Finite Group Databasebeta

← #28483178

Subgroups — #28483178

Order 15104.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432839C211yesmaximal, characteristic
2#21424869C211yesmaximal, characteristic, complemented
3#21422743C211yesmaximal, characteristic
4#21422546C211yesmaximal, characteristic
5#21421932C211yesmaximal, characteristic, complemented
6#21421930C211yesmaximal, characteristic, complemented
7C59×#1570C211yesmaximal, characteristic, complemented
8C2×#14273733—21no—
9#16837952C2211yescharacteristic
10#16837479—21no—
11#16837472—22no—
12#16837467—21no—
13C2×#14273662—21no—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837368—21no—
16C4×#11745664C2211yescharacteristic
17C2×C4×C59⋊C8C2211yescharacteristic
18#16837166—21no—
19#16837147C411yescharacteristic
20#16837147C411yescharacteristic
21#16837145C411yescharacteristic
22#16837145C411yescharacteristic
23#16837129—22no—
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#334—21no—
28C59×#334—21no—
29C2×#11745685—21no—
30#14273733—42no—
31#14273733—42no—
32C2×#11745664D411yescharacteristic
33C2×#11745664D411yescharacteristic
34C2×#11745659—22no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C2×C8×D59—22no—
38#14273727C2×C411yescharacteristic
39#14273724C2×C411yescharacteristic
40#14273715—21no—
41#14273701—21no—
42#14273701Q812yes—
43C4×D236—21no—
44C4×D236D412yes—
45#14273662—42no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)C2×C411yescharacteristic
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49(C2×D59)⋊C8—22no—
50(C2×D59)⋊C8—22no—
51#14273638—21no—
52C4×C59⋊C8—21no—
53C4×C59⋊C8—21no—
54C4×C59⋊C8—21no—
55C2×M16×C59—21no—
56C22×C8×C59—21no—
57C2×C42×C59C2311yescharacteristic
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59C2×C411yescharacteristic
62C4⋊C8×C59C2×C411yescharacteristic
63C4⋊C8×C59D411yescharacteristic
64C4⋊C8×C59D411yescharacteristic
65C22⋊C8×C59—21no—
66C4×C8×C59—21no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×Q811yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C2×C22⋊D59—22no—
71#11745685—41no—
72#11745685—41no—
73#11745685—41no—
74#11745685—41no—
75C2×D236—21no—
76C2×D236—21no—
77C2×D236C4⋊C412yes—
78C2×C4×D59—22no—
79C2×Dic118—21no—
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—21no—
83C2×C8×C59—21no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C42×C59C22×C411yescharacteristic
90C42×C59C2×D411yescharacteristic
91C42×C59C2×D411yescharacteristic
92C42×C59#4111yescharacteristic
93(C2×D59)⋊C4—22no—
94(C2×D59)⋊C4—22no—
95C4⋊Dic59—21no—
96C4⋊Dic59C4⋊C412yes—
97#11745664—21no—
98#11745664—21no—
99#11745664—21no—
100#11745664SD1612yes—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—22no—
103C2×C59⋊C8—21no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110#11745659—44no—
111C8×D59—44no—
112C23×C59C22⋊Q811yescharacteristic
113C22×D59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59C2×C4⋊C411yescharacteristic
118C2×C4×C59C4×D411yescharacteristic
119C2×C4×C59C4×D411yescharacteristic
120C2×C4×C59#12011yescharacteristic
121C2×C4×C59C22⋊Q811yescharacteristic
122C2×C4×C59#12311yescharacteristic
123C2×C4×C59C2×SD1611yescharacteristic
124C2×C4×C59#13511yescharacteristic
125C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
127C22⋊D59—42no—
128C22⋊D59—42no—
129C2×Dic59—22no—
130D236—42no—
131D236—41no—
132D236—41no—
133D236—41no—
134D236—41no—
135C4×D59—42no—
136C4×D59—42no—
137Dic118—41no—
138Dic118—41no—
139C8×C59—41no—
140C8×C59—41no—
141C8×C59—41no—
142C8×C59—41no—
143C8×C59—81no—
144C59⋊C8—22no—
145C59⋊C8—42no—
146C59⋊C8—41no—
147C59⋊C8—41no—
148C59⋊C8—41no—
149#8932—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#38511yescharacteristic
153C22×C59#47611yescharacteristic
154C22×C59#47811yescharacteristic
155C2×D59—42no—
156C2×D59—42no—
157C2×D59—42no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59C4×SD1611yescharacteristic
162C4×C59#44211yescharacteristic
163C4×C59#44311yescharacteristic
164C4×C59#44411yescharacteristic
165C4×C59#44911yescharacteristic
166C4×C59#45011yescharacteristic
167Dic59—42no—
168#2935—591no—
169#1950—591no—
170#1570D5911yescharacteristic, complemented
171#1540—591no—
172#1492—591no—
173#1278—591no—
174#1277—591no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59#185811yescharacteristic
178C2×C59#186411yescharacteristic
179C2×C59#186611yescharacteristic
180D59—82no—
181C2×#131—1181no—
182#532—591no—
183#436—1181no—
184#435—1182no—
185C8×D4—1181no—
186#431C2×D5911yescharacteristic
187C2×#108—1181no—
188#423C2×D5911yescharacteristic
189C2×C4⋊C8—591no—
190C2×C4⋊C8C2×D5911yescharacteristic
191#408—1181no—
192C4×M16—591no—
193C2×C4×C8—591no—
194#341—1181no—
195#334—21no—
196#334—21no—
197#332—591no—
198#332—591no—
199#331—591no—
200#331—591no—
201D4⋊C8—1182no—
202C59#893211yescharacteristic, complemented, Sylow
203C2×#41—1181no—
204#131—2362no—
205#131—2362no—
206C2×M16—21no—
207C2×M16—591no—
208C2×M16—591no—
209C2×M16—1182no—
210C22×C8—21no—
211C22×C8—1182no—
212C22×C8—1181no—
213C22×C8—1181no—
214C4⋊Q8—591no—
215#127—591no—
216#124—1181no—
217#121—592no—
218#121—1181no—
219C4×D4—592no—
220C4×D4—1181no—
221C2×C42C22×D5911yescharacteristic
222#108—2362no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—591no—
227C4⋊C8—591no—
228C4⋊C8—1181no—
229C4⋊C8C4×D5911yescharacteristic
230C4⋊C8C4×D5911yescharacteristic
231C4⋊C8C22⋊D5911yescharacteristic
232C4⋊C8C22⋊D5911yescharacteristic
233C22⋊C8—21no—
234C22⋊C8—1182no—
235C22⋊C8—1182no—
236#97—1181no—
237C4×C8—21no—
238C4×C8—1181no—
239C4×C8—1181no—
240C4×C8—1181no—
241#41—2361no—
242#41—2361no—
243#41—2361no—
244#41—2361no—
245C2×Q8—1181no—
246C2×D4—592no—
247C2×D4—1182no—
248C2×D4—1181no—
249C2×D4—1181no—
250C22×C4—1182no—
251C22×C4#1174568511yescharacteristic
252C22×C4D4×D5911yescharacteristic
253C22×C4Q8×D5911yescharacteristic
254M16—41no—
255M16—41no—
256M16—592no—
257M16—1181no—
258M16—1181no—
259M16—1181no—
260M16—2364no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—1181no—
269C2×C8—1182no—
270C2×C8—1181no—
271C2×C8—1181no—
272C2×C8—1181no—
273C2×C8—2364no—
274C2×C8—2361no—
275C2×C8—2361no—
276C2×C8—2361no—
277C2×C8—2361no—
278C4⋊C4—592no—
279C4⋊C4—1181no—
280C22⋊C4—1182no—
281C22⋊C4—1182no—
282C42C2×C4×D5911yescharacteristic
283C42D4×D5911yescharacteristic
284C42#1174568911yescharacteristic
285C42C2×C22⋊D5911yescharacteristic
286C23—1182no—
287C23#1427370511yescharacteristic
288Q8—2361no—
289Q8—2361no—
290D4—2362no—
291D4—2362no—
292D4—2362no—
293D4—2361no—
294D4—2361no—
295D4—2361no—
296D4—2361no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—1182no—
301C2×C4—2362no—
302C2×C4—2362no—
303C2×C4C4⋊C4×D5911yescharacteristic
304C2×C4C4×C22⋊D5911yescharacteristic
305C2×C4#1427368911yescharacteristic
306C2×C4#1427369511yescharacteristic
307C2×C4#1427370311yescharacteristic
308C2×C4#1427371211yescharacteristic
309C2×C4SD16×D5911yescharacteristic
310C2×C4#1427374811yescharacteristic
311C2×C4#1427375411yescharacteristic
312C2×C4#1427375411yescharacteristic
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—81no—
318C8—1182no—
319C8—2362no—
320C8—2361no—
321C8—2361no—
322C8—2361no—
323C22—21no—
324C22—21no—
325C22—2362no—
326C22—2362no—
327C22—2362no—
328C22#1683727311yescharacteristic
329C22#1683769111yescharacteristic
330C22#1683770111yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—2362no—
335C4#1683748911yescharacteristic
336C4#1683751311yescharacteristic
337C4#1683751711yescharacteristic
338C4#1683752211yescharacteristic
339C4#1683754611yescharacteristic
340C4#1683755111yescharacteristic
341C2—21no—
342C2—21no—
343C2—4722no—
344C2#2142431411yescharacteristic
345C2#2142435411yescharacteristic
346C2#2142437011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1570C2
socleC22×C59#385
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1570D59
hypocenterC59#8932
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. #21424869C2semidirect
5. #21421932C2semidirect
6. #21421930C2semidirect
7. C59×#1570C2semidirect
170. #1570D59semidirect
202. C59#8932semidirect

Maximal subgroups

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

subgroupindex
#214328392
#214248692
#214227432
#214225462
#214219322
#214219302
C59×#15702
#893259