Finite Group Databasebeta

← #28499079

Subgroups — #28499079

Order 15104.

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

normal subgroups only

This group has 400 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#21429956C211yesmaximal, characteristic
2C4×#14273646C211yesmaximal, characteristic, complemented
3#21422127C212yesmaximal, complemented
4#21422100C212yesmaximal
5C59×#1461C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×#14273646—21no—
8C4×#11745664—21no—
9#16837337—21no—
10#16837322C411yescharacteristic
11#16837322C2211yescharacteristic
12#16837319C411yescharacteristic
13#16837319C2211yescharacteristic
14#16837313—21no—
15#16837293—21no—
16#16837293—21no—
17#16837270C411yescharacteristic
18#16837171C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C2×#11745664—22no—
26#14273715—41no—
27#14273715—41no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36#14273646—24no—
37#14273638—41no—
38C4×C59⋊C8—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×Q8×C59D411yescharacteristic
46C4×Q8×C59—41no—
47C4×Q8×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C22⋊C8×C59—22no—
56C4×C8×C59—41no—
57C23×D59—21no—
58C2×Q8×C59—22no—
59C2×Q8×C59—22no—
60C2×Q8×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×D236—41no—
72C2×D236—81no—
73C2×Dic118—41no—
74C2×Dic118—81no—
75C2×C8×C59—42no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85(C2×D59)⋊C4—42no—
86(C2×D59)⋊C4—41no—
87(C2×D59)⋊C4—41no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—81no—
91(C2×D59)⋊C4—81no—
92Dic59⋊C4—81no—
93Dic59⋊C4—81no—
94#11745664—24no—
95#11745664—42no—
96C2×C59⋊C8—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98C22×D59—41no—
99C22×D59—41no—
100C22×D59—41no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59—41no—
105Q8×C59—44no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
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—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#10412yes—
123C2×C4×C59#13512yes—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133D236—82no—
134Dic118—82no—
135C8×C59—82no—
136C59⋊C8—44no—
137#11738—591nomaximal, Sylow
138C22×C59—21no—
139C22×C59—21no—
140C22×C59C2×#9911yescharacteristic
141C22×C59#41611yescharacteristic
142C22×C59C2×#10411yescharacteristic
143C2×D59—81no—
144C2×D59—81no—
145C2×D59—81no—
146C2×D59—81no—
147C2×D59—81no—
148C2×D59—81no—
149C2×D59—81no—
150C4×C59—21no—
151C4×C59—21no—
152C4×C59—22no—
153C4×C59—22no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158Dic59—81no—
159Dic59—81no—
160Dic59—81no—
161#2632—591no—
162C4×#101—591no—
163#1461D5911yescharacteristic, complemented
164#1354—592no—
165#1345—592no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59#150111yescharacteristic
169C2×C59#212211yescharacteristic
170C2×C59#213311yescharacteristic
171D59—161no—
172C2×#124—591no—
173C2×C4×Q8C2×D5911yescharacteristic
174#431—21no—
175C2×#101—1181no—
176C4×M16—1181no—
177#399—1181no—
178#395—591no—
179#395—591no—
180#394—591no—
181#394—591no—
182#393—1181no—
183#388—1181no—
184#388—1181no—
185#383—591no—
186#343—592no—
187#324—21no—
188#324C2×D5912yes—
189C59#1173811yescharacteristic, complemented, Sylow
190C22×Q8—591no—
191C22×Q8D23611yescharacteristic
192C22×D4—591no—
193C2×M16—1182no—
194#124—2361no—
195#124—2361no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4C4×D5911yescharacteristic
201C2×C4⋊C4C22⋊D5911yescharacteristic
202C2×C4⋊C4C22×D5911yescharacteristic
203C2×C22⋊C4—1181no—
204C2×C22⋊C4—1181no—
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C22⋊C4—1181no—
208C2×C22⋊C4—1181no—
209C2×C42C4×D5911yescharacteristic
210C2×C42D23611yescharacteristic
211C2×C42C22⋊D5911yescharacteristic
212C4⋊C8—41no—
213#101—1184no—
214C22⋊C8—22no—
215#97—2361no—
216C4×C8—41no—
217C4×C8—2361no—
218#95—591no—
219#95—591no—
220#95—1181no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224C24—1181no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—4721no—
229C2×Q8—2361no—
230C2×D4—4721no—
231C2×D4—2361no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4(C2×D59)⋊C411yescharacteristic
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—2362no—
244C2×C8—42no—
245C2×C8—2362no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—4721no—
250C4⋊C4—4721no—
251C22⋊C4—4721no—
252C22⋊C4—4721no—
253C22⋊C4—2362no—
254C22⋊C4—2361no—
255C22⋊C4—2361no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—2361no—
265C23—2361no—
266C23—2361no—
267C23—2361no—
268C23—2361no—
269C23—2361no—
270C23—2361no—
271C23C2×(C2×D59)⋊C411yescharacteristic
272Q8—44no—
273Q8—4722no—
274D4—4722no—
275C2×C4—21no—
276C2×C4—21no—
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—2361no—
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#1427364111yescharacteristic
299C2×C4#1427364111yescharacteristic
300C2×C4#1427365312yes—
301C2×C4#1427374912yes—
302C8—82no—
303C8—2364no—
304C22—21no—
305C22—21no—
306C22—4721no—
307C22—4721no—
308C22—4721no—
309C22—4721no—
310C22—4721no—
311C22—4721no—
312C22—4721no—
313C22C2×#1427364111yescharacteristic
314C22#1683739911yescharacteristic
315C22C2×#1427365311yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4721no—
325C4—4721no—
326C4—4721no—
327C2—21no—
328C2—21no—
329C2—9441no—
330C2#2142258811yescharacteristic
331C2#2142574511yescharacteristic
332C2#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#11738
LCSG ▷ C22×C4×C59 ▷ C23×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. C4×#14273646C2semidirect
3. #21422127C2semidirect
5. C59×#1461C2semidirect
163. #1461D59semidirect
189. C59#11738semidirect

Maximal subgroups

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

subgroupindex
#214299562
C4×#142736462
#214221272
#214221002
C59×#14612
#1173859