Finite Group Databasebeta

← #28477210

Subgroups — #28477210

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424164C211yesmaximal, characteristic
2#21423790C211yesmaximal, characteristic
3#21421922C212yesmaximal, complemented
4#21421919C212yesmaximal, complemented
5C59×#2855C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×C22⋊(C59⋊C8)—21no—
8#16837157—21no—
9#16837157—21no—
10#16837157C2212yes—
11#16837157C2212yes—
12C2×C22⋊Q8×C59C411yescharacteristic
13C22×C42×C59C2211yescharacteristic
14C59×#391C411yescharacteristic
15C59×#391C2211yescharacteristic
16C59×#389C411yescharacteristic
17C59×#389C2211yescharacteristic
18C59×#387—21no—
19C59×#387—21no—
20C59×#385—21no—
21C59×#381—21no—
22C59×#380C411yescharacteristic
23C2×#11745664—24no—
24C2×#11745664—42no—
25C2×#11745664—42no—
26C22×C59⋊C8—22no—
27C22×C59⋊C8—22no—
28C22⋊(C59⋊C8)—42no—
29C22×Q8×C59C2×C411yescharacteristic
30C23×C4×C59C2×C411yescharacteristic
31C23×C4×C59D411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C22⋊Q8×C59—42no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59C2×C411yescharacteristic
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59—21no—
42C2×C42×C59—42no—
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59D411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C2×C42×C59C2311yescharacteristic
47#95×C59—21no—
48#95×C59—21no—
49#95×C59—21no—
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54C24×C59C22⋊C411yescharacteristic
55C2×Q8×C59—22no—
56C2×Q8×C59—41no—
57C2×Q8×C59—41no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C4⋊C4×C59—22no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C42×C59—41no—
88C42×C59—41no—
89C42×C59—41no—
90C42×C59—41no—
91#11745664—48no—
92C2×C59⋊C8—24no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8—42no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8—42no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59#9912yes—
105C23×C59C2×C22⋊C411yescharacteristic
106Q8×C59—82no—
107C2×C4×C59—24no—
108C2×C4×C59—24no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—44no—
124C2×C4×C59—44no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59#10412yes—
138C2×C4×C59#13112yes—
139C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C59#13711yescharacteristic
141C2×C4×C59#13712yes—
142C59⋊C8—44no—
143C59⋊C8—82no—
144#7695—591nomaximal, Sylow
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—22no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59#40811yescharacteristic
153C22×C59C2×#9911yescharacteristic
154C22×C59#41311yescharacteristic
155C22×C59#41711yescharacteristic
156C22×C59#41911yescharacteristic
157C22×C59C2×#10411yescharacteristic
158C22×C59#42111yescharacteristic
159C4×C59—22no—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—21no—
163C4×C59—22no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59—81no—
169C4×C59—81no—
170#2855D5911yescharacteristic
171#1817—591no—
172#1728—591no—
173#1272—592no—
174#1271—592no—
175C2×C59—42no—
176C2×C59#146811yescharacteristic
177C2×C59#148511yescharacteristic
178C2×C59#149511yescharacteristic
179C2×C59#150411yescharacteristic
180C2×C59#151111yescharacteristic
181C2×C59#152911yescharacteristic
182C2×C59#155911yescharacteristic
183C22×M16—1181no—
184C2×C22⋊Q8Dic5911yescharacteristic
185C22×C42C2×D5911yescharacteristic
186C2×C22⋊C8—1181no—
187#391Dic5911yescharacteristic
188#391C2×D5911yescharacteristic
189#389Dic5911yescharacteristic
190#389C2×D5911yescharacteristic
191#387—21no—
192#387—21no—
193#385—21no—
194#381—21no—
195#380Dic5911yescharacteristic
196#336—592no—
197#336—592no—
198#336—1181no—
199#336—1181no—
200C59#769511yescharacteristic, complemented, Sylow
201C22×Q8C2×Dic5911yescharacteristic
202C23×C4C2×Dic5911yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C23×C4C22⋊D5911yescharacteristic
205C2×M16—1184no—
206C2×M16—2362no—
207C2×M16—2362no—
208C22×C8—1182no—
209C22×C8—1182no—
210C22⋊Q8—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5911yescharacteristic
215C2×C4⋊C4C2×Dic5911yescharacteristic
216C2×C4⋊C4C2×Dic5911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—42no—
220C2×C42C2×Dic5911yescharacteristic
221C2×C42C22⋊D5911yescharacteristic
222C2×C42C22⋊D5911yescharacteristic
223C2×C42C22×D5911yescharacteristic
224C22⋊C8—2362no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24C22⋊Dic5911yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C22⋊Dic5911yescharacteristic
251C22×C4C22⋊Dic5911yescharacteristic
252C22×C4C22⋊Dic5911yescharacteristic
253C22×C4C22×Dic5911yescharacteristic
254C22×C4C2×C22⋊D5911yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256M16—2368no—
257C2×C8—1184no—
258C2×C8—2362no—
259C2×C8—2362no—
260C2×C8—2362no—
261C2×C8—2362no—
262C2×C8—2362no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—41no—
273C42—41no—
274C42—41no—
275C42—41no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23#1427364212yes—
283C23C2×C22⋊Dic5911yescharacteristic
284Q8—82no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#1427365412yes—
316C2×C4#1427373412yes—
317C2×C4#1427375211yescharacteristic
318C2×C4#1427375711yescharacteristic
319C2×C4#1427375812yes—
320C8—4722no—
321C8—2364no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22#1683736911yescharacteristic
330C22C2×#1427364211yescharacteristic
331C22#1683738611yescharacteristic
332C22#1683740311yescharacteristic
333C22#1683741111yescharacteristic
334C22C2×#1427365411yescharacteristic
335C22#1683741911yescharacteristic
336C4—21no—
337C4—22no—
338C4—21no—
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—42no—
348C2#2142244411yescharacteristic
349C2#2142251711yescharacteristic
350C2#2142256111yescharacteristic
351C2#2142260011yescharacteristic
352C2#2142262911yescharacteristic
353C2#2142270311yescharacteristic
354C2#2142282611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2855C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2855D59
hypocenterC59#7695
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #21421922C2semidirect
4. #21421919C2semidirect
200. C59#7695semidirect

Maximal subgroups

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

subgroupindex
#214241642
#214237902
#214219222
#214219192
C59×#28552
#769559