Finite Group Databasebeta

← #28483362

Subgroups — #28483362

Order 15104.

694 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 119 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#21433137C211yesmaximal, characteristic, complemented
2C4×#14273723C211yesmaximal, characteristic
3#21422488C212yesmaximal, complemented
4#21421928C212yesmaximal, complemented
5C59×#1460C211yesmaximal, characteristic, complemented
6C2×#14273723C411yescharacteristic
7C2×C4×D236C2211yescharacteristic
8C2×C4×C22⋊D59—22no—
9#16837480—24no—
10#16837455C2212yes—
11#16837366—22no—
12#16837305C411yescharacteristic
13#16837303C411yescharacteristic
14#16837302—22no—
15#16837271C411yescharacteristic
16C42×Dic59C2211yescharacteristic
17#16837131—24no—
18C2×C4×D4×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#323—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—22no—
23C22×C4×D59—21no—
24C2×#11745664—24no—
25#14273723—22no—
26#14273723—22no—
27C4×D236—22no—
28C4×D236—22no—
29C4×C22⋊D59—48no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—22no—
32C2×C22⋊Dic59—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59C2×C411yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39C4⋊(C59⋊C8)D414yescomplemented
40C22⋊(C59⋊C8)—24no—
41C22⋊(C59⋊C8)—22no—
42C4×C59⋊C8—22no—
43C22×D4×C59C2×C411yescharacteristic
44C23×C4×C59D412yes—
45C4×D4×C59—22no—
46C4×D4×C59—22no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D412yes—
49C2×C42×C59C2311yescharacteristic
50C4⋊C8×C59—22no—
51C22⋊C8×C59—22no—
52C4×C8×C59—22no—
53C24×C59C22⋊C412yes—
54C23×D59—21no—
55C2×D4×C59—22no—
56C2×D4×C59—21no—
57C2×D4×C59—21no—
58C2×D4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—44no—
62C22×C4×C59C22⋊C412yes—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C2×C22⋊D59—28no—
67C2×C22⋊D59—44no—
68C2×C22⋊D59—44no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×D236—22no—
73C2×D236—22no—
74C2×D236—21no—
75C2×D236M1612yescomplemented
76C2×C4×D59—44no—
77C2×C4×D59—41no—
78C2×C4×D59—41no—
79C2×C8×C59—42no—
80C4⋊C4×C59—22no—
81C22⋊C4×C59—24no—
82C42×C59C2×D412yes—
83C42×C59#4112yes—
84C22⋊Dic59—44no—
85(C2×D59)⋊C4—44no—
86(C2×D59)⋊C4—42no—
87C4⋊Dic59—22no—
88Dic59⋊C4—44no—
89C4×Dic59—24no—
90C4×Dic59—21no—
91C4×Dic59—22no—
92C4×Dic59—42no—
93C4×Dic59M1612yescomplemented
94#11745664—84no—
95C2×C59⋊C8—24no—
96C23×C59—22no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59C2×C22⋊C411yescharacteristic
102C22×D59—22no—
103C22×D59—42no—
104C22×D59—42no—
105C22×D59—41no—
106C22×D59—41no—
107D4×C59—42no—
108D4×C59—42no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—41no—
112C2×C4×C59—44no—
113C2×C4×C59—42no—
114C2×C4×C59—82no—
115C2×C4×C59C4×D412yes—
116C2×C4×C59#12011yescharacteristic
117C2×C4×C59#12112yes—
118C2×C4×C59#12311yescharacteristic
119C2×C4×C59C2×M1611yescharacteristic
120C2×C4×C59C2×M1611yescharacteristic
121C2×C4×C59#13112yes—
122C22⋊D59—416no—
123C2×Dic59—22no—
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—42no—
130C2×Dic59—41no—
131D236—24no—
132D236—42no—
133C4×D59—82no—
134C4×D59—82no—
135C8×C59—82no—
136C59⋊C8—44no—
137#8963—591nomaximal, Sylow
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—44no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59—82no—
145C22×C59#38611yescharacteristic
146C22×C59#40711yescharacteristic
147C22×C59#40811yescharacteristic
148C2×D59—44no—
149C2×D59—42no—
150C2×D59—82no—
151C2×D59—81no—
152C2×D59—81no—
153C4×C59—22no—
154C4×C59—21no—
155C4×C59—42no—
156C4×C59#43512yes—
157C4×C59#43612yes—
158C4×C59#45312yes—
159Dic59—24no—
160Dic59—42no—
161Dic59—81no—
162#2975—591no—
163C4×#127—591no—
164#1478—592no—
165#1460D5911yescharacteristic, complemented
166#1276—592no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59—82no—
170C2×C59#187011yescharacteristic
171C2×C59#188411yescharacteristic
172C2×C59#188711yescharacteristic
173D59—82no—
174C2×#127—591no—
175C2×C4×D4—591no—
176C2×C4×D4—1182no—
177C2×C4×D4C2×D5911yescharacteristic
178#436—1184no—
179#431—592no—
180#431C2×D5912yes—
181#407—1182no—
182#390—591no—
183#390—591no—
184#390—1182no—
185#384—591no—
186C43—591no—
187D4⋊C8—1184no—
188#323—22no—
189C59#896311yescharacteristic, complemented, Sylow
190C22×D4—591no—
191C22×D4—1182no—
192C22×D4C4×D5911yescharacteristic
193C23×C4—1181no—
194C23×C4D23612yes—
195C2×M16—1184no—
196#127—1182no—
197#127—1182no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—1182no—
201C4×D4—1182no—
202C4×D4—2368no—
203C2×C4⋊C4—591no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4C4×D5911yescharacteristic
206C2×C22⋊C4—1182no—
207C2×C22⋊C4—1182no—
208C2×C22⋊C4—1181no—
209C2×C22⋊C4C22⋊D5912yes—
210C2×C42—591no—
211C2×C42—591no—
212C2×C42—1181no—
213C2×C42—1181no—
214C2×C42C22×D5911yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—594no—
217C22⋊C8—22no—
218C22⋊C8—1184no—
219C22⋊C8—1182no—
220C4×C8—22no—
221C4×C8—1182no—
222C24—1181no—
223C24(C2×D59)⋊C412yes—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—592no—
229C2×D4—1188no—
230C2×D4—1182no—
231C2×D4—1181no—
232C2×D4—1182no—
233C2×D4—2364no—
234C2×D4—2364no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—2364no—
242C22×C4—2361no—
243C22×C4—2361no—
244C22×C4(C2×D59)⋊C412yes—
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4C2×D23611yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248M16—4724no—
249C2×C8—42no—
250C2×C8—1184no—
251C4⋊C4—22no—
252C4⋊C4—1182no—
253C4⋊C4—2364no—
254C22⋊C4—24no—
255C22⋊C4—2364no—
256C22⋊C4—2364no—
257C22⋊C4—2362no—
258C42—592no—
259C42—1184no—
260C42—1181no—
261C42—1182no—
262C42—2362no—
263C42D4×D5912yes—
264C42#1174568712yes—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—1182no—
271C23—2362no—
272C23—2362no—
273C23—2361no—
274C23—2361no—
275C23C2×(C2×D59)⋊C411yescharacteristic
276D4—42no—
277D4—42no—
278D4—1184no—
279D4—23616no—
280D4—2362no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—1182no—
288C2×C4—1182no—
289C2×C4—1182no—
290C2×C4—1182no—
291C2×C4—4722no—
292C2×C4—4722no—
293C2×C4—2362no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4(C22⋊D59)⋊C412yes—
298C2×C4#1427369711yescharacteristic
299C2×C4#1427369912yes—
300C2×C4#1427371111yescharacteristic
301C2×C4M16×D5911yescharacteristic
302C2×C4M16×D5911yescharacteristic
303C2×C4#1427373312yes—
304C8—82no—
305C8—2364no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—82no—
313C22—4722no—
314C22—4721no—
315C22—4721no—
316C22—2364no—
317C22—2362no—
318C22#1683727911yescharacteristic
319C22#1683736511yescharacteristic
320C22#1683736811yescharacteristic
321C4—22no—
322C4—21no—
323C4—42no—
324C4—1184no—
325C4—4721no—
326C4—2362no—
327C4#1683747612yes—
328C4#1683747712yes—
329C4#1683757212yes—
330C2—21no—
331C2—21no—
332C2—82no—
333C2—4722no—
334C2#2142439111yescharacteristic
335C2#2142447711yescharacteristic
336C2#2142449211yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1460C2
socleC22×C59#386
prime radicalC2×#14273723C4
cosocleC2×C42×C59C23
hypercenter#1460D59
hypocenterC59#8963
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
1. #21433137C2semidirect
3. #21422488C2semidirect
4. #21421928C2semidirect
5. C59×#1460C2semidirect
39. C4⋊(C59⋊C8)D4semidirect
75. C2×D236M16semidirect
93. C4×Dic59M16semidirect
165. #1460D59semidirect
189. C59#8963semidirect

Maximal subgroups

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

subgroupindex
#214331372
C4×#142737232
#214224882
#214219282
C59×#14602
#896359