Finite Group Databasebeta

← #28487016

Subgroups — #28487016

Order 15104.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425023C211yesmaximal, characteristic
2#21424902C211yesmaximal, characteristic
3#21424854C211yesmaximal, characteristic, complemented
4C2×#16837145C211yesmaximal, characteristic
5#21421980C211yesmaximal, characteristic, complemented
6#21421931C211yesmaximal, characteristic, complemented
7C8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
8C2×#14273724C2211yescharacteristic
9C2×#14273660—21no—
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×#14273649—21no—
13C2×#14273649—21no—
14C2×#14273647—21no—
15C2×#14273647—21no—
16#16837272C2211yescharacteristic
17#16837164—21no—
18#16837145C414yescomplemented
19C2×C4⋊C8×C59C2211yescharacteristic
20C2×C4×C8×C59—21no—
21C2×C4×C8×C59C2211yescharacteristic
22C4×C4⋊C4×C59C2211yescharacteristic
23C59×#336—21no—
24C22×D236—21no—
25C22×D236—21no—
26C22×D236—21no—
27C22×C59⋊C8—21no—
28C22×C59⋊C8—21no—
29#14273724—21no—
30#14273724—21no—
31#14273724—21no—
32#14273724C2×C412yes—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59—21no—
36#14273660—42no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)C2×C412yes—
41#14273649—42no—
42#14273649—42no—
43#14273647—42no—
44#14273647—42no—
45C22×C8×C59—21no—
46C22×C8×C59D411yescharacteristic
47C22×C8×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C42×C59—21no—
51C2×C42×C59C2311yescharacteristic
52C4⋊C8×C59—21no—
53C4⋊C8×C59C2×C412yes—
54C4×C8×C59—21no—
55C4×C8×C59—21no—
56C4×C8×C59—42no—
57#95×C59—21no—
58C23×D59—41no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59#4111yescharacteristic
64C22×Dic59—41no—
65C2×D236—22no—
66C2×D236—22no—
67C2×D236—22no—
68C2×D236—41no—
69C2×D236—41no—
70C2×D236—41no—
71C2×D236—41no—
72C2×D236—41no—
73C2×D236—41no—
74C2×D236—41no—
75C2×D236—41no—
76C2×D236—41no—
77C2×D236—41no—
78C2×D236—41no—
79C2×D236—81no—
80C2×D236—81no—
81C2×C8×C59—21no—
82C2×C8×C59—21no—
83C2×C8×C59—22no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59D812yes—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59D812yes—
93C4⋊C4×C59SD1612yes—
94C42×C59—41no—
95C42×C59—41no—
96C42×C59C22×C411yescharacteristic
97C42×C59#4111yescharacteristic
98C42×C59#4111yescharacteristic
99C42×C59#4111yescharacteristic
100C4⋊Dic59—41no—
101C4⋊Dic59—41no—
102C4⋊Dic59—41no—
103C4⋊Dic59—41no—
104C4⋊Dic59—81no—
105C2×C59⋊C8—22no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C23×C59#12111yescharacteristic
112C22×D59—42no—
113C22×D59—81no—
114C22×D59—81no—
115C22×D59—81no—
116C22×D59—81no—
117C22×D59—81no—
118C22×D59—81no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59#11711yescharacteristic
128C2×C4×C59C4×D411yescharacteristic
129C2×C4×C59C4×D411yescharacteristic
130C2×C4×C59#12311yescharacteristic
131C2×C4×C59#12411yescharacteristic
132C2×C4×C59#12611yescharacteristic
133C2×C4×C59C2×D811yescharacteristic
134C2×C4×C59C2×D811yescharacteristic
135C2×C4×C59C2×SD1611yescharacteristic
136C2×C4×C59#13511yescharacteristic
137C2×C4×C59#13511yescharacteristic
138C2×C4×C59#13511yescharacteristic
139C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C59#13711yescharacteristic
141C2×Dic59—81no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144D236—42no—
145D236—42no—
146D236—42no—
147D236—42no—
148D236—82no—
149D236—82no—
150D236—81no—
151D236—81no—
152D236—81no—
153D236—81no—
154C8×C59—22no—
155C8×C59—42no—
156C8×C59—41no—
157C59⋊C8—42no—
158C59⋊C8—81no—
159#9563—591nomaximal, Sylow
160C22×C59#38811yescharacteristic
161C22×C59#45911yescharacteristic
162C22×C59#46411yescharacteristic
163C22×C59#46511yescharacteristic
164C22×C59#46611yescharacteristic
165C22×C59#47111yescharacteristic
166C22×C59#47211yescharacteristic
167C2×D59—82no—
168C2×D59—82no—
169C2×D59—82no—
170C2×D59—161no—
171C2×D59—161no—
172C2×D59—161no—
173C2×D59—161no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—42no—
177C4×C59—42no—
178C4×C59C4×D811yescharacteristic
179C4×C59C4×D811yescharacteristic
180C4×C59C4×SD1611yescharacteristic
181C4×C59#44011yescharacteristic
182C4×C59#48011yescharacteristic
183C4×C59#48211yescharacteristic
184C4×C59#48611yescharacteristic
185C4×C59#48811yescharacteristic
186Dic59—161no—
187#1982—591no—
188#1957—591no—
189#1947—591no—
190C8×C4⋊C4D5911yescharacteristic, complemented
191C2×#331—591no—
192#1296—591no—
193#1277—591no—
194C2×C59#166511yescharacteristic
195C2×C59#166711yescharacteristic
196C2×C59#167811yescharacteristic
197C2×C59#168211yescharacteristic
198C2×C59#192511yescharacteristic
199C2×C59#193111yescharacteristic
200C2×C59#193411yescharacteristic
201D59—162no—
202C2×#127—591no—
203C2×#107—1181no—
204C2×C4⋊C8—591no—
205C2×C4⋊C8—591no—
206C2×C4⋊C8C2×D5911yescharacteristic
207C2×#102—1181no—
208C2×#102—1181no—
209C2×#102—1181no—
210C2×#102—1181no—
211C2×C4×C8—21no—
212C2×C4×C8C2×D5911yescharacteristic
213#384—591no—
214C4×C4⋊C4C2×D5911yescharacteristic
215#340—1181no—
216#336—21no—
217#331—594no—
218C59#956311yescharacteristic, complemented, Sylow
219C22×D4—1181no—
220C22×D4—1181no—
221C22×D4—1181no—
222C22×C8—21no—
223C22×C8—1181no—
224C22×C8—1181no—
225C22×C8C22⋊D5911yescharacteristic
226C22×C8C22⋊D5911yescharacteristic
227#127—592no—
228#127—1181no—
229#127—1181no—
230#127—1181no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4—1181no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4D23611yescharacteristic
235C2×C4⋊C4D23611yescharacteristic
236C2×C42—21no—
237C2×C42C22×D5911yescharacteristic
238#107—2362no—
239C4⋊C8—21no—
240C4⋊C8—592no—
241C4⋊C8—1181no—
242C4⋊C8—1181no—
243C4⋊C8—1181no—
244C4⋊C8C4×D5912yes—
245#102—2362no—
246#102—2362no—
247#102—2362no—
248#102—2362no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—42no—
252#95—21no—
253C24—2361no—
254C2×D4—1182no—
255C2×D4—1182no—
256C2×D4—1182no—
257C2×D4—4721no—
258C2×D4—4721no—
259C2×D4—2361no—
260C2×D4—2361no—
261C2×D4—2361no—
262C2×D4—2361no—
263C2×D4—2361no—
264C2×D4—2361no—
265C2×D4—2361no—
266C2×D4—2361no—
267C2×D4—2361no—
268C2×D4—2361no—
269C2×D4—2361no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—2361no—
273C22×C4C2×D23611yescharacteristic
274C22×C4#1174568511yescharacteristic
275C22×C4C2×C22⋊D5911yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—1182no—
285C2×C8—2361no—
286C2×C8—2361no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8#1174566912yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—4721no—
294C4⋊C4—2361no—
295C4⋊C4—2361no—
296C4⋊C4—2361no—
297C4⋊C4—2361no—
298C4⋊C4#1174566012yes—
299C4⋊C4D47212yes—
300C42—41no—
301C42—41no—
302C42C2×C4×D5911yescharacteristic
303C42#1174568511yescharacteristic
304C42#1174568511yescharacteristic
305C42#1174568511yescharacteristic
306C23—4721no—
307C23—4721no—
308C23—4721no—
309C23—4721no—
310C23—4721no—
311C23—4721no—
312C23—2362no—
313C23#1427370111yescharacteristic
314D4—4722no—
315D4—4722no—
316D4—4721no—
317D4—4721no—
318D4—4721no—
319D4—4721no—
320D4—2362no—
321D4—2362no—
322D4—2362no—
323D4—2362no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—4721no—
333C2×C4—4721no—
334C2×C4—4721no—
335C2×C4#1427368411yescharacteristic
336C2×C4C4×C22⋊D5911yescharacteristic
337C2×C4C4×D23611yescharacteristic
338C2×C4#1427371011yescharacteristic
339C2×C4#1427371511yescharacteristic
340C2×C4#1427372111yescharacteristic
341C2×C4C2×#1174566911yescharacteristic
342C2×C4C2×D47211yescharacteristic
343C2×C4C2×#1174566011yescharacteristic
344C2×C4#1427374711yescharacteristic
345C2×C4#1427374911yescharacteristic
346C2×C4#1427374911yescharacteristic
347C2×C4#1427375211yescharacteristic
348C2×C4#1427375811yescharacteristic
349C8—22no—
350C8—41no—
351C8—42no—
352C8—4721no—
353C8—2362no—
354C22—9441no—
355C22—9441no—
356C22—9441no—
357C22—9441no—
358C22—4722no—
359C22—4722no—
360C22—4722no—
361C22#1683729311yescharacteristic
362C22#1683759711yescharacteristic
363C22#1683762411yescharacteristic
364C22#1683763111yescharacteristic
365C22#1683763811yescharacteristic
366C22#1683766511yescharacteristic
367C22#1683767211yescharacteristic
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—9441no—
373C4C4×#1174566911yescharacteristic
374C4C4×D47211yescharacteristic
375C4C4×#1174566011yescharacteristic
376C4#1683750011yescharacteristic
377C4#1683771511yescharacteristic
378C4#1683772611yescharacteristic
379C4#1683774911yescharacteristic
380C4#1683775711yescharacteristic
381C2—9442no—
382C2#2142343711yescharacteristic
383C2#2142345111yescharacteristic
384C2#2142351211yescharacteristic
385C2#2142354011yescharacteristic
386C2#2142471611yescharacteristic
387C2#2142475811yescharacteristic
388C2#2142477911yescharacteristic
subgroupG/H
centerC23#14273701
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC8×C4⋊C4×C59C2
socleC23×C59#121
prime radicalC2×#14273724C22
cosocleC2×C42×C59C23
hypercenterC8×C4⋊C4D59
hypocenterC59#9563
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21424854C2semidirect
5. #21421980C2semidirect
6. #21421931C2semidirect
7. C8×C4⋊C4×C59C2semidirect
18. #16837145C4semidirect
190. C8×C4⋊C4D59semidirect
218. C59#9563semidirect

Maximal subgroups

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

subgroupindex
#214250232
#214249022
#214248542
C2×#168371452
#214219802
#214219312
C8×C4⋊C4×C592
#956359