Finite Group Databasebeta

← #28486431

Subgroups — #28486431

Order 15104.

610 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 99 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#21428563C211yesmaximal, characteristic
2#21426699C211yesmaximal, characteristic, complemented
3#21422626C212yesmaximal
4#21421950C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273710—21no—
8C2×#14273701C2211yescharacteristic
9#16837337—22no—
10#16837303C412yescomplemented
11#16837288—24no—
12#16837271C412yescomplemented
13#16837265C2212yes—
14#16837262—21no—
15C4×C22⋊Dic59—21no—
16C4×C22⋊Dic59C2211yescharacteristic
17#16837137—24no—
18C22×C4⋊C4×C59C2211yescharacteristic
19C59×#407C2212yes—
20C59×#343—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23#14273710—44no—
24#14273710—44no—
25#14273701—22no—
26#14273701—22no—
27C2×C4⋊Dic59—22no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—22no—
35C2×(C2×D59)⋊C4—21no—
36C2×C4×Dic59—22no—
37C2×C4×Dic59C2×C412yes—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95—22no—
42C59⋊#95D414yes—
43C23×C4×C59D411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C2×C4⋊C4×C59—22no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59C2×C412yes—
50C2×C4⋊C4×C59D412yes—
51C22⋊C8×C59—22no—
52C22⋊C8×C59—22no—
53C24×C59C2×D411yescharacteristic
54C23×D59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C412yes—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D412yes—
67C22×C4×C59#4112yes—
68C2×C22⋊D59—42no—
69C2×C22⋊D59—42no—
70C2×C22⋊D59—41no—
71C2×C22⋊D59—41no—
72C22×Dic59—22no—
73C22×Dic59—21no—
74C22×Dic59—24no—
75C2×D236—22no—
76C2×D236—42no—
77M16×C59—44no—
78C2×C8×C59—42no—
79C4⋊C4×C59—24no—
80C4⋊C4×C59—24no—
81C22⋊Dic59—24no—
82C22⋊Dic59—44no—
83(C2×D59)⋊C4—44no—
84(C2×D59)⋊C4—44no—
85(C2×D59)⋊C4—42no—
86C4⋊Dic59—22no—
87C4⋊Dic59—82no—
88Dic59⋊C4—44no—
89Dic59⋊C4—44no—
90C4×Dic59—42no—
91C4×Dic59—82no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—41no—
95C23×C59C2×C22⋊C411yescharacteristic
96C23×C59#12011yescharacteristic
97C23×C59#12311yescharacteristic
98C22×D59—41no—
99C22×D59—41no—
100C22×D59—42no—
101C22×D59—41no—
102C22×D59—42no—
103C2×C4×C59—22no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—81no—
114C2×C4×C59—81no—
115C2×C4×C59C4×D412yes—
116C2×C4×C59#12112yes—
117C22⋊D59—82no—
118C22⋊D59—82no—
119C2×Dic59—24no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—44no—
125C2×Dic59—44no—
126C2×Dic59—41no—
127C2×Dic59—44no—
128D236—82no—
129C8×C59—82no—
130#9464—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—44no—
139C22×C59#38611yescharacteristic
140C22×C59#41012yes—
141C22×C59#45312yes—
142C2×D59—82no—
143C2×D59—82no—
144C2×D59—81no—
145C2×D59—81no—
146C2×D59—81no—
147C4×C59—22no—
148C4×C59—22no—
149C4×C59—42no—
150C4×C59—42no—
151Dic59—44no—
152Dic59—84no—
153Dic59—81no—
154#2501—591no—
155#2313—591no—
156#1821D5911yescharacteristic, complemented
157#1510—592no—
158#1284—592no—
159C2×C59—22no—
160C2×C59—22no—
161C2×C59—41no—
162C2×C59#188711yescharacteristic
163C2×C59#189911yescharacteristic
164C2×C59#191211yescharacteristic
165D59—161no—
166C2×#123—1181no—
167C2×#123—1181no—
168C2×#121—591no—
169C22×C4⋊C4C2×D5911yescharacteristic
170#407C2×D5912yes—
171#399—1182no—
172#390—592no—
173#387—1184no—
174#384—592no—
175#381—592no—
176#380—1181no—
177C4×C22⋊C4—591no—
178C4×C22⋊C4—1181no—
179#343—22no—
180#327—1184no—
181C59#946411yescharacteristic, complemented, Sylow
182C22×D4—591no—
183C22×D4—1181no—
184C23×C4D23611yescharacteristic
185C23×C4D23611yescharacteristic
186C23×C4C22×D5911yescharacteristic
187C2×M16—22no—
188#123—2364no—
189#123—2364no—
190#121—1182no—
191#121—1182no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1182no—
196C2×C4⋊C4—1182no—
197C2×C4⋊C4—1182no—
198C2×C4⋊C4C4×D5912yes—
199C2×C4⋊C4C22⋊D5912yes—
200C2×C22⋊C4—1181no—
201C2×C22⋊C4—1182no—
202C2×C22⋊C4—1182no—
203C2×C22⋊C4—1182no—
204C2×C22⋊C4—1181no—
205C2×C42—592no—
206C2×C42—1182no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—594no—
210#95—1182no—
211#95—1182no—
212#95—1182no—
213#95—1181no—
214C24—1181no—
215C24C2×D23611yescharacteristic
216C2×D4—1182no—
217C2×D4—2362no—
218C2×D4—2362no—
219C2×D4—2362no—
220C2×D4—2361no—
221C2×D4—2361no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—1182no—
230C22×C4—1181no—
231C22×C4—1184no—
232C22×C4(C2×D59)⋊C412yes—
233C22×C4(C2×D59)⋊C412yes—
234C22×C4C2×C4×D5911yescharacteristic
235C22×C4D4×D5912yes—
236C22×C4#1174568712yes—
237C22×C4C2×C22⋊D5911yescharacteristic
238M16—44no—
239C2×C8—42no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—1182no—
243C4⋊C4—4722no—
244C4⋊C4—2364no—
245C4⋊C4—2364no—
246C22⋊C4—1184no—
247C22⋊C4—2364no—
248C22⋊C4—2364no—
249C22⋊C4—2364no—
250C22⋊C4—2362no—
251C42—4722no—
252C42—2362no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—2361no—
257C23—2361no—
258C23—2362no—
259C23—2361no—
260C23—2362no—
261C23C2×(C2×D59)⋊C411yescharacteristic
262C23#1427369711yescharacteristic
263C23#1427371111yescharacteristic
264D4—4722no—
265D4—4722no—
266D4—4722no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—1184no—
280C2×C4—2364no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2364no—
284C2×C4—2361no—
285C2×C4—2364no—
286C2×C4—2362no—
287C2×C4—2362no—
288C2×C4(C22⋊D59)⋊C412yes—
289C2×C4#1427369912yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—4722no—
300C22—4722no—
301C22—4721no—
302C22—4721no—
303C22—4721no—
304C22#1683727911yescharacteristic
305C22#1683737512yes—
306C22#1683757212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—4724no—
312C4—4721no—
313C4—2364no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—9441no—
318C2#2142449211yescharacteristic
319C2#2142455211yescharacteristic
320C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1821C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1821D59
hypocenterC59#9464
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #21426699C2semidirect
4. #21421950C2semidirect
5. C59×#1821C2semidirect
10. #16837303C4semidirect
12. #16837271C4semidirect
156. #1821D59semidirect
181. C59#9464semidirect

Maximal subgroups

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

subgroupindex
#214285632
#214266992
#214226262
#214219502
C59×#18212
#946459