Finite Group Databasebeta

← #28560717

Subgroups — #28560717

Order 15104.

564 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434678C211yesmaximal, characteristic
2#21433197C211yesmaximal, characteristic
3#21423003C212yesmaximal, complemented
4#21422946C212yesmaximal, complemented
5#21422944C212yesmaximal, complemented
6#21422913C212yesmaximal, complemented
7#21422842C211yesmaximal, characteristic
8#21422517C211yesmaximal, characteristic
9#21422516C211yesmaximal, characteristic
10#21422386C211yesmaximal, characteristic
11C59×#2986C211yesmaximal, characteristic
12#16837959C2211yescharacteristic
13#16837805C411yescharacteristic
14#16837781C411yescharacteristic
15#16837754C411yescharacteristic
16#16837752C411yescharacteristic
17#16837641C412yes—
18#16837641C412yes—
19#16837532C2212yes—
20#16837529—21no—
21C8×Dic118C2212yes—
22#16837526—21no—
23C4×#11745672C2212yes—
24C4×#11745672C2212yes—
25#16837460—22no—
26#16837455C2211yescharacteristic
27C4×#11745664C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29#16837154C2211yescharacteristic
30#16837154C2211yescharacteristic
31#16837152C2211yescharacteristic
32#16837152C2211yescharacteristic
33Q8⋊(C59⋊C8)C2212yes—
34Q8⋊(C59⋊C8)C2212yes—
35#16837132C2212yes—
36#16837132C2212yes—
37#16837124C2212yes—
38#16837123C2212yes—
39C59×#533C2211yescharacteristic
40C59×#446—21no—
41C59×#446C2212yes—
42C8×Q8×C59—21no—
43C8×Q8×C59C2212yes—
44C59×#433—22no—
45C59×#431C2211yescharacteristic
46C59×#431C2211yescharacteristic
47C2×#11745672C2×C412yes—
48C2×#11745672C2×C412yes—
49C2×#11745664—21no—
50C22×C59⋊C8C2×C411yescharacteristic
51C22×C59⋊C8C2×C411yescharacteristic
52#14273727C2×C411yescharacteristic
53#14273727C2×C411yescharacteristic
54#14273719—21no—
55#14273706—21no—
56#14273706C2×C412yes—
57C4×Dic118—21no—
58C4×Dic118C2312yes—
59#14273686—21no—
60#14273660C2×C412yes—
61#14273660C2×C412yes—
62C4⋊(C59⋊C8)C2311yescharacteristic
63C4⋊(C59⋊C8)C2311yescharacteristic
64Dic59⋊C8—22no—
65Dic59⋊C8—22no—
66#14273651C2×C412yes—
67#14273651C2×C412yes—
68#14273650C2×C412yes—
69#14273650C2×C412yes—
70C22⋊(C59⋊C8)—21no—
71#14273638—21no—
72#14273637—22no—
73C4×C59⋊C8—21no—
74C4×C59⋊C8C2×C411yescharacteristic
75C4×C59⋊C8C2×C411yescharacteristic
76C4×C59⋊C8C2312yes—
77C4×C59⋊C8C2311yescharacteristic
78C4×C59⋊C8C2311yescharacteristic
79C8×Dic59—22no—
80C4⋊Q8×C59C2×C411yescharacteristic
81C4⋊Q8×C59C2×C411yescharacteristic
82#125×C59—21no—
83C22⋊Q8×C59—21no—
84C22⋊Q8×C59C2×C412yes—
85C4×Q8×C59—21no—
86C4×Q8×C59C2312yes—
87#117×C59—21no—
88C2×C42×C59C2311yescharacteristic
89C4⋊C8×C59—22no—
90C4⋊C8×C59—22no—
91C4⋊C8×C59D411yescharacteristic
92C4⋊C8×C59D411yescharacteristic
93C4⋊C8×C59C2311yescharacteristic
94C4⋊C8×C59C2311yescharacteristic
95C22⋊C8×C59—21no—
96C22⋊C8×C59D412yes—
97#97×C59—22no—
98C4×C8×C59—22no—
99C4×C8×C59—21no—
100C4×C8×C59C2312yes—
101C2×Q8×C59—21no—
102C2×Q8×C59C22×C412yes—
103C22×C4×C59C22×C411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59#4111yescharacteristic
106C2×Dic118—21no—
107C2×Dic118C22×C412yes—
108C2×C8×C59—21no—
109C2×C8×C59—21no—
110C2×C8×C59C2×D412yes—
111C2×C8×C59C2×D412yes—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—22no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59C22×C412yes—
118C22⋊C4×C59—22no—
119C42×C59—22no—
120C42×C59C22×C411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C42×C59C2411yescharacteristic
124C22⋊Dic59—22no—
125#11745672—24no—
126#11745672—42no—
127C4⋊Dic59—21no—
128C4⋊Dic59—21no—
129C4⋊Dic59—21no—
130C4⋊Dic59C22×C412yes—
131Dic59⋊C4—22no—
132Dic59⋊C4—22no—
133C4×Dic59—22no—
134#11745664—42no—
135C2×C59⋊C8—22no—
136C2×C59⋊C8—21no—
137C2×C59⋊C8—21no—
138C2×C59⋊C8—22no—
139C2×C59⋊C8—41no—
140C2×C59⋊C8C22×C411yescharacteristic
141C2×C59⋊C8C22×C411yescharacteristic
142C2×C59⋊C8C22×C411yescharacteristic
143C2×C59⋊C8C22×C411yescharacteristic
144C23×C59C4×D411yescharacteristic
145Q8×C59—22no—
146Q8×C59—22no—
147Q8×C59—42no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—22no—
153C2×C4×C59—22no—
154C2×C4×C59—22no—
155C2×C4×C59C4×D411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59C4×D411yescharacteristic
158C2×C4×C59C23×C411yescharacteristic
159C2×C4×C59C22×D411yescharacteristic
160C2×C4×C59C2×#4111yescharacteristic
161C2×Dic59—22no—
162C2×Dic59—22no—
163Dic118—22no—
164Dic118—22no—
165Dic118—42no—
166C8×C59—22no—
167C8×C59—22no—
168C8×C59—41no—
169C8×C59—41no—
170C59⋊C8—21no—
171C59⋊C8—21no—
172C59⋊C8—21no—
173C59⋊C8—42no—
174C59⋊C8—41no—
175C59⋊C8—41no—
176C59⋊C8#13112yes—
177#18090—591nomaximal, Sylow
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59C2×C4×D411yescharacteristic
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—42no—
186C4×C59—42no—
187C4×C59#44412yes—
188C4×C59C2×#13111yescharacteristic
189C4×C59#56811yescharacteristic
190C4×C59#57811yescharacteristic
191C4×C59#57811yescharacteristic
192Dic59—42no—
193Dic59—42no—
194#3170—591no—
195#2986—591no—
196#2986D5911yescharacteristic
197#1593—592no—
198#1584—592no—
199#1584—592no—
200#1579—592no—
201#1564—591no—
202#1485—591no—
203#1485—591no—
204#1451—591no—
205C2×C59—41no—
206C2×C59#293511yescharacteristic
207C2×C59#295011yescharacteristic
208C2×C59C2×#44411yescharacteristic
209#533—591no—
210#533C2×D5911yescharacteristic
211#500—591no—
212#494—591no—
213#487—591no—
214#487—591no—
215#467—592no—
216#467—592no—
217#446—21no—
218#446—592no—
219#446—1181no—
220#446C2×D5912yes—
221C8×Q8—21no—
222C8×Q8—592no—
223C8×Q8—1181no—
224C8×Q8C2×D5912yes—
225C4×Q16—592no—
226C4×Q16—592no—
227#433—22no—
228#433—1182no—
229#431—591no—
230#431C2×D5911yescharacteristic
231#431C2×D5911yescharacteristic
232C4×M16—591no—
233C2×C4×C8—591no—
234#335—591no—
235#335—591no—
236#334—591no—
237#334—591no—
238Q8⋊C8—592no—
239Q8⋊C8—592no—
240Q8⋊C8—592no—
241Q8⋊C8—592no—
242#322—592no—
243#322—592no—
244C59#1809011yescharacteristic, complemented, Sylow
245C2×Q16—592no—
246C2×Q16—592no—
247C2×M16—1181no—
248C22×C8—591no—
249C22×C8—591no—
250C4⋊Q8—591no—
251C4⋊Q8—591no—
252C4⋊Q8C4×D5911yescharacteristic
253C4⋊Q8C4×D5911yescharacteristic
254#125—21no—
255#125—1181no—
256C22⋊Q8—21no—
257C22⋊Q8—592no—
258C22⋊Q8—1181no—
259C22⋊Q8C4×D5912yes—
260C4×Q8—21no—
261C4×Q8—592no—
262C4×Q8—1181no—
263C4×Q8C22×D5912yes—
264#117—21no—
265#117—1181no—
266C2×C42C22×D5911yescharacteristic
267#107—592no—
268#107—592no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—591no—
272C4⋊C8—591no—
273C4⋊C8—1182no—
274C4⋊C8—1182no—
275C4⋊C8C22⋊D5911yescharacteristic
276C4⋊C8C22⋊D5911yescharacteristic
277C4⋊C8C22×D5911yescharacteristic
278C4⋊C8C22×D5911yescharacteristic
279#103—592no—
280#103—592no—
281#103—592no—
282#103—592no—
283C22⋊C8—21no—
284C22⋊C8—1181no—
285C22⋊C8C22⋊D5912yes—
286#97—22no—
287#97—1182no—
288#97—1181no—
289C4×C8—22no—
290C4×C8—21no—
291C4×C8—591no—
292C4×C8—591no—
293C4×C8—591no—
294C4×C8—591no—
295C4×C8—592no—
296C4×C8—1182no—
297C4×C8—1181no—
298C4×C8C22×D5912yes—
299C2×Q8—21no—
300C2×Q8—592no—
301C2×Q8—1181no—
302C2×Q8C2×C4×D5912yes—
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4#1174568511yescharacteristic
305C22×C4C2×C22⋊D5911yescharacteristic
306Q16—1184no—
307Q16—2362no—
308M16—2362no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—591no—
312C2×C8—591no—
313C2×C8—591no—
314C2×C8—591no—
315C2×C8—1182no—
316C2×C8—1181no—
317C2×C8—1181no—
318C2×C8—1182no—
319C2×C8—2361no—
320C2×C8C2×C22⋊D5912yes—
321C2×C8C2×C22⋊D5912yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—592no—
328C4⋊C4—1181no—
329C4⋊C4—1181no—
330C4⋊C4—1182no—
331C4⋊C4—1181no—
332C4⋊C4—1182no—
333C4⋊C4C2×C4×D5912yes—
334C22⋊C4—22no—
335C22⋊C4—1182no—
336C42—22no—
337C42—1182no—
338C42C2×C4×D5911yescharacteristic
339C42#1174568511yescharacteristic
340C42C2×C22⋊D5911yescharacteristic
341C42C23×D5911yescharacteristic
342C23C4×C22⋊D5911yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—42no—
346Q8—1182no—
347Q8—1182no—
348Q8—2362no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—1182no—
357C2×C4—1182no—
358C2×C4C4×C22⋊D5911yescharacteristic
359C2×C4C4×C22⋊D5911yescharacteristic
360C2×C4C4×C22⋊D5911yescharacteristic
361C2×C4C22×C4×D5911yescharacteristic
362C2×C4C22×C22⋊D5911yescharacteristic
363C2×C4C2×#1174568511yescharacteristic
364C8—22no—
365C8—22no—
366C8—41no—
367C8—41no—
368C8—592no—
369C8—1181no—
370C8—1181no—
371C8—1181no—
372C8—2362no—
373C8—2361no—
374C8—2361no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22C2×C4×C22⋊D5911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—2362no—
385C4—2362no—
386C4#1683752212yes—
387C4#131×D5911yescharacteristic
388C4#1683816711yescharacteristic
389C4#1683821711yescharacteristic
390C4#1683821711yescharacteristic
391C2—41no—
392C2#2143283811yescharacteristic
393C2#2143296211yescharacteristic
394C2C2×#1683752211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#2986C2
socleC22×C59C2×C4×D4
prime radicalC23×C59C4×D4
cosocleC42×C59C24
hypercenter#2986D59
hypocenterC59#18090
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2986

Complements

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

NH ≅ G/Ntype
3. #21423003C2semidirect
4. #21422946C2semidirect
5. #21422944C2semidirect
6. #21422913C2semidirect
244. C59#18090semidirect

Maximal subgroups

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

subgroupindex
#214346782
#214331972
#214230032
#214229462
#214229442
#214229132
#214228422
#214225172
#214225162
#214223862
C59×#29862
#1809059