Finite Group Databasebeta

← #28502630

Subgroups — #28502630

Order 15104.

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

normal subgroups only

This group has 290 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
2#21426248C211yesmaximal, characteristic, complemented
3#21423614C211yesmaximal, characteristic, complemented
4#21422139C211yesmaximal, characteristic
5#21422078C211yesmaximal, characteristic, complemented
6#21422066C211yesmaximal, characteristic, complemented
7C59×#3402C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837612—21no—
10#16837227—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837206—21no—
13#16837206—21no—
14#16837203—21no—
15#16837178C2211yescharacteristic
16#16837153C2211yescharacteristic
17#16837147—21no—
18C59×#556—21no—
19C59×#497—21no—
20C59×#495C2211yescharacteristic
21C59×#464—21no—
22C59×#463—21no—
23C59×#462—21no—
24C59×#446C2211yescharacteristic
25C59×#440—21no—
26C4×Q16×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×C59—21no—
36C2×Q16×C59D411yescharacteristic
37C2×Q16×C59D411yescharacteristic
38C2×SD16×C59—21no—
39C2×SD16×C59—21no—
40C2×D8×C59D411yescharacteristic
41C4⋊Q8×C59—21no—
42#126×C59—41no—
43#126×C59—41no—
44#124×C59—21no—
45#124×C59—21no—
46C22⋊Q8×C59—41no—
47C4×Q8×C59—21no—
48C4×Q8×C59—21no—
49C4×Q8×C59D411yescharacteristic
50C4×D4×C59—21no—
51#117×C59—41no—
52#107×C59D411yescharacteristic
53C4⋊C8×C59—21no—
54C4⋊C8×C59D411yescharacteristic
55#103×C59—21no—
56#103×C59—21no—
57#102×C59—21no—
58#97×C59—21no—
59C4×C8×C59C2311yescharacteristic
60C2×Q8×C59—21no—
61C2×Q8×C59—21no—
62C2×Q8×C59D811yescharacteristic
63C2×D4×C59—21no—
64C22×C4×C59—41no—
65C2×Dic118—41no—
66Q16×C59—21no—
67Q16×C59—21no—
68Q16×C59—41no—
69Q16×C59—81no—
70SD16×C59—41no—
71SD16×C59—41no—
72SD16×C59—81no—
73D8×C59—41no—
74C2×C8×C59—21no—
75C2×C8×C59C2×D411yescharacteristic
76C2×C8×C59C2×D411yescharacteristic
77C4⋊C4×C59—21no—
78C4⋊C4×C59—21no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
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—
89C42×C59—21no—
90C42×C59—41no—
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—
100Q8×C59—41no—
101Q8×C59—41no—
102Q8×C59—41no—
103Q8×C59—41no—
104Q8×C59—41no—
105D4×C59—41no—
106D4×C59—41no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—82no—
113C2×C4×C59#12011yescharacteristic
114C2×C4×C59C2×D811yescharacteristic
115C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
116C2×Dic59—81no—
117Dic118—81no—
118Dic118—81no—
119C8×C59—21no—
120C8×C59—41no—
121C8×C59#13511yescharacteristic
122C8×C59#13711yescharacteristic
123C59⋊C8—81no—
124#12391—591nomaximal, Sylow
125C22×C59—81no—
126C22×C59—81no—
127C22×C59—81no—
128C22×C59#44711yescharacteristic
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—41no—
132C4×C59—41no—
133C4×C59—81no—
134C4×C59—81no—
135C4×C59—81no—
136C4×C59#45211yescharacteristic
137C4×C59#45511yescharacteristic
138Dic59—161no—
139#3402D5911yescharacteristic
140#2243—591no—
141#2227—591no—
142#1694—591no—
143#1360—591no—
144#1338—591no—
145#1334—591no—
146C2×C59—161no—
147C2×C59#164111yescharacteristic
148C2×C59#219011yescharacteristic
149C2×C59#219511yescharacteristic
150#556—21no—
151#500—591no—
152#497—21no—
153#495C2×D5911yescharacteristic
154#464—21no—
155#463—21no—
156#462—21no—
157#462—1181no—
158#446C2×D5911yescharacteristic
159#440—21no—
160C4×Q16C2×D5911yescharacteristic
161#366—1181no—
162C4⋊C16—591no—
163#358—1181no—
164#358—1181no—
165#357—1181no—
166#346—591no—
167#334—591no—
168#332—1181no—
169C59#1239111yescharacteristic, complemented, Sylow
170C2×Q16—21no—
171C2×Q16—2361no—
172C2×Q16C22⋊D5911yescharacteristic
173C2×Q16C22⋊D5911yescharacteristic
174C2×SD16—21no—
175C2×SD16—21no—
176C2×D8C22⋊D5911yescharacteristic
177C4⋊Q8—21no—
178C4⋊Q8—1181no—
179#126—41no—
180#126—41no—
181#124—21no—
182#124—21no—
183C22⋊Q8—41no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8C22⋊D5911yescharacteristic
187C4×D4—21no—
188#117—41no—
189C2×C16—1181no—
190C2×C16—1181no—
191#107—1181no—
192#107—1181no—
193#107C22⋊D5911yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—1181no—
196C4⋊C8C22⋊D5911yescharacteristic
197#103—21no—
198#103—21no—
199#103—2361no—
200#102—21no—
201#97—21no—
202C4×C8C22×D5911yescharacteristic
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—2361no—
206C2×Q8#1174566911yescharacteristic
207C2×D4—21no—
208C22×C4—41no—
209Q16—21no—
210Q16—21no—
211Q16—41no—
212Q16—81no—
213Q16—4722no—
214SD16—41no—
215SD16—41no—
216SD16—81no—
217D8—41no—
218C2×C8—21no—
219C2×C8—2361no—
220C2×C8C2×C22⋊D5911yescharacteristic
221C2×C8C2×C22⋊D5911yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—2361no—
229C4⋊C4—2361no—
230C4⋊C4#1174566911yescharacteristic
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—21no—
237C42—41no—
238C42—41no—
239C42C2×C22⋊D5911yescharacteristic
240C16—2361no—
241C16—2361no—
242C23—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—4721no—
249Q8—4721no—
250D4—41no—
251D4—41no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—4721no—
259C2×C4#1427369611yescharacteristic
260C2×C4C2×#1174566911yescharacteristic
261C2×C4#1427375311yescharacteristic
262C8—21no—
263C8—41no—
264C8—4721no—
265C8#1427374711yescharacteristic
266C8#1427375711yescharacteristic
267C22—81no—
268C22—81no—
269C22—81no—
270C22#1683753611yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—9441no—
279C4#1683756411yescharacteristic
280C4#1683758011yescharacteristic
281C2—161no—
282C2#2142328911yescharacteristic
283C2#2142604211yescharacteristic
284C2#2142607411yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3402C2
socleC22×C59#447
prime radicalC2×D8×C59D4
hypercenter#3402D59
hypocenterC59#12391
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #446 ◁ #3402

Complements

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

NH ≅ G/Ntype
2. #21426248C2semidirect
3. #21423614C2semidirect
5. #21422078C2semidirect
6. #21422066C2semidirect
169. C59#12391semidirect

Maximal subgroups

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

subgroupindex
#214263322
#214262482
#214236142
#214221392
#214220782
#214220662
C59×#34022
#1239159