Finite Group Databasebeta

← #28499216

Subgroups — #28499216

Order 15104.

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

normal subgroups only

This group has 420 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#21429794C211yesmaximal, characteristic
2#21423835C211yesmaximal, characteristic
3#21422124C212yesmaximal, complemented
4#21422103C212yesmaximal, complemented
5C59×#1464C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273719C411yescharacteristic
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837327C411yescharacteristic
12#16837297—21no—
13#16837296—21no—
14#16837286—21no—
15#16837275—21no—
16#16837248C411yescharacteristic
17#16837171—21no—
18#16837171C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#432—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C2×#11745664—24no—
24C2×#11745664—22no—
25C2×#11745664—41no—
26C2×#11745664—42no—
27C22×C59⋊C8—41no—
28#14273719—41no—
29#14273719—41no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C2×C4×Dic59—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47#117×C59—41no—
48#117×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C4⋊C8×C59—41no—
56C22⋊C8×C59—22no—
57#97×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C22×C4×C59—22no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×C8×C59—42no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85C4⋊Dic59—41no—
86C4⋊Dic59—41no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89Dic59⋊C4—41no—
90Dic59⋊C4—41no—
91Dic59⋊C4—41no—
92Dic59⋊C4—41no—
93C4×Dic59—42no—
94#11745664—24no—
95#11745664—44no—
96#11745664—42no—
97C2×C59⋊C8—44no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59#10412yes—
104C23×C59C2×C22⋊C411yescharacteristic
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—21no—
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—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C59#13711yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C8×C59—82no—
138C59⋊C8—44no—
139#11767—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—22no—
143C22×C59—42no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—81no—
147C22×C59C2×#9911yescharacteristic
148C22×C59#41711yescharacteristic
149C22×C59C2×#10411yescharacteristic
150C4×C59—21no—
151C4×C59—21no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158Dic59—42no—
159Dic59—81no—
160Dic59—81no—
161Dic59—81no—
162#2611—591no—
163#1742—591no—
164#1464D5911yescharacteristic
165#1353—592no—
166#1346—592no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59—42no—
170C2×C59#150411yescharacteristic
171C2×C59#212311yescharacteristic
172C2×C59#213211yescharacteristic
173C22×M16—1181no—
174C2×#125—591no—
175C2×#117C2×D5911yescharacteristic
176#432—21no—
177#400—591no—
178#399—591no—
179#398—591no—
180#397—591no—
181#389—1181no—
182#388—1181no—
183#387—1181no—
184#385—1181no—
185#376—591no—
186#343—592no—
187#343—1181no—
188#324C2×D5912yes—
189#323—21no—
190C59#1176711yescharacteristic, complemented, Sylow
191C23×C4D23611yescharacteristic
192C2×M16—1184no—
193C2×M16—1182no—
194C2×M16—2361no—
195C2×M16—2362no—
196C22×C8—2361no—
197#125—2361no—
198#125—2361no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—591no—
202C2×C4⋊C4—591no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4D23611yescharacteristic
208C2×C4⋊C4C22×D5911yescharacteristic
209C2×C22⋊C4—1181no—
210C2×C22⋊C4—1181no—
211C2×C22⋊C4C4×D5911yescharacteristic
212C2×C22⋊C4C22⋊D5911yescharacteristic
213C2×C42—1181no—
214C2×C42C4×D5911yescharacteristic
215C2×C42C22⋊D5911yescharacteristic
216C4⋊C8—41no—
217C22⋊C8—22no—
218#97—41no—
219#95—591no—
220#95—591no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226C24(C2×D59)⋊C411yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—1181no—
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C2×D23611yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—1184no—
243M16—2364no—
244M16—2362no—
245C2×C8—42no—
246C2×C8—2364no—
247C2×C8—2362no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—2361no—
251C4⋊C4—2361no—
252C4⋊C4—2362no—
253C4⋊C4—2362no—
254C4⋊C4—2361no—
255C4⋊C4—2361no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—2362no—
261C22⋊C4—2362no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—2362no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1427365312yes—
272C23C2×(C2×D59)⋊C411yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—1182no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4#1427364111yescharacteristic
302C2×C4#1427364111yescharacteristic
303C2×C4#1427375511yescharacteristic
304C2×C4#1427376011yescharacteristic
305C8—82no—
306C8—2364no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1427364111yescharacteristic
315C22#1683740411yescharacteristic
316C22C2×#1427365311yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—4721no—
326C4—4721no—
327C4—4721no—
328C4—2362no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2142260111yescharacteristic
333C2#2142574911yescharacteristic
334C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1464C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1464D59
hypocenterC59#11767
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #21422124C2semidirect
4. #21422103C2semidirect
190. C59#11767semidirect

Maximal subgroups

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

subgroupindex
#214297942
#214238352
#214221242
#214221032
C59×#14642
#1176759