Finite Group Databasebeta

← #28505211

Subgroups — #28505211

Order 15104.

424 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432684C211yesmaximal, characteristic
2#21425885C211yesmaximal, characteristic, complemented
3#21425884C211yesmaximal, characteristic, complemented
4C2×#16837183C211yesmaximal, characteristic
5#21424126C211yesmaximal, characteristic
6#21422203C211yesmaximal, characteristic, complemented
7C59×#1386C211yesmaximal, characteristic, complemented
8C22×#11745659C2211yescharacteristic
9#117×D59—21no—
10C2×#14273667C2211yescharacteristic
11#16837443—21no—
12#16837443—21no—
13#16837441—21no—
14#16837441—21no—
15#16837440C2211yescharacteristic
16#16837438—21no—
17C2×#14273661C2211yescharacteristic
18#16837259—21no—
19#16837230—21no—
20#16837229—21no—
21#16837228—21no—
22#16837227—21no—
23#16837183C414yescomplemented
24#16837164—21no—
25C2×M32×C59C2211yescharacteristic
26C59×#427C2211yescharacteristic
27C2×#106×C59C2211yescharacteristic
28C22×C4×D59D411yescharacteristic
29C2×#11745659—21no—
30C2×#11745659—21no—
31C2×#11745659—21no—
32C2×#11745659—21no—
33C2×#11745659C2×C412yes—
34C2×#11745659Q812yes—
35C22×C59⋊C8D411yescharacteristic
36#14273686—21no—
37#14273685—21no—
38#14273685—21no—
39#14273684—41no—
40#14273683—41no—
41C2×C4⋊Dic59—21no—
42C4⋊C4×D59—21no—
43C4⋊C4×D59—21no—
44C2×(C2×D59)⋊C4—41no—
45C22⋊C4×D59—41no—
46C42×D59—41no—
47#14273667—21no—
48#14273667C2×C412yescomplemented
49C2×C59⋊C16—21no—
50#14273661—21no—
51#14273661—21no—
52#14273661D411yescharacteristic
53#14273661D411yescharacteristic
54#14273660—21no—
55#14273660—21no—
56#14273659—21no—
57#14273658—21no—
58#14273658—21no—
59C59⋊#95—41no—
60C22×C8×C59C2311yescharacteristic
61#117×C59—21no—
62C2×C4⋊C4×C59—21no—
63M32×C59—21no—
64M32×C59C2×C412yescomplemented
65C2×C16×C59—21no—
66#107×C59D411yescharacteristic
67#107×C59D411yescharacteristic
68#106×C59—21no—
69#106×C59—21no—
70#106×C59—21no—
71C23×D59D811yescharacteristic
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—41no—
75C22×Dic59D811yescharacteristic
76C2×C4×D59—21no—
77C2×C4×D59—21no—
78C2×C4×D59—21no—
79C2×C4×D59—21no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C2×C4×D59C4⋊C412yes—
83C2×C4×D59Q1612yes—
84C2×C8×C59C22×C411yescharacteristic
85C2×C8×C59C2×D411yescharacteristic
86C2×C8×C59C2×D411yescharacteristic
87C2×C8×C59C2×Q811yescharacteristic
88C2×C8×C59#4111yescharacteristic
89C2×C8×C59#4111yescharacteristic
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C42×C59—41no—
96C22⋊Dic59—41no—
97(C2×D59)⋊C4—41no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—82no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—41no—
103C4⋊Dic59—41no—
104Dic59⋊C4—41no—
105Dic59⋊C4—41no—
106C4×Dic59—41no—
107C4×Dic59—41no—
108C4×Dic59—41no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8C4⋊C412yes—
112#11745659—22no—
113#11745659—22no—
114#11745659—22no—
115#11745659—22no—
116C16×C59—22no—
117C59⋊C16—22no—
118C23×C59C2×D811yescharacteristic
119C22×D59—21no—
120C22×D59—21no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59#10712yes—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×C4×C59—81no—
129C2×C4×C59C2×C4⋊C411yescharacteristic
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12311yescharacteristic
132C2×C4×C59C2×Q1611yescharacteristic
133C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C59#13711yescharacteristic
135C2×Dic59—21no—
136C2×Dic59—21no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141C2×Dic59—81no—
142C2×Dic59#10712yes—
143C4×D59—22no—
144C4×D59—22no—
145C4×D59—22no—
146C4×D59—22no—
147C4×D59—82no—
148C4×D59—82no—
149C8×C59C4×D411yescharacteristic
150C8×C59C4×D411yescharacteristic
151C8×C59C22⋊Q811yescharacteristic
152C8×C59C22⋊Q811yescharacteristic
153C59⋊C8—22no—
154#12923—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59C2×#10711yescharacteristic
158C22×C59#44711yescharacteristic
159C22×C59#48011yescharacteristic
160C2×D59—22no—
161C2×D59—41no—
162C2×D59—41no—
163C2×D59—42no—
164C2×D59—81no—
165C4×C59—81no—
166C4×C59—81no—
167C4×C59—161no—
168C4×C59#38511yescharacteristic
169C4×C59#42811yescharacteristic
170C4×C59#45111yescharacteristic
171C4×C59#48411yescharacteristic
172Dic59—22no—
173Dic59—81no—
174Dic59—81no—
175Dic59—161no—
176#2911—591no—
177#2159—591no—
178#2159—591no—
179C2×#349—591no—
180#1809—591no—
181#1386—591no—
182#1386D5911yescharacteristic, complemented
183C2×C59—21no—
184C2×C59—21no—
185C2×C59#185211yescharacteristic
186C2×C59#219011yescharacteristic
187C2×C59#219111yescharacteristic
188D59—42no—
189C22×M16—591no—
190C2×#117—1181no—
191C2×M32—591no—
192C2×M32C2×D5911yescharacteristic
193#428—1181no—
194#428—1181no—
195#428—1181no—
196#428—1181no—
197#427—591no—
198#427—1181no—
199#427C2×D5911yescharacteristic
200C2×#107—591no—
201C2×#106C2×D5911yescharacteristic
202#380—1181no—
203#367—1181no—
204#367—1181no—
205#366—1181no—
206#366—1181no—
207#349—594no—
208#340—1181no—
209C59#1292311yescharacteristic, complemented, Sylow
210C23×C4—591no—
211C2×M16—592no—
212C2×M16—592no—
213C2×M16—1181no—
214C2×M16—1181no—
215C2×M16—1181no—
216C2×M16—1181no—
217C22×C8—591no—
218C22×C8C22×D5911yescharacteristic
219#117—21no—
220#117—1181no—
221#117—1181no—
222#117—1181no—
223#117—2361no—
224#117—2361no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C22⋊C4—2361no—
230C2×C22⋊C4—2361no—
231C2×C42—2361no—
232M32—21no—
233M32—592no—
234M32—1181no—
235M32C4×D5912yescomplemented
236C2×C16—21no—
237C2×C16—1181no—
238#107—591no—
239#107—591no—
240#107—1181no—
241#107—1181no—
242#107—1181no—
243#107—1181no—
244#107D23611yescharacteristic
245#107D23611yescharacteristic
246#106—21no—
247#106—21no—
248#106—21no—
249#106—1181no—
250#106—1181no—
251#106—1181no—
252#95—2361no—
253C24—591no—
254C22×C4—41no—
255C22×C4—591no—
256C22×C4—592no—
257C22×C4—592no—
258C22×C4—1181no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—2361no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4D4×D5911yescharacteristic
266M16—1182no—
267M16—1182no—
268M16—1182no—
269M16—1182no—
270C2×C8—592no—
271C2×C8—1181no—
272C2×C8—1181no—
273C2×C8C2×C4×D5911yescharacteristic
274C2×C8C2×D23611yescharacteristic
275C2×C8#1174568511yescharacteristic
276C2×C8D4×D5911yescharacteristic
277C2×C8#1174568711yescharacteristic
278C2×C8Q8×D5911yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—1181no—
284C4⋊C4—1181no—
285C4⋊C4—2361no—
286C4⋊C4—2361no—
287C4⋊C4—2361no—
288C4⋊C4—2361no—
289C22⋊C4—41no—
290C22⋊C4—4722no—
291C22⋊C4—2361no—
292C22⋊C4—2361no—
293C22⋊C4—2361no—
294C42—41no—
295C42—2361no—
296C42—2361no—
297C42—2361no—
298C16—22no—
299C16—1182no—
300C23—592no—
301C23—1181no—
302C23—1181no—
303C23—2361no—
304C23—2361no—
305C23D8×D5911yescharacteristic
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—592no—
312C2×C4—1181no—
313C2×C4—1181no—
314C2×C4—1182no—
315C2×C4—1182no—
316C2×C4—1182no—
317C2×C4—1182no—
318C2×C4—4722no—
319C2×C4—4722no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4—2361no—
324C2×C4—2361no—
325C2×C4C4⋊C4×D5911yescharacteristic
326C2×C4#1427369711yescharacteristic
327C2×C4#1427370911yescharacteristic
328C2×C4Q16×D5911yescharacteristic
329C2×C4#1427375411yescharacteristic
330C2×C4#1427375611yescharacteristic
331C8—1182no—
332C8(C22⋊D59)⋊C411yescharacteristic
333C8C4×D23611yescharacteristic
334C8#1427370511yescharacteristic
335C8#1427370711yescharacteristic
336C22—21no—
337C22—21no—
338C22—1182no—
339C22—4721no—
340C22—2362no—
341C22—2361no—
342C22—2361no—
343C22#107×D5911yescharacteristic
344C22#1683753711yescharacteristic
345C22#1683771411yescharacteristic
346C4—81no—
347C4—81no—
348C4—161no—
349C4—1182no—
350C4—9441no—
351C4—4721no—
352C4—4721no—
353C4#1683727611yescharacteristic
354C4#1683744111yescharacteristic
355C4#1683755911yescharacteristic
356C4#1683773611yescharacteristic
357C2—21no—
358C2—21no—
359C2—2362no—
360C2#2142428511yescharacteristic
361C2#2142604311yescharacteristic
362C2#2142604811yescharacteristic
subgroupG/H
centerC22#16837537
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1386C2
socleC22×C59#447
prime radicalC23×D59D8
cosocleC22×C8×C59C23
hypercenter#1386D59
hypocenterC59#12923
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1386

Complements

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

NH ≅ G/Ntype
2. #21425885C2semidirect
3. #21425884C2semidirect
6. #21422203C2semidirect
7. C59×#1386C2semidirect
23. #16837183C4semidirect
48. #14273667C2×C4semidirect
64. M32×C59C2×C4semidirect
182. #1386D59semidirect
209. C59#12923semidirect
235. M32C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214326842
#214258852
#214258842
C2×#168371832
#214241262
#214222032
C59×#13862
#1292359