Finite Group Databasebeta

← #28497834

Subgroups — #28497834

Order 15104.

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

normal subgroups only

This group has 388 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
1#424×D59C211yesmaximal, characteristic
2#21425705C211yesmaximal, characteristic
3C22⋊C16×D59C211yesmaximal, characteristic
4#21422038C211yesmaximal, characteristic, complemented
5#21422037C211yesmaximal, characteristic, complemented
6#21422037C211yesmaximal, characteristic, complemented
7C59×#1323C211yesmaximal, characteristic, complemented
8C2×M16×D59C411yescharacteristic
9C22×C8×D59C2211yescharacteristic
10#117×D59C411yescharacteristic
11C2×#14273666—21no—
12C2×C16×D59—21no—
13#16837430C411yescharacteristic
14#16837429C411yescharacteristic
15#16837428C2211yescharacteristic
16#16837427—21no—
17#16837427—21no—
18C4⋊C8×D59—21no—
19C4⋊C8×D59—21no—
20C22⋊(C59⋊C16)C2211yescharacteristic
21C22⋊(C59⋊C16)C2211yescharacteristic
22(C2×D59)⋊C16—21no—
23(C2×D59)⋊C16—21no—
24C59×#424C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×C22⋊C16C2211yescharacteristic
27C22×C4×D59C2×C411yescharacteristic
28C2×#11745664C2×C411yescharacteristic
29C2×#11745659—21no—
30M16×D59—42no—
31C22×C59⋊C8C2×C411yescharacteristic
32C2×C8×D59—21no—
33C2×C8×D59—42no—
34C2×C8×D59D411yescharacteristic
35C2×C8×D59D411yescharacteristic
36C2×C8×D59D411yescharacteristic
37C2×C8×D59D411yescharacteristic
38#14273686C2×C411yescharacteristic
39#14273685C811yescharacteristic
40#14273685C811yescharacteristic
41#14273684—21no—
42#14273683—21no—
43C4⋊C4×D59C811yescharacteristic
44C4⋊C4×D59C811yescharacteristic
45C22⋊C4×D59—21no—
46C42×D59—21no—
47#14273666—42no—
48C2×C59⋊C16—21no—
49C2×C59⋊C16—21no—
50C16×D59—42no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)—21no—
53Dic59⋊C8—21no—
54Dic59⋊C8—21no—
55Dic59⋊C8—21no—
56Dic59⋊C8—21no—
57C2×M16×C59C2×C411yescharacteristic
58C22×C8×C59C2311yescharacteristic
59#117×C59C2×C411yescharacteristic
60C2×C16×C59—21no—
61C2×C16×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59—21no—
64C23×D59M1611yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×Dic59M1611yescharacteristic
67C2×C4×D59—22no—
68C2×C4×D59—22no—
69C2×C4×D59—21no—
70C2×C4×D59—21no—
71C2×C4×D59C22⋊C411yescharacteristic
72C2×C4×D59C22⋊C411yescharacteristic
73C2×C4×D59C2×C811yescharacteristic
74C2×C4×D59C2×C811yescharacteristic
75M16×C59—41no—
76C2×C8×C59—21no—
77C2×C8×C59—41no—
78C2×C8×C59C2×D411yescharacteristic
79C2×C8×C59C2×D411yescharacteristic
80C4⋊C4×C59C2×C811yescharacteristic
81C4⋊C4×C59C2×C811yescharacteristic
82C22⋊C4×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—21no—
85(C2×D59)⋊C4—21no—
86(C2×D59)⋊C4—21no—
87C4⋊Dic59C2×C811yescharacteristic
88C4⋊Dic59C2×C811yescharacteristic
89Dic59⋊C4—21no—
90Dic59⋊C4—21no—
91C4×Dic59—21no—
92C4×Dic59—21no—
93C4×Dic59—21no—
94#11745664—41no—
95C2×C59⋊C8—21no—
96C2×C59⋊C8—41no—
97C2×C59⋊C8C22⋊C411yescharacteristic
98C2×C59⋊C8C22⋊C411yescharacteristic
99#11745659—42no—
100C8×D59—22no—
101C8×D59—22no—
102C8×D59—41no—
103C8×D59—41no—
104C8×D59—42no—
105C16×C59—41no—
106C16×C59—41no—
107C59⋊C16—41no—
108C59⋊C16—41no—
109C23×C59C2×M1611yescharacteristic
110C22×D59—22no—
111C22×D59—22no—
112C22×D59—41no—
113C22×D59C22⋊C811yescharacteristic
114C22×D59C22⋊C811yescharacteristic
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59C2×C22⋊C411yescharacteristic
120C2×C4×C59C22×C811yescharacteristic
121C2×Dic59—22no—
122C2×Dic59—21no—
123C2×Dic59—21no—
124C2×Dic59C22⋊C811yescharacteristic
125C2×Dic59C22⋊C811yescharacteristic
126C4×D59—22no—
127C4×D59—21no—
128C4×D59—21no—
129C4×D59—22no—
130C4×D59—42no—
131C4×D59—42no—
132C4×D59#9912yes—
133C4×D59#10112yes—
134C8×C59—21no—
135C8×C59—21no—
136C8×C59—41no—
137C59⋊C8—21no—
138C59⋊C8—21no—
139C59⋊C8—41no—
140#11444—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59C2×C22⋊C811yescharacteristic
145C2×D59—21no—
146C2×D59—21no—
147C2×D59—21no—
148C2×D59—21no—
149C2×D59—44no—
150C2×D59#32412yes—
151C2×D59#35012yes—
152C4×C59—21no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59C2×#9911yescharacteristic
156C4×C59C2×#10111yescharacteristic
157Dic59—21no—
158Dic59—41no—
159Dic59—41no—
160Dic59#32412yes—
161C2×#424—591no—
162#2112—591no—
163C2×C22⋊C16—591no—
164#1323—591no—
165#1323—591no—
166#1323—591no—
167#1323D5911yescharacteristic, complemented
168C2×C59—41no—
169C2×C59C2×#32411yescharacteristic
170C2×C59C2×#35011yescharacteristic
171C2×C59#211711yescharacteristic
172D59—22no—
173D59—22no—
174C22×M16—591no—
175C23×C8—591no—
176C2×#117—591no—
177C2×M32—1181no—
178C22×C16—1181no—
179#424—591no—
180#424—591no—
181#424—591no—
182#424—1181no—
183#424—1181no—
184#424C2×D5911yescharacteristic
185C2×C4⋊C8—1181no—
186C2×C4⋊C8—1181no—
187C22⋊C16—591no—
188C22⋊C16—591no—
189C22⋊C16—1181no—
190C22⋊C16—1181no—
191C22⋊C16C2×D5911yescharacteristic
192C22⋊C16C2×D5911yescharacteristic
193C59#1144411yescharacteristic, complemented, Sylow
194C23×C4—591no—
195C2×M16—591no—
196C2×M16—1181no—
197C2×M16—2362no—
198C2×M16C4×D5911yescharacteristic
199C22×C8—591no—
200C22×C8—591no—
201C22×C8—591no—
202C22×C8—591no—
203C22×C8—591no—
204C22×C8—1181no—
205C22×C8—2362no—
206C22×C8C22×D5911yescharacteristic
207#117—591no—
208#117—591no—
209#117—591no—
210#117—1181no—
211#117—1181no—
212#117C4×D5911yescharacteristic
213C2×C4⋊C4—591no—
214C2×C4⋊C4—591no—
215C2×C22⋊C4—1181no—
216C2×C42—1181no—
217M32—2362no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—1181no—
221C2×C16—1181no—
222C2×C16—2362no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—1181no—
226C4⋊C8—1181no—
227C4⋊C8—1181no—
228C4⋊C8—1181no—
229C4⋊C8—1181no—
230C4⋊C8—1181no—
231C24—591no—
232C22×C4—591no—
233C22×C4—591no—
234C22×C4—591no—
235C22×C4—591no—
236C22×C4—591no—
237C22×C4—1182no—
238C22×C4—1182no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4C2×C4×D5911yescharacteristic
242M16—41no—
243M16—2362no—
244M16—2361no—
245C2×C8—21no—
246C2×C8—41no—
247C2×C8—591no—
248C2×C8—591no—
249C2×C8—1182no—
250C2×C8—1182no—
251C2×C8—1181no—
252C2×C8—2361no—
253C2×C8—2361no—
254C2×C8—2362no—
255C2×C8—2361no—
256C2×C8D4×D5911yescharacteristic
257C2×C8D4×D5911yescharacteristic
258C4⋊C4—591no—
259C4⋊C4—591no—
260C4⋊C4—1181no—
261C4⋊C4—1181no—
262C4⋊C4C8×D5911yescharacteristic
263C4⋊C4C8×D5911yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—1181no—
266C22⋊C4—1181no—
267C22⋊C4—1181no—
268C42—21no—
269C42—1181no—
270C42—1181no—
271C42—1181no—
272C16—41no—
273C16—41no—
274C16—2361no—
275C16—2361no—
276C23—591no—
277C23—591no—
278C23—1182no—
279C23—1182no—
280C23—2361no—
281C23M16×D5911yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—591no—
287C2×C4—592no—
288C2×C4—592no—
289C2×C4—591no—
290C2×C4—1182no—
291C2×C4—1182no—
292C2×C4—1182no—
293C2×C4—1181no—
294C2×C4—1181no—
295C2×C4—1181no—
296C2×C4—1181no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4C22⋊C4×D5911yescharacteristic
300C2×C4C2×C8×D5911yescharacteristic
301C8—21no—
302C8—21no—
303C8—41no—
304C8—1181no—
305C8—1181no—
306C8—2361no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—592no—
311C22—592no—
312C22—1181no—
313C22—1181no—
314C22—1181no—
315C22—1181no—
316C22—2364no—
317C22C22⋊C8×D5911yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—592no—
322C4—1181no—
323C4—2361no—
324C4—2361no—
325C4#99×D5911yescharacteristic
326C4#101×D5911yescharacteristic
327C2—41no—
328C2—1182no—
329C2—1182no—
330C2#324×D5911yescharacteristic
331C2#350×D5911yescharacteristic
332C2#2142572211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC22×C8C22×D59
FittingC59×#1323C2
socleC22×C59C2×C22⋊C8
prime radicalC23×D59M16
cosocleC22×C8×C59C23
hypercenter#1323D59
hypocenterC59#11444
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
4. #21422038C2semidirect
5. #21422037C2semidirect
6. #21422037C2semidirect
7. C59×#1323C2semidirect
167. #1323D59semidirect
193. C59#11444semidirect

Maximal subgroups

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

subgroupindex
#424×D592
#214257052
C22⋊C16×D592
#214220382
#214220372
C59×#13232
#1144459