Finite Group Databasebeta

← #28558468

Subgroups — #28558468

Order 15104.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433062C212yesmaximal, complemented
2C4×#14273749C211yesmaximal, characteristic
3#21423011C212yesmaximal, complemented
4#21423006C212yesmaximal, complemented
5#21423000C214yesmaximal, complemented
6#21422844C211yesmaximal, characteristic, complemented
7#21422479C212yesmaximal, complemented
8C2×C59×#322C211yesmaximal, characteristic, complemented
9C2×#14273749C411yescharacteristic
10C4×#11745685C2212yes—
11#16837532C2214yes—
12C4×Dic236C411yescharacteristic
13C4×Dic236C2211yescharacteristic
14C4×#11745660C412yes—
15C4×#11745660C2212yes—
16C4×D472C411yescharacteristic
17C4×D472C2211yescharacteristic
18#16837481C2214yes—
19#16837473—24no—
20#16837455C2212yes—
21#16837453—22no—
22#16837440C411yescharacteristic
23#16837399C412yes—
24C4×#11745659—22no—
25#16837155C2212yes—
26#16837153C2212yes—
27#16837134C2214yes—
28#16837131C2214yes—
29C2×C4×C8×C59C2211yescharacteristic
30C2×C4×C8×C59C2211yescharacteristic
31C2×C4×C8×C59C2211yescharacteristic
32C59×#322C2212yes—
33C59×#322C2212yes—
34C2×#11745685C2×C412yes—
35#14273749—24no—
36C2×Dic236C2×C411yescharacteristic
37C2×#11745660C2×C412yes—
38C2×D472C2×C411yescharacteristic
39C2×#11745659—24no—
40C4×Dic118C2×C412yes—
41C4×Dic118C2312yes—
42C4×D236C2×C412yes—
43C4×D236C2312yes—
44C4×C22⋊D59—22no—
45C4×C22⋊D59—22no—
46#14273686C2×C412yes—
47#14273684—22no—
48C42×D59—22no—
49#14273661C2×C411yescharacteristic
50#14273661C2×C411yescharacteristic
51#14273659C2×C412yes—
52C4⋊(C59⋊C8)C2314yes—
53Dic59⋊C8—24no—
54#14273652C2×C412yes—
55#14273652C2×C412yes—
56#14273649C2×C412yes—
57#14273649C2×C412yes—
58C22⋊(C59⋊C8)—22no—
59(C2×D59)⋊C8—24no—
60#14273637—24no—
61C4×C59⋊C8—22no—
62C22×C8×C59—21no—
63C22×C8×C59C2×C411yescharacteristic
64C22×C8×C59C2×C411yescharacteristic
65C22×C8×C59D412yes—
66C2×C42×C59C2311yescharacteristic
67C4×C8×C59—21no—
68C4×C8×C59C2×C411yescharacteristic
69C4×C8×C59C2×C411yescharacteristic
70C4×C8×C59D412yes—
71C4×C8×C59C2312yes—
72C4×C8×C59C2311yescharacteristic
73C4×C8×C59C2311yescharacteristic
74C4×C8×C59C2312yes—
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C2×C22⋊D59—22no—
79#11745685—24no—
80#11745685—24no—
81C2×D236C22×C412yes—
82C2×C4×D59—22no—
83C2×C4×D59—22no—
84C2×C4×D59—22no—
85C2×Dic118C22×C412yes—
86C2×C8×C59—22no—
87C2×C8×C59—22no—
88C2×C8×C59—42no—
89C2×C8×C59—42no—
90C2×C8×C59M1614yescomplemented
91C2×C8×C59C22×C411yescharacteristic
92C2×C8×C59C22×C411yescharacteristic
93C2×C8×C59C22×C411yescharacteristic
94C2×C8×C59C22×C411yescharacteristic
95C2×C8×C59C2×D414yes—
96C42×C59C22×C411yescharacteristic
97C42×C59C2×D411yescharacteristic
98C42×C59#4111yescharacteristic
99C42×C59C2411yescharacteristic
100C22⋊Dic59—22no—
101(C2×D59)⋊C4—22no—
102(C2×D59)⋊C4—22no—
103C4⋊Dic59C22×C412yes—
104C4⋊Dic59C22×C412yes—
105Dic59⋊C4—22no—
106Dic59⋊C4—22no—
107C4×Dic59—22no—
108C4×Dic59—22no—
109C4×Dic59—22no—
110C2×C59⋊C8—24no—
111Dic236—22no—
112D472—22no—
113#11745660—24no—
114#11745659—48no—
115C23×C59C4×D411yescharacteristic
116C22×D59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59C4×D411yescharacteristic
120C2×C4×C59C4×D411yescharacteristic
121C2×C4×C59C4×D411yescharacteristic
122C2×C4×C59C2×M1612yes—
123C2×C4×C59#13112yes—
124C2×C4×C59C23×C411yescharacteristic
125C2×C4×C59C22×D411yescharacteristic
126C2×C4×C59C2×#4111yescharacteristic
127C22⋊D59—44no—
128C2×Dic59—22no—
129C2×Dic59—22no—
130C2×Dic59—22no—
131D236—22no—
132D236—22no—
133C4×D59—44no—
134C4×D59—44no—
135C4×D59—44no—
136Dic118—22no—
137Dic118—22no—
138C8×C59—21no—
139C8×C59—21no—
140C8×C59—24no—
141C8×C59—42no—
142C8×C59C2×M1612yes—
143C8×C59C2×M1612yes—
144C59⋊C8—44no—
145#17890—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59#43612yes—
149C22×C59C2×C4×D411yescharacteristic
150C2×D59—42no—
151C2×D59—42no—
152C2×D59—42no—
153C4×C59—21no—
154C4×C59—21no—
155C4×C59#43612yes—
156C4×C59#44412yes—
157C4×C59C22×M1611yescharacteristic
158C4×C59C2×#13111yescharacteristic
159C4×C59#57512yes—
160Dic59—42no—
161Dic59—42no—
162Dic59—42no—
163#2965—592no—
164C4×#135—591no—
165#1594—592no—
166#1593—592no—
167#1592—594no—
168#1564—591no—
169#1476—592no—
170C2×#322D5911yescharacteristic, complemented
171C2×C59—22no—
172C2×C59C2×#43611yescharacteristic
173C2×C59#294311yescharacteristic
174C2×C59C2×#44411yescharacteristic
175D59—82no—
176C2×#135—591no—
177C4×#41—592no—
178#446—594no—
179C4×Q16—591no—
180C4×Q16—591no—
181C4×SD16—592no—
182C4×SD16—592no—
183C4×D8—591no—
184C4×D8—591no—
185#436—594no—
186#435—1184no—
187#431—592no—
188#431—1182no—
189#427—591no—
190#416—592no—
191C4×M16—1182no—
192C2×C4×C8C2×D5911yescharacteristic
193C2×C4×C8C2×D5911yescharacteristic
194C2×C4×C8C2×D5911yescharacteristic
195#335—592no—
196#334—592no—
197Q8⋊C8—594no—
198D4⋊C8—594no—
199#322C2×D5912yes—
200#322C2×D5912yes—
201C59#1789011yescharacteristic, complemented, Sylow
202C2×#41—592no—
203#135—1184no—
204C2×Q16—591no—
205C2×SD16—592no—
206C2×D8—591no—
207C2×M16—1184no—
208C22×C8—21no—
209C22×C8C4×D5911yescharacteristic
210C22×C8C4×D5911yescharacteristic
211C22×C8D23612yes—
212C4×Q8—592no—
213C4×Q8—592no—
214C4×D4—592no—
215C4×D4—592no—
216C4×D4—1182no—
217C4×D4—1182no—
218#117—592no—
219#117—1182no—
220C2×C42—1182no—
221C2×C42C22×D5911yescharacteristic
222#107—591no—
223#107—591no—
224#106—592no—
225C4⋊C8—594no—
226C4⋊C8—1184no—
227#103—592no—
228#103—592no—
229#102—592no—
230#102—592no—
231C22⋊C8—1182no—
232C22⋊C8—1184no—
233#97—1184no—
234C4×C8—21no—
235C4×C8—1182no—
236C4×C8C4×D5911yescharacteristic
237C4×C8C4×D5911yescharacteristic
238C4×C8D23612yes—
239C4×C8C22×D5912yes—
240C4×C8C22×D5911yescharacteristic
241C4×C8C22×D5911yescharacteristic
242C4×C8C22×D5912yes—
243#41—1184no—
244#41—1184no—
245C2×Q8—592no—
246C2×D4—592no—
247C2×D4—1182no—
248C22×C4—1182no—
249C22×C4—1182no—
250C22×C4—1182no—
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4#1174568511yescharacteristic
254Q16—1182no—
255SD16—1184no—
256D8—1182no—
257M16—2368no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—1184no—
263C2×C8#1174565914yescomplemented
264C2×C8C2×C4×D5911yescharacteristic
265C2×C8C2×C4×D5911yescharacteristic
266C2×C8C2×C4×D5911yescharacteristic
267C2×C8C2×C4×D5911yescharacteristic
268C2×C8C2×D23614yes—
269C4⋊C4—592no—
270C4⋊C4—592no—
271C4⋊C4—1182no—
272C4⋊C4—1182no—
273C22⋊C4—1182no—
274C22⋊C4—1182no—
275C22⋊C4—1182no—
276C42—1182no—
277C42—1182no—
278C42—1182no—
279C42C2×C4×D5911yescharacteristic
280C42C2×D23611yescharacteristic
281C42#1174568511yescharacteristic
282C42C23×D5911yescharacteristic
283C23—1182no—
284C23C4×D23611yescharacteristic
285Q8—1182no—
286Q8—1182no—
287D4—1182no—
288D4—1182no—
289D4—2364no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—1182no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—2364no—
296C2×C4—2364no—
297C2×C4—2364no—
298C2×C4C4×D23611yescharacteristic
299C2×C4C4×D23611yescharacteristic
300C2×C4C4×D23611yescharacteristic
301C2×C4C2×#1174565912yes—
302C2×C4#1427373312yes—
303C2×C4C22×C4×D5911yescharacteristic
304C2×C4C22×D23611yescharacteristic
305C2×C4C2×#1174568511yescharacteristic
306C8—21no—
307C8—21no—
308C8—24no—
309C8—42no—
310C8—2364no—
311C8C2×#1174565912yes—
312C8C2×#1174565912yes—
313C22—21no—
314C22—21no—
315C22—2362no—
316C22—2362no—
317C22—2362no—
318C22#1683748112yes—
319C22C2×C4×D23611yescharacteristic
320C4—21no—
321C4—21no—
322C4—2362no—
323C4—2362no—
324C4—2362no—
325C4#1683748112yes—
326C4#1683752512yes—
327C4C22×#1174565911yescharacteristic
328C4C2×#1427373311yescharacteristic
329C4#1683820212yes—
330C2—22no—
331C2—4722no—
332C2C2×#1683748111yescharacteristic
333C2#2143289811yescharacteristic
334C2C2×#1683752511yescharacteristic
subgroupG/H
centerC42C2×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC2×C59×#322C2
socleC22×C59C2×C4×D4
prime radicalC2×#14273749C4
cosocleC42×C59C24
hypercenterC2×#322D59
hypocenterC59#17890
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ C2×#322

Complements

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

NH ≅ G/Ntype
1. #21433062C2semidirect
3. #21423011C2semidirect
4. #21423006C2semidirect
5. #21423000C2semidirect
6. #21422844C2semidirect
7. #21422479C2semidirect
8. C2×C59×#322C2semidirect
90. C2×C8×C59M16semidirect
170. C2×#322D59semidirect
201. C59#17890semidirect
263. C2×C8#11745659semidirect

Maximal subgroups

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

subgroupindex
#214330622
C4×#142737492
#214230112
#214230062
#214230002
#214228442
#214224792
C2×C59×#3222
#1789059