Finite Group Databasebeta

← #28485364

Subgroups — #28485364

Order 15104.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#21435240C211yesmaximal, characteristic
2#21433033C211yesmaximal, characteristic
3#21424917C211yesmaximal, characteristic
4#21422661C211yesmaximal, characteristic
5#21421956C211yesmaximal, characteristic
6#21421926C211yesmaximal, characteristic
7C59×#1546C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837798C412yescomplemented
10#16837754C412yescomplemented
11#16837732—22no—
12#16837684—22no—
13#16837531—22no—
14#16837531—22no—
15#16837455C2211yescharacteristic
16#16837430—21no—
17C2×#14273651—21no—
18C2×#14273651—21no—
19#16837360—21no—
20C2×#14273638C2211yescharacteristic
21#16837268C2211yescharacteristic
22#16837140—22no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#332—22no—
27C22×C59⋊C8D411yescharacteristic
28C22×C59⋊C8D411yescharacteristic
29#14273727C2×C411yescharacteristic
30#14273727C2×C411yescharacteristic
31#14273719—21no—
32#14273706—21no—
33#14273706D412yes—
34C4×Dic118—21no—
35C4×Dic118D412yes—
36#14273686—21no—
37C2×C4⋊Dic59—21no—
38#14273658—24no—
39C4⋊(C59⋊C8)—21no—
40Dic59⋊C8—24no—
41#14273651—42no—
42#14273651—42no—
43#14273650—24no—
44C22⋊(C59⋊C8)—21no—
45#14273638—21no—
46#14273638—21no—
47#14273637—22no—
48C4×C59⋊C8C2×C412yes—
49C8×Dic59—22no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×Q8×C59—21no—
53C2×M16×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4⋊Q8×C59C2×C412yes—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D412yes—
64C22⋊C8×C59—21no—
65C4×C8×C59—21no—
66C2×Q8×C59—22no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59#4111yescharacteristic
76C22×Dic59—41no—
77C2×Dic118—21no—
78C2×Dic118C22⋊C412yes—
79M16×C59—41no—
80M16×C59—41no—
81C2×C8×C59—22no—
82C2×C8×C59—41no—
83C4⋊C4×C59—22no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C42×C59C22×C411yescharacteristic
88C42×C59C2×D411yescharacteristic
89C42×C59C2×D411yescharacteristic
90C42×C59#4111yescharacteristic
91C22⋊Dic59—22no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—41no—
96C4⋊Dic59—41no—
97C4⋊Dic59C22⋊C412yes—
98Dic59⋊C4—22no—
99Dic59⋊C4—22no—
100C4×Dic59—22no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—21no—
103C2×C59⋊C8—22no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8SD1612yes—
108C23×C59#12111yescharacteristic
109Q8×C59—42no—
110Q8×C59—42no—
111Q8×C59—81no—
112Q8×C59—81no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—81no—
117C2×C4×C59—81no—
118C2×C4×C59C2×C22⋊C411yescharacteristic
119C2×C4×C59C4×D411yescharacteristic
120C2×C4×C59C4×D411yescharacteristic
121C2×C4×C59#12011yescharacteristic
122C2×C4×C59#12111yescharacteristic
123C2×C4×C59#12311yescharacteristic
124C2×C4×C59C2×SD1611yescharacteristic
125C2×C4×C59#13511yescharacteristic
126C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C59#13711yescharacteristic
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59—81no—
131C2×Dic59—81no—
132C2×Dic59—81no—
133Dic118—42no—
134Dic118—81no—
135C8×C59—42no—
136C8×C59—81no—
137C59⋊C8—22no—
138C59⋊C8—42no—
139C59⋊C8—41no—
140#9292—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59#38611yescharacteristic
144C22×C59#46011yescharacteristic
145C22×C59#46411yescharacteristic
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—82no—
151C4×C59C4×SD1611yescharacteristic
152C4×C59#44011yescharacteristic
153C4×C59#44811yescharacteristic
154C4×C59#44911yescharacteristic
155Dic59—42no—
156Dic59—42no—
157Dic59—161no—
158#3229—591no—
159#2961—591no—
160#1960—591no—
161#1546D5911yescharacteristic
162#1518—591no—
163#1286—591no—
164#1275—591no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59#188011yescharacteristic
168C2×C59#190211yescharacteristic
169C2×C59#190911yescharacteristic
170#533—591no—
171C2×C4⋊Q8C2×D5911yescharacteristic
172#498—592no—
173#487—592no—
174#483—1182no—
175#475—1182no—
176#446—1182no—
177#446—1182no—
178#431—591no—
179#431C2×D5911yescharacteristic
180#424—1181no—
181#423C2×D5911yescharacteristic
182C2×#103—1181no—
183C2×#103—1181no—
184#405—1181no—
185C2×#97—591no—
186#382—591no—
187#332—22no—
188#328—1182no—
189C59#929211yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C2×M16—21no—
192C22×C8—591no—
193C22×C8—591no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—591no—
198C4⋊Q8—591no—
199C4⋊Q8C4×D5912yes—
200#125—1181no—
201C22⋊Q8—592no—
202C22⋊Q8—1181no—
203C4×Q8—592no—
204C4×Q8—1181no—
205#117—1181no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—1181no—
209C2×C42C22×D5911yescharacteristic
210#106—1184no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—1184no—
214C4⋊C8—1181no—
215C4⋊C8C22⋊D5912yes—
216#103—1184no—
217#103—2362no—
218#103—2362no—
219C22⋊C8—21no—
220C22⋊C8—1181no—
221#97—1182no—
222#97—1181no—
223#97—1181no—
224C4×C8—21no—
225C4×C8—592no—
226C4×C8—1182no—
227#95—1181no—
228#95—1181no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—592no—
236C2×Q8—1181no—
237C22×C4—41no—
238C22×C4—2361no—
239C22×C4#1174568511yescharacteristic
240C22×C4D4×D5911yescharacteristic
241C22×C4D4×D5911yescharacteristic
242M16—41no—
243M16—41no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—592no—
247C2×C8—1181no—
248C2×C8—1181no—
249C2×C8—1182no—
250C2×C8—1181no—
251C2×C8—1181no—
252C2×C8—1181no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—592no—
258C4⋊C4—1181no—
259C4⋊C4—1181no—
260C4⋊C4—1181no—
261C4⋊C4—1182no—
262C4⋊C4—1182no—
263C4⋊C4—2361no—
264C4⋊C4—2361no—
265C22⋊C4—1182no—
266C42—1182no—
267C42C2×C4×D5911yescharacteristic
268C42D4×D5911yescharacteristic
269C42#1174568711yescharacteristic
270C42C2×C22⋊D5911yescharacteristic
271C23#1427370011yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—4721no—
277Q8—2362no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—1182no—
284C2×C4—1182no—
285C2×C4—4721no—
286C2×C4—4721no—
287C2×C4—4721no—
288C2×C4C22⋊C4×D5911yescharacteristic
289C2×C4(C22⋊D59)⋊C411yescharacteristic
290C2×C4C4×C22⋊D5911yescharacteristic
291C2×C4#1427369511yescharacteristic
292C2×C4#1427369811yescharacteristic
293C2×C4#1427370911yescharacteristic
294C2×C4SD16×D5911yescharacteristic
295C2×C4#1427374611yescharacteristic
296C2×C4#1427375111yescharacteristic
297C2×C4#1427375611yescharacteristic
298C8—42no—
299C8—81no—
300C8—1182no—
301C8—2362no—
302C8—2361no—
303C22—21no—
304C22—21no—
305C22#1683727811yescharacteristic
306C22#1683760111yescharacteristic
307C22#1683762311yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—9441no—
314C4—2362no—
315C4—2362no—
316C4#1683749311yescharacteristic
317C4#1683749911yescharacteristic
318C4#1683753911yescharacteristic
319C4#1683754411yescharacteristic
320C2—21no—
321C2—21no—
322C2#2142444811yescharacteristic
323C2#2142457211yescharacteristic
324C2#2142461511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1546C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1546D59
hypocenterC59#9292
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
9. #16837798C4semidirect
10. #16837754C4semidirect
189. C59#9292semidirect

Maximal subgroups

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

subgroupindex
#214352402
#214330332
#214249172
#214226612
#214219562
#214219262
C59×#15462
#929259