Finite Group Databasebeta

← #28499322

Subgroups — #28499322

Order 15104.

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

normal subgroups only

This group has 572 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
2#21422434C211yesmaximal, characteristic
3#21422125C214yesmaximal, complemented
4C4×#100×C59C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic
6C2×#14273706—21no—
7#16837366—21no—
8#16837322C2212yes—
9#16837314C412yes—
10#16837308—21no—
11#16837282—21no—
12#16837274—21no—
13#16837262C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#100×C59—21no—
18C4×M16×C59—21no—
19C59×#343C2212yes—
20C22×Dic118—21no—
21C22×Dic118—44no—
22C22×Dic118—41no—
23C22×Dic118—41no—
24C22×Dic118C2×C412yes—
25C23×Dic59—21no—
26C2×#11745664—41no—
27#14273706—44no—
28C2×C4⋊Dic59—21no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—41no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95C2×C412yes—
37C22×D4×C59D411yescharacteristic
38C23×C4×C59C2×C411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C2×M16×C59—22no—
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2311yescharacteristic
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C42×C59D411yescharacteristic
47#100×C59—24no—
48#97×C59—41no—
49C4×C8×C59—41no—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C22⋊C411yescharacteristic
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×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—22no—
67C22×Dic59—22no—
68C22×Dic59—44no—
69C22×Dic59—42no—
70C2×Dic118—24no—
71C2×Dic118—42no—
72C2×Dic118—48no—
73C2×Dic118—44no—
74C2×Dic118—41no—
75C2×Dic118—41no—
76C2×Dic118—42no—
77C2×Dic118—44no—
78C2×Dic118—44no—
79M16×C59—24no—
80M16×C59—42no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—44no—
88C22⋊Dic59—42no—
89C4⋊Dic59—42no—
90Dic59⋊C4—44no—
91#11745664—82no—
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
104D4×C59—44no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—81no—
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#13712yes—
121C2×Dic59—24no—
122C2×Dic59—42no—
123C2×Dic59—48no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128Dic118—44no—
129Dic118—44no—
130Dic118—44no—
131Dic118—44no—
132C8×C59—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—41no—
154C4×C59—41no—
155C4×C59AΣL(1,𝔽16)12yes—
156Dic59—44no—
157Dic59—82no—
158#2409—591no—
159C4×#100D5911yescharacteristic
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×Q8—591no—
170C2×C22⋊Q8—1181no—
171C2×C4×D4C2×D5911yescharacteristic
172C2×#100—21no—
173#407—1181no—
174C4×M16—21no—
175#395—592no—
176#393—592no—
177#391—1181no—
178#386—1181no—
179#385—1181no—
180#380—591no—
181#343C2×D5912yes—
182#324—592no—
183#323—1181no—
184C59#1179111yescharacteristic, complemented, Sylow
185C22×Q8—592no—
186C22×Q8—1181no—
187C22×Q8—2361no—
188C22×Q8—2361no—
189C22×Q8—2364no—
190C22×D4C22⋊D5911yescharacteristic
191C23×C4—1181no—
192C23×C4C4×D5911yescharacteristic
193C23×C4D23611yescharacteristic
194C2×M16—22no—
195C2×M16—2361no—
196C22⋊Q8—2364no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1182no—
201C2×C4⋊C4C22×D5911yescharacteristic
202C2×C22⋊C4—1182no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4C4×D5911yescharacteristic
205C2×C22⋊C4D23611yescharacteristic
206C2×C42C22⋊D5911yescharacteristic
207#100—24no—
208C22⋊C8—1182no—
209C22⋊C8—2361no—
210#97—41no—
211C4×C8—41no—
212#95—592no—
213#95—1182no—
214#95—1182no—
215C24(C2×D59)⋊C411yescharacteristic
216C24(C2×D59)⋊C411yescharacteristic
217C2×Q8—1184no—
218C2×Q8—2362no—
219C2×Q8—2362no—
220C2×Q8—2364no—
221C2×Q8—2364no—
222C2×Q8—2368no—
223C2×Q8—2364no—
224C2×Q8—2361no—
225C2×Q8—2361no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4—2364no—
238C22×C4—2362no—
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4C2×C4×D5911yescharacteristic
242C22×C4C2×D23611yescharacteristic
243C22×C4C2×C22⋊D5911yescharacteristic
244M16—24no—
245M16—42no—
246M16—4722no—
247C2×C8—42no—
248C2×C8—2362no—
249C4⋊C4—41no—
250C4⋊C4—2362no—
251C4⋊C4—2364no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—2364no—
255C22⋊C4—2362no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1427365512yes—
268C23C2×(C2×D59)⋊C411yescharacteristic
269Q8—2364no—
270Q8—2364no—
271Q8—2364no—
272Q8—2364no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—1184no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4—2362no—
292C2×C4—2368no—
293C2×C4—2362no—
294C2×C4#1427364111yescharacteristic
295C2×C4#1427364111yescharacteristic
296C2×C4#1427375712yes—
297C8—44no—
298C8—4722no—
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×#1427364111yescharacteristic
312C22#1683741011yescharacteristic
313C22C2×#1427365511yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—4722no—
320C4—2364no—
321C4#1683719012yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#2142261511yescharacteristic
327C2C2×#1683719011yescharacteristic
328C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC4×#100×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C4⋊C4×C59C23
hypercenterC4×#100D59
hypocenterC59#11791
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
184. C59#11791semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214224342
#214221252
C4×#100×C592
#1179159