Finite Group Databasebeta

← #28483006

Subgroups — #28483006

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#21432661C211yesmaximal, characteristic
2#21424117C211yesmaximal, characteristic
3#21422866C211yesmaximal, characteristic, complemented
4#21422852C211yesmaximal, characteristic, complemented
5#21422479C211yesmaximal, characteristic
6#21421999C211yesmaximal, characteristic, complemented
7C59×#1279C211yesmaximal, characteristic, complemented
8C2×#14273733—21no—
9C4×#11745685C2211yescharacteristic
10#16837455C2211yescharacteristic
11#16837428—21no—
12#16837427—22no—
13#16837426C2211yescharacteristic
14#16837426C2211yescharacteristic
15#16837424—21no—
16C4⋊C8×D59—21no—
17C2×#14273655—21no—
18#16837162—21no—
19#16837153—22no—
20#16837152—21no—
21#16837152—21no—
22#16837134C411yescharacteristic, complemented
23#16837134C411yescharacteristic, complemented
24#16837131C411yescharacteristic, complemented
25#16837131C411yescharacteristic, complemented
26C2×C4⋊C8×C59C2211yescharacteristic
27C2×C4×C8×C59C2211yescharacteristic
28C4×C4⋊C4×C59C2211yescharacteristic
29C2×#11745685D411yescharacteristic
30#14273733—42no—
31#14273733—42no—
32C2×#11745664—21no—
33C2×#11745664—21no—
34C2×#11745659—22no—
35C2×C8×D59—22no—
36C4×Dic118C2×C411yescharacteristic
37C4×Dic118Q811yescharacteristic
38C4×D236C2×C411yescharacteristic
39C4×D236Q811yescharacteristic
40C4×C22⋊D59—21no—
41C4×C22⋊D59—21no—
42#14273686D411yescharacteristic
43#14273684—21no—
44C42×D59—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50C4⋊(C59⋊C8)D412yes—
51Dic59⋊C8—22no—
52Dic59⋊C8—22no—
53#14273655—42no—
54C22⋊(C59⋊C8)—21no—
55C4×C59⋊C8—21no—
56C22×C8×C59—21no—
57C22×C8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C42×C59—21no—
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—21no—
62C4⋊C8×C59D411yescharacteristic
63C4⋊C8×C59D411yescharacteristic
64C4×C8×C59—21no—
65C4×C8×C59C2×C411yescharacteristic
66C4×C8×C59C2×C411yescharacteristic
67#95×C59—21no—
68C22×C4×C59—41no—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×C4×C59#4111yescharacteristic
72C2×C22⋊D59—21no—
73#11745685—21no—
74#11745685—21no—
75#11745685SD1612yes—
76#11745685SD1612yes—
77C2×D236C4⋊C411yescharacteristic
78C2×C4×D59—21no—
79C2×C4×D59—21no—
80C2×C4×D59—21no—
81C2×Dic118C4⋊C411yescharacteristic
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—42no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—41no—
91C42×C59—41no—
92C42×C59—41no—
93C42×C59C22×C411yescharacteristic
94C42×C59C2×D411yescharacteristic
95C42×C59C2×D411yescharacteristic
96C42×C59C2×Q811yescharacteristic
97C22⋊Dic59—21no—
98(C2×D59)⋊C4—21no—
99(C2×D59)⋊C4—21no—
100C4⋊Dic59C4⋊C411yescharacteristic
101C4⋊Dic59C4⋊C411yescharacteristic
102Dic59⋊C4—21no—
103Dic59⋊C4—21no—
104C4×Dic59—21no—
105C4×Dic59—21no—
106C4×Dic59—21no—
107#11745664—41no—
108#11745664—41no—
109#11745664—41no—
110#11745664—41no—
111C2×C59⋊C8—21no—
112C2×C59⋊C8—21no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—41no—
115#11745659—44no—
116C8×D59—44no—
117C23×C59#12311yescharacteristic
118C22×D59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—81no—
124C2×C4×C59—81no—
125C2×C4×C59C2×C4⋊C411yescharacteristic
126C2×C4×C59C4×D411yescharacteristic
127C2×C4×C59C4×D411yescharacteristic
128C2×C4×C59#12011yescharacteristic
129C2×C4×C59C22⋊Q811yescharacteristic
130C2×C4×C59C22⋊Q811yescharacteristic
131C2×C4×C59C2×SD1611yescharacteristic
132C2×C4×C59C2×SD1611yescharacteristic
133C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C59#13711yescharacteristic
135C22⋊D59—22no—
136C22⋊D59—22no—
137C2×Dic59—21no—
138C2×Dic59—21no—
139C2×Dic59—21no—
140D236—21no—
141D236#10612yes—
142C4×D59—22no—
143C4×D59—22no—
144C4×D59—41no—
145C4×D59—41no—
146C4×D59—81no—
147Dic118—21no—
148Dic118#10612yes—
149C8×C59—22no—
150C8×C59—41no—
151C8×C59—41no—
152C8×C59—41no—
153C59⋊C8—41no—
154C59⋊C8—41no—
155C59⋊C8—42no—
156C59⋊C8—81no—
157#8904—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59#38511yescharacteristic
161C22×C59#48111yescharacteristic
162C22×C59#48311yescharacteristic
163C2×D59—22no—
164C2×D59—41no—
165C2×D59—41no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—82no—
170C4×C59C2×#10611yescharacteristic
171C4×C59#42811yescharacteristic
172C4×C59#44311yescharacteristic
173C4×C59#44411yescharacteristic
174C4×C59#44811yescharacteristic
175C4×C59#45011yescharacteristic
176Dic59—22no—
177Dic59—41no—
178Dic59—41no—
179#2906—591no—
180#1806—591no—
181#1570—591no—
182#1566—591no—
183#1476—591no—
184#1303—591no—
185#1279D5911yescharacteristic, complemented
186C2×C59—21no—
187C2×C59—21no—
188C2×C59#185311yescharacteristic
189C2×C59#186111yescharacteristic
190C2×C59#186611yescharacteristic
191D59—42no—
192C2×#131—1181no—
193C4×#41—591no—
194#431—591no—
195#424—1182no—
196#424—1181no—
197#423—591no—
198#423—591no—
199#423—1181no—
200C2×C4⋊C8—1181no—
201C2×C4⋊C8C2×D5911yescharacteristic
202C2×#104—1181no—
203C2×C4×C8C2×D5911yescharacteristic
204C4×C4⋊C4C2×D5911yescharacteristic
205#339—1181no—
206#334—1182no—
207#334—1181no—
208#334—1181no—
209Q8⋊C8—591no—
210Q8⋊C8—591no—
211D4⋊C8—591no—
212D4⋊C8—591no—
213C59#890411yescharacteristic, complemented, Sylow
214C2×#41—591no—
215#131—2362no—
216#131—2362no—
217C2×M16—1182no—
218C2×M16—1181no—
219C2×M16—1181no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—1182no—
223C4×Q8—591no—
224C4×Q8—591no—
225C4×D4—591no—
226C4×D4—591no—
227C4×D4—1181no—
228C4×D4—1181no—
229#117—591no—
230#117—1181no—
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—1181no—
234C2×C42C22×D5911yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—591no—
237C4⋊C8—591no—
238C4⋊C8—592no—
239C4⋊C8—1182no—
240C4⋊C8—1182no—
241C4⋊C8—1181no—
242C4⋊C8—1181no—
243C4⋊C8—1181no—
244C4⋊C8D23611yescharacteristic
245C4⋊C8D23611yescharacteristic
246#104—2362no—
247C22⋊C8—1181no—
248C4×C8—21no—
249C4×C8—1181no—
250C4×C8C4×D5911yescharacteristic
251C4×C8C4×D5911yescharacteristic
252#95—21no—
253#41—592no—
254#41—592no—
255#41—1181no—
256#41—1181no—
257C2×Q8—591no—
258C2×D4—591no—
259C2×D4—1181no—
260C22×C4—41no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4#1174568511yescharacteristic
265C22×C4D4×D5911yescharacteristic
266C22×C4#1174568711yescharacteristic
267M16—2364no—
268M16—2361no—
269M16—2361no—
270M16—2361no—
271M16—2361no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—1181no—
280C2×C8—1181no—
281C2×C8—1182no—
282C2×C8—2364no—
283C2×C8—2361no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—591no—
287C4⋊C4—591no—
288C4⋊C4—1181no—
289C4⋊C4—1181no—
290C22⋊C4—1181no—
291C22⋊C4—1181no—
292C22⋊C4—1181no—
293C42—41no—
294C42—41no—
295C42—1181no—
296C42—1181no—
297C42—1181no—
298C42C2×C4×D5911yescharacteristic
299C42C2×D23611yescharacteristic
300C42D4×D5911yescharacteristic
301C42Q8×D5911yescharacteristic
302C23—1181no—
303C23#1427370911yescharacteristic
304Q8—592no—
305Q8—1181no—
306D4—592no—
307D4—1182no—
308D4—1182no—
309D4—1181no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—1181no—
317C2×C4—1182no—
318C2×C4—1182no—
319C2×C4—1181no—
320C2×C4—1181no—
321C2×C4—4721no—
322C2×C4—2361no—
323C2×C4—2361no—
324C2×C4C4⋊C4×D5911yescharacteristic
325C2×C4(C22⋊D59)⋊C411yescharacteristic
326C2×C4C4×D23611yescharacteristic
327C2×C4#1427369711yescharacteristic
328C2×C4#1427370511yescharacteristic
329C2×C4#1427370711yescharacteristic
330C2×C4SD16×D5911yescharacteristic
331C2×C4SD16×D5911yescharacteristic
332C2×C4#1427375111yescharacteristic
333C2×C4#1427375911yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—4721no—
339C8—2361no—
340C8—2361no—
341C8—2362no—
342C22—21no—
343C22—21no—
344C22—1182no—
345C22—2361no—
346C22—2361no—
347C22#1683727611yescharacteristic
348C22#1683771911yescharacteristic
349C22#1683773111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—1182no—
355C4—2361no—
356C4—2361no—
357C4#106×D5911yescharacteristic
358C4#1683744311yescharacteristic
359C4#1683752011yescharacteristic
360C4#1683752511yescharacteristic
361C4#1683754211yescharacteristic
362C4#1683755411yescharacteristic
363C2—21no—
364C2—21no—
365C2—2362no—
366C2#2142429011yescharacteristic
367C2#2142433711yescharacteristic
368C2#2142437411yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1279C2
socleC22×C59#385
prime radicalC2×#11745685D4
cosocleC2×C42×C59C23
hypercenter#1279D59
hypocenterC59#8904
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #21422866C2semidirect
4. #21422852C2semidirect
6. #21421999C2semidirect
7. C59×#1279C2semidirect
22. #16837134C4semidirect
23. #16837134C4semidirect
24. #16837131C4semidirect
25. #16837131C4semidirect
185. #1279D59semidirect
213. C59#8904semidirect

Maximal subgroups

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

subgroupindex
#214326612
#214241172
#214228662
#214228522
#214224792
#214219992
C59×#12792
#890459