Finite Group Databasebeta

← #28742031

Subgroups — #28742031

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837838C211yesmaximal, characteristic
2#21432973C211yesmaximal, characteristic, complemented
3#21425970C212yesmaximal, complemented
4#21425960C212yesmaximal, complemented
5C4×#14273671C212yesmaximal, complemented
6C4×#14273668C212yesmaximal, complemented
7#21425885C211yesmaximal, characteristic, complemented
8#21425801C211yesmaximal, characteristic, complemented
9C2×#16837203C211yesmaximal, characteristic, complemented
10#21425693C211yesmaximal, characteristic, complemented
11C59×#2948C211yesmaximal, characteristic, complemented
12C2×#14273749C2211yescharacteristic
13#16837882—21no—
14#16837838C418yescomplemented
15C2×#14273671C2212yes—
16C2×#14273668C2212yes—
17C2×#14273667C2211yescharacteristic
18C22×C59⋊C16C2211yescharacteristic
19#16837504—22no—
20C4×Dic236C2212yescomplemented
21C4×D472C2212yescomplemented
22C2×#14273661C2211yescharacteristic
23#16837404—21no—
24#16837229C2212yescomplemented
25#16837227C2212yescomplemented
26#16837205C2211yescharacteristic, complemented
27#16837205C2211yescharacteristic, complemented
28#16837203C2212yes—
29#16837203C2212yes—
30#16837202C2211yescharacteristic, complemented
31#16837202C2211yescharacteristic, complemented
32#16837178C2212yescomplemented
33C4×C59⋊C16C2212yescomplemented
34C22×D8×C59C2211yescharacteristic
35C59×#518—21no—
36C59×#442—22no—
37C4×D8×C59C2212yes—
38C4×D8×C59C2212yes—
39C59×#427C2211yescharacteristic
40C59×#418—21no—
41C59×#405C2211yescharacteristic
42C2×#11745685—21no—
43#14273749—22no—
44#14273749C2×C414yescomplemented
45C2×Dic236C2311yescharacteristic
46C2×#11745660—21no—
47C2×D472C2311yescharacteristic
48C4×Dic118—22no—
49C4×D236—22no—
50(C22⋊D59)⋊C4—42no—
51#14273685—41no—
52C2×C4⋊Dic59—21no—
53C4⋊C4×D59—41no—
54#14273671C2×C418yescomplemented
55#14273668C2×C418yescomplemented
56#14273667C2×C414yescomplemented
57C2×C59⋊C16—22no—
58C2×C59⋊C16C2311yescharacteristic
59C2×C59⋊C16C2311yescharacteristic
60C2×C59⋊C16C2312yes—
61#14273661—21no—
62#14273661C2312yes—
63#14273652—22no—
64#14273649—22no—
65C22×D4×C59—21no—
66C2×D8×C59—21no—
67C2×D8×C59—22no—
68C2×D8×C59C2×C414yescomplemented
69C2×D8×C59C2312yes—
70C2×M16×C59D411yescharacteristic
71C22×C8×C59C2311yescharacteristic
72C4×D4×C59—22no—
73C4×D4×C59—22no—
74C4×D4×C59—41no—
75C4×D4×C59—41no—
76#117×C59—21no—
77#117×C59D411yescharacteristic
78C2×C22⋊C4×C59—42no—
79#107×C59C2311yescharacteristic
80#107×C59C2311yescharacteristic
81#106×C59—21no—
82#102×C59—22no—
83#102×C59—22no—
84#97×C59D412yes—
85C4×C8×C59C2312yes—
86C24×C59—41no—
87C2×D4×C59—24no—
88C2×D4×C59—22no—
89C2×D4×C59—41no—
90C2×D4×C59—42no—
91C22×C4×C59—42no—
92C22×C4×C59—42no—
93C22×C4×C59C2×D411yescharacteristic
94C2×C22⋊D59—41no—
95C22×Dic59—41no—
96#11745685—24no—
97#11745685—42no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×C4×D59—42no—
101C2×Dic118—21no—
102D8×C59—24no—
103D8×C59—22no—
104D8×C59C22×C414yes—
105M16×C59—22no—
106C2×C8×C59C22×C412yes—
107C2×C8×C59C2×D412yes—
108C2×C8×C59C2×D411yescharacteristic
109C2×C8×C59#4112yes—
110C2×C8×C59C2411yescharacteristic
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59D811yescharacteristic
114C4⋊C4×C59D811yescharacteristic
115C22⋊C4×C59—41no—
116C22⋊C4×C59—42no—
117C22⋊C4×C59—42no—
118C22⋊C4×C59D812yes—
119C42×C59—41no—
120C42×C59C2×D412yes—
121(C2×D59)⋊C4—42no—
122C4⋊Dic59—22no—
123C4⋊Dic59—41no—
124Dic59⋊C4—42no—
125C4×Dic59—42no—
126Dic236—21no—
127Dic236C22×C412yes—
128D472—21no—
129D472C22×C412yes—
130#11745660—24no—
131C59⋊C16—21no—
132C59⋊C16—21no—
133C59⋊C16C22×C414yes—
134C23×C59—42no—
135C23×C59—44no—
136C23×C59—161no—
137C23×C59—161no—
138C23×C59C2×D811yescharacteristic
139C22×D59—41no—
140D4×C59—24no—
141D4×C59—42no—
142D4×C59—44no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—162no—
146C2×C4×C59—162no—
147C2×C4×C59C4×D412yes—
148C2×C4×C59C2×D812yes—
149C2×C4×C59C2×D812yes—
150C2×C4×C59C2×D811yescharacteristic
151C2×C4×C59#13512yes—
152C2×C4×C59C22×D411yescharacteristic
153C22⋊D59—44no—
154C2×Dic59—41no—
155C2×Dic59—42no—
156C2×Dic59—161no—
157D236—22no—
158D236—41no—
159C4×D59—44no—
160C4×D59—162no—
161Dic118—22no—
162Dic118—41no—
163C8×C59—22no—
164C8×C59C4×D412yes—
165C8×C59C23×C411yescharacteristic
166C8×C59C2×#4111yescharacteristic
167#33454—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—44no—
171C22×C59—82no—
172C22×C59—82no—
173C22×C59—164no—
174C22×C59C4×D812yes—
175C22×C59C22×D811yescharacteristic
176C2×D59—42no—
177C2×D59—81no—
178C2×D59—81no—
179C4×C59—22no—
180C4×C59—22no—
181C4×C59—81no—
182C4×C59—81no—
183C4×C59C4×D812yes—
184C4×C59C2×C4×D411yescharacteristic
185C4×C59C2×#13511yescharacteristic
186Dic59—42no—
187Dic59—82no—
188C2×#509—591no—
189#2951—591no—
190#2948D5911yescharacteristic, complemented
191#2177—592no—
192#2175—592no—
193C4×SD32—592no—
194C4×D16—592no—
195#2159—591no—
196#2138—591no—
197C2×#357—591no—
198#2108—591no—
199C2×C59—22no—
200C2×C59—84no—
201C2×C59C2×C4×D811yescharacteristic
202C2×C59#341511yescharacteristic
203C2×C59#341611yescharacteristic
204D59—82no—
205C2×#135—591no—
206C22×D8C2×D5911yescharacteristic
207#520—1181no—
208#518—21no—
209#509—598no—
210C2×SD32—592no—
211C2×D16—592no—
212C2×M32—591no—
213C22×C16—591no—
214#442—22no—
215#440—1182no—
216C4×Q16—592no—
217C4×D8—592no—
218C4×D8C2×D5912yes—
219C4×D8C2×D5912yes—
220#427C2×D5911yescharacteristic
221C2×#107—591no—
222#418—21no—
223#417—1181no—
224#405C2×D5911yescharacteristic
225#367—592no—
226#366—592no—
227#358—591no—
228#358—591no—
229#357—592no—
230#357—592no—
231#357—591no—
232#357—591no—
233#346—592no—
234C4×C16—592no—
235C59#3345411yescharacteristic, complemented, Sylow
236C2×#41—1181no—
237C22×D4—21no—
238#135—594no—
239#135—1182no—
240C2×Q16—591no—
241C2×SD16—1181no—
242C2×D8—22no—
243C2×D8—21no—
244C2×D8—591no—
245C2×D8C4×D5914yescomplemented
246C2×D8C22×D5912yes—
247C2×M16C22⋊D5911yescharacteristic
248C22×C8C22×D5911yescharacteristic
249C4×Q8—1182no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—1182no—
255C4×D4—2362no—
256#117—21no—
257#117—2361no—
258#117C22⋊D5911yescharacteristic
259C2×C4⋊C4—1181no—
260C2×C4⋊C4—2361no—
261C2×C22⋊C4—42no—
262SD32—598no—
263D16—598no—
264M32—594no—
265C2×C16—592no—
266C2×C16—591no—
267C2×C16—591no—
268C2×C16—1182no—
269#107—592no—
270#107—1181no—
271#107C22×D5911yescharacteristic
272#107C22×D5911yescharacteristic
273#106—21no—
274#103—1182no—
275#102—22no—
276#102—22no—
277#102—1182no—
278#97C22⋊D5912yes—
279C4×C8C22×D5912yes—
280C24—41no—
281#41—1184no—
282#41—2362no—
283C2×Q8—1181no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—1181no—
289C2×D4—2361no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—2361no—
293C22×C4—2362no—
294C22×C4—2361no—
295C22×C4C2×C22⋊D5911yescharacteristic
296Q16—592no—
297Q16—1181no—
298SD16—1184no—
299D8—24no—
300D8—22no—
301D8—592no—
302D8—1181no—
303D8C2×C4×D5914yes—
304M16—22no—
305C2×C8C2×C4×D5912yes—
306C2×C8#1174568512yes—
307C2×C8C2×C22⋊D5912yes—
308C2×C8C2×C22⋊D5911yescharacteristic
309C2×C8C23×D5911yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—1182no—
313C4⋊C4—2361no—
314C4⋊C4—2362no—
315C4⋊C4#1174566911yescharacteristic
316C4⋊C4#1174566911yescharacteristic
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—2362no—
321C22⋊C4#1174566912yes—
322C42—41no—
323C42—2362no—
324C42C2×C22⋊D5912yes—
325C16—594no—
326C16—1181no—
327C16—1181no—
328C23—42no—
329C23—44no—
330C23—161no—
331C23—161no—
332C23—2361no—
333C23C2×#1174566911yescharacteristic
334Q8—1182no—
335Q8—2361no—
336D4—24no—
337D4—42no—
338D4—44no—
339D4—1182no—
340D4—2364no—
341D4—2361no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—162no—
345C2×C4—162no—
346C2×C4—9442no—
347C2×C4—9441no—
348C2×C4—2362no—
349C2×C4—2361no—
350C2×C4—2364no—
351C2×C4C4×C22⋊D5912yes—
352C2×C4C2×#1174566912yes—
353C2×C4C2×#1174566912yes—
354C2×C4C2×#1174566911yescharacteristic
355C2×C4#1427374712yes—
356C2×C4C22×C22⋊D5911yescharacteristic
357C8—22no—
358C8C4×C22⋊D5912yes—
359C8C22×C4×D5911yescharacteristic
360C8C2×#1174568511yescharacteristic
361C22—21no—
362C22—21no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—164no—
367C22—4721no—
368C22—4721no—
369C22—2362no—
370C22C4×#1174566912yes—
371C22C22×#1174566911yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—81no—
376C4—4722no—
377C4—2362no—
378C4C4×#1174566912yes—
379C4C2×C4×C22⋊D5911yescharacteristic
380C4C2×#1427374711yescharacteristic
381C2—22no—
382C2—84no—
383C2—4722no—
384C2C2×C4×#1174566911yescharacteristic
385C2#2143723511yescharacteristic
386C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2948C2
socleC22×C59C22×D8
prime radicalC2×#16837838C2
cosocleC2×C8×C59C24
hypercenter#2948D59
hypocenterC59#33454
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #21432973C2semidirect
3. #21425970C2semidirect
4. #21425960C2semidirect
5. C4×#14273671C2semidirect
6. C4×#14273668C2semidirect
7. #21425885C2semidirect
8. #21425801C2semidirect
9. C2×#16837203C2semidirect
10. #21425693C2semidirect
11. C59×#2948C2semidirect
14. #16837838C4semidirect
20. C4×Dic236C22semidirect
21. C4×D472C22semidirect
24. #16837229C22semidirect
25. #16837227C22semidirect
26. #16837205C22semidirect
27. #16837205C22semidirect
30. #16837202C22semidirect
31. #16837202C22semidirect
32. #16837178C22semidirect
33. C4×C59⋊C16C22semidirect
44. #14273749C2×C4semidirect
54. #14273671C2×C4semidirect
55. #14273668C2×C4semidirect
56. #14273667C2×C4semidirect
68. C2×D8×C59C2×C4semidirect
190. #2948D59semidirect
235. C59#33454semidirect
245. C2×D8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378382
#214329732
#214259702
#214259602
C4×#142736712
C4×#142736682
#214258852
#214258012
C2×#168372032
#214256932
C59×#29482
#3345459