Finite Group Databasebeta

← #28573858

Subgroups — #28573858

Order 15104.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435023C211yesmaximal, characteristic, complemented
2C2×#16837682C211yesmaximal, characteristic, complemented
3#21433560C211yesmaximal, characteristic
4#21423519C214yesmaximal, complemented
5#21423437C214yesmaximal, complemented
6#21422867C211yesmaximal, characteristic, complemented
7#21422780C211yesmaximal, characteristic, complemented
8#21422743C211yesmaximal, characteristic, complemented
9C59×#2972C211yesmaximal, characteristic, complemented
10C2×#14273753—21no—
11#16837952C2211yescharacteristic
12C2×#14273727C2211yescharacteristic
13#16837775C2212yescomplemented
14#16837753C2212yescomplemented
15#16837682—22no—
16#16837682C2214yes—
17#16837617—22no—
18#16837608—22no—
19#16837607C2212yes—
20#16837597C2212yes—
21#16837543—22no—
22C2×#14273660—21no—
23#16837426C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25#16837401—21no—
26C2×#14273648—21no—
27#16837153C2214yescomplemented
28#16837149C2214yescomplemented
29#16837147C2212yescomplemented
30#16837145C2212yescomplemented
31C2×C4×D4×C59C2211yescharacteristic
32C8×D4×C59—22no—
33C8×D4×C59C2214yes—
34C59×#431C2211yescharacteristic
35C59×#431C2211yescharacteristic
36C2×C22⋊C8×C59—22no—
37C2×#11745685—21no—
38C22×Dic118—21no—
39#14273753—44no—
40C2×#11745660—24no—
41C2×#11745670—22no—
42C2×#11745669—22no—
43C2×#11745664—21no—
44C22×C59⋊C8—21no—
45#14273727—21no—
46#14273727—21no—
47#14273727—21no—
48#14273727C2311yescharacteristic
49#14273727C2312yes—
50#14273724C2311yescharacteristic
51#14273715—21no—
52#14273701—21no—
53#14273701—21no—
54C4×D236—21no—
55C4×D236—21no—
56C2×C4⋊Dic59—21no—
57C2×C4⋊Dic59—21no—
58#14273660—24no—
59C4⋊(C59⋊C8)—21no—
60C4⋊(C59⋊C8)—21no—
61C4⋊(C59⋊C8)C2312yes—
62C4⋊(C59⋊C8)C2312yes—
63#14273652—24no—
64#14273650—22no—
65#14273648—24no—
66#14273647—22no—
67C22×D4×C59D411yescharacteristic
68C23×C4×C59—21no—
69C22×C8×C59—24no—
70C4×D4×C59—21no—
71C4×D4×C59—21no—
72C4×D4×C59D412yes—
73C4×D4×C59C2312yes—
74C2×C4⋊C4×C59D411yescharacteristic
75C2×C22⋊C4×C59—21no—
76C2×C42×C59C2311yescharacteristic
77C4⋊C8×C59D412yes—
78C4⋊C8×C59C2312yes—
79C22⋊C8×C59—21no—
80C22⋊C8×C59—24no—
81C22⋊C8×C59D412yes—
82C4×C8×C59—21no—
83C4×C8×C59C2312yes—
84C24×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—22no—
87C2×D4×C59SD1614yescomplemented
88C2×D4×C59C2×D412yes—
89C22×C4×C59—21no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—82no—
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59#4111yescharacteristic
96C2×C22⋊D59—41no—
97C22×Dic59—41no—
98#11745685—42no—
99#11745685—42no—
100C2×D236—21no—
101C2×D236—21no—
102C2×D236—21no—
103C2×C4×D59—41no—
104C2×Dic118—21no—
105C2×Dic118—22no—
106C2×Dic118—41no—
107C2×Dic118—82no—
108C2×C8×C59—22no—
109C2×C8×C59—84no—
110C2×C8×C59C2×D414yes—
111C4⋊C4×C59—21no—
112C4⋊C4×C59C2×D412yes—
113C22⋊C4×C59—22no—
114C22⋊C4×C59—22no—
115C42×C59C2×D411yescharacteristic
116C42×C59C2×D411yescharacteristic
117C42×C59#4111yescharacteristic
118C42×C59C2411yescharacteristic
119#11745670—44no—
120#11745669—44no—
121(C2×D59)⋊C4—41no—
122(C2×D59)⋊C4—41no—
123C4⋊Dic59—22no—
124C4⋊Dic59—21no—
125C4⋊Dic59—22no—
126C4⋊Dic59—21no—
127C4⋊Dic59—41no—
128C4⋊Dic59—41no—
129#11745664—42no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—42no—
133#11745660—84no—
134C23×C59—22no—
135C23×C59—24no—
136C23×C59—81no—
137C23×C59—81no—
138C23×C59#12111yescharacteristic
139C22×D59—41no—
140D4×C59—22no—
141D4×C59—24no—
142D4×C59C2×SD1614yes—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—41no—
146C2×C4×C59—44no—
147C2×C4×C59—82no—
148C2×C4×C59—81no—
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59C2×SD1612yes—
153C2×C4×C59#13512yes—
154C2×C4×C59C22×D411yescharacteristic
155C2×C4×C59C22×D411yescharacteristic
156C2×C4×C59C2×#4111yescharacteristic
157C22⋊D59—82no—
158C2×Dic59—42no—
159C2×Dic59—41no—
160C2×Dic59—82no—
161D236—41no—
162D236—41no—
163D236—42no—
164D236—81no—
165C4×D59—82no—
166Dic118—41no—
167Dic118—41no—
168Dic118—44no—
169Dic118—82no—
170C8×C59—24no—
171C8×C59—42no—
172C59⋊C8—42no—
173C59⋊C8—42no—
174#19385—591nomaximal, Sylow
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—24no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—84no—
181C22×C59#46512yes—
182C22×C59C2×#12111yescharacteristic
183C2×D59—81no—
184C2×D59—81no—
185C2×D59—81no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—42no—
190C4×C59#46512yes—
191C4×C59C22×SD1611yescharacteristic
192C4×C59C2×#13511yescharacteristic
193C4×C59#57612yes—
194Dic59—82no—
195Dic59—81no—
196#3206—591no—
197C2×#474—591no—
198#3036—591no—
199#2972D5911yescharacteristic, complemented
200#1679—594no—
201#1665—594no—
202#1570—591no—
203#1549—591no—
204#1540—591no—
205C2×C59—22no—
206C2×C59—44no—
207C2×C59#304111yescharacteristic
208C2×C59C2×#46511yescharacteristic
209C2×C59#306911yescharacteristic
210D59—161no—
211C2×AGL(1,ℤ/8)—1181no—
212#532—591no—
213C2×C4⋊Q8—591no—
214C2×C4×D4C2×D5911yescharacteristic
215#492—592no—
216#487—592no—
217#474—594no—
218#474—1182no—
219#463—1182no—
220#461—592no—
221#461—1182no—
222#459—592no—
223#448—1182no—
224C8×D4—22no—
225C8×D4C2×D5914yes—
226#431C2×D5911yescharacteristic
227#431C2×D5911yescharacteristic
228C2×#107—1181no—
229#423—591no—
230C2×C4⋊C8—591no—
231#417—1181no—
232C2×#102—1181no—
233C2×C22⋊C8—22no—
234#334—594no—
235#332—592no—
236#332—594no—
237#331—592no—
238C59#1938511yescharacteristic, complemented, Sylow
239C2×#41—1181no—
240C22×Q8—1181no—
241C22×D4D23611yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—2364no—
244C2×SD16—1182no—
245C2×SD16—1184no—
246C2×D8—1182no—
247C2×M16—1181no—
248C22×C8—24no—
249C22×C8—1181no—
250C4⋊Q8—592no—
251C4⋊Q8—591no—
252C4⋊Q8—1181no—
253C4⋊Q8—1181no—
254C4⋊Q8—1181no—
255#127—591no—
256#124—1181no—
257#121—1181no—
258#121—1181no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—1181no—
262C4×D4—1181no—
263C4×D4D23612yes—
264C4×D4C22×D5912yes—
265C2×C4⋊C4—1181no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4D23611yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D5911yescharacteristic
270#107—1184no—
271C4⋊C8—592no—
272C4⋊C8—592no—
273C4⋊C8—1181no—
274C4⋊C8—1181no—
275C4⋊C8C22⋊D5912yes—
276C4⋊C8C22×D5912yes—
277#103—1184no—
278#103—1182no—
279#102—1184no—
280#102—1182no—
281C22⋊C8—21no—
282C22⋊C8—24no—
283C22⋊C8C22⋊D5912yes—
284C4×C8—21no—
285C4×C8C22×D5912yes—
286C24—21no—
287#41—2362no—
288#41—2362no—
289C2×Q8—1182no—
290C2×Q8—1181no—
291C2×Q8—4722no—
292C2×Q8—2361no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—1181no—
298C2×D4—2361no—
299C2×D4#1174566014yescomplemented
300C2×D4C2×D23612yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—82no—
305C22×C4—2361no—
306C22×C4—2361no—
307C22×C4C2×D23611yescharacteristic
308C22×C4#1174568511yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310SD16—4724no—
311SD16—2364no—
312D8—2364no—
313M16—2362no—
314C2×C8—22no—
315C2×C8—84no—
316C2×C8—1182no—
317C2×C8—1182no—
318C2×C8—2362no—
319C2×C8C2×C22⋊D5914yes—
320C4⋊C4—21no—
321C4⋊C4—1181no—
322C4⋊C4—1182no—
323C4⋊C4—1182no—
324C4⋊C4—1181no—
325C4⋊C4—2361no—
326C4⋊C4—2361no—
327C4⋊C4C2×D23612yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—2361no—
331C22⋊C4—2361no—
332C42C2×D23611yescharacteristic
333C42#1174568511yescharacteristic
334C42C2×C22⋊D5911yescharacteristic
335C42C23×D5911yescharacteristic
336C23—22no—
337C23—24no—
338C23—81no—
339C23—81no—
340C23—2361no—
341C23#1427370111yescharacteristic
342Q8—4722no—
343Q8—2361no—
344Q8—2361no—
345Q8—2364no—
346D4—22no—
347D4—24no—
348D4—4721no—
349D4—4722no—
350D4—2361no—
351D4—2361no—
352D4—2362no—
353D4C2×#1174566014yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—4722no—
361C2×C4—4722no—
362C2×C4—2361no—
363C2×C4—2362no—
364C2×C4#1427370111yescharacteristic
365C2×C4#1427370111yescharacteristic
366C2×C4#1427370111yescharacteristic
367C2×C4C2×#1174566012yes—
368C2×C4#1427374912yes—
369C2×C4C22×C22⋊D5911yescharacteristic
370C2×C4C22×D23611yescharacteristic
371C2×C4C2×#1174568511yescharacteristic
372C8—24no—
373C8—42no—
374C8—2362no—
375C8—2362no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22#1683763112yes—
386C22C2×#1427370111yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—4721no—
392C4—4722no—
393C4#1683763112yes—
394C4C22×#1174566011yescharacteristic
395C4C2×#1427374911yescharacteristic
396C4#1683820512yes—
397C2—22no—
398C2—44no—
399C2—9441no—
400C2#2143359911yescharacteristic
401C2C2×#1683763111yescharacteristic
402C2#2143383311yescharacteristic
subgroupG/H
centerC22C2×#14273701
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#2972C2
socleC22×C59C2×#121
prime radical#21433560C2
hypercenter#2972D59
hypocenterC59#19385
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21435023C2semidirect
2. C2×#16837682C2semidirect
4. #21423519C2semidirect
5. #21423437C2semidirect
6. #21422867C2semidirect
7. #21422780C2semidirect
8. #21422743C2semidirect
9. C59×#2972C2semidirect
13. #16837775C22semidirect
14. #16837753C22semidirect
27. #16837153C22semidirect
28. #16837149C22semidirect
29. #16837147C22semidirect
30. #16837145C22semidirect
87. C2×D4×C59SD16semidirect
199. #2972D59semidirect
238. C59#19385semidirect
299. C2×D4#11745660semidirect

Maximal subgroups

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

subgroupindex
#214350232
C2×#168376822
#214335602
#214235192
#214234372
#214228672
#214227802
#214227432
C59×#29722
#1938559