Finite Group Databasebeta

← #28482997

Subgroups — #28482997

Order 15104.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432764C211yesmaximal, characteristic
2#21424117C211yesmaximal, characteristic
3#21422867C211yesmaximal, characteristic, complemented
4#21422854C211yesmaximal, characteristic, complemented
5#21422476C211yesmaximal, characteristic
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1303C211yesmaximal, characteristic, complemented
8C4×#11745685C2211yescharacteristic
9#16837462—22no—
10#16837458—21no—
11#16837454—21no—
12#16837426C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273655—21no—
15#16837370—21no—
16#16837360—21no—
17C2×C4×C59⋊C8C2211yescharacteristic
18#16837162—21no—
19#16837154—22no—
20#16837153—21no—
21#16837153—21no—
22#16837132C411yescharacteristic, complemented
23#16837132C411yescharacteristic, complemented
24#16837129C411yescharacteristic, complemented
25#16837129C411yescharacteristic, complemented
26C59×#431C2211yescharacteristic
27C59×#423C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C2×#11745685D411yescharacteristic
30C2×#11745664—21no—
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33C4×Dic118C2×C411yescharacteristic
34C4×Dic118Q811yescharacteristic
35C4×D236C2×C411yescharacteristic
36C4×D236Q811yescharacteristic
37C4×C22⋊D59—21no—
38C4×C22⋊D59—21no—
39#14273686D411yescharacteristic
40#14273684—21no—
41C42×D59—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D411yescharacteristic
47#14273655—42no—
48(C2×D59)⋊C8—22no—
49(C2×D59)⋊C8—22no—
50#14273637—22no—
51C4×C59⋊C8—21no—
52C4×C59⋊C8C2×C411yescharacteristic
53C4×C59⋊C8C2×C411yescharacteristic
54C8×Dic59—22no—
55C2×M16×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C42×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C4⋊C8×C59C2×C411yescharacteristic
61C4⋊C8×C59C2×C411yescharacteristic
62C4⋊C8×C59D412yes—
63C22⋊C8×C59—21no—
64C4×C8×C59—21no—
65#95×C59—21no—
66C22×C4×C59—41no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59#4111yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C2×C22⋊D59—21no—
71#11745685—21no—
72#11745685—21no—
73#11745685—21no—
74#11745685—21no—
75C2×D236C4⋊C411yescharacteristic
76C2×C4×D59—21no—
77C2×C4×D59—21no—
78C2×C4×D59—21no—
79C2×Dic118C4⋊C411yescharacteristic
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—22no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—41no—
87C42×C59—41no—
88C42×C59—41no—
89C42×C59C22×C411yescharacteristic
90C42×C59C2×D411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59C2×Q811yescharacteristic
93C22⋊Dic59—21no—
94(C2×D59)⋊C4—21no—
95(C2×D59)⋊C4—21no—
96C4⋊Dic59C4⋊C411yescharacteristic
97C4⋊Dic59C4⋊C411yescharacteristic
98Dic59⋊C4—21no—
99Dic59⋊C4—21no—
100C4×Dic59—21no—
101C4×Dic59—21no—
102C4×Dic59—21no—
103#11745664—41no—
104#11745664—41no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—42no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C23×C59#12311yescharacteristic
114C22×D59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—81no—
120C2×C4×C59—81no—
121C2×C4×C59C2×C4⋊C411yescharacteristic
122C2×C4×C59C4×D411yescharacteristic
123C2×C4×C59C4×D411yescharacteristic
124C2×C4×C59#12011yescharacteristic
125C2×C4×C59C22⋊Q811yescharacteristic
126C2×C4×C59C22⋊Q811yescharacteristic
127C2×C4×C59#13511yescharacteristic
128C2×C4×C59#13511yescharacteristic
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C59#13711yescharacteristic
131C22⋊D59—41no—
132C22⋊D59—41no—
133C2×Dic59—21no—
134C2×Dic59—21no—
135C2×Dic59—21no—
136D236—21no—
137D236—21no—
138C4×D59—41no—
139C4×D59—41no—
140C4×D59—41no—
141C4×D59—41no—
142C4×D59—81no—
143Dic118—21no—
144Dic118—21no—
145C8×C59—41no—
146C8×C59—41no—
147C8×C59—42no—
148C59⋊C8—22no—
149C59⋊C8—41no—
150C59⋊C8—41no—
151C59⋊C8—41no—
152C59⋊C8—81no—
153#8903—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59#38511yescharacteristic
157C22×C59#48211yescharacteristic
158C22×C59#48511yescharacteristic
159C2×D59—41no—
160C2×D59—41no—
161C2×D59—41no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—82no—
166C4×C59#42711yescharacteristic
167C4×C59#42811yescharacteristic
168C4×C59#44311yescharacteristic
169C4×C59#44411yescharacteristic
170C4×C59#44911yescharacteristic
171C4×C59#45211yescharacteristic
172Dic59—41no—
173Dic59—41no—
174Dic59—41no—
175#2925—591no—
176#1806—591no—
177#1570—591no—
178#1567—591no—
179#1476—591no—
180#1303D5911yescharacteristic, complemented
181#1279—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59#185411yescharacteristic
185C2×C59#186111yescharacteristic
186C2×C59#186611yescharacteristic
187D59—81no—
188C4×#41—591no—
189#433—1182no—
190#432—1181no—
191#431—1181no—
192#431C2×D5911yescharacteristic
193#423—591no—
194#423C2×D5911yescharacteristic
195C2×C4⋊C8—591no—
196C2×#104—1181no—
197#408—1181no—
198#405—1181no—
199C2×C4×C8—591no—
200C4×C4⋊C4C2×D5911yescharacteristic
201#339—1181no—
202#335—1182no—
203#334—1181no—
204#334—1181no—
205Q8⋊C8—591no—
206Q8⋊C8—591no—
207D4⋊C8—591no—
208D4⋊C8—591no—
209C59#890311yescharacteristic, complemented, Sylow
210C2×#41—591no—
211C2×M16—21no—
212C2×M16—1181no—
213C22×C8—1181no—
214C22×C8—1181no—
215C4×Q8—591no—
216C4×Q8—591no—
217C4×D4—591no—
218C4×D4—591no—
219C4×D4—1181no—
220C4×D4—1181no—
221#117—591no—
222#117—1181no—
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—1181no—
226C2×C42C22×D5911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—591no—
229C4⋊C8—591no—
230C4⋊C8—1181no—
231C4⋊C8—1181no—
232C4⋊C8—1181no—
233C4⋊C8C4×D5911yescharacteristic
234C4⋊C8C4×D5911yescharacteristic
235C4⋊C8D23612yes—
236#104—2362no—
237C22⋊C8—21no—
238C22⋊C8—1182no—
239C22⋊C8—1182no—
240#97—1182no—
241C4×C8—21no—
242C4×C8—591no—
243C4×C8—591no—
244C4×C8—1182no—
245C4×C8—1181no—
246#95—21no—
247#41—1181no—
248#41—1181no—
249#41—1181no—
250#41—1181no—
251C2×Q8—591no—
252C2×D4—591no—
253C2×D4—1181no—
254C22×C4—41no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—1181no—
258C22×C4#1174568511yescharacteristic
259C22×C4D4×D5911yescharacteristic
260C22×C4#1174568711yescharacteristic
261M16—41no—
262M16—41no—
263M16—2361no—
264M16—2361no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—1181no—
269C2×C8—1181no—
270C2×C8—1181no—
271C2×C8—1181no—
272C2×C8—2362no—
273C2×C8—2361no—
274C2×C8—2361no—
275C2×C8—2361no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—591no—
279C4⋊C4—591no—
280C4⋊C4—1181no—
281C4⋊C4—1181no—
282C22⋊C4—1181no—
283C22⋊C4—1181no—
284C22⋊C4—1181no—
285C42—41no—
286C42—41no—
287C42—1181no—
288C42—1181no—
289C42—1181no—
290C42C2×C4×D5911yescharacteristic
291C42C2×D23611yescharacteristic
292C42D4×D5911yescharacteristic
293C42Q8×D5911yescharacteristic
294C23—1181no—
295C23#1427370911yescharacteristic
296Q8—1181no—
297Q8—1181no—
298D4—1181no—
299D4—1181no—
300D4—2361no—
301D4—2361no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—1181no—
309C2×C4—1181no—
310C2×C4—1181no—
311C2×C4—4721no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4C4⋊C4×D5911yescharacteristic
317C2×C4(C22⋊D59)⋊C411yescharacteristic
318C2×C4C4×D23611yescharacteristic
319C2×C4#1427369711yescharacteristic
320C2×C4#1427370511yescharacteristic
321C2×C4#1427370711yescharacteristic
322C2×C4#1427374811yescharacteristic
323C2×C4#1427374811yescharacteristic
324C2×C4#1427375111yescharacteristic
325C2×C4#1427375911yescharacteristic
326C8—41no—
327C8—41no—
328C8—42no—
329C8—1182no—
330C8—4721no—
331C8—2361no—
332C8—2361no—
333C8—2361no—
334C22—21no—
335C22—21no—
336C22—2361no—
337C22—2361no—
338C22—2361no—
339C22#1683727611yescharacteristic
340C22#1683772511yescharacteristic
341C22#1683774111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—82no—
346C4—2361no—
347C4—2361no—
348C4—2361no—
349C4#1683743711yescharacteristic
350C4#1683744311yescharacteristic
351C4#1683751611yescharacteristic
352C4#1683752111yescharacteristic
353C4#1683754811yescharacteristic
354C4#1683756711yescharacteristic
355C2—21no—
356C2—21no—
357C2—4721no—
358C2#2142429611yescharacteristic
359C2#2142433911yescharacteristic
360C2#2142437211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1303C2
socleC22×C59#385
prime radicalC2×#11745685D4
cosocleC2×C42×C59C23
hypercenter#1303D59
hypocenterC59#8903
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #21422867C2semidirect
4. #21422854C2semidirect
6. #21421937C2semidirect
7. C59×#1303C2semidirect
22. #16837132C4semidirect
23. #16837132C4semidirect
24. #16837129C4semidirect
25. #16837129C4semidirect
180. #1303D59semidirect
209. C59#8903semidirect

Maximal subgroups

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

subgroupindex
#214327642
#214241172
#214228672
#214228542
#214224762
#214219372
C59×#13032
#890359