Finite Group Databasebeta

← #28508368

Subgroups — #28508368

Order 15104.

484 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 81 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#21424376C212yesmaximal
2#21424305C211yesmaximal, characteristic
3#21422221C211yesmaximal, characteristic
4#21422214C212yesmaximal
5C59×#2804C211yesmaximal, characteristic
6#16837877C2211yescharacteristic
7#16837442—21no—
8C2×#14273644C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837275—22no—
11#16837168—22no—
12#16837162—21no—
13C59×#400—21no—
14C59×#393—22no—
15C59×#391—21no—
16C59×#390C2211yescharacteristic
17C59×#386C2212yes—
18C59×#385—22no—
19C2×#11745664—21no—
20D4×Dic59—41no—
21D4×Dic59—41no—
22#14273686—21no—
23C2×Dic59⋊C4—42no—
24C2×C22⋊Dic59—22no—
25C2×C22⋊Dic59—22no—
26#14273660—41no—
27#14273658—41no—
28#14273644—21no—
29#14273644Q812yescomplemented
30#14273642—22no—
31#14273642Q814yescomplemented
32C59⋊#95—42no—
33C22×Q8×C59—21no—
34C22×D4×C59C2311yescharacteristic
35C23×C4×C59D412yes—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59D412yes—
42C2×C42×C59D411yescharacteristic
43#95×C59—22no—
44#95×C59—22no—
45#95×C59—21no—
46#95×C59—22no—
47C24×C59C2×D412yes—
48C2×Q8×C59—42no—
49C2×Q8×C59—81no—
50C2×Q8×C59—81no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59C2×Q811yescharacteristic
54C2×D4×C59C2×Q812yes—
55C2×D4×C59#4111yescharacteristic
56C22×C4×C59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—42no—
61C22×C4×C59—82no—
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—22no—
64C22×Dic59—22no—
65C22×Dic59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—81no—
68C4⋊C4×C59—81no—
69C22⋊C4×C59—44no—
70C22⋊C4×C59—42no—
71C22⋊C4×C59—82no—
72C42×C59—41no—
73C22⋊Dic59—24no—
74C22⋊Dic59—42no—
75C22⋊Dic59—42no—
76C22⋊Dic59—41no—
77C4⋊Dic59—21no—
78C4⋊Dic59—21no—
79Dic59⋊C4—84no—
80C4×Dic59—41no—
81#11745664—22no—
82#11745664—41no—
83C2×C59⋊C8—41no—
84C23×C59—22no—
85C23×C59—22no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59#12011yescharacteristic
89C23×C59C22⋊Q812yes—
90C23×C59C4⋊Q812yes—
91Q8×C59—84no—
92D4×C59—41no—
93D4×C59—42no—
94D4×C59—41no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—44no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—82no—
110C2×C4×C59C22⋊Q811yescharacteristic
111C2×C4×C59C4⋊Q811yescharacteristic
112C2×Dic59—44no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—82no—
116C2×Dic59—82no—
117C2×Dic59—82no—
118C2×Dic59—82no—
119C59⋊C8—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—82no—
137C4×C59—81no—
138C4×C59—82no—
139Dic59—84no—
140Dic59—81no—
141Dic59—81no—
142#2804D5911yescharacteristic
143#1867—592no—
144#1856—591no—
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#518—591no—
155#428—1181no—
156C2×#100—591no—
157C2×#99—592no—
158#400—21no—
159#393—22no—
160#391—21no—
161#390C2×D5911yescharacteristic
162#386C2×D5912yes—
163#385—22no—
164#385—1182no—
165#342—1182no—
166#339—1181no—
167C59#1346311yescharacteristic, complemented, Sylow
168C22×Q8—21no—
169C22×D4C22×D5911yescharacteristic
170C23×C4C22⋊D5912yes—
171C2×M16—1181no—
172C4×D4—2361no—
173C4×D4—2361no—
174#117—1181no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—2362no—
178C2×C4⋊C4C22⋊D5911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—1182no—
182C2×C22⋊C4—1182no—
183C2×C22⋊C4C22⋊D5912yes—
184C2×C42C22⋊D5911yescharacteristic
185#107—2361no—
186#106—2361no—
187#100—592no—
188#100—1181no—
189#99—594no—
190#99—1182no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—22no—
195#95—2362no—
196C24C2×C22⋊D5912yes—
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Q8×D5911yescharacteristic
203C2×D4Q8×D5912yes—
204C2×D4#1174568911yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4—2362no—
214C22×C4C2×C22⋊D5911yescharacteristic
215M16—1182no—
216M16—2361no—
217C2×C8—2361no—
218C4⋊C4—41no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—1181no—
222C4⋊C4—1181no—
223C4⋊C4—4724no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—1184no—
228C22⋊C4—2362no—
229C22⋊C4—2362no—
230C22⋊C4—2361no—
231C42—41no—
232C42—2361no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23#1427369611yescharacteristic
238C23#1427370312yes—
239C23#1427372512yes—
240Q8—84no—
241D4—41no—
242D4—42no—
243D4—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—4722no—
260C2×C4—4722no—
261C2×C4—4722no—
262C2×C4—4722no—
263C2×C4—2364no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4#1427370311yescharacteristic
267C2×C4#1427372511yescharacteristic
268C8—2362no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—82no—
278C22#1683732111yescharacteristic
279C22#1683758911yescharacteristic
280C22#1683759211yescharacteristic
281C4—21no—
282C4—21no—
283C4—42no—
284C4—82no—
285C4—81no—
286C4—82no—
287C4—4724no—
288C4—4721no—
289C4—4721no—
290C2—21no—
291C2—21no—
292C2—42no—
293C2—42no—
294C2#2142544811yescharacteristic
295C2#2142609211yescharacteristic
296C2#2142609711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, prime radical, cosocleC22×D4×C59C23
FrattiniC22×D4C22×D59
FittingC59×#2804C2
socleC22×C59B(4,𝔽2)
hypercenter#2804D59
hypocenterC59#13463
LCSG ▷ C22×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2804

Complements

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

NH ≅ G/Ntype
29. #14273644Q8semidirect
31. #14273642Q8semidirect
167. C59#13463semidirect

Maximal subgroups

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

subgroupindex
#214243762
#214243052
#214222212
#214222142
C59×#28042
#1346359