Finite Group Databasebeta

← #28742141

Subgroups — #28742141

Order 15104.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437232C211yesmaximal, characteristic
2C2×#16837525C211yesmaximal, characteristic, complemented
3#21432898C211yesmaximal, characteristic, complemented
4#21425996C214yesmaximal, complemented
5#21425989C214yesmaximal, complemented
6#21425869C211yesmaximal, characteristic, complemented
7#21425846C211yesmaximal, characteristic, complemented
8#21425833C211yesmaximal, characteristic, complemented
9C2×C59×#347C211yesmaximal, characteristic, complemented
10C2×#14273749C2211yescharacteristic
11C2×#14273749C2211yescharacteristic
12C2×#14273733—21no—
13C4×#11745685—21no—
14#16837845C412yescomplemented
15#16837845C2212yes—
16#16837840C412yescomplemented
17#16837840C2212yes—
18#16837834C414yescomplemented
19#16837525—22no—
20#16837525C2214yes—
21#16837515C2212yescomplemented
22#16837510C2212yescomplemented
23#16837504—22no—
24C2×#14273663C2211yescharacteristic
25#16837440C2211yescharacteristic
26C2×#14273655—21no—
27#16837399—21no—
28#16837225C2214yescomplemented
29#16837219C2212yescomplemented
30#16837216C2212yescomplemented
31#16837213C2214yes—
32C2×M32×C59C2211yescharacteristic
33C2×M32×C59C2211yescharacteristic
34C2×#97×C59C2211yescharacteristic
35C59×#347C2214yescomplemented
36C2×#11745685—21no—
37C2×#11745685—21no—
38#14273749—21no—
39#14273749—21no—
40#14273749—21no—
41#14273749—21no—
42#14273749C2×C412yes—
43#14273749C2×C412yes—
44#14273749C2×C412yescomplemented
45#14273749C2312yes—
46C2×Dic236C2×C411yescharacteristic, complemented
47C2×Dic236C2311yescharacteristic
48C2×#11745660—21no—
49C2×#11745660—21no—
50C2×D472C2×C411yescharacteristic, complemented
51C2×D472C2311yescharacteristic
52#14273733—24no—
53#14273733—42no—
54C2×#11745664—21no—
55C2×#11745659—41no—
56C2×C8×D59—41no—
57C4×Dic118—22no—
58C4×D236—22no—
59C4×C22⋊D59—42no—
60#14273686—21no—
61#14273684—41no—
62C42×D59—41no—
63Dic472C2×C414yescomplemented
64#14273672C2×C414yescomplemented
65#14273672C2×C414yescomplemented
66D944C2×C414yescomplemented
67#14273663—21no—
68#14273663C2312yes—
69#14273661—21no—
70#14273659C2312yes—
71#14273655—24no—
72#14273652—22no—
73#14273649—22no—
74C22×C8×C59D411yescharacteristic
75C22×C8×C59C2311yescharacteristic
76C2×C42×C59D411yescharacteristic
77M32×C59—21no—
78M32×C59C2×C412yescomplemented
79M32×C59C2312yes—
80M32×C59C2312yes—
81C2×C16×C59—21no—
82C2×C16×C59C2×C412yescomplemented
83#97×C59D412yes—
84#97×C59C2312yes—
85C22×C4×C59—21no—
86C22×C4×C59C2×D411yescharacteristic
87C2×C22⋊D59—41no—
88C2×C22⋊D59—41no—
89#11745685—22no—
90#11745685—22no—
91#11745685—22no—
92#11745685—22no—
93#11745685—41no—
94#11745685—41no—
95#11745685—41no—
96#11745685—41no—
97C2×D236—21no—
98C2×D236—21no—
99C2×C4×D59—41no—
100C2×C4×D59—41no—
101C2×C4×D59—42no—
102C2×Dic118—21no—
103C2×Dic118—21no—
104C2×C8×C59—21no—
105C2×C8×C59—21no—
106C2×C8×C59C22×C411yescharacteristic
107C2×C8×C59C22×C411yescharacteristic
108C2×C8×C59C2×D412yes—
109C2×C8×C59C2×D411yescharacteristic
110C2×C8×C59#4111yescharacteristic
111C2×C8×C59#4111yescharacteristic
112C2×C8×C59C2411yescharacteristic
113C42×C59—21no—
114C42×C59C2×D412yes—
115C22⋊Dic59—41no—
116(C2×D59)⋊C4—42no—
117C4⋊Dic59—22no—
118Dic59⋊C4—42no—
119C4×Dic59—42no—
120C4×Dic59—41no—
121#11745664—22no—
122#11745664—41no—
123C2×C59⋊C8—41no—
124Dic236—21no—
125Dic236—21no—
126Dic236C22×C412yes—
127Dic236C22×C412yes—
128D472—21no—
129D472—21no—
130D472C22×C412yes—
131D472C22×C412yes—
132#11745660—22no—
133#11745660—22no—
134#11745660—22no—
135#11745660—22no—
136#11745659—44no—
137C8×D59—44no—
138C16×C59—22no—
139C16×C59C22×C414yes—
140C23×C59AGL(1,ℤ/8)11yescharacteristic
141C22×D59—41no—
142C22×D59—41no—
143C2×C4×C59—22no—
144C2×C4×C59—81no—
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C59C22×D411yescharacteristic
151C22⋊D59—42no—
152C22⋊D59—42no—
153C22⋊D59—42no—
154C22⋊D59—42no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—42no—
158D236—22no—
159D236—22no—
160D236—41no—
161D236—41no—
162C4×D59—42no—
163C4×D59—42no—
164C4×D59—42no—
165C4×D59—42no—
166C4×D59—84no—
167Dic118—22no—
168Dic118—22no—
169Dic118—41no—
170Dic118—41no—
171C8×C59—22no—
172C8×C59C4×D411yescharacteristic
173C8×C59C4×D411yescharacteristic
174C8×C59C23×C411yescharacteristic
175C8×C59C2×#4111yescharacteristic
176C59⋊C8—42no—
177#33468—591nomaximal, Sylow
178C22×C59—41no—
179C22×C59#44211yescharacteristic
180C22×C59#44211yescharacteristic
181C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
182C2×D59—42no—
183C2×D59—42no—
184C2×D59—82no—
185C2×D59—82no—
186C4×C59—42no—
187C4×C59#44211yescharacteristic
188C4×C59#44211yescharacteristic
189C4×C59C2×C4×D411yescharacteristic
190C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
191Dic59—42no—
192Dic59—42no—
193Dic59—82no—
194#3414—591no—
195C2×#444—591no—
196#2943—591no—
197AGL(1,ℤ/16)—594no—
198#2180—594no—
199#2155—591no—
200#2148—591no—
201#2145—591no—
202C2×#347D5911yescharacteristic, complemented
203C2×C59—21no—
204C2×C59—21no—
205C2×C59—21no—
206C2×C59C2×#44211yescharacteristic
207D59—82no—
208D59—82no—
209C2×#135—591no—
210C2×#135—591no—
211C2×#131—1181no—
212C4×#41—1181no—
213#510—592no—
214#510—592no—
215#509—592no—
216#509—592no—
217#508—594no—
218C2×M32C2×D5911yescharacteristic
219C2×M32C2×D5911yescharacteristic
220#444—594no—
221#444—1182no—
222#442—592no—
223#441—592no—
224#440—1182no—
225C2×#108—591no—
226#427—591no—
227C2×#104—1181no—
228#416—1181no—
229C2×#97C2×D5911yescharacteristic
230#365—594no—
231#362—592no—
232#361—592no—
233#360—594no—
234#347C2×D5914yescomplemented
235C59#3346811yescharacteristic, complemented, Sylow
236C2×#41—1181no—
237C2×#41—1181no—
238#135—592no—
239#135—592no—
240#135—592no—
241#135—592no—
242#135—1181no—
243#135—1181no—
244#135—1181no—
245#135—1181no—
246C2×Q16—591no—
247C2×Q16—591no—
248C2×SD16—1181no—
249C2×SD16—1181no—
250C2×D8—591no—
251C2×D8—591no—
252#131—1184no—
253#131—2362no—
254C2×M16—1181no—
255C2×M16—2361no—
256C22×C8—2361no—
257C22×C8D23611yescharacteristic
258C22×C8C22×D5911yescharacteristic
259C4×Q8—1182no—
260C4×D4—1182no—
261C4×D4—2362no—
262#117—1181no—
263#117—2361no—
264C2×C42—2361no—
265C2×C42D23611yescharacteristic
266Q32—594no—
267SD32—594no—
268SD32—594no—
269D16—594no—
270M32—21no—
271M32C4×D5912yescomplemented
272M32C22×D5912yes—
273M32C22×D5912yes—
274C2×C16—21no—
275C2×C16C4×D5912yescomplemented
276#108—592no—
277#108—1181no—
278#107—1181no—
279#106—592no—
280#104—1184no—
281#103—1182no—
282#102—1182no—
283#97D23612yes—
284#97C22×D5912yes—
285#41—1182no—
286#41—1182no—
287#41—1182no—
288#41—1182no—
289#41—2361no—
290#41—2361no—
291#41—2361no—
292#41—2361no—
293C2×Q8—1181no—
294C2×Q8—1181no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—2361no—
298C2×D4—2361no—
299C22×C4—21no—
300C22×C4—2362no—
301C22×C4—2361no—
302C22×C4—2361no—
303C22×C4C2×D23611yescharacteristic
304Q16—592no—
305Q16—592no—
306Q16—1181no—
307Q16—1181no—
308SD16—1182no—
309SD16—1182no—
310SD16—1182no—
311SD16—1182no—
312D8—592no—
313D8—592no—
314D8—1181no—
315D8—1181no—
316M16—1182no—
317M16—2364no—
318M16—2361no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—2364no—
322C2×C8—2361no—
323C2×C8C2×C4×D5911yescharacteristic
324C2×C8C2×C4×D5911yescharacteristic
325C2×C8C2×D23612yes—
326C2×C8C2×D23611yescharacteristic
327C2×C8#1174568511yescharacteristic
328C2×C8#1174568511yescharacteristic
329C2×C8C23×D5911yescharacteristic
330C4⋊C4—1182no—
331C4⋊C4—2362no—
332C22⋊C4—2362no—
333C22⋊C4—2361no—
334C42—21no—
335C42—2362no—
336C42—2361no—
337C42C2×D23612yes—
338C16—22no—
339C16C2×C4×D5914yes—
340C23—2361no—
341C23—2361no—
342C23#1427375511yescharacteristic
343Q8—1182no—
344Q8—1182no—
345Q8—2361no—
346Q8—2361no—
347D4—1182no—
348D4—1182no—
349D4—2362no—
350D4—2362no—
351D4—2362no—
352D4—2362no—
353D4—2361no—
354D4—2361no—
355C2×C4—22no—
356C2×C4—81no—
357C2×C4—4724no—
358C2×C4—2362no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2362no—
362C2×C4—2362no—
363C2×C4—2362no—
364C2×C4—2362no—
365C2×C4C4×D23611yescharacteristic
366C2×C4C4×D23611yescharacteristic
367C2×C4#1427375511yescharacteristic
368C2×C4#1427375511yescharacteristic
369C2×C4#1427375511yescharacteristic
370C2×C4C22×D23611yescharacteristic
371C8—22no—
372C8—2362no—
373C8C4×D23611yescharacteristic
374C8C4×D23611yescharacteristic
375C8C22×C4×D5911yescharacteristic
376C8C2×#1174568511yescharacteristic
377C22—41no—
378C22—4722no—
379C22—4722no—
380C22—2362no—
381C22—2362no—
382C22#1683751511yescharacteristic
383C22#1683751511yescharacteristic
384C22C2×#1427375511yescharacteristic
385C4—42no—
386C4—4722no—
387C4—2362no—
388C4—2362no—
389C4#1683751511yescharacteristic
390C4#1683751511yescharacteristic
391C4C2×C4×D23611yescharacteristic
392C4C2×#1427375511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—4722no—
397C2—4722no—
398C2C2×#1683751511yescharacteristic
subgroupG/H
centerC4#16837515
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC2×C59×#347C2
socleC2×C59C2×#442
prime radical#21437232C2
cosocleC2×C8×C59C24
hypercenterC2×#347D59
hypocenterC59#33468
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#347

Complements

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

NH ≅ G/Ntype
2. C2×#16837525C2semidirect
3. #21432898C2semidirect
4. #21425996C2semidirect
5. #21425989C2semidirect
6. #21425869C2semidirect
7. #21425846C2semidirect
8. #21425833C2semidirect
9. C2×C59×#347C2semidirect
14. #16837845C4semidirect
16. #16837840C4semidirect
18. #16837834C4semidirect
21. #16837515C22semidirect
22. #16837510C22semidirect
28. #16837225C22semidirect
29. #16837219C22semidirect
30. #16837216C22semidirect
35. C59×#347C22semidirect
44. #14273749C2×C4semidirect
46. C2×Dic236C2×C4semidirect
50. C2×D472C2×C4semidirect
63. Dic472C2×C4semidirect
64. #14273672C2×C4semidirect
65. #14273672C2×C4semidirect
66. D944C2×C4semidirect
78. M32×C59C2×C4semidirect
82. C2×C16×C59C2×C4semidirect
202. C2×#347D59semidirect
234. #347C2×D59semidirect
235. C59#33468semidirect
271. M32C4×D59semidirect
275. C2×C16C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214372322
C2×#168375252
#214328982
#214259962
#214259892
#214258692
#214258462
#214258332
C2×C59×#3472
#3346859