Finite Group Databasebeta

← #28500790

Subgroups — #28500790

Order 15104.

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

normal subgroups only

This group has 476 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#21434871C211yesmaximal, characteristic, complemented
2#21426025C212yesmaximal, complemented
3#21422107C212yesmaximal, complemented
4#21422020C211yesmaximal, characteristic, complemented
5C59×#1999C211yesmaximal, characteristic
6C2×#14273706—21no—
7C2×#14273671—22no—
8#16837738—22no—
9#16837643—21no—
10#16837643C2212yes—
11#16837560—22no—
12#16837203—22no—
13C22⋊(C59⋊C16)C2212yes—
14C22×D8×C59C2211yescharacteristic
15C2×#121×C59—21no—
16C2×#102×C59—21no—
17C2×C22⋊C8×C59C2211yescharacteristic
18C59×#340C2211yescharacteristic
19C22×Dic118—21no—
20C2×Dic236—21no—
21C2×Dic236D412yescomplemented
22#14273706—22no—
23#14273706—22no—
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59—21no—
27C2×Dic59⋊C4—41no—
28C2×C22⋊Dic59—41no—
29#14273671—44no—
30C2×C59⋊C16—22no—
31#14273661—21no—
32#14273661D412yescomplemented
33#14273652—22no—
34#14273652—22no—
35C22×D4×C59—21no—
36C22×D4×C59—41no—
37C23×C4×C59D411yescharacteristic
38C2×D8×C59—22no—
39C2×D8×C59—42no—
40C22×C8×C59D411yescharacteristic
41C22×C8×C59C2311yescharacteristic
42#121×C59—42no—
43#121×C59—42no—
44C2×C4⋊C4×C59—21no—
45C2×C22⋊C4×C59—41no—
46#102×C59—42no—
47C22⋊C8×C59—22no—
48C24×C59—41no—
49C24×C59D811yescharacteristic
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59—44no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59—82no—
58C2×D4×C59—82no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59D811yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×Dic59—41no—
67C22×Dic59—41no—
68C2×Dic118—22no—
69C2×Dic118—41no—
70C2×Dic118—81no—
71C2×Dic118—81no—
72D8×C59—24no—
73D8×C59—42no—
74C2×C8×C59—21no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C2×C8×C59C2×D412yes—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—44no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C4⋊Dic59—22no—
84C4⋊Dic59—41no—
85Dic59⋊C4—42no—
86Dic59⋊C4—42no—
87Dic236—42no—
88Dic236—81no—
89C59⋊C16—42no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—42no—
95C23×C59—81no—
96C23×C59—81no—
97C23×C59—81no—
98C23×C59—81no—
99C23×C59—81no—
100C23×C59—81no—
101C23×C59SD3212yescomplemented
102C23×C59C2×D811yescharacteristic
103D4×C59—42no—
104D4×C59—42no—
105D4×C59—81no—
106D4×C59—88no—
107D4×C59—81no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—42no—
115C2×C4×C59—81no—
116C2×C4×C59—81no—
117C2×C4×C59—81no—
118C2×C4×C59#12011yescharacteristic
119C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—82no—
123C2×Dic59—82no—
124Dic118—42no—
125Dic118—42no—
126Dic118—82no—
127C8×C59—22no—
128C8×C59—41no—
129#12103—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—42no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59—82no—
138C22×C59—82no—
139C22×C59—82no—
140C22×C59—161no—
141C22×C59—161no—
142C22×C59—161no—
143C22×C59—161no—
144C22×C59#44711yescharacteristic
145C22×C59C2×SD3211yescharacteristic
146C22×C59#51011yescharacteristic
147C4×C59—21no—
148C4×C59—41no—
149C4×C59—82no—
150C4×C59#45311yescharacteristic
151C4×C59B(4,𝔽2)11yescharacteristic
152Dic59—82no—
153Dic59—82no—
154#3189—591no—
155#2187—592no—
156#1999D5911yescharacteristic
157#1348—592no—
158#1314—591no—
159C2×C59—21no—
160C2×C59—21no—
161C2×C59—42no—
162C2×C59—162no—
163C2×C59#159511yescharacteristic
164C2×C59#218511yescharacteristic
165C2×C59#219011yescharacteristic
166C22×D8C2×D5911yescharacteristic
167C2×C22⋊Q8—1181no—
168C2×#121—21no—
169C2×SD32—1182no—
170#484—1182no—
171#467—592no—
172#467—1181no—
173#451—1182no—
174C2×#102—21no—
175C2×C22⋊C8C2×D5911yescharacteristic
176#357—1182no—
177C22⋊C16—592no—
178#340C2×D5911yescharacteristic
179C59#1210311yescharacteristic, complemented, Sylow
180C22×Q8—1181no—
181C22×D4—21no—
182C22×D4—41no—
183C23×C4C22⋊D5911yescharacteristic
184C2×Q16—592no—
185C2×Q16—1181no—
186C2×D8—22no—
187C2×D8—42no—
188C22×C8C22⋊D5911yescharacteristic
189C22×C8C22×D5911yescharacteristic
190C22⋊Q8—1182no—
191C22⋊Q8—1182no—
192C22⋊Q8—2361no—
193C22⋊Q8—2361no—
194#121—42no—
195#121—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—2361no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—2361no—
201SD32—2364no—
202C2×C16—1182no—
203#107—592no—
204#107—1181no—
205#103—1182no—
206#103—1182no—
207#102—42no—
208C22⋊C8—22no—
209C24—41no—
210C24#1174566911yescharacteristic
211C2×Q8—1182no—
212C2×Q8—4721no—
213C2×Q8—4721no—
214C2×Q8—2361no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—2361no—
230C22×C4—2361no—
231C22×C4#1174566911yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233Q16—4721no—
234Q16—2362no—
235D8—24no—
236D8—42no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8D4×D5912yes—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—1182no—
244C4⋊C4—2361no—
245C4⋊C4—2362no—
246C4⋊C4—2362no—
247C22⋊C4—44no—
248C22⋊C4—2362no—
249C22⋊C4—2362no—
250C16—2362no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23#1427367112yescomplemented
263C23C2×#1174566911yescharacteristic
264Q8—4722no—
265Q8—2362no—
266Q8—2362no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—88no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—4722no—
283C2×C4—4722no—
284C2×C4—2362no—
285C2×C4—2362no—
286C2×C4#1427369511yescharacteristic
287C2×C4#1427375211yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22#1683753511yescharacteristic
305C22C2×#1427367111yescharacteristic
306C22#1683784211yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—4722no—
311C4—4722no—
312C4#1683756911yescharacteristic
313C4#1683758811yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—162no—
318C2#2142301311yescharacteristic
319C2#2142601211yescharacteristic
320C2#2142604111yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#1999C2
socleC22×C59#447
hypercenter#1999D59
hypocenterC59#12103
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. #21434871C2semidirect
2. #21426025C2semidirect
3. #21422107C2semidirect
4. #21422020C2semidirect
21. C2×Dic236D4semidirect
32. #14273661D4semidirect
101. C23×C59SD32semidirect
179. C59#12103semidirect
262. C23#14273671semidirect

Maximal subgroups

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

subgroupindex
#214348712
#214260252
#214221072
#214220202
C59×#19992
#1210359