Finite Group Databasebeta

← #28593772

Subgroups — #28593772

Order 15104.

1170 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
1C2×#16837455C211yesmaximal, characteristic
2C4×#14273686C211yesmaximal, characteristic
3#21425049C212yesmaximal, complemented
4#21425020C212yesmaximal, complemented
5#21424409C214yesmaximal
6C4×#14273651C214yesmaximal, complemented
7C2×C59×#533C211yesmaximal, characteristic
8C2×#14273686C2212yes—
9#16837455—24no—
10C2×C4⋊(C59⋊C8)C2212yes—
11C2×#14273651C2218yes—
12C2×C22⋊(C59⋊C8)C2212yes—
13C2×C4×C59⋊C8C2212yes—
14C4×C4⋊Dic59C2212yes—
15C4×C4⋊Dic59C2212yes—
16C4×C22⋊Dic59—21no—
17C4×C22⋊Dic59—21no—
18#16837248—21no—
19C42×Dic59—21no—
20#16837164C2218yes—
21C59×#533—24no—
22C59×#533C418yescomplemented
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×C4⋊Q8×C59C2211yescharacteristic
25C2×#125×C59—21no—
26C2×C22⋊Q8×C59—21no—
27C2×C22⋊Q8×C59C2212yes—
28C2×C4×Q8×C59—21no—
29C2×C4×Q8×C59C2212yes—
30C2×#117×C59—21no—
31C22×C42×C59C2211yescharacteristic
32C22×C59⋊C8C2314yes—
33#14273686—28no—
34C2×C4⋊Dic59C2314yes—
35C2×C22⋊Dic59—22no—
36C2×C4×Dic59—22no—
37C2×C4×Dic59—22no—
38C2×C4×Dic59—22no—
39C4⋊(C59⋊C8)—24no—
40#14273651—216no—
41C22⋊(C59⋊C8)—24no—
42C4×C59⋊C8—24no—
43C59⋊#95—22no—
44C59⋊#95—22no—
45C22×Q8×C59—21no—
46C22×Q8×C59C2312yes—
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59C2311yescharacteristic
50C4⋊Q8×C59—22no—
51C4⋊Q8×C59—22no—
52C4⋊Q8×C59C2×C414yes—
53C4⋊Q8×C59C2×C414yes—
54#125×C59—28no—
55C22⋊Q8×C59—28no—
56C22⋊Q8×C59—24no—
57C22⋊Q8×C59C2×C418yes—
58C4×Q8×C59—28no—
59C4×Q8×C59—24no—
60C4×Q8×C59C2×C418yes—
61#117×C59—28no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—22no—
66C2×C4⋊C4×C59—22no—
67C2×C4⋊C4×C59C2312yes—
68C2×C22⋊C4×C59—22no—
69C2×C42×C59—22no—
70C2×C42×C59C2×C414yes—
71C2×C42×C59D411yescharacteristic
72C2×C42×C59D411yescharacteristic
73C2×C42×C59D414yes—
74C2×C42×C59C2311yescharacteristic
75C2×C42×C59C2311yescharacteristic
76C24×C59C2×D411yescharacteristic
77C2×Q8×C59—24no—
78C2×Q8×C59—22no—
79C2×Q8×C59—28no—
80C2×Q8×C59—44no—
81C2×Q8×C59C22×C414yes—
82C22×C4×C59—24no—
83C22×C4×C59—24no—
84C22×C4×C59—24no—
85C22×C4×C59—22no—
86C22×C4×C59—22no—
87C22×C4×C59C22⋊C414yes—
88C22×C4×C59C22⋊C414yes—
89C22×C4×C59C22×C412yes—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D412yes—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2411yescharacteristic
97C22×Dic59—24no—
98C4⋊C4×C59—24no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59—24no—
101C4⋊C4×C59—28no—
102C4⋊C4×C59—28no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59C22×C414yes—
105C22⋊C4×C59—28no—
106C42×C59—28no—
107C42×C59C22⋊C414yes—
108C42×C59C22⋊C414yes—
109C42×C59C22×C412yes—
110C42×C59C22×C412yes—
111C42×C59C2×D412yes—
112C42×C59C2×D412yes—
113C22⋊Dic59—28no—
114C4⋊Dic59—24no—
115C4⋊Dic59#4118yes—
116C4×Dic59—28no—
117C4×Dic59—44no—
118C4×Dic59—44no—
119C2×C59⋊C8—24no—
120C2×C59⋊C8—28no—
121C23×C59—22no—
122C23×C59—22no—
123C23×C59—22no—
124C23×C59C2×C22⋊C412yes—
125C23×C59C22×D411yescharacteristic
126Q8×C59—28no—
127Q8×C59—44no—
128Q8×C59—48no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—24no—
133C2×C4×C59—24no—
134C2×C4×C59—28no—
135C2×C4×C59—28no—
136C2×C4×C59—24no—
137C2×C4×C59—24no—
138C2×C4×C59—44no—
139C2×C4×C59—44no—
140C2×C4×C59C2×C22⋊C412yes—
141C2×C4×C59C2×C22⋊C412yes—
142C2×C4×C59C2×C22⋊C412yes—
143C2×C4×C59C2×C22⋊C412yes—
144C2×C4×C59C2×C22⋊C412yes—
145C2×C4×C59#13514yes—
146C2×C4×C59#13714yes—
147C2×C4×C59C23×C411yescharacteristic
148C2×C4×C59C22×D411yescharacteristic
149C2×C4×C59C2×#4114yes—
150C2×Dic59—28no—
151C2×Dic59—44no—
152C2×Dic59—44no—
153C59⋊C8—44no—
154#21389—591nomaximal, Sylow
155C22×C59—22no—
156C22×C59—22no—
157C22×C59—22no—
158C22×C59—44no—
159C22×C59C22×C22⋊C411yescharacteristic
160C22×C59#52512yes—
161C22×C59C2×#13512yes—
162C22×C59C2×#13712yes—
163C4×C59—22no—
164C4×C59—22no—
165C4×C59—22no—
166C4×C59—44no—
167C4×C59—44no—
168C4×C59#41614yes—
169C4×C59#41914yes—
170C4×C59C4×#4112yes—
171C4×C59#53212yes—
172Dic59—48no—
173C2×#533D5911yescharacteristic
174C2×#431—591no—
175C4×#117—591no—
176#1988—592no—
177#1981—592no—
178#1873—594no—
179C4×#103—594no—
180C2×C59—42no—
181C2×C59#229711yescharacteristic
182C2×C59C2×#41611yescharacteristic
183C2×C59C2×#41911yescharacteristic
184C2×C59#310714yes—
185#533—24no—
186#533Dic5918yescomplemented
187C2×C4⋊Q8C2×D5911yescharacteristic
188C2×C4⋊Q8C2×D5911yescharacteristic
189C2×#125—21no—
190C2×C22⋊Q8—21no—
191C2×C22⋊Q8C2×D5912yes—
192C2×C4×Q8—21no—
193C2×C4×Q8C2×D5912yes—
194C2×#117—21no—
195C2×#117—592no—
196C22×C42C2×D5911yescharacteristic
197#431—1184no—
198C2×C4⋊C8—592no—
199C2×#103—598no—
200C2×C22⋊C8—592no—
201C2×C4×C8—592no—
202C4×C4⋊C4—592no—
203C4×C4⋊C4—592no—
204C4×C22⋊C4—1181no—
205C4×C22⋊C4—1181no—
206#376—1181no—
207C43—1181no—
208#340—598no—
209C59#2138911yescharacteristic, complemented, Sylow
210C22×Q8—21no—
211C22×Q8C22×D5912yes—
212C23×C4C22⋊D5911yescharacteristic
213C23×C4C22⋊D5911yescharacteristic
214C23×C4C22×D5911yescharacteristic
215C22×C8—594no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8C2×Dic5914yes—
219C4⋊Q8C2×Dic5914yes—
220#125—28no—
221C22⋊Q8—28no—
222C22⋊Q8—24no—
223C22⋊Q8C2×Dic5918yes—
224C4×Q8—28no—
225C4×Q8—24no—
226C4×Q8C2×Dic5918yes—
227#117—28no—
228#117—1188no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—594no—
235C2×C4⋊C4C22×D5912yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—1182no—
238C2×C42—22no—
239C2×C42—1182no—
240C2×C42—1182no—
241C2×C42—1182no—
242C2×C42C2×Dic5914yes—
243C2×C42C22⋊D5911yescharacteristic
244C2×C42C22⋊D5911yescharacteristic
245C2×C42C22⋊D5914yes—
246C2×C42C22×D5911yescharacteristic
247C2×C42C22×D5911yescharacteristic
248C4⋊C8—1184no—
249#103—11816no—
250C22⋊C8—1184no—
251C4×C8—1184no—
252#95—1182no—
253#95—1182no—
254C24C2×C22⋊D5911yescharacteristic
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—28no—
258C2×Q8—44no—
259C2×Q8C22×Dic5914yes—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—1184no—
266C22×C4C22⋊Dic5914yes—
267C22×C4C22⋊Dic5914yes—
268C22×C4C22×Dic5912yes—
269C22×C4C2×C22⋊D5911yescharacteristic
270C22×C4C2×C22⋊D5911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272C22×C4C2×C22⋊D5912yes—
273C22×C4C2×C22⋊D5911yescharacteristic
274C22×C4C2×C22⋊D5911yescharacteristic
275C22×C4C23×D5911yescharacteristic
276C2×C8—1184no—
277C2×C8—1188no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—22no—
284C4⋊C4—598no—
285C4⋊C4—1184no—
286C4⋊C4C22×Dic5914yes—
287C22⋊C4—28no—
288C22⋊C4—1188no—
289C42—28no—
290C42—1188no—
291C42—2364no—
292C42—2364no—
293C42C22⋊Dic5914yes—
294C42C22⋊Dic5914yes—
295C42C22×Dic5912yes—
296C42C22×Dic5912yes—
297C42C2×C22⋊D5912yes—
298C42C2×C22⋊D5912yes—
299C23—22no—
300C23—22no—
301C23—22no—
302C23C2×C22⋊Dic5912yes—
303C23C22×C22⋊D5911yescharacteristic
304Q8—28no—
305Q8—48no—
306Q8—44no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—1188no—
319C2×C4—2364no—
320C2×C4—2364no—
321C2×C4C2×C22⋊Dic5912yes—
322C2×C4C2×C22⋊Dic5912yes—
323C2×C4C2×C22⋊Dic5912yes—
324C2×C4C2×C22⋊Dic5912yes—
325C2×C4C2×C22⋊Dic5912yes—
326C2×C4#1427374714yes—
327C2×C4#1427375814yes—
328C2×C4C23×Dic5911yescharacteristic
329C2×C4C22×C22⋊D5911yescharacteristic
330C2×C4#41×D5914yes—
331C8—2364no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—44no—
336C22C22×C22⋊Dic5911yescharacteristic
337C22#1683790912yes—
338C22C2×#1427374712yes—
339C22C2×#1427375812yes—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—44no—
344C4—44no—
345C4—2368no—
346C4#1683739814yes—
347C4#1683741114yes—
348C4#41×Dic5912yes—
349C4#1683794812yes—
350C2—42no—
351C2#2142658711yescharacteristic
352C2C2×#1683739811yescharacteristic
353C2C2×#1683741111yescharacteristic
354C2#2143416114yes—
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#533C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#533D59
hypocenterC59#21389
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
3. #21425049C2semidirect
4. #21425020C2semidirect
6. C4×#14273651C2semidirect
22. C59×#533C4semidirect
186. #533Dic59semidirect
209. C59#21389semidirect

Maximal subgroups

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

subgroupindex
C2×#168374552
C4×#142736862
#214250492
#214250202
#214244092
C4×#142736512
C2×C59×#5332
#2138959