Finite Group Databasebeta

← #28498317

Subgroups — #28498317

Order 15104.

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

normal subgroups only

This group has 412 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#21429072C211yesmaximal, characteristic
2#21422440C211yesmaximal, characteristic, complemented
3#21422029C212yesmaximal, complemented
4#21422027C212yesmaximal
5C59×#1461C211yesmaximal, characteristic, complemented
6C2×#14273725C411yescharacteristic
7C2×#14273716C411yescharacteristic
8C2×#14273715C411yescharacteristic
9#16837459—21no—
10#16837324—21no—
11#16837324—21no—
12#16837307C2211yescharacteristic
13#16837303C2211yescharacteristic
14#16837293—21no—
15#16837293—21no—
16C4×C4⋊Dic59C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25#14273725—42no—
26#14273716—42no—
27#14273715—41no—
28#14273715—41no—
29C2×C4⋊Dic59—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C4×Dic59C2×C411yescharacteristic
40C4⋊(C59⋊C8)—41no—
41C22⋊(C59⋊C8)—22no—
42#14273638—41no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×Q8×C59C2×C411yescharacteristic
47C4×Q8×C59—41no—
48C4×Q8×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C4⋊C8×C59—41no—
56C22⋊C8×C59—22no—
57C4×C8×C59—41no—
58C23×D59—21no—
59C2×Q8×C59—22no—
60C2×Q8×C59—22no—
61C2×Q8×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×D236—22no—
73C2×D236—41no—
74C2×D236—41no—
75C2×Dic118—22no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—42no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—41no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—42no—
95C4⋊Dic59—42no—
96Dic59⋊C4—42no—
97Dic59⋊C4—42no—
98C4×Dic59—41no—
99C4×Dic59—41no—
100C2×C59⋊C8—42no—
101C23×C59C2×C22⋊C411yescharacteristic
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105C22×D59—41no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109Q8×C59—44no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59#10012yes—
127C2×C4×C59#13112yes—
128C2×Dic59—22no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137D236—82no—
138Dic118—82no—
139C8×C59—82no—
140C59⋊C8—82no—
141#11565—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59#40811yescharacteristic
145C22×C59C2×#9911yescharacteristic
146C22×C59C2×#10011yescharacteristic
147C2×D59—81no—
148C2×D59—81no—
149C2×D59—81no—
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—81no—
153C2×D59—81no—
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—22no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162Dic59—42no—
163Dic59—81no—
164Dic59—81no—
165#2545—591no—
166#1467—591no—
167#1461D5911yescharacteristic, complemented
168#1319—592no—
169#1318—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59#147111yescharacteristic
173C2×C59#212611yescharacteristic
174C2×C59#213311yescharacteristic
175D59—161no—
176C2×C4⋊Q8—591no—
177C2×#124—591no—
178C2×#124—591no—
179C2×C4×Q8C2×D5911yescharacteristic
180#432—1181no—
181#431—21no—
182#396—1181no—
183#396—1181no—
184#391—591no—
185#390—591no—
186#388—1181no—
187#388—1181no—
188C4×C4⋊C4—591no—
189#324—21no—
190#324—592no—
191#324—1181no—
192#324C2×D5912yes—
193C59#1156511yescharacteristic, complemented, Sylow
194C22×Q8—591no—
195C22×Q8C4×D5911yescharacteristic
196C22×D4—591no—
197C4⋊Q8—2362no—
198#124—2361no—
199#124—2361no—
200#124—2362no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4D23611yescharacteristic
207C2×C4⋊C4C22⋊D5911yescharacteristic
208C2×C4⋊C4C22×D5911yescharacteristic
209C2×C22⋊C4—1181no—
210C2×C22⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C42—591no—
216C2×C42—591no—
217C2×C42C4×D5911yescharacteristic
218C2×C42D23611yescharacteristic
219C2×C42C22⋊D5911yescharacteristic
220C4⋊C8—41no—
221C4⋊C8—2361no—
222C22⋊C8—22no—
223C22⋊C8—1182no—
224#97—2361no—
225C4×C8—41no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229C24—1181no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—1182no—
234C2×Q8—2361no—
235C2×Q8—2361no—
236C2×D4—1182no—
237C2×D4—2361no—
238C2×D4—2361no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4C2×C4×D5911yescharacteristic
247C22×C4C2×D23611yescharacteristic
248C22×C4C2×C22⋊D5911yescharacteristic
249C2×C8—42no—
250C2×C8—2362no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2362no—
255C4⋊C4—2362no—
256C4⋊C4—2362no—
257C22⋊C4—2362no—
258C22⋊C4—2362no—
259C22⋊C4—2362no—
260C22⋊C4—2362no—
261C22⋊C4—2361no—
262C22⋊C4—2361no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—2361no—
271C42—2361no—
272C23—2361no—
273C23—2361no—
274C23—2361no—
275C23—2361no—
276C23—2361no—
277C23—2361no—
278C23—2361no—
279C23C2×(C2×D59)⋊C411yescharacteristic
280Q8—44no—
281Q8—4722no—
282D4—4722no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—1182no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4#1427364111yescharacteristic
307C2×C4#1427364111yescharacteristic
308C2×C4#1427364312yes—
309C2×C4#1427373312yes—
310C8—82no—
311C8—4722no—
312C22—21no—
313C22—21no—
314C22—4721no—
315C22—4721no—
316C22—4721no—
317C22—4721no—
318C22—4721no—
319C22—4721no—
320C22—4721no—
321C22#1683736811yescharacteristic
322C22C2×#1427364111yescharacteristic
323C22C2×#1427364311yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—4721no—
333C4—4721no—
334C4—2362no—
335C2—21no—
336C2—21no—
337C2—9441no—
338C2#2142245511yescharacteristic
339C2#2142575911yescharacteristic
340C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1461C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1461D59
hypocenterC59#11565
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21422440C2semidirect
3. #21422029C2semidirect
5. C59×#1461C2semidirect
167. #1461D59semidirect
193. C59#11565semidirect

Maximal subgroups

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

subgroupindex
#214290722
#214224402
#214220292
#214220272
C59×#14612
#1156559