Finite Group Databasebeta

← #29453037

Subgroups — #29453037

Order 15104.

2604 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21438870C211yesmaximal, characteristic
2#21438807C212yesmaximal, complemented
3#21433014C216yesmaximal, complemented
4#21433007C216yesmaximal, complemented
5#21433000C216yesmaximal, complemented
6C2×#16837525C216yesmaximal, complemented
7#21432707C211yesmaximal, characteristic, complemented
8#21432644C212yesmaximal, complemented
9C59×#2875C211yesmaximal, characteristic, complemented
10#16838238C2212yes—
11#16838219C414yescomplemented
12#16838214C418yescomplemented
13#16838207C414yescomplemented
14#16838202C2216yes—
15C2×#14273749C2216yes—
16#16838169C2212yes—
17#16838168C22112yescomplemented
18#16838167—28no—
19C2×#14273733C22112yescomplemented
20#131×D59—28no—
21#16837525C2214yescomplemented
22#16837525C22112yescomplemented
23#16837525C22112yescomplemented
24#16837525C22112yescomplemented
25#16837520C22124yescomplemented
26#16837451C2216yescomplemented
27C2×#14273663C2216yescomplemented
28#16837420C22112yescomplemented
29C2×#14273655C22112yescomplemented
30C2×#131×C59C2211yescharacteristic
31C2×#131×C59C2211yescharacteristic
32C4×#41×C59C2211yescharacteristic
33C59×#405C2213yescomplemented
34C59×#405C2213yescomplemented
35C4×M16×C59C2213yescomplemented
36C2×#97×C59C2213yescomplemented
37#14273775—22no—
38#14273774C2×C418yes—
39#14273773C2×C418yes—
40#14273772—26no—
41C2×#11745685C23112yes—
42#41×D59—28no—
43#14273760C2×C4124yescomplemented
44#14273755C2×C4124yescomplemented
45#14273749C2×C4112yescomplemented
46#14273749C2×C4112yescomplemented
47#14273749C2314yes—
48#14273749C23112yes—
49C2×Dic236C2×C416yescomplemented
50C2×#11745660C2×C4112yescomplemented
51C2×D472C2×C416yescomplemented
52#14273735—224no—
53#14273734—28no—
54#14273733—212no—
55#14273733—212no—
56#14273733C2318yes—
57#14273733C23124yes—
58C2×#11745664C23112yes—
59C2×#11745659—212no—
60M16×D59—224no—
61C2×C8×D59—212no—
62#14273663C2314yes—
63#14273663C23112yes—
64#14273655C2318yes—
65#14273655C23124yes—
66C2×#41×C59C2311yescharacteristic
67#131×C59C2×C414yes—
68#131×C59C2×C414yes—
69#131×C59D418yes—
70C2×M16×C59C2313yes—
71C2×M16×C59C2313yes—
72C22×C8×C59C2313yes—
73C22×C8×C59C2313yes—
74C4×Q8×C59—21no—
75C4×D4×C59—23no—
76#117×C59C2313yes—
77C2×C42×C59C2313yes—
78#97×C59C2311yescharacteristic
79#97×C59C2313yes—
80#97×C59C2313yes—
81#97×C59C2313yes—
82C4×C8×C59C2316yes—
83#41×C59C22×C414yes—
84#41×C59C2×D414yes—
85C2×Q8×C59#4111yescharacteristic
86C2×D4×C59#4113yes—
87C22×C4×C59—23no—
88C22×C4×C59C2413yes—
89C2×C22⋊D59—212no—
90#11745689—28no—
91Q8×D59—28no—
92#11745687—224no—
93D4×D59—224no—
94#11745685—212no—
95#11745685—212no—
96#11745685C22×C4124yes—
97#11745685C2418yes—
98C2×D236C22×C4112yes—
99C2×C4×D59—212no—
100C2×Dic118C22×C4112yes—
101M16×C59C22×C4112yes—
102M16×C59C2×D4112yes—
103C2×C8×C59C22×C416yes—
104C2×C8×C59C22×C416yes—
105C2×C8×C59C2×D4112yes—
106C2×C8×C59C2413yes—
107C2×C8×C59C2411yescharacteristic
108C2×C8×C59C2413yes—
109C2×C8×C59C2411yescharacteristic
110C4⋊C4×C59—23no—
111C22⋊C4×C59—23no—
112C42×C59—23no—
113C42×C59C2413yes—
114C42×C59C2411yescharacteristic
115#11745664—212no—
116#11745664C2418yes—
117C2×C59⋊C8—212no—
118Dic236C22×C414yes—
119Dic236C22×C4112yes—
120D472C22×C414yes—
121D472C22×C4112yes—
122#11745660C22×C418yes—
123#11745660C22×C4124yes—
124#11745659—224no—
125#11745659—28no—
126C8×D59—28no—
127C8×D59—224no—
128C23×C59C2×#4113yes—
129C22×D59—212no—
130Q8×C59C4×D414yes—
131D4×C59C4×D4112yes—
132C2×C4×C59—21no—
133C2×C4×C59—23no—
134C2×C4×C59—83no—
135C2×C4×C59C23×C416yes—
136C2×C4×C59C22×D416yes—
137C2×C4×C59C2×#4113yes—
138C2×C4×C59C2511yescharacteristic
139C22⋊D59—224no—
140C22⋊D59—28no—
141C2×Dic59—212no—
142D236—212no—
143D236C23×C418yes—
144C4×D59—224no—
145C4×D59—28no—
146Dic118—212no—
147Dic118C23×C418yes—
148C8×C59C23×C411yescharacteristic
149C8×C59C23×C416yes—
150C8×C59C23×C411yescharacteristic
151C8×C59C22×D412yes—
152C8×C59C22×D416yes—
153C59⋊C8—28no—
154#60184—591nomaximal, Sylow
155C22×C59—43no—
156C22×C59C2×C4×D416yes—
157C22×C59C22×#4111yescharacteristic
158C2×D59—28no—
159C2×D59—812no—
160C4×C59—43no—
161C4×C59—41no—
162C4×C59C2×C4×D416yes—
163C4×C59C24×C411yescharacteristic
164C4×C59C23×D411yescharacteristic
165Dic59—28no—
166#3584—591no—
167#3573—592no—
168#2957—596no—
169#2956—596no—
170#2955—596no—
171C2×#444—596no—
172#2916—591no—
173#2901—592no—
174#2875D5911yescharacteristic, complemented
175C2×C59—21no—
176C2×C59—26no—
177C2×C59C22×C4×D411yescharacteristic
178D59—48no—
179#585—592no—
180#578—594no—
181#577—598no—
182#576—594no—
183#575—596no—
184C2×#135—596no—
185#568—592no—
186#568—5912no—
187#568—1188no—
188C2×#131—5912no—
189C2×#131—1188no—
190C2×#131C2×D5911yescharacteristic
191C2×#131C2×D5911yescharacteristic
192C4×#41C2×D5911yescharacteristic
193#444—594no—
194#444—5912no—
195#444—5912no—
196#444—5912no—
197#443—5924no—
198#430—596no—
199C2×#108—596no—
200#421—5912no—
201C2×#104—5912no—
202#405C2×D5913yescomplemented
203#405C2×D5913yescomplemented
204C4×M16C2×D5913yescomplemented
205C2×#97C2×D5913yescomplemented
206C59#6018411yescharacteristic, complemented, Sylow
2072−1+4—598no—
2082−1+4—1182no—
209H(5,𝔽2)—598no—
210H(5,𝔽2)—1186no—
211C2×#41—5912no—
212C2×#41—1188no—
213C2×#41C22×D5911yescharacteristic
214#137—5924no—
215AGL(1,ℤ/8)—5924no—
216#135—5912no—
217#135—5912no—
218#135—594no—
219#135—5912no—
220C2×Q16—596no—
221C2×SD16—5912no—
222C2×D8—596no—
223#131—598no—
224#131—5924no—
225#131—1188no—
226#131—11824no—
227#131—11812no—
228#131—11812no—
229#131C4×D5914yes—
230#131C4×D5914yes—
231#131D23618yes—
232C2×M16—5912no—
233C2×M16—11824no—
234C2×M16—11812no—
235C2×M16C22×D5913yes—
236C2×M16C22×D5913yes—
237C22×C8—11812no—
238C22×C8C22×D5913yes—
239C22×C8C22×D5913yes—
240C4×Q8—21no—
241C4×D4—23no—
242#117C22×D5913yes—
243C2×C42C22×D5913yes—
244#108—594no—
245#108—5912no—
246#104—598no—
247#104—5924no—
248#97C22×D5911yescharacteristic
249#97C22×D5913yes—
250#97C22×D5913yes—
251#97C22×D5913yes—
252C4×C8C22×D5916yes—
253#41—5924no—
254#41—598no—
255#41—1188no—
256#41—11824no—
257#41—11812no—
258#41—11812no—
259#41C2×C4×D5914yes—
260#41C2×D23614yes—
261C2×Q8—5912no—
262C2×Q8—1188no—
263C2×Q8#1174568511yescharacteristic
264C2×D4—5912no—
265C2×D4—11812no—
266C2×D4—11824no—
267C2×D4#1174568513yes—
268C22×C4—23no—
269C22×C4—11812no—
270C22×C4C23×D5913yes—
271Q16—594no—
272Q16—5912no—
273SD16—598no—
274SD16—5924no—
275D8—594no—
276D8—5912no—
277M16—598no—
278M16—11812no—
279M16—1188no—
280M16—11824no—
281M16C2×C4×D59112yes—
282M16C2×D236112yes—
283C2×C8—11812no—
284C2×C8—1188no—
285C2×C8—11824no—
286C2×C8C2×C4×D5916yes—
287C2×C8C2×C4×D5916yes—
288C2×C8C2×D236112yes—
289C2×C8C23×D5913yes—
290C2×C8C23×D5911yescharacteristic
291C2×C8C23×D5913yes—
292C2×C8C23×D5911yescharacteristic
293C4⋊C4—23no—
294C22⋊C4—23no—
295C42—23no—
296C42C23×D5913yes—
297C42C23×D5911yescharacteristic
298C23—11812no—
299C23C2×#1174568513yes—
300Q8—598no—
301Q8—11812no—
302Q8C4×D23614yes—
303D4—598no—
304D4—1188no—
305D4—11824no—
306D4—11812no—
307D4C4×D236112yes—
308C2×C4—21no—
309C2×C4—23no—
310C2×C4—83no—
311C2×C4—11812no—
312C2×C4—1188no—
313C2×C4—11824no—
314C2×C4C22×C4×D5916yes—
315C2×C4C22×D23616yes—
316C2×C4C2×#1174568513yes—
317C2×C4C24×D5911yescharacteristic
318C8—1188no—
319C8C22×C4×D5911yescharacteristic
320C8C22×C4×D5911yescharacteristic
321C8C22×C4×D5916yes—
322C8C22×D23612yes—
323C8C22×D23616yes—
324C22—43no—
325C22—1188no—
326C22—47212no—
327C22C2×C4×D23616yes—
328C22C22×#1174568511yescharacteristic
329C4—41no—
330C4—43no—
331C4—1188no—
332C4C2×C4×D23616yes—
333C4C23×C4×D5911yescharacteristic
334C4C23×D23611yescharacteristic
335C2—21no—
336C2—26no—
337C2—2368no—
338C2C22×C4×D23611yescharacteristic
subgroupG/H
centerC4C23×D236
commutatorC4×C59C24×C4
FrattiniC2×C4C24×D59
FittingC59×#2875C2
socleC2×C59C22×C4×D4
prime radical#21438870C2
cosocleC2×C4×C59C25
hypercenter#2875D59
hypocenterC59#60184
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×#131 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21438807C2semidirect
3. #21433014C2semidirect
4. #21433007C2semidirect
5. #21433000C2semidirect
6. C2×#16837525C2semidirect
7. #21432707C2semidirect
8. #21432644C2semidirect
9. C59×#2875C2semidirect
11. #16838219C4semidirect
12. #16838214C4semidirect
13. #16838207C4semidirect
17. #16838168C22semidirect
19. C2×#14273733C22semidirect
21. #16837525C22semidirect
22. #16837525C22semidirect
23. #16837525C22semidirect
24. #16837525C22semidirect
25. #16837520C22semidirect
26. #16837451C22semidirect
27. C2×#14273663C22semidirect
28. #16837420C22semidirect
29. C2×#14273655C22semidirect
33. C59×#405C22semidirect
34. C59×#405C22semidirect
35. C4×M16×C59C22semidirect
36. C2×#97×C59C22semidirect
43. #14273760C2×C4semidirect
44. #14273755C2×C4semidirect
45. #14273749C2×C4semidirect
46. #14273749C2×C4semidirect
49. C2×Dic236C2×C4semidirect
50. C2×#11745660C2×C4semidirect
51. C2×D472C2×C4semidirect
174. #2875D59semidirect
202. #405C2×D59semidirect
203. #405C2×D59semidirect
204. C4×M16C2×D59semidirect
205. C2×#97C2×D59semidirect
206. C59#60184semidirect

Maximal subgroups

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

subgroupindex
#214388702
#214388072
#214330142
#214330072
#214330002
C2×#168375252
#214327072
#214326442
C59×#28752
#6018459