Finite Group Databasebeta

← #28499323

Subgroups — #28499323

Order 15104.

572 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14273644C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422126C214yesmaximal, complemented
4C59×#2409C211yesmaximal, characteristic
5C2×#14273644—21no—
6#16837366—21no—
7C4×#11745664—21no—
8#16837171C2212yes—
9#16837128C2212yes—
10#16837126—21no—
11C23×Q8×C59C411yescharacteristic
12C2×C22⋊Q8×C59—21no—
13C2×C4×D4×C59C2211yescharacteristic
14C59×#395C2212yes—
15C59×#393C412yes—
16C59×#391—21no—
17C59×#386—21no—
18C59×#385—21no—
19C59×#380C411yescharacteristic
20C2×#11745664—22no—
21C2×#11745664—41no—
22#14273644—24no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—41no—
25#14273638—41no—
26C4×C59⋊C8—41no—
27C22×Q8×C59—21no—
28C22×Q8×C59—41no—
29C22×Q8×C59—41no—
30C22×Q8×C59—44no—
31C22×Q8×C59C2×C412yes—
32C22×D4×C59D411yescharacteristic
33C23×C4×C59—21no—
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C22⋊Q8×C59—44no—
37C4×D4×C59—41no—
38C4×D4×C59—41no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47#95×C59—22no—
48#95×C59—22no—
49#95×C59C2×C412yes—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C22⋊C411yescharacteristic
52C2×Q8×C59—24no—
53C2×Q8×C59—42no—
54C2×Q8×C59—44no—
55C2×Q8×C59—42no—
56C2×Q8×C59—48no—
57C2×Q8×C59—44no—
58C2×Q8×C59—44no—
59C2×Q8×C59—41no—
60C2×Q8×C59—41no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—22no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—44no—
71C22×C4×C59—42no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C4⋊C4×C59—41no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—44no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—44no—
85C22⋊C4×C59—42no—
86C42×C59—21no—
87C42×C59—21no—
88#11745664—24no—
89#11745664—42no—
90#11745664—82no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—42no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59#10412yes—
103C23×C59C2×C22⋊C411yescharacteristic
104Q8×C59—44no—
105Q8×C59—44no—
106Q8×C59—44no—
107Q8×C59—44no—
108D4×C59—44no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—24no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—48no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—81no—
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#9911yescharacteristic
131C2×C4×C59#13712yes—
132C59⋊C8—44no—
133C59⋊C8—82no—
134#11791—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—22no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—81no—
147C22×C59C2×#9911yescharacteristic
148C22×C59#41911yescharacteristic
149C22×C59C2×#10411yescharacteristic
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—41no—
153C4×C59—44no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—82no—
157C4×C59AΣL(1,𝔽16)12yes—
158#2409D5911yescharacteristic
159C4×#100—591no—
160#1465—591no—
161#1354—594no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—42no—
165C2×C59—81no—
166C2×C59#150811yescharacteristic
167C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
168C2×C59#213511yescharacteristic
169C23×Q8Dic5911yescharacteristic
170C2×C22⋊Q8—21no—
171C2×C4×D4C2×D5911yescharacteristic
172C2×#100—1181no—
173#407—1181no—
174C4×M16—1181no—
175#395C2×D5912yes—
176#393Dic5912yes—
177#391—21no—
178#386—21no—
179#385—21no—
180#380Dic5911yescharacteristic
181#343—592no—
182#324—592no—
183#323—1181no—
184C59#1179111yescharacteristic, complemented, Sylow
185C22×Q8—21no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8C2×Dic5912yes—
190C22×D4C22⋊D5911yescharacteristic
191C23×C4—21no—
192C23×C4C2×Dic5911yescharacteristic
193C23×C4C22⋊D5911yescharacteristic
194C2×M16—1182no—
195C2×M16—2361no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22×D5911yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×Dic5911yescharacteristic
205C2×C22⋊C4C22⋊D5911yescharacteristic
206C2×C42C22⋊D5911yescharacteristic
207#100—1184no—
208C22⋊C8—1182no—
209C22⋊C8—2361no—
210#97—2361no—
211C4×C8—2361no—
212#95—22no—
213#95—22no—
214#95C2×Dic5912yes—
215C24C22⋊Dic5911yescharacteristic
216C24C22⋊Dic5911yescharacteristic
217C2×Q8—24no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic5911yescharacteristic
240C22×C4C22⋊Dic5911yescharacteristic
241C22×C4C22×Dic5911yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243C22×C4C2×C22⋊D5911yescharacteristic
244M16—1184no—
245M16—4722no—
246M16—2362no—
247C2×C8—2362no—
248C2×C8—2362no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1427365412yes—
268C23C2×C22⋊Dic5911yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—44no—
272Q8—44no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#1427364211yescharacteristic
295C2×C4#1427364211yescharacteristic
296C2×C4#1427375812yes—
297C8—4722no—
298C8—2364no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1427364211yescharacteristic
312C22#1683741111yescharacteristic
313C22C2×#1427365411yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4#1683718912yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#2142261611yescharacteristic
327C2C2×#1683718911yescharacteristic
328C2#2142579111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2409C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C4⋊C4×C59C23
hypercenter#2409D59
hypocenterC59#11791
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #21422126C2semidirect
184. C59#11791semidirect

Maximal subgroups

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

subgroupindex
C4×#142736442
#214224342
#214221262
C59×#24092
#1179159