Finite Group Databasebeta

← #28508359

Subgroups — #28508359

Order 15104.

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

normal subgroups only

This group has 540 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#21430468C211yesmaximal, characteristic
2#21424377C212yesmaximal
3#21422221C211yesmaximal, characteristic
4#21422215C212yesmaximal
5C59×#1856C211yesmaximal, characteristic
6C2×#14273704—22no—
7C2×#14273642C2212yes—
8#16837325—22no—
9#16837322—21no—
10#16837300—21no—
11#16837282C2212yes—
12#16837274—22no—
13#16837169—22no—
14C59×#518C2211yescharacteristic
15C59×#428—21no—
16C2×#100×C59C2211yescharacteristic
17C59×#390C2211yescharacteristic
18C59×#339—21no—
19C22×Dic118—21no—
20C23×Dic59D412yes—
21#14273704—48no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×Dic59⋊C4—22no—
25C2×Dic59⋊C4—22no—
26C2×Dic59⋊C4—42no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59D412yes—
30#14273642—22no—
31#14273642Q814yescomplemented
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—42no—
35C22×D4×C59C2311yescharacteristic
36C2×M16×C59—21no—
37C4×D4×C59—41no—
38C4×D4×C59—41no—
39#117×C59—21no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C42×C59D411yescharacteristic
44#107×C59—41no—
45#106×C59—41no—
46#100×C59—21no—
47#100×C59Q812yescomplemented
48C24×C59C2×D412yes—
49C2×D4×C59—41no—
50C2×D4×C59—41no—
51C2×D4×C59C2×Q811yescharacteristic
52C2×D4×C59C2×Q812yes—
53C2×D4×C59#4111yescharacteristic
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—22no—
60C22×Dic59—22no—
61C22×Dic59—44no—
62C22×Dic59—42no—
63C22×Dic59—42no—
64C2×Dic118—44no—
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—44no—
79C22⋊Dic59—42no—
80C22⋊Dic59—42no—
81C4⋊Dic59—24no—
82C4⋊Dic59—81no—
83Dic59⋊C4—44no—
84Dic59⋊C4—44no—
85Dic59⋊C4—84no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59#12011yescharacteristic
91C23×C59C22⋊Q812yes—
92C23×C59C4⋊Q812yes—
93D4×C59—41no—
94D4×C59—42no—
95D4×C59—41no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—82no—
106C2×C4×C59—82no—
107C2×C4×C59C22⋊Q811yescharacteristic
108C2×C4×C59C4⋊Q811yescharacteristic
109C2×Dic59—24no—
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—44no—
113C2×Dic59—42no—
114C2×Dic59—44no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—82no—
118C2×Dic59—82no—
119C2×Dic59—82no—
120Dic118—84no—
121C8×C59—42no—
122#13461—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—22no—
126C22×C59—22no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—42no—
131C22×C59—82no—
132C22×C59#39511yescharacteristic
133C22×C59B(4,𝔽2)11yescharacteristic
134C22×C59#45811yescharacteristic
135C4×C59—21no—
136C4×C59—21no—
137C4×C59—42no—
138C4×C59—81no—
139C4×C59—81no—
140C4×C59—81no—
141Dic59—44no—
142Dic59—82no—
143Dic59—84no—
144#2683—591no—
145#1867—592no—
146#1856D5911yescharacteristic
147#1393—591no—
148#1390—592no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—42no—
152C2×C59—42no—
153C2×C59#206211yescharacteristic
154C2×C59#219611yescharacteristic
155C2×C59#220111yescharacteristic
156C2×C22⋊Q8—1182no—
157#518C2×D5911yescharacteristic
158#428—21no—
159C2×#100C2×D5911yescharacteristic
160C2×#99—592no—
161#396—1182no—
162#395—1181no—
163#390C2×D5911yescharacteristic
164#389—1181no—
165#386—592no—
166#385—1182no—
167#342—1182no—
168#339—21no—
169C59#1346111yescharacteristic, complemented, Sylow
170C22×Q8—1181no—
171C22×D4C22×D5911yescharacteristic
172C23×C4—592no—
173C2×M16—21no—
174C22⋊Q8—2368no—
175C4×D4—41no—
176C4×D4—41no—
177#117—21no—
178C2×C4⋊C4—1181no—
179C2×C4⋊C4—1181no—
180C2×C4⋊C4—1182no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4—2362no—
183C2×C4⋊C4C22⋊D5911yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—592no—
187C2×C22⋊C4—1182no—
188C2×C22⋊C4—1182no—
189C2×C42C22⋊D5911yescharacteristic
190#107—41no—
191#106—41no—
192#100—21no—
193#100Dic11812yescomplemented
194#99—594no—
195#99—1182no—
196#95—1181no—
197#95—1182no—
198#95—2362no—
199C24D4×D5912yes—
200C2×Q8—4721no—
201C2×Q8—2364no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic11811yescharacteristic
205C2×D4#1174568511yescharacteristic
206C2×D4Q8×D5912yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4—2364no—
214C22×C4—2362no—
215C22×C4—2362no—
216C22×C4C2×C22⋊D5911yescharacteristic
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—1184no—
224C4⋊C4—4721no—
225C4⋊C4—4724no—
226C4⋊C4—2364no—
227C4⋊C4—2364no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C22⋊C4—41no—
231C22⋊C4—1184no—
232C22⋊C4—2364no—
233C22⋊C4—2362no—
234C22⋊C4—2362no—
235C42—41no—
236C42—41no—
237C23—22no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23#1427369511yescharacteristic
242C23#1427370512yes—
243C23#1427372612yes—
244Q8—4724no—
245D4—41no—
246D4—41no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—1184no—
260C2×C4—4722no—
261C2×C4—4722no—
262C2×C4—4722no—
263C2×C4—2364no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4—2362no—
269C2×C4—2364no—
270C2×C4#1427370611yescharacteristic
271C2×C4#1427372511yescharacteristic
272C8—42no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22#1683732211yescharacteristic
283C22#1683758811yescharacteristic
284C22#1683759111yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—4722no—
292C4—4724no—
293C4—2364no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#2142544911yescharacteristic
299C2#2142607911yescharacteristic
300C2#2142610811yescharacteristic
subgroupG/H
centerC22#16837588
commutator, prime radical, cosocleC22×D4×C59C23
FrattiniC22×D4C22×D59
FittingC59×#1856C2
socleC22×C59B(4,𝔽2)
hypercenter#1856D59
hypocenterC59#13461
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
31. #14273642Q8semidirect
47. #100×C59Q8semidirect
169. C59#13461semidirect
193. #100Dic118semidirect

Maximal subgroups

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

subgroupindex
#214304682
#214243772
#214222212
#214222152
C59×#18562
#1346159