Finite Group Databasebeta

← #28483166

Subgroups — #28483166

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#21432842C211yesmaximal, characteristic
2#21424871C211yesmaximal, characteristic, complemented
3#21422871C211yesmaximal, characteristic, complemented
4#21422743C211yesmaximal, characteristic
5#21422548C211yesmaximal, characteristic
6#21421931C211yesmaximal, characteristic, complemented
7C59×#1278C211yesmaximal, characteristic, complemented
8C2×#14273735—21no—
9#16837952C2211yescharacteristic
10#16837481—21no—
11#16837475—22no—
12C8×D236—21no—
13C2×#14273662—21no—
14#16837426C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17#16837370—21no—
18#16837166—21no—
19#16837155—21no—
20#16837155—21no—
21#16837147C411yescharacteristic
22#16837147C411yescharacteristic
23#16837145C411yescharacteristic
24#16837145C411yescharacteristic
25#16837131—22no—
26C59×#431C2211yescharacteristic
27C4×M16×C59C2211yescharacteristic
28C2×C4×C8×C59C2211yescharacteristic
29C2×#11745685—21no—
30#14273735—42no—
31#14273735—42no—
32C2×#11745664—21no—
33C2×#11745659—22no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C2×C8×D59—22no—
37#14273727C2×C411yescharacteristic
38#14273724C2×C411yescharacteristic
39#14273715—21no—
40#14273701—21no—
41#14273701Q812yes—
42C4×D236—21no—
43C4×D236D412yes—
44#14273662—42no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50C4⋊(C59⋊C8)C2×C411yescharacteristic
51C4⋊(C59⋊C8)D411yescharacteristic
52C4⋊(C59⋊C8)D411yescharacteristic
53(C2×D59)⋊C8—22no—
54(C2×D59)⋊C8—22no—
55C2×M16×C59D411yescharacteristic
56C2×M16×C59D411yescharacteristic
57C22×C8×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59C2×C411yescharacteristic
60C4⋊C8×C59C2×C411yescharacteristic
61C22⋊C8×C59—21no—
62#97×C59—21no—
63C4×C8×C59—21no—
64C4×C8×C59—21no—
65C4×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—21no—
81M16×C59—21no—
82M16×C59—21no—
83M16×C59D812yes—
84C2×C8×C59—21no—
85C2×C8×C59—22no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C42×C59C22×C411yescharacteristic
92C42×C59C2×D411yescharacteristic
93C42×C59C2×D411yescharacteristic
94C42×C59#4111yescharacteristic
95(C2×D59)⋊C4—22no—
96(C2×D59)⋊C4—22no—
97C4⋊Dic59—21no—
98C4⋊Dic59C4⋊C412yes—
99#11745664—41no—
100#11745664—41no—
101C2×C59⋊C8—21no—
102C2×C59⋊C8—21no—
103C2×C59⋊C8—21no—
104C2×C59⋊C8—21no—
105C2×C59⋊C8—41no—
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×D811yescharacteristic
124C2×C4×C59#13511yescharacteristic
125C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C59#13711yescharacteristic
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—22no—
140C8×C59—42no—
141C8×C59—41no—
142C8×C59—41no—
143C8×C59—41no—
144C59⋊C8—41no—
145C59⋊C8—41no—
146C59⋊C8—41no—
147C59⋊C8—41no—
148C59⋊C8—81no—
149#8930—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59#38511yescharacteristic
153C22×C59#47411yescharacteristic
154C22×C59#47811yescharacteristic
155C2×D59—42no—
156C2×D59—42no—
157C2×D59—42no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59C4×D811yescharacteristic
162C4×C59#44011yescharacteristic
163C4×C59#44311yescharacteristic
164C4×C59#44411yescharacteristic
165C4×C59#44711yescharacteristic
166C4×C59#45211yescharacteristic
167Dic59—42no—
168#2935—591no—
169#1950—591no—
170#1571—591no—
171#1540—591no—
172#1492—591no—
173#1278D5911yescharacteristic, complemented
174#1277—591no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59#185811yescharacteristic
178C2×C59#186211yescharacteristic
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#423—591no—
189C2×C4⋊C8—591no—
190C2×C4⋊C8—591no—
191#408—1181no—
192C4×M16C2×D5911yescharacteristic
193C2×C4×C8C2×D5911yescharacteristic
194#341—1181no—
195#335—1181no—
196#335—1181no—
197#332—591no—
198#332—591no—
199#331—591no—
200#331—591no—
201D4⋊C8—1182no—
202C59#893011yescharacteristic, complemented, Sylow
203C2×#41—1181no—
204#131—2362no—
205#131—2362no—
206C2×M16—1182no—
207C2×M16—1181no—
208C2×M16D23611yescharacteristic
209C2×M16D23611yescharacteristic
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—591no—
224C4⋊C8—591no—
225C4⋊C8—591no—
226C4⋊C8—591no—
227C4⋊C8—1181no—
228C4⋊C8—1181no—
229C4⋊C8—1181no—
230C4⋊C8—1181no—
231C4⋊C8C4×D5911yescharacteristic
232C4⋊C8C4×D5911yescharacteristic
233C22⋊C8—21no—
234C22⋊C8—1182no—
235C22⋊C8—1182no—
236#97—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
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×C4C2×D23611yescharacteristic
252C22×C4#1174568711yescharacteristic
253C22×C4Q8×D5911yescharacteristic
254M16—21no—
255M16—21no—
256M16—21no—
257M16—2364no—
258M16—2361no—
259M16—2361no—
260M16D47212yes—
261C2×C8—21no—
262C2×C8—22no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—1181no—
269C2×C8—1181no—
270C2×C8—1181no—
271C2×C8—1181no—
272C2×C8—2364no—
273C2×C8—2361no—
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
283C42#1174568511yescharacteristic
284C42D4×D5911yescharacteristic
285C42D4×D5911yescharacteristic
286C23—1182no—
287C23#1427370711yescharacteristic
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×C4(C22⋊D59)⋊C411yescharacteristic
305C2×C4C4×D23611yescharacteristic
306C2×C4#1427369711yescharacteristic
307C2×C4#1427370511yescharacteristic
308C2×C4#1427370911yescharacteristic
309C2×C4C2×D47211yescharacteristic
310C2×C4#1427374911yescharacteristic
311C2×C4#1427375511yescharacteristic
312C2×C4#1427376011yescharacteristic
313C8—22no—
314C8—42no—
315C8—41no—
316C8—41no—
317C8—41no—
318C8—4721no—
319C8—2361no—
320C8—2361no—
321C8—2361no—
322C8—2361no—
323C22—21no—
324C22—21no—
325C22—2362no—
326C22—2362no—
327C22—2362no—
328C22#1683727611yescharacteristic
329C22#1683768311yescharacteristic
330C22#1683770311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—2362no—
335C4C4×D47211yescharacteristic
336C4#1683750411yescharacteristic
337C4#1683751611yescharacteristic
338C4#1683752111yescharacteristic
339C4#1683753811yescharacteristic
340C4#1683756611yescharacteristic
341C2—21no—
342C2—21no—
343C2—4722no—
344C2#2142431711yescharacteristic
345C2#2142434611yescharacteristic
346C2#2142437211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1278C2
socleC22×C59#385
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#1278D59
hypocenterC59#8930
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. #21424871C2semidirect
3. #21422871C2semidirect
6. #21421931C2semidirect
7. C59×#1278C2semidirect
173. #1278D59semidirect
202. C59#8930semidirect

Maximal subgroups

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

subgroupindex
#214328422
#214248712
#214228712
#214227432
#214225482
#214219312
C59×#12782
#893059