Finite Group Databasebeta

← #28508371

Subgroups — #28508371

Order 15104.

484 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 79 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#21432042C211yesmaximal, characteristic
2#21424377C212yesmaximal
3#21422221C211yesmaximal, characteristic
4#21422215C212yesmaximal
5C59×#1856C211yesmaximal, characteristic
6C2×#14273642C2212yes—
7#16837341—21no—
8#16837314—22no—
9#16837308—21no—
10#16837282C2212yes—
11#16837277—22no—
12#16837274—22no—
13#16837169—22no—
14C59×#518C2211yescharacteristic
15C59×#428—21no—
16C2×#100×C59C2211yescharacteristic
17C59×#390C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118—21no—
20C23×Dic59D412yes—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×Dic59⋊C4—42no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—22no—
26C2×C22⋊Dic59D412yes—
27#14273642—22no—
28#14273642Q814yescomplemented
29C59⋊#95—22no—
30C59⋊#95—22no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C59⋊#95—42no—
34C22×D4×C59C2311yescharacteristic
35C2×M16×C59—21no—
36C4×D4×C59—41no—
37C4×D4×C59—41no—
38#117×C59—21no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C42×C59D411yescharacteristic
43#107×C59—41no—
44#106×C59—41no—
45#100×C59—21no—
46#100×C59Q812yescomplemented
47C24×C59C2×D412yes—
48C2×D4×C59—41no—
49C2×D4×C59—41no—
50C2×D4×C59C2×Q811yescharacteristic
51C2×D4×C59C2×Q812yes—
52C2×D4×C59#4111yescharacteristic
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59C2×D411yescharacteristic
58C22×Dic59—22no—
59C22×Dic59—22no—
60C22×Dic59—42no—
61C22×Dic59—42no—
62C22×Dic59—82no—
63C2×Dic118—42no—
64C2×Dic118—81no—
65C2×Dic118—81no—
66M16×C59—22no—
67M16×C59—41no—
68C2×C8×C59—41no—
69C4⋊C4×C59—21no—
70C4⋊C4×C59—21no—
71C4⋊C4×C59—41no—
72C22⋊C4×C59—44no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—41no—
75C42×C59—41no—
76C42×C59—41no—
77C22⋊Dic59—24no—
78C22⋊Dic59—42no—
79C22⋊Dic59—42no—
80C22⋊Dic59—82no—
81C4⋊Dic59—81no—
82C4⋊Dic59—81no—
83Dic59⋊C4—84no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59#12011yescharacteristic
89C23×C59C22⋊Q812yes—
90C23×C59C4⋊Q812yes—
91D4×C59—41no—
92D4×C59—42no—
93D4×C59—41no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—42no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—82no—
104C2×C4×C59—82no—
105C2×C4×C59C22⋊Q811yescharacteristic
106C2×C4×C59C4⋊Q811yescharacteristic
107C2×Dic59—42no—
108C2×Dic59—44no—
109C2×Dic59—42no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—44no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115C2×Dic59—82no—
116C2×Dic59—82no—
117C2×Dic59—82no—
118Dic118—84no—
119C8×C59—42no—
120#13463—591nomaximal, Sylow
121C22×C59—21no—
122C22×C59—21no—
123C22×C59—22no—
124C22×C59—22no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—82no—
130C22×C59#39511yescharacteristic
131C22×C59B(4,𝔽2)11yescharacteristic
132C22×C59#45811yescharacteristic
133C4×C59—21no—
134C4×C59—21no—
135C4×C59—42no—
136C4×C59—81no—
137C4×C59—81no—
138C4×C59—81no—
139Dic59—82no—
140Dic59—84no—
141Dic59—82no—
142#2804—591no—
143#1867—592no—
144#1856D5911yescharacteristic
145#1393—591no—
146#1390—592no—
147C2×C59—21no—
148C2×C59—21no—
149C2×C59—42no—
150C2×C59—42no—
151C2×C59#206211yescharacteristic
152C2×C59#219811yescharacteristic
153C2×C59#219911yescharacteristic
154#518C2×D5911yescharacteristic
155#428—21no—
156C2×#100C2×D5911yescharacteristic
157C2×#99—592no—
158#400—1181no—
159#393—1182no—
160#391—1181no—
161#390C2×D5911yescharacteristic
162#386—592no—
163#385—1182no—
164#385—1182no—
165#342—1182no—
166#339—21no—
167C59#1346311yescharacteristic, complemented, Sylow
168C22×Q8—1181no—
169C22×D4C22×D5911yescharacteristic
170C23×C4—592no—
171C2×M16—21no—
172C4×D4—41no—
173C4×D4—41no—
174#117—21no—
175C2×C4⋊C4—1181no—
176C2×C4⋊C4—1181no—
177C2×C4⋊C4—2362no—
178C2×C4⋊C4C22⋊D5911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—592no—
182C2×C22⋊C4—1182no—
183C2×C22⋊C4—1182no—
184C2×C42C22⋊D5911yescharacteristic
185#107—41no—
186#106—41no—
187#100—21no—
188#100Dic11812yescomplemented
189#99—594no—
190#99—1182no—
191#95—1182no—
192#95—1182no—
193#95—1181no—
194#95—1182no—
195#95—2362no—
196C24D4×D5912yes—
197C2×Q8—4721no—
198C2×Q8—4721no—
199C2×Q8—2362no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4C2×Dic11811yescharacteristic
203C2×D4#1174568511yescharacteristic
204C2×D4Q8×D5912yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—1182no—
210C22×C4—1182no—
211C22×C4—4722no—
212C22×C4—2362no—
213C22×C4—2362no—
214C22×C4C2×C22⋊D5911yescharacteristic
215M16—22no—
216M16—41no—
217C2×C8—41no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—4721no—
222C4⋊C4—4721no—
223C4⋊C4—4724no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—1184no—
228C22⋊C4—4722no—
229C22⋊C4—2362no—
230C22⋊C4—2362no—
231C42—41no—
232C42—41no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1427369511yescharacteristic
238C23#1427370512yes—
239C23#1427372612yes—
240Q8—4724no—
241D4—41no—
242D4—41no—
243D4—42no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—4722no—
256C2×C4—4722no—
257C2×C4—4722no—
258C2×C4—2364no—
259C2×C4—2362no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4—2364no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4#1427370611yescharacteristic
267C2×C4#1427372511yescharacteristic
268C8—42no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1683732211yescharacteristic
279C22#1683758811yescharacteristic
280C22#1683759111yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—4722no—
288C4—4724no—
289C4—4722no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2142544911yescharacteristic
295C2#2142609111yescharacteristic
296C2#2142609611yescharacteristic
subgroupG/H
centerC22#16837588
commutator, prime radical, cosocleC22×D4×C59C23
FrattiniC22×D4C22×D59
FittingC59×#1856C2
socleC22×C59B(4,𝔽2)
hypercenter#1856D59
hypocenterC59#13463
LCSG ▷ C22×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1856

Complements

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

NH ≅ G/Ntype
28. #14273642Q8semidirect
46. #100×C59Q8semidirect
167. C59#13463semidirect
188. #100Dic118semidirect

Maximal subgroups

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

subgroupindex
#214320422
#214243772
#214222212
#214222152
C59×#18562
#1346359