Databasebeta

← #28498322

Subgroups — #28498322

Order 15104.

494 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429063C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic, complemented
3#21422029C212yesmaximal, complemented
4#21422027C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic, complemented
6C2×#14273723C411yescharacteristic
7C2×#14273715C411yescharacteristic
8C2×#14273714C411yescharacteristic
9#16837366—21no—
10#16837318—21no—
11#16837318—21no—
12#16837307C2211yescharacteristic
13#16837303C2211yescharacteristic
14#16837281—21no—
15#16837281—21no—
16C4×C4⋊Dic59C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C22×C22⋊D59—21no—
27C2×#11745664—41no—
28#14273723—42no—
29#14273715—41no—
30#14273715—41no—
31#14273714—42no—
32C2×C4⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×C4×Dic59C2×C411yescharacteristic
40C2×C4×Dic59C2×C411yescharacteristic
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—41no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×D4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×C59—41no—
50C4×D4×C59—41no—
51C4×D4×C59—41no—
52C2×C4⋊C4×C59C2311yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C42×C59C2×C411yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C22⋊C411yescharacteristic
60C23×D59—21no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—81no—
69C22×C4×C59—81no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—42no—
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—81no—
80C2×C22⋊D59—81no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C2×D236—22no—
85C2×D236—41no—
86C2×D236—41no—
87C2×Dic118—22no—
88C2×Dic118—41no—
89C2×Dic118—41no—
90C2×C8×C59—42no—
91C2×C8×C59—82no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C42×C59—21no—
96C42×C59—21no—
97C22⋊Dic59—42no—
98C22⋊Dic59—42no—
99(C2×D59)⋊C4—42no—
100(C2×D59)⋊C4—41no—
101(C2×D59)⋊C4—41no—
102(C2×D59)⋊C4—81no—
103(C2×D59)⋊C4—81no—
104C4⋊Dic59—42no—
105C4×Dic59—41no—
106C4×Dic59—41no—
107#11745664—82no—
108C2×C59⋊C8—42no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59#10012yes—
119C23×C59C2×C22⋊C411yescharacteristic
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127D4×C59—44no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—41no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—81no—
140C2×C4×C59—81no—
141C2×C4×C59#9911yescharacteristic
142C2×C4×C59#9911yescharacteristic
143C2×C4×C59#13112yes—
144C22⋊D59—84no—
145C22⋊D59—84no—
146C2×Dic59—22no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155D236—82no—
156Dic118—82no—
157C8×C59—82no—
158C59⋊C8—82no—
159#11566—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—22no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59#40811yescharacteristic
173C22×C59C2×#9911yescharacteristic
174C22×C59C2×#10011yescharacteristic
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—81no—
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—22no—
185C4×C59—41no—
186C4×C59—41no—
187C4×C59—41no—
188Dic59—42no—
189Dic59—81no—
190Dic59—81no—
191#2544—591no—
192#1465—591no—
193#1459D5911yescharacteristic, complemented
194#1319—592no—
195#1318—592no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—42no—
199C2×C59—81no—
200C2×C59#147111yescharacteristic
201C2×C59#212711yescharacteristic
202C2×C59#213211yescharacteristic
203D59—161no—
204C2×#127—591no—
205C2×#124—591no—
206C2×#124—591no—
207C2×C4×D4C2×D5911yescharacteristic
208#407—1181no—
209C2×C22⋊C8—21no—
210#394—1181no—
211#394—1181no—
212#391—591no—
213#390—591no—
214#386—1181no—
215#386—1181no—
216C4×C4⋊C4—591no—
217#324—592no—
218#324C2×D5912yes—
219#323—21no—
220#323—1181no—
221C59#1156611yescharacteristic, complemented, Sylow
222C22×Q8—591no—
223C22×D4—591no—
224C22×D4—1181no—
225C22×D4—1181no—
226C22×D4C4×D5911yescharacteristic
227C23×C4D23611yescharacteristic
228C23×C4C22⋊D5911yescharacteristic
229C2×M16—2361no—
230C22×C8—41no—
231#127—2362no—
232#124—2361no—
233#124—2361no—
234#124—2362no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—1181no—
238C2×C4⋊C4C22×D5911yescharacteristic
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C22⋊C4—1181no—
245C2×C22⋊C4D23611yescharacteristic
246C2×C22⋊C4C22⋊D5911yescharacteristic
247C2×C42—591no—
248C2×C42—591no—
249C2×C42C4×D5911yescharacteristic
250C22⋊C8—22no—
251C22⋊C8—41no—
252C22⋊C8—1182no—
253C22⋊C8—2361no—
254#95—1181no—
255#95—1181no—
256#95—1181no—
257C24—1181no—
258C24(C2×D59)⋊C411yescharacteristic
259C24(C2×D59)⋊C411yescharacteristic
260C2×Q8—1182no—
261C2×Q8—2361no—
262C2×Q8—2361no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—1182no—
267C2×D4—4721no—
268C2×D4—4721no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2362no—
272C2×D4—2362no—
273C2×D4—2362no—
274C2×D4—2362no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4(C2×D59)⋊C411yescharacteristic
285C22×C4(C2×D59)⋊C411yescharacteristic
286C22×C4C2×C4×D5911yescharacteristic
287C22×C4C2×D23611yescharacteristic
288C22×C4C2×C22⋊D5911yescharacteristic
289M16—4722no—
290C2×C8—42no—
291C2×C8—82no—
292C2×C8—2362no—
293C4⋊C4—41no—
294C4⋊C4—2362no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—4721no—
298C22⋊C4—4721no—
299C22⋊C4—2362no—
300C22⋊C4—2362no—
301C22⋊C4—2362no—
302C22⋊C4—2361no—
303C22⋊C4—2361no—
304C42—21no—
305C42—21no—
306C42—2361no—
307C42—2361no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—2361no—
318C23—2361no—
319C23—2361no—
320C23—2361no—
321C23—2361no—
322C23—2361no—
323C23—2361no—
324C23#1427364312yes—
325C23C2×(C2×D59)⋊C411yescharacteristic
326Q8—4722no—
327D4—44no—
328D4—4722no—
329D4—4724no—
330D4—4724no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—1182no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4#1427364111yescharacteristic
354C2×C4#1427364111yescharacteristic
355C2×C4#1427373312yes—
356C8—82no—
357C8—4722no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—4721no—
371C22—4721no—
372C22—4721no—
373C22—4721no—
374C22—4721no—
375C22—4721no—
376C22—4721no—
377C22#1683736811yescharacteristic
378C22C2×#1427364111yescharacteristic
379C22C2×#1427364311yescharacteristic
380C4—21no—
381C4—21no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—4721no—
387C4—4721no—
388C4—2362no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—81no—
393C2—9441no—
394C2#2142245511yescharacteristic
395C2#2142576311yescharacteristic
396C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC2×#14273723C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11566
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21422434C2semidirect
3. #21422029C2semidirect
4. #21422027C2semidirect
5. C59×#1459C2semidirect
193. #1459D59semidirect
221. C59#11566semidirect

Maximal subgroups

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

subgroupindex
#214290632
#214224342
#214220292
#214220272
C59×#14592
#1156659