Finite Group Databasebeta

← #28482996

Subgroups — #28482996

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#21424116C211yesmaximal, characteristic
2#21422866C211yesmaximal, characteristic, complemented
3#21422854C211yesmaximal, characteristic, complemented
4#21422478C211yesmaximal, characteristic
5#21421997C211yesmaximal, characteristic, complemented
6#21421935C211yesmaximal, characteristic, complemented
7C59×#2925C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9#16837426C2211yescharacteristic
10C2×#14273654—21no—
11C2×C4×C59⋊C8C2211yescharacteristic
12C4×C4⋊Dic59C2211yescharacteristic
13#16837161—21no—
14#16837154—22no—
15#16837152—21no—
16#16837152—21no—
17Q8⋊(C59⋊C8)C411yescharacteristic, complemented
18Q8⋊(C59⋊C8)C411yescharacteristic, complemented
19D4⋊(C59⋊C8)C411yescharacteristic, complemented
20D4⋊(C59⋊C8)C411yescharacteristic, complemented
21C4×#41×C59C2211yescharacteristic
22C59×#433—22no—
23C59×#432—21no—
24C59×#431—21no—
25C59×#423C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#408—21no—
28C59×#405—21no—
29C2×#11745664—21no—
30C22×C59⋊C8—21no—
31C2×C4⋊Dic59—21no—
32C2×C4×Dic59—21no—
33C4⋊(C59⋊C8)—21no—
34C4⋊(C59⋊C8)—21no—
35C4⋊(C59⋊C8)C2×C411yescharacteristic
36C4⋊(C59⋊C8)C2×C411yescharacteristic
37#14273654—42no—
38C22⋊(C59⋊C8)—21no—
39C4×C59⋊C8—21no—
40C4×C59⋊C8—21no—
41C4×C59⋊C8C2×C411yescharacteristic
42C4×C59⋊C8C2×C411yescharacteristic
43C59⋊#95—21no—
44C2×#41×C59D411yescharacteristic
45C2×M16×C59—21no—
46C22×C8×C59—21no—
47C4×Q8×C59C2×C411yescharacteristic
48C4×Q8×C59Q811yescharacteristic
49C4×D4×C59—21no—
50C4×D4×C59—21no—
51C4×D4×C59C2×C411yescharacteristic
52C4×D4×C59Q811yescharacteristic
53#117×C59—21no—
54#117×C59D411yescharacteristic
55C2×C42×C59—21no—
56C2×C42×C59C2311yescharacteristic
57C4⋊C8×C59—21no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59D411yescharacteristic
60C4⋊C8×C59D411yescharacteristic
61C4⋊C8×C59D412yes—
62C22⋊C8×C59—22no—
63C22⋊C8×C59—22no—
64#97×C59—22no—
65C4×C8×C59—22no—
66#41×C59—21no—
67#41×C59—21no—
68#41×C59—21no—
69#41×C59—21no—
70C2×Q8×C59C4⋊C411yescharacteristic
71C2×D4×C59—21no—
72C2×D4×C59C4⋊C411yescharacteristic
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59#4111yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×Dic59—41no—
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—22no—
84C2×C8×C59—21no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—21no—
89C4⋊C4×C59C4⋊C411yescharacteristic
90C4⋊C4×C59C4⋊C411yescharacteristic
91C22⋊C4×C59—21no—
92C22⋊C4×C59—21no—
93C22⋊C4×C59—21no—
94C42×C59—21no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59C22×C411yescharacteristic
98C42×C59C2×D411yescharacteristic
99C42×C59C2×D411yescharacteristic
100C42×C59C2×Q811yescharacteristic
101C4⋊Dic59—22no—
102C4⋊Dic59—41no—
103C4×Dic59—41no—
104C4×Dic59—41no—
105#11745664—41no—
106#11745664—41no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—42no—
112C2×C59⋊C8—41no—
113C23×C59—21no—
114C23×C59#12311yescharacteristic
115Q8×C59—21no—
116Q8×C59—21no—
117D4×C59—21no—
118D4×C59—21no—
119D4×C59—41no—
120D4×C59—41no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—81no—
132C2×C4×C59C2×C4⋊C411yescharacteristic
133C2×C4×C59C4×D411yescharacteristic
134C2×C4×C59C4×D411yescharacteristic
135C2×C4×C59#12011yescharacteristic
136C2×C4×C59C22⋊Q811yescharacteristic
137C2×C4×C59C22⋊Q811yescharacteristic
138C2×C4×C59#13511yescharacteristic
139C2×C4×C59#13511yescharacteristic
140C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C59#13711yescharacteristic
142C2×Dic59—42no—
143C2×Dic59—81no—
144C2×Dic59—81no—
145C8×C59—42no—
146C8×C59—41no—
147C8×C59—41no—
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—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59#38511yescharacteristic
160C22×C59#48211yescharacteristic
161C22×C59#48511yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59#42711yescharacteristic
169C4×C59#42811yescharacteristic
170C4×C59#44311yescharacteristic
171C4×C59#44411yescharacteristic
172C4×C59#44911yescharacteristic
173C4×C59#45211yescharacteristic
174Dic59—82no—
175#2925D5911yescharacteristic
176#1806—591no—
177#1570—591no—
178#1567—591no—
179#1476—591no—
180#1303—591no—
181#1279—591no—
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—81no—
185C2×C59#185411yescharacteristic
186C2×C59#186111yescharacteristic
187C2×C59#186611yescharacteristic
188C4×#41C2×D5911yescharacteristic
189#433—22no—
190#432—21no—
191#431—21no—
192#431—591no—
193#423—591no—
194#423C2×D5911yescharacteristic
195C2×C4⋊C8C2×D5911yescharacteristic
196C2×#104—1181no—
197#408—21no—
198#405—21no—
199C2×C4×C8—591no—
200C4×C4⋊C4—591no—
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×#41C22⋊D5911yescharacteristic
211C2×M16—21no—
212C2×M16—1181no—
213C22×C8—21no—
214C22×C8—1181no—
215C4×Q8Dic11811yescharacteristic
216C4×Q8C4×D5911yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic11811yescharacteristic
220C4×D4C4×D5911yescharacteristic
221#117—21no—
222#117C22⋊D5911yescharacteristic
223C2×C4⋊C4—1181no—
224C2×C42—21no—
225C2×C42—1181no—
226C2×C42C22×D5911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—591no—
230C4⋊C8—591no—
231C4⋊C8—1181no—
232C4⋊C8—1181no—
233C4⋊C8C22⋊D5911yescharacteristic
234C4⋊C8C22⋊D5911yescharacteristic
235C4⋊C8C22⋊D5912yes—
236#104—2362no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1181no—
240#97—22no—
241C4×C8—22no—
242C4×C8—591no—
243C4×C8—591no—
244C4×C8—1181no—
245C4×C8—1181no—
246#95—1181no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic59⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic59⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—2361no—
258C22×C4#1174568511yescharacteristic
259C22×C4#1174568511yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261M16—41no—
262M16—41no—
263M16—2361no—
264M16—2361no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—1181no—
271C2×C8—1181no—
272C2×C8—1181no—
273C2×C8—1181no—
274C2×C8—2362no—
275C2×C8—2361no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—1182no—
279C4⋊C4—2361no—
280C4⋊C4Dic59⋊C411yescharacteristic
281C4⋊C4Dic59⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—2361no—
289C42—2361no—
290C42C2×Dic11811yescharacteristic
291C42C2×C4×D5911yescharacteristic
292C42C2×C22⋊D5911yescharacteristic
293C42C2×C22⋊D5911yescharacteristic
294C23—21no—
295C23#1427371011yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—4721no—
314C2×C4—4721no—
315C2×C4—2362no—
316C2×C4C2×Dic59⋊C411yescharacteristic
317C2×C4C4×C22⋊D5911yescharacteristic
318C2×C4C4×C22⋊D5911yescharacteristic
319C2×C4#1427369611yescharacteristic
320C2×C4#1427370611yescharacteristic
321C2×C4#1427370611yescharacteristic
322C2×C4#1427374711yescharacteristic
323C2×C4#1427374711yescharacteristic
324C2×C4#1427375211yescharacteristic
325C2×C4#1427375711yescharacteristic
326C8—42no—
327C8—41no—
328C8—41no—
329C8—1182no—
330C8—4721no—
331C8—2361no—
332C8—2361no—
333C8—2361no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1683727511yescharacteristic
340C22#1683772611yescharacteristic
341C22#1683774311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—4722no—
349C4#1683743811yescharacteristic
350C4#1683744511yescharacteristic
351C4#1683751711yescharacteristic
352C4#1683752211yescharacteristic
353C4#1683754711yescharacteristic
354C4#1683756411yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2142429511yescharacteristic
359C2#2142433611yescharacteristic
360C2#2142437111yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2925C2
socleC22×C59#385
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#2925D59
hypocenterC59#8903
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #21422866C2semidirect
3. #21422854C2semidirect
5. #21421997C2semidirect
6. #21421935C2semidirect
17. Q8⋊(C59⋊C8)C4semidirect
18. Q8⋊(C59⋊C8)C4semidirect
19. D4⋊(C59⋊C8)C4semidirect
20. D4⋊(C59⋊C8)C4semidirect
209. C59#8903semidirect

Maximal subgroups

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

subgroupindex
#214241162
#214228662
#214228542
#214224782
#214219972
#214219352
C59×#29252
#890359