Finite Group Databasebeta

← #28499299

Subgroups — #28499299

Order 15104.

590 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427461C211yesmaximal, characteristic
2C4×#14273642C211yesmaximal, characteristic
3#21422118C214yesmaximal, complemented
4C59×#1466C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic, complemented
6C2×#14273706—21no—
7C2×#14273642—21no—
8#16837322C412yescomplemented
9#16837314C2212yes—
10#16837308—21no—
11#16837282—21no—
12#16837274—21no—
13#16837262C411yescharacteristic, complemented
14C4×C22⋊Dic59—21no—
15#16837169C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#432—21no—
18C59×#323—21no—
19C59×#323C2212yes—
20C22×Dic118—21no—
21C22×Dic118—44no—
22C22×Dic118—41no—
23C22×Dic118—41no—
24C22×Dic118C2×C412yes—
25C23×Dic59—21no—
26#14273706—44no—
27C2×C4⋊Dic59—21no—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C4×Dic59—41no—
33#14273642—24no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—41no—
37C59⋊#95C2×C412yes—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59D411yescharacteristic
47C4⋊C8×C59—41no—
48C22⋊C8×C59—22no—
49#97×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—41no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—22no—
67C22×Dic59—22no—
68C22×Dic59—44no—
69C22×Dic59—42no—
70C22×Dic59—42no—
71C2×Dic118—24no—
72C2×Dic118—42no—
73C2×Dic118—48no—
74C2×Dic118—44no—
75C2×Dic118—41no—
76C2×Dic118—41no—
77C2×Dic118—42no—
78C2×Dic118—44no—
79C2×Dic118—44no—
80C2×C8×C59—42no—
81C4⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—24no—
87C22⋊Dic59—44no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—42no—
91Dic59⋊C4—44no—
92C4×Dic59—82no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59#10412yes—
102C23×C59C2×C22⋊C411yescharacteristic
103C23×C59#13712yes—
104D4×C59—44no—
105C2×C4×C59—22no—
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—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—81no—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C2×Dic59—24no—
122C2×Dic59—42no—
123C2×Dic59—48no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—44no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—82no—
130C2×Dic59—82no—
131Dic118—44no—
132Dic118—44no—
133Dic118—44no—
134Dic118—44no—
135C8×C59—82no—
136#11785—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—22no—
140C22×C59—22no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59C2×#9911yescharacteristic
150C22×C59#41911yescharacteristic
151C22×C59C2×#10411yescharacteristic
152C4×C59—21no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59AΣL(1,𝔽16)12yes—
158Dic59—44no—
159Dic59—82no—
160Dic59—84no—
161#2409—591no—
162C4×#99—591no—
163#1466D5911yescharacteristic
164#1351—594no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—42no—
168C2×C59—42no—
169C2×C59#150811yescharacteristic
170C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
171C2×C59#212911yescharacteristic
172C23×Q8—591no—
173C2×C22⋊Q8—1181no—
174C2×C4×D4C2×D5911yescharacteristic
175#432—21no—
176C2×#99—1181no—
177#395—592no—
178#393—592no—
179#391—1181no—
180#386—1181no—
181#385—1181no—
182#380—591no—
183C4×C22⋊C4—1181no—
184#342—592no—
185#323—21no—
186#323C2×D5912yes—
187C59#1178511yescharacteristic, complemented, Sylow
188C22×Q8—592no—
189C22×Q8—1181no—
190C22×Q8—2361no—
191C22×Q8—2361no—
192C22×Q8—2364no—
193C22×D4D23611yescharacteristic
194C23×C4—1181no—
195C23×C4C4×D5911yescharacteristic
196C23×C4D23611yescharacteristic
197C22⋊Q8—2364no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—1182no—
201C2×C4⋊C4—1181no—
202C2×C4⋊C4C4×D5911yescharacteristic
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1181no—
205C2×C22⋊C4—1182no—
206C2×C22⋊C4C22⋊D5911yescharacteristic
207C2×C22⋊C4C22×D5911yescharacteristic
208C2×C42—2361no—
209C2×C42C22⋊D5911yescharacteristic
210C4⋊C8—41no—
211#99—1184no—
212C22⋊C8—22no—
213#97—41no—
214#95—592no—
215#95—1182no—
216#95—1182no—
217#95—2361no—
218C24(C2×D59)⋊C411yescharacteristic
219C24C2×D23611yescharacteristic
220C2×Q8—1184no—
221C2×Q8—2362no—
222C2×Q8—2362no—
223C2×Q8—2364no—
224C2×Q8—2364no—
225C2×Q8—2368no—
226C2×Q8—2364no—
227C2×Q8—2361no—
228C2×Q8—2361no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1182no—
239C22×C4—1182no—
240C22×C4—2364no—
241C22×C4—2362no—
242C22×C4—2362no—
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4C2×C4×D5911yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—2362no—
251C4⋊C4—2364no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—1184no—
255C22⋊C4—2364no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C42—21no—
259C42—21no—
260C42—4722no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1427365312yes—
270C23C2×(C2×D59)⋊C411yescharacteristic
271C23#1427376012yes—
272Q8—2364no—
273Q8—2364no—
274Q8—2364no—
275Q8—2364no—
276D4—44no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—1184no—
292C2×C4—4722no—
293C2×C4—4722no—
294C2×C4—2364no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2368no—
300C2×C4—2362no—
301C2×C4#1427364111yescharacteristic
302C2×C4#1427364111yescharacteristic
303C8—82no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1427364111yescharacteristic
317C22#1683741211yescharacteristic
318C22C2×#1427365311yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4722no—
325C4—4724no—
326C4—2364no—
327C4#1683719012yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#2142261711yescharacteristic
333C2C2×#1683719011yescharacteristic
334C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1466C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1466D59
hypocenterC59#11785
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #21422118C2semidirect
5. C23×Dic118C4semidirect
8. #16837322C4semidirect
13. #16837262C4semidirect
187. C59#11785semidirect

Maximal subgroups

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

subgroupindex
#214274612
C4×#142736422
#214221182
C59×#14662
#1178559