Finite Group Databasebeta

← #28502623

Subgroups — #28502623

Order 15104.

302 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426332C211yesmaximal, characteristic, complemented
2#21426243C211yesmaximal, characteristic, complemented
3#21423614C211yesmaximal, characteristic, complemented
4#21422139C211yesmaximal, characteristic, complemented
5#21422078C211yesmaximal, characteristic, complemented
6#21422063C211yesmaximal, characteristic, complemented
7C59×#3403C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837612—21no—
10#16837229—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837206—21no—
13#16837203—21no—
14#16837203—21no—
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17#16837147—21no—
18C59×#559—21no—
19C59×#496—21no—
20C59×#495C2211yescharacteristic
21C59×#464—21no—
22C59×#463—21no—
23C59×#459—21no—
24C59×#446C2211yescharacteristic
25C59×#440—21no—
26C4×D8×C59C2211yescharacteristic
27C2×#11745672—41no—
28#14273727—21no—
29C2×C59⋊C16—21no—
30C2×C59⋊C16—21no—
31#14273661—21no—
32#14273661—21no—
33C4⋊(C59⋊C8)—21no—
34#14273650—41no—
35C2×Q16×C59D411yescharacteristic
36C2×SD16×C59—21no—
37C2×SD16×C59—21no—
38C2×D8×C59—21no—
39C2×D8×C59D411yescharacteristic
40C2×D8×C59D411yescharacteristic
41#127×C59—21no—
42#124×C59—21no—
43#124×C59—21no—
44#124×C59—41no—
45#123×C59—41no—
46#123×C59—41no—
47C4×Q8×C59—21no—
48C4×Q8×C59D411yescharacteristic
49C4×D4×C59—21no—
50C4×D4×C59—21no—
51C4×D4×C59—41no—
52#107×C59D411yescharacteristic
53C4⋊C8×C59—21no—
54C4⋊C8×C59D411yescharacteristic
55#103×C59—21no—
56#102×C59—21no—
57#102×C59—21no—
58#97×C59—21no—
59C4×C8×C59C2311yescharacteristic
60C2×Q8×C59—21no—
61C2×Q8×C59D811yescharacteristic
62C2×D4×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59—41no—
68C2×Dic118—41no—
69Q16×C59—41no—
70SD16×C59—41no—
71SD16×C59—41no—
72SD16×C59—81no—
73D8×C59—21no—
74D8×C59—21no—
75D8×C59—41no—
76D8×C59—81no—
77C2×C8×C59—21no—
78C2×C8×C59C2×D411yescharacteristic
79C2×C8×C59C2×D411yescharacteristic
80C4⋊C4×C59—21no—
81C4⋊C4×C59—21no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59D811yescharacteristic
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C42×C59—21no—
91C42×C59—41no—
92C42×C59C2×D411yescharacteristic
93#11745672—82no—
94C4⋊Dic59—41no—
95C4⋊Dic59—41no—
96C2×C59⋊C8—41no—
97C59⋊C16—41no—
98C59⋊C16—41no—
99C23×C59—41no—
100C23×C59—41no—
101Q8×C59—41no—
102Q8×C59—41no—
103Q8×C59—41no—
104D4×C59—41no—
105D4×C59—41no—
106D4×C59—41no—
107D4×C59—41no—
108D4×C59—82no—
109D4×C59—161no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—161no—
115C2×C4×C59—161no—
116C2×C4×C59#12011yescharacteristic
117C2×C4×C59C2×D811yescharacteristic
118C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
119C2×Dic59—81no—
120Dic118—81no—
121Dic118—81no—
122C8×C59—21no—
123C8×C59—41no—
124C8×C59#13511yescharacteristic
125C8×C59#13711yescharacteristic
126C59⋊C8—81no—
127#12390—591nomaximal, Sylow
128C22×C59—81no—
129C22×C59—81no—
130C22×C59—81no—
131C22×C59—81no—
132C22×C59—81no—
133C22×C59—81no—
134C22×C59#44711yescharacteristic
135C4×C59—21no—
136C4×C59—21no—
137C4×C59—41no—
138C4×C59—41no—
139C4×C59—81no—
140C4×C59—81no—
141C4×C59#45211yescharacteristic
142C4×C59#45511yescharacteristic
143Dic59—161no—
144#3403D5911yescharacteristic
145#2243—591no—
146#2226—591no—
147#1694—591no—
148#1360—591no—
149#1338—591no—
150#1333—591no—
151C2×C59—161no—
152C2×C59—161no—
153C2×C59#164111yescharacteristic
154C2×C59#219011yescharacteristic
155C2×C59#219411yescharacteristic
156#559—21no—
157#500—591no—
158#496—21no—
159#495C2×D5911yescharacteristic
160#464—21no—
161#463—21no—
162#462—1181no—
163#459—21no—
164#446C2×D5911yescharacteristic
165#440—21no—
166C4×D8C2×D5911yescharacteristic
167#367—1181no—
168C4⋊C16—591no—
169#358—1181no—
170#357—1181no—
171#357—1181no—
172#346—591no—
173#334—591no—
174#332—1181no—
175C59#1239011yescharacteristic, complemented, Sylow
176C2×Q16—2361no—
177C2×Q16C22⋊D5911yescharacteristic
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—21no—
181C2×D8C22⋊D5911yescharacteristic
182C2×D8C22⋊D5911yescharacteristic
183C4⋊Q8—1181no—
184#127—21no—
185#124—21no—
186#124—21no—
187#124—41no—
188#123—41no—
189#123—41no—
190C4×Q8—21no—
191C4×Q8C22⋊D5911yescharacteristic
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—41no—
195C2×C16—1181no—
196C2×C16—1181no—
197#107—1181no—
198#107—1181no—
199#107C22⋊D5911yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—1181no—
202C4⋊C8C22⋊D5911yescharacteristic
203#103—21no—
204#103—2361no—
205#102—21no—
206#102—21no—
207#97—21no—
208C4×C8C22×D5911yescharacteristic
209C2×Q8—21no—
210C2×Q8—2361no—
211C2×Q8#1174566911yescharacteristic
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—41no—
215C2×D4—41no—
216C22×C4—41no—
217C22×C4—41no—
218Q16—41no—
219Q16—4722no—
220SD16—41no—
221SD16—41no—
222SD16—81no—
223D8—21no—
224D8—21no—
225D8—41no—
226D8—81no—
227C2×C8—21no—
228C2×C8—2361no—
229C2×C8C2×C22⋊D5911yescharacteristic
230C2×C8C2×C22⋊D5911yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—2361no—
235C4⋊C4—2361no—
236C4⋊C4#1174566911yescharacteristic
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C42—21no—
244C42—41no—
245C42C2×C22⋊D5911yescharacteristic
246C16—2361no—
247C16—2361no—
248C23—41no—
249C23—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—4721no—
254Q8—4721no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—82no—
260D4—161no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—161no—
266C2×C4—161no—
267C2×C4—4721no—
268C2×C4#1427369611yescharacteristic
269C2×C4C2×#1174566911yescharacteristic
270C2×C4#1427375311yescharacteristic
271C8—21no—
272C8—41no—
273C8—4721no—
274C8#1427374711yescharacteristic
275C8#1427375711yescharacteristic
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#1683753611yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—9441no—
290C4#1683756411yescharacteristic
291C4#1683758011yescharacteristic
292C2—161no—
293C2—161no—
294C2#2142328911yescharacteristic
295C2#2142604211yescharacteristic
296C2#2142606711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
Fitting, prime radicalC59×#3403C2
socleC22×C59#447
hypercenter#3403D59
hypocenterC59#12390
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3403

Complements

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

NH ≅ G/Ntype
1. #21426332C2semidirect
2. #21426243C2semidirect
3. #21423614C2semidirect
4. #21422139C2semidirect
5. #21422078C2semidirect
6. #21422063C2semidirect
175. C59#12390semidirect

Maximal subgroups

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

subgroupindex
#214263322
#214262432
#214236142
#214221392
#214220782
#214220632
C59×#34032
#1239059