Finite Group Databasebeta

← #28506891

Subgroups — #28506891

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432790C211yesmaximal, characteristic
2#21425861C211yesmaximal, characteristic, complemented
3#21425725C211yesmaximal, characteristic, complemented
4#21425723C211yesmaximal, characteristic, complemented
5#21425707C211yesmaximal, characteristic, complemented
6#21425700C211yesmaximal, characteristic, complemented
7C59×#2112C211yesmaximal, characteristic, complemented
8C2×#14273733C2211yescharacteristic
9#16837913C411yescharacteristic
10C2×#14273667C2211yescharacteristic
11C2×#14273667C2211yescharacteristic
12#16837473—22no—
13C8×C22⋊D59—22no—
14#16837428C411yescharacteristic
15#16837368C411yescharacteristic
16#16837366C411yescharacteristic
17#16837357C2211yescharacteristic
18#16837222—22no—
19#16837187—22no—
20#16837185—22no—
21C22⋊(C59⋊C16)—22no—
22#16837179—22no—
23C23×C8×C59C2211yescharacteristic
24C2×M32×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C59×C22⋊C16—22no—
27C2×#11745685C2×C411yescharacteristic
28#14273733—21no—
29#14273733—21no—
30#14273733—21no—
31#14273733—21no—
32C2×#11745664C2×C411yescharacteristic
33C2×#11745664C2×C411yescharacteristic
34C2×#11745659—21no—
35C2×C8×D59—21no—
36#14273710—21no—
37#14273706—21no—
38#14273701—21no—
39#14273696—21no—
40C4×C22⋊D59—22no—
41#14273686C2×C411yescharacteristic
42#14273667—21no—
43#14273667—21no—
44#14273667—21no—
45#14273667—21no—
46C2×C59⋊C16—21no—
47C2×C59⋊C16D412yescomplemented
48Dic59⋊C8—22no—
49C22⋊(C59⋊C8)—22no—
50(C2×D59)⋊C8—22no—
51#14273637D412yes—
52C8×Dic59D412yes—
53C23×C4×C59C2×C411yescharacteristic
54C22×C8×C59—21no—
55C22×C8×C59—21no—
56C22×C8×C59—21no—
57C22×C8×C59—21no—
58C22×C8×C59—21no—
59C22×C8×C59—21no—
60C22×C8×C59C2×C411yescharacteristic
61C22×C8×C59C2311yescharacteristic
62M32×C59—21no—
63M32×C59—21no—
64M32×C59—21no—
65M32×C59—21no—
66C2×C16×C59—21no—
67C2×C16×C59D412yescomplemented
68C24×C59M1611yescharacteristic
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59M1611yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C2×C22⋊D59—22no—
78C2×C22⋊D59—21no—
79#11745685—21no—
80#11745685—21no—
81#11745685—21no—
82#11745685—21no—
83C2×D236M1611yescharacteristic
84C2×C4×D59—21no—
85C2×Dic118M1611yescharacteristic
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59C2×D411yescharacteristic
94C2×C8×C59C2×D411yescharacteristic
95C2×C8×C59C2×D411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59#4111yescharacteristic
98C2×C8×C59#4111yescharacteristic
99C22⋊Dic59—22no—
100C22⋊Dic59—21no—
101(C2×D59)⋊C4—22no—
102C4⋊Dic59—21no—
103Dic59⋊C4—22no—
104C4×Dic59C22⋊C412yes—
105#11745664—21no—
106#11745664—21no—
107#11745664—21no—
108#11745664—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8C22⋊C412yes—
111#11745659—41no—
112#11745659—41no—
113C8×D59—41no—
114C8×D59—41no—
115C16×C59—22no—
116C16×C59—41no—
117C59⋊C16—22no—
118C59⋊C16—41no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—42no—
123C23×C59C2×M1611yescharacteristic
124C22×D59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59C2×C22⋊C411yescharacteristic
133C2×C4×C59C4×D411yescharacteristic
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59C2×M1611yescharacteristic
136C2×C4×C59#13111yescharacteristic
137C2×C4×C59#13111yescharacteristic
138C22⋊D59—44no—
139C22⋊D59—41no—
140C22⋊D59—41no—
141C2×Dic59—22no—
142C2×Dic59—21no—
143D236—21no—
144D236—21no—
145C4×D59—41no—
146C4×D59—41no—
147Dic118—21no—
148Dic118—21no—
149C8×C59—41no—
150C8×C59#12111yescharacteristic
151C8×C59#12111yescharacteristic
152C8×C59#12411yescharacteristic
153C8×C59#12711yescharacteristic
154C59⋊C8—22no—
155C59⋊C8—41no—
156#13200—591nomaximal, Sylow
157C22×C59—22no—
158C22×C59—22no—
159C22×C59—22no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59#40711yescharacteristic
166C22×C59#43511yescharacteristic
167C22×C59#43511yescharacteristic
168C2×D59—42no—
169C2×D59—41no—
170C4×C59—41no—
171C4×C59#39011yescharacteristic
172C4×C59#40811yescharacteristic
173C4×C59#43511yescharacteristic
174C4×C59#43511yescharacteristic
175Dic59—42no—
176Dic59—41no—
177#2929—591no—
178#2153—591no—
179#2117—591no—
180#2117—591no—
181#2112—591no—
182#2112D5911yescharacteristic, complemented
183#2110—591no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59—42no—
188C2×C59#195511yescharacteristic
189D59—81no—
190C2×#131—591no—
191C23×C8C2×D5911yescharacteristic
192#525—591no—
193C2×M32—591no—
194C2×M32—591no—
195C2×M32C2×D5911yescharacteristic
196C2×M32C2×D5911yescharacteristic
197#435—1182no—
198C8×D4—1182no—
199#424—591no—
200#408—591no—
201#407—591no—
202#405—591no—
203#364—1182no—
204#350—1182no—
205#350—1182no—
206C22⋊C16—22no—
207C22⋊C16—1182no—
208#347—1182no—
209C59#1320011yescharacteristic, complemented, Sylow
210C2×#41—591no—
211C23×C4C4×D5911yescharacteristic
212#131—1181no—
213#131—1181no—
214#131—1181no—
215#131—1181no—
216C2×M16—591no—
217C2×M16—591no—
218C2×M16—1181no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—1181no—
226C22×C8C4×D5911yescharacteristic
227C22×C8C22×D5911yescharacteristic
228#123—1181no—
229C22⋊Q8—1181no—
230#121—1181no—
231#120—1181no—
232C4×D4—1182no—
233#117—591no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1181no—
239M32—1181no—
240M32—1181no—
241M32—1181no—
242C2×C16—21no—
243C2×C16—592no—
244C2×C16—1181no—
245C2×C16C22⋊D5912yescomplemented
246C4⋊C8—1182no—
247C22⋊C8—1182no—
248C22⋊C8—1182no—
249#97—592no—
250C4×C8—592no—
251C24#1174565911yescharacteristic
252#41—1181no—
253#41—1181no—
254#41—1181no—
255#41—1181no—
256C2×Q8—591no—
257C2×D4—591no—
258C2×D4—1182no—
259C2×D4—1181no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1181no—
267C22×C4#1174565911yescharacteristic
268C22×C4C2×C4×D5911yescharacteristic
269M16—1181no—
270M16—1181no—
271M16—1181no—
272M16—1181no—
273M16—2361no—
274M16—2361no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—592no—
283C2×C8—1181no—
284C2×C8—2361no—
285C2×C8—2361no—
286C2×C8#1174568511yescharacteristic
287C2×C8D4×D5911yescharacteristic
288C2×C8D4×D5911yescharacteristic
289C2×C8D4×D5911yescharacteristic
290C2×C8#1174568711yescharacteristic
291C2×C8C2×C22⋊D5911yescharacteristic
292C4⋊C4—1182no—
293C4⋊C4—1181no—
294C22⋊C4—1182no—
295C22⋊C4—1182no—
296C22⋊C4—1181no—
297C42—592no—
298C16—22no—
299C16—41no—
300C16—1182no—
301C16—2361no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—1181no—
307C23C2×#1174565911yescharacteristic
308Q8—1181no—
309Q8—1181no—
310D4—1181no—
311D4—1181no—
312D4—2364no—
313D4—2361no—
314D4—2361no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—1182no—
323C2×C4—1181no—
324C2×C4—2361no—
325C2×C4—2361no—
326C2×C4C22⋊C4×D5911yescharacteristic
327C2×C4(C22⋊D59)⋊C411yescharacteristic
328C2×C4C4×C22⋊D5911yescharacteristic
329C2×C4M16×D5911yescharacteristic
330C2×C4#1427373311yescharacteristic
331C2×C4#1427373511yescharacteristic
332C8—41no—
333C8—1182no—
334C8—2361no—
335C8#1427369911yescharacteristic
336C8#1427370011yescharacteristic
337C8#1427371311yescharacteristic
338C8#1427372311yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2362no—
348C22—2361no—
349C22#1683736411yescharacteristic
350C22#1683747011yescharacteristic
351C22#1683747311yescharacteristic
352C4—41no—
353C4—2362no—
354C4—2361no—
355C4#1683730211yescharacteristic
356C4#1683736711yescharacteristic
357C4#1683747111yescharacteristic
358C4#1683747611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—4721no—
364C2#2142489211yescharacteristic
subgroupG/H
centerC8#14273723
commutatorC22×C4×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2112C2
socleC2×C59#1955
prime radical#16837913C4
cosocleC22×C8×C59C23
hypercenter#2112D59
hypocenterC59#13200
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #21425861C2semidirect
3. #21425725C2semidirect
4. #21425723C2semidirect
5. #21425707C2semidirect
6. #21425700C2semidirect
7. C59×#2112C2semidirect
47. C2×C59⋊C16D4semidirect
67. C2×C16×C59D4semidirect
182. #2112D59semidirect
209. C59#13200semidirect
245. C2×C16C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214327902
#214258612
#214257252
#214257232
#214257072
#214257002
C59×#21122
#1320059