Finite Group Databasebeta

← #28498618

Subgroups — #28498618

Order 15104.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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#436×D59C211yesmaximal, characteristic
2#21425897C211yesmaximal, characteristic, complemented
3#21425894C211yesmaximal, characteristic, complemented
4#21422055C211yesmaximal, characteristic, complemented
5#21422054C211yesmaximal, characteristic, complemented
6#21422053C211yesmaximal, characteristic, complemented
7C59×#1330C211yesmaximal, characteristic, complemented
8C2×M16×D59—21no—
9C4×D4×D59C411yescharacteristic
10#16837481C2211yescharacteristic
11#16837480C411yescharacteristic
12#16837479C411yescharacteristic
13#16837478—21no—
14#16837477—21no—
15C4⋊C8×D59C411yescharacteristic
16C22⋊C8×D59—21no—
17C4×C8×D59C2211yescharacteristic
18C4⋊(C59⋊C16)C2211yescharacteristic, complemented
19Dic59⋊C16—21no—
20(C2×D59)⋊C16—21no—
21(C2×D59)⋊C16—21no—
22#16837178C2211yescharacteristic, complemented
23C16×Dic59—21no—
24C59×#436C2211yescharacteristic
25C59×C4⋊C16C2211yescharacteristic, complemented
26C59×#346C2211yescharacteristic, complemented
27C2×D4×D59C811yescharacteristic
28C22×C4×D59—21no—
29C2×#11745664—21no—
30C2×#11745659—21no—
31C2×#11745659—21no—
32M16×D59—42no—
33M16×D59—42no—
34C2×C8×D59—21no—
35C2×C8×D59D411yescharacteristic
36C2×C8×D59D411yescharacteristic
37C4×D236C2×C411yescharacteristic
38D4×Dic59C811yescharacteristic
39#14273689C811yescharacteristic
40C4×C22⋊D59—21no—
41(C22⋊D59)⋊C4—21no—
42C4⋊C4×D59C811yescharacteristic
43C22⋊C4×D59—21no—
44C42×D59C2×C411yescharacteristic
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C4⋊(C59⋊C8)C2×C411yescharacteristic
48Dic59⋊C8—21no—
49C22⋊(C59⋊C8)—21no—
50(C2×D59)⋊C8—21no—
51(C2×D59)⋊C8—21no—
52C4×C59⋊C8C2×C411yescharacteristic
53C8×Dic59D411yescharacteristic
54C8×Dic59D411yescharacteristic
55C2×M16×C59—21no—
56C4×D4×C59C2×C411yescharacteristic
57C2×C16×C59—21no—
58C2×C16×C59—21no—
59C4⋊C8×C59C2×C411yescharacteristic
60C22⋊C8×C59—21no—
61C4×C8×C59C2311yescharacteristic
62C23×D59—21no—
63C2×D4×C59C2×C811yescharacteristic
64C22×C4×C59—21no—
65C2×C22⋊D59—21no—
66C22×Dic59—21no—
67D4×D59—42no—
68C2×D236C2×C811yescharacteristic
69C2×C4×D59—21no—
70C2×C4×D59—21no—
71C2×C4×D59—21no—
72C2×C4×D59—42no—
73C2×C4×D59—42no—
74C2×C4×D59C22⋊C411yescharacteristic
75C2×C4×D59C2×C811yescharacteristic
76C2×C4×D59M1611yescharacteristic
77M16×C59—41no—
78M16×C59—41no—
79C2×C8×C59—21no—
80C2×C8×C59C2×D411yescharacteristic
81C2×C8×C59C2×D411yescharacteristic
82C4⋊C4×C59C2×C811yescharacteristic
83C22⋊C4×C59—21no—
84C42×C59C22×C411yescharacteristic
85C22⋊Dic59—21no—
86(C2×D59)⋊C4—21no—
87(C2×D59)⋊C4—21no—
88C4⋊Dic59C2×C811yescharacteristic
89Dic59⋊C4—21no—
90C4×Dic59C22⋊C411yescharacteristic
91C4×Dic59C2×C811yescharacteristic
92C4×Dic59M1611yescharacteristic
93#11745664—42no—
94C2×C59⋊C8—21no—
95C2×C59⋊C8C22⋊C411yescharacteristic
96C2×C59⋊C8C22⋊C411yescharacteristic
97#11745659—42no—
98#11745659—41no—
99#11745659—41no—
100C8×D59—21no—
101C8×D59—21no—
102C8×D59—41no—
103C8×D59—42no—
104C16×C59—41no—
105C16×C59—41no—
106C59⋊C16—41no—
107C59⋊C16—41no—
108C23×C59—21no—
109C22×D59—21no—
110C22×D59—42no—
111C22×D59—42no—
112C22×D59—42no—
113C22×D59C22⋊C811yescharacteristic
114D4×C59—41no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59C2×C22⋊C411yescharacteristic
119C2×C4×C59C22×C811yescharacteristic
120C2×C4×C59C2×M1611yescharacteristic
121C22⋊D59—42no—
122C2×Dic59—21no—
123C2×Dic59—42no—
124C2×Dic59C22⋊C811yescharacteristic
125C2×Dic59C22⋊C811yescharacteristic
126C2×Dic59C22⋊C811yescharacteristic
127D236—41no—
128C4×D59—22no—
129C4×D59—21no—
130C4×D59—21no—
131C4×D59—22no—
132C4×D59—42no—
133C4×D59—42no—
134C4×D59—41no—
135C4×D59—41no—
136C8×C59—21no—
137C8×C59—41no—
138C8×C59AGL(1,ℤ/8)11yescharacteristic
139C8×C59AGL(1,ℤ/8)11yescharacteristic
140C59⋊C8—21no—
141C59⋊C8—21no—
142C59⋊C8—41no—
143#11634—591nomaximal, Sylow
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59C2×C22⋊C811yescharacteristic
148C2×D59—21no—
149C2×D59—21no—
150C2×D59—41no—
151C2×D59—41no—
152C2×D59—41no—
153C2×D59—84no—
154C2×D59#35012yes—
155C4×C59—21no—
156C4×C59—21no—
157C4×C59—41no—
158C4×C59#41811yescharacteristic
159C4×C59#42111yescharacteristic
160Dic59—21no—
161Dic59—21no—
162Dic59—41no—
163Dic59#34912yes—
164C2×#436—591no—
165#2163—591no—
166#2162—591no—
167#1330—591no—
168#1330—591no—
169#1330—591no—
170#1330D5911yescharacteristic, complemented
171C2×C59—81no—
172C2×C59#147811yescharacteristic
173C2×C59C2×#34911yescharacteristic
174C2×C59C2×#35011yescharacteristic
175D59—22no—
176D59—81no—
177C22×M16—1181no—
178C2×C4×D4—591no—
179#436—591no—
180#436—591no—
181#436—591no—
182#436—1181no—
183#436—1181no—
184#436C2×D5911yescharacteristic
185C2×C4⋊C8—591no—
186C2×C22⋊C8—1181no—
187C2×C4×C8—591no—
188C4⋊C16—591no—
189C4⋊C16—1181no—
190C4⋊C16C2×D5911yescharacteristic, complemented
191C22⋊C16—1181no—
192C22⋊C16—1181no—
193#346—591no—
194#346C2×D5911yescharacteristic, complemented
195C4×C16—1181no—
196C59#1163411yescharacteristic, complemented, Sylow
197C22×D4—591no—
198C23×C4—1181no—
199C2×M16—21no—
200C2×M16—1181no—
201C2×M16—1181no—
202C2×M16—1181no—
203C2×M16—2362no—
204C2×M16—2362no—
205C22×C8—591no—
206C22×C8—591no—
207C22×C8—1181no—
208C4×D4—591no—
209C4×D4—591no—
210C4×D4—591no—
211C4×D4—1181no—
212C4×D4—1181no—
213C4×D4C4×D5911yescharacteristic
214C2×C4⋊C4—591no—
215C2×C22⋊C4—1181no—
216C2×C42—591no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—1181no—
220C2×C16—1181no—
221C4⋊C8—591no—
222C4⋊C8—1181no—
223C4⋊C8C4×D5911yescharacteristic
224C22⋊C8—21no—
225C22⋊C8—1181no—
226C22⋊C8—1181no—
227C22⋊C8—1181no—
228C4×C8—591no—
229C4×C8—591no—
230C4×C8—591no—
231C4×C8C22×D5911yescharacteristic
232C24—1181no—
233C2×D4—591no—
234C2×D4—1181no—
235C2×D4—2362no—
236C2×D4C8×D5911yescharacteristic
237C22×C4—21no—
238C22×C4—591no—
239C22×C4—591no—
240C22×C4—591no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—1181no—
245C22×C4—2362no—
246C22×C4—2362no—
247M16—41no—
248M16—41no—
249M16—2362no—
250M16—2362no—
251M16—2361no—
252M16—2361no—
253C2×C8—21no—
254C2×C8—591no—
255C2×C8—591no—
256C2×C8—1181no—
257C2×C8—1181no—
258C2×C8—1181no—
259C2×C8—2361no—
260C2×C8—2362no—
261C2×C8D4×D5911yescharacteristic
262C2×C8D4×D5911yescharacteristic
263C4⋊C4—591no—
264C4⋊C4—1181no—
265C4⋊C4C8×D5911yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—1181no—
268C22⋊C4—1181no—
269C22⋊C4—1181no—
270C42—591no—
271C42—591no—
272C42—591no—
273C42C2×C4×D5911yescharacteristic
274C16—41no—
275C16—41no—
276C16—2361no—
277C16—2361no—
278C23—21no—
279C23—591no—
280C23—1181no—
281C23—2362no—
282C23—2362no—
283C23—2362no—
284D4—41no—
285D4—2361no—
286D4—2362no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—591no—
291C2×C4—591no—
292C2×C4—591no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—1181no—
296C2×C4—1181no—
297C2×C4—1181no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4C22⋊C4×D5911yescharacteristic
304C2×C4C2×C8×D5911yescharacteristic
305C2×C4M16×D5911yescharacteristic
306C8—21no—
307C8—41no—
308C8—1181no—
309C8—1181no—
310C8—2361no—
311C8#1427375111yescharacteristic
312C8#1427375411yescharacteristic
313C22—41no—
314C22—41no—
315C22—41no—
316C22—592no—
317C22—1181no—
318C22—1181no—
319C22—4724no—
320C22—2361no—
321C22—2361no—
322C22—2361no—
323C22C22⋊C8×D5911yescharacteristic
324C4—21no—
325C4—21no—
326C4—41no—
327C4—592no—
328C4—1181no—
329C4—1181no—
330C4—2361no—
331C4#1683740511yescharacteristic
332C4#1683741711yescharacteristic
333C2—81no—
334C2—1182no—
335C2—4721no—
336C2#2142248511yescharacteristic
337C2#349×D5911yescharacteristic
338C2#350×D5911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC4×C8C22×D59
FittingC59×#1330C2
socleC22×C59C2×C22⋊C8
prime radicalC2×D4×D59C8
cosocleC4×C8×C59C23
hypercenter#1330D59
hypocenterC59#11634
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1330

Complements

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

NH ≅ G/Ntype
2. #21425897C2semidirect
3. #21425894C2semidirect
4. #21422055C2semidirect
5. #21422054C2semidirect
6. #21422053C2semidirect
7. C59×#1330C2semidirect
18. C4⋊(C59⋊C16)C22semidirect
22. #16837178C22semidirect
25. C59×C4⋊C16C22semidirect
26. C59×#346C22semidirect
170. #1330D59semidirect
190. C4⋊C16C2×D59semidirect
194. #346C2×D59semidirect
196. C59#11634semidirect

Maximal subgroups

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

subgroupindex
#436×D592
#214258972
#214258942
#214220552
#214220542
#214220532
C59×#13302
#1163459