Finite Group Databasebeta

← #28482987

Subgroups — #28482987

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432664C211yesmaximal, characteristic
2#21422872C211yesmaximal, characteristic, complemented
3#21422854C211yesmaximal, characteristic, complemented
4#21422478C211yesmaximal, characteristic
5#21421999C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1806C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837455C2211yescharacteristic
10#16837429—22no—
11#16837429—21no—
12#16837426—21no—
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)—21no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×C4×C59⋊C8C2211yescharacteristic
17#16837154—22no—
18#16837154—21no—
19#16837154—21no—
20Q8⋊(C59⋊C8)C411yescharacteristic, complemented
21Q8⋊(C59⋊C8)C411yescharacteristic, complemented
22D4⋊(C59⋊C8)C411yescharacteristic, complemented
23D4⋊(C59⋊C8)C411yescharacteristic, complemented
24C4×#41×C59C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×#104×C59—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C59×#339—21no—
29#14273734—42no—
30#14273734—42no—
31C2×#11745664—22no—
32C2×#11745664—21no—
33C22×C59⋊C8—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)C2×C411yescharacteristic
41C4⋊(C59⋊C8)C2×C411yescharacteristic
42C4⋊(C59⋊C8)D411yescharacteristic
43C4⋊(C59⋊C8)D411yescharacteristic
44C4⋊(C59⋊C8)D412yes—
45C22⋊(C59⋊C8)—21no—
46C4×C59⋊C8—21no—
47C4×C59⋊C8—21no—
48C4×C59⋊C8C2×C411yescharacteristic
49C4×C59⋊C8C2×C411yescharacteristic
50C2×#41×C59D411yescharacteristic
51C2×M16×C59—21no—
52C4×Q8×C59C2×C411yescharacteristic
53C4×Q8×C59Q811yescharacteristic
54C4×D4×C59—21no—
55C4×D4×C59—21no—
56C4×D4×C59C2×C411yescharacteristic
57C4×D4×C59Q811yescharacteristic
58#117×C59—21no—
59#117×C59D411yescharacteristic
60C2×C4⋊C4×C59—21no—
61C2×C42×C59—21no—
62C2×C42×C59—21no—
63C2×C42×C59C2311yescharacteristic
64C4⋊C8×C59—21no—
65C4⋊C8×C59—21no—
66#104×C59—42no—
67#95×C59—21no—
68#41×C59—21no—
69#41×C59—21no—
70#41×C59D812yes—
71#41×C59Q1612yes—
72C2×Q8×C59C4⋊C411yescharacteristic
73C2×D4×C59—21no—
74C2×D4×C59C4⋊C411yescharacteristic
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×C4×C59#4111yescharacteristic
82M16×C59—41no—
83M16×C59—41no—
84C2×C8×C59—41no—
85C4⋊C4×C59—21no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—41no—
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×C59—41no—
98C42×C59—41no—
99C42×C59C22×C411yescharacteristic
100C42×C59C2×D411yescharacteristic
101C42×C59C2×D411yescharacteristic
102C42×C59C2×Q811yescharacteristic
103#11745664—44no—
104#11745664—41no—
105#11745664—41no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—21no—
108C2×C59⋊C8—21no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—42no—
114C2×C59⋊C8—44no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C23×C59—21no—
118C23×C59#12311yescharacteristic
119Q8×C59—21no—
120Q8×C59#10712yes—
121D4×C59—22no—
122D4×C59—22no—
123D4×C59—21no—
124D4×C59#10712yes—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—42no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59C2×C4⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59C4×D411yescharacteristic
142C2×C4×C59#12011yescharacteristic
143C2×C4×C59C22⋊Q811yescharacteristic
144C2×C4×C59C22⋊Q811yescharacteristic
145C2×C4×C59C2×D811yescharacteristic
146C2×C4×C59C2×Q1611yescharacteristic
147C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C59#13711yescharacteristic
149C8×C59—81no—
150C59⋊C8—22no—
151C59⋊C8—41no—
152C59⋊C8—41no—
153C59⋊C8—41no—
154C59⋊C8—41no—
155C59⋊C8—42no—
156C59⋊C8—41no—
157#8902—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—22no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59#38511yescharacteristic
164C22×C59#48011yescharacteristic
165C22×C59#48411yescharacteristic
166C4×C59—22no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—82no—
173C4×C59C2×#10711yescharacteristic
174C4×C59#42811yescharacteristic
175C4×C59#44311yescharacteristic
176C4×C59#44411yescharacteristic
177C4×C59#44711yescharacteristic
178C4×C59#45111yescharacteristic
179#2906—591no—
180#1806D5911yescharacteristic
181#1571—591no—
182#1567—591no—
183#1476—591no—
184#1303—591no—
185#1279—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—42no—
189C2×C59#185211yescharacteristic
190C2×C59#186111yescharacteristic
191C2×C59#186611yescharacteristic
192C2×#131—1181no—
193C4×#41C2×D5911yescharacteristic
194#431—591no—
195#424—1182no—
196#424—1181no—
197#423—591no—
198#423—1181no—
199#423C2×D5911yescharacteristic
200C2×C4⋊C8—591no—
201C2×C4⋊C8—1181no—
202C2×#104—21no—
203C2×C4×C8—591no—
204C4×C4⋊C4C2×D5911yescharacteristic
205#339—21no—
206#335—1182no—
207#335—1181no—
208#335—1181no—
209Q8⋊C8—591no—
210Q8⋊C8—591no—
211D4⋊C8—591no—
212D4⋊C8—591no—
213C59#890211yescharacteristic, complemented, Sylow
214C2×#41C22⋊D5911yescharacteristic
215#131—2362no—
216#131—2362no—
217C2×M16—21no—
218C2×M16—1182no—
219C2×M16—1181no—
220C22×C8—1182no—
221C22×C8—1181no—
222C22×C8—1181no—
223C4×Q8Dic11811yescharacteristic
224C4×Q8C4×D5911yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic11811yescharacteristic
228C4×D4C4×D5911yescharacteristic
229#117—21no—
230#117C22⋊D5911yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—591no—
238C4⋊C8—591no—
239C4⋊C8—591no—
240C4⋊C8—591no—
241C4⋊C8—592no—
242C4⋊C8—1182no—
243C4⋊C8—1182no—
244C4⋊C8—1181no—
245C4⋊C8—1181no—
246#104—42no—
247C22⋊C8—1181no—
248C4×C8—591no—
249C4×C8—591no—
250C4×C8—1181no—
251C4×C8—1181no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1174566912yes—
256#41#1174567212yes—
257C2×Q8Dic59⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4Dic59⋊C411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4#1174568711yescharacteristic
265C22×C4#1174568711yescharacteristic
266C22×C4C2×C22⋊D5911yescharacteristic
267M16—41no—
268M16—41no—
269M16—2364no—
270M16—2361no—
271M16—2361no—
272C2×C8—41no—
273C2×C8—1181no—
274C2×C8—1181no—
275C2×C8—1181no—
276C2×C8—1181no—
277C2×C8—1181no—
278C2×C8—1181no—
279C2×C8—1182no—
280C2×C8—2362no—
281C2×C8—2364no—
282C2×C8—2361no—
283C2×C8—2361no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4Dic59⋊C411yescharacteristic
289C4⋊C4Dic59⋊C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×Dic11811yescharacteristic
299C42C2×C4×D5911yescharacteristic
300C42D4×D5911yescharacteristic
301C42D4×D5911yescharacteristic
302C23—21no—
303C23#1427370811yescharacteristic
304Q8—21no—
305Q8#1427366012yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1427366012yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×Dic59⋊C411yescharacteristic
325C2×C4(C22⋊D59)⋊C411yescharacteristic
326C2×C4(C22⋊D59)⋊C411yescharacteristic
327C2×C4#1427369511yescharacteristic
328C2×C4#1427370411yescharacteristic
329C2×C4#1427370411yescharacteristic
330C2×C4C2×#1174566911yescharacteristic
331C2×C4C2×#1174567211yescharacteristic
332C2×C4#1427375211yescharacteristic
333C2×C4#1427375711yescharacteristic
334C8—81no—
335C8—1182no—
336C8—2361no—
337C8—2361no—
338C8—2361no—
339C8—2361no—
340C8—2362no—
341C8—2361no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1683727411yescharacteristic
348C22#1683771311yescharacteristic
349C22#1683773511yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1427366011yescharacteristic
358C4#1683744511yescharacteristic
359C4#1683751611yescharacteristic
360C4#1683752111yescharacteristic
361C4#1683753511yescharacteristic
362C4#1683755711yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2142428311yescharacteristic
367C2#2142433411yescharacteristic
368C2#2142436911yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1806C2
socleC22×C59#385
prime radicalC2×#41×C59D4
cosocleC2×C42×C59C23
hypercenter#1806D59
hypocenterC59#8902
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21422872C2semidirect
3. #21422854C2semidirect
5. #21421999C2semidirect
6. #21421937C2semidirect
20. Q8⋊(C59⋊C8)C4semidirect
21. Q8⋊(C59⋊C8)C4semidirect
22. D4⋊(C59⋊C8)C4semidirect
23. D4⋊(C59⋊C8)C4semidirect
213. C59#8902semidirect

Maximal subgroups

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

subgroupindex
#214326642
#214228722
#214228542
#214224782
#214219992
#214219372
C59×#18062
#890259