Finite Group Databasebeta

← #28484192

Subgroups — #28484192

Order 15104.

630 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433120C211yesmaximal, characteristic, complemented
2#21426818C211yesmaximal, characteristic
3#21422486C212yesmaximal, complemented
4#21421924C212yesmaximal, complemented
5C59×#1460C211yesmaximal, characteristic, complemented
6C2×#14273723C411yescharacteristic
7C2×C4×D236C2211yescharacteristic
8D4×C59⋊C8—24no—
9#16837455C2212yes—
10C2×C22⋊(C59⋊C8)—22no—
11#16837305C411yescharacteristic
12#16837303C411yescharacteristic
13#16837281—22no—
14#16837278—22no—
15#16837267C411yescharacteristic
16#16837248C2211yescharacteristic
17#16837129—24no—
18C2×C4×D4×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#323—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—22no—
23C22×C4×D59—21no—
24C22×C59⋊C8—24no—
25#14273723—22no—
26#14273723C814yescomplemented
27C4×D236—22no—
28C4×D236—22no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C22⋊Dic59—22no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—22no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—24no—
37C22⋊(C59⋊C8)—22no—
38C4×C59⋊C8D414yescomplemented
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×D4×C59C2×C411yescharacteristic
43C23×C4×C59D412yes—
44C4×D4×C59—22no—
45C4×D4×C59—22no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59C2311yescharacteristic
49C4⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C4×C8×C59—22no—
52C24×C59C22⋊C412yes—
53C23×D59—21no—
54C2×D4×C59—22no—
55C2×D4×C59—22no—
56C2×D4×C59C2×C812yes—
57C2×D4×C59M1612yes—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—82no—
61C22×C4×C59C22⋊C412yes—
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C2×C22⋊D59—28no—
66C2×C22⋊D59—82no—
67C2×C22⋊D59—82no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×D236—22no—
72C2×D236—22no—
73C2×D236—21no—
74C2×D236C2×C812yescomplemented
75C2×C4×D59—42no—
76C2×C4×D59—82no—
77C2×C8×C59—42no—
78C4⋊C4×C59—22no—
79C22⋊C4×C59—24no—
80C42×C59C2×D412yes—
81C42×C59#4112yes—
82C22⋊Dic59—82no—
83(C2×D59)⋊C4—41no—
84(C2×D59)⋊C4—41no—
85(C2×D59)⋊C4—82no—
86C4⋊Dic59—22no—
87C4×Dic59—21no—
88C4×Dic59—22no—
89C4×Dic59C2×C812yescomplemented
90C2×C59⋊C8—24no—
91C2×C59⋊C8—84no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59C22⋊C814yes—
97C23×C59C2×C22⋊C411yescharacteristic
98C22×D59—22no—
99C22×D59—42no—
100C22×D59—42no—
101C22×D59—41no—
102C22×D59—41no—
103D4×C59—24no—
104D4×C59—24no—
105C2×C4×C59—22no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—44no—
109C2×C4×C59—42no—
110C2×C4×C59—82no—
111C2×C4×C59C4×D412yes—
112C2×C4×C59#12011yescharacteristic
113C2×C4×C59#12112yes—
114C2×C4×C59#12311yescharacteristic
115C2×C4×C59C22×C811yescharacteristic
116C2×C4×C59C2×M1611yescharacteristic
117C2×C4×C59#13112yes—
118C22⋊D59—88no—
119C2×Dic59—22no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—42no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127D236—24no—
128D236—42no—
129C4×D59—84no—
130C8×C59—82no—
131C59⋊C8—44no—
132#9101—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—24no—
136C22×C59—44no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—82no—
140C22×C59#38611yescharacteristic
141C22×C59C2×C22⋊C811yescharacteristic
142C22×C59#40811yescharacteristic
143C2×D59—44no—
144C2×D59—42no—
145C2×D59—82no—
146C2×D59—81no—
147C2×D59—81no—
148C4×C59—22no—
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—42no—
152C4×C59C8×D412yes—
153C4×C59#43512yes—
154Dic59—42no—
155Dic59—81no—
156Dic59—81no—
157#2972—591no—
158#2328—591no—
159#1478—592no—
160#1460D5911yescharacteristic, complemented
161#1274—592no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—44no—
165C2×C59#186911yescharacteristic
166C2×C59#191011yescharacteristic
167C2×C59#191111yescharacteristic
168D59—82no—
169C2×#127—591no—
170C2×C4×D4—591no—
171C2×C4×D4C2×D5911yescharacteristic
172C8×D4—1184no—
173#431—592no—
174#431C2×D5912yes—
175C2×C22⋊C8—1182no—
176#390—591no—
177#390—591no—
178#386—1182no—
179#386—1182no—
180#382—591no—
181#376—591no—
182D4⋊C8—1184no—
183#323—22no—
184C59#910111yescharacteristic, complemented, Sylow
185C22×D4—591no—
186C22×D4—1182no—
187C22×D4C4×D5911yescharacteristic
188C23×C4—1181no—
189C23×C4D23612yes—
190C22×C8—1184no—
191#127—594no—
192#127—1182no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—1182no—
196C4×D4—1182no—
197C2×C4⋊C4—591no—
198C2×C4⋊C4C4×D5911yescharacteristic
199C2×C22⋊C4—1182no—
200C2×C22⋊C4—1182no—
201C2×C22⋊C4—1181no—
202C2×C22⋊C4C22⋊D5912yes—
203C2×C42—591no—
204C2×C42—591no—
205C2×C42C22×D5911yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—1182no—
208C22⋊C8—22no—
209C22⋊C8—1184no—
210C22⋊C8—1182no—
211C4×C8—22no—
212C4×C8—594no—
213#95—1182no—
214#95—1181no—
215#95—1181no—
216C24—1181no—
217C24(C2×D59)⋊C412yes—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—592no—
221C2×D4—1188no—
222C2×D4—1182no—
223C2×D4—1181no—
224C2×D4—1182no—
225C2×D4—4722no—
226C2×D4—4722no—
227C2×D4C8×D5912yes—
228C2×D4#1174565912yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—4722no—
236C22×C4—2362no—
237C22×C4(C2×D59)⋊C412yes—
238C22×C4C2×C4×D5911yescharacteristic
239C22×C4C2×D23611yescharacteristic
240C22×C4C2×C22⋊D5911yescharacteristic
241C2×C8—42no—
242C2×C8—1184no—
243C2×C8—4724no—
244C4⋊C4—22no—
245C4⋊C4—1182no—
246C22⋊C4—24no—
247C22⋊C4—4722no—
248C22⋊C4—4722no—
249C22⋊C4—2361no—
250C22⋊C4—2361no—
251C42—592no—
252C42—1181no—
253C42—1182no—
254C42D4×D5912yes—
255C42#1174568712yes—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—1182no—
261C23—2362no—
262C23—2362no—
263C23—2361no—
264C23—2361no—
265C23(C2×D59)⋊C814yes—
266C23C2×(C2×D59)⋊C411yescharacteristic
267D4—24no—
268D4—24no—
269D4—1184no—
270D4—4728no—
271D4—2362no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—1182no—
279C2×C4—4724no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2362no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4(C22⋊D59)⋊C412yes—
288C2×C4#1427369711yescharacteristic
289C2×C4#1427369912yes—
290C2×C4#1427371111yescharacteristic
291C2×C4C2×C8×D5911yescharacteristic
292C2×C4C2×#1174565911yescharacteristic
293C2×C4#1427373512yes—
294C8—82no—
295C8—2364no—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—4722no—
304C22—4721no—
305C22—4721no—
306C22—2364no—
307C22—2362no—
308C22#1683727911yescharacteristic
309C22C2×(C2×D59)⋊C811yescharacteristic
310C22#1683737011yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—4721no—
316C4—4721no—
317C4—2362no—
318C4(C22⋊D59)⋊C812yes—
319C4#1683747012yes—
320C2—21no—
321C2—21no—
322C2—44no—
323C2—4722no—
324C2#2142438611yescharacteristic
325C2#2142462111yescharacteristic
326C2#2142462611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC2×C4×C59C22×C8
FrattiniC2×C42C22×D59
FittingC59×#1460C2
socleC22×C59#386
prime radicalC2×#14273723C4
cosocleC2×C42×C59C23
hypercenter#1460D59
hypocenterC59#9101
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
1. #21433120C2semidirect
3. #21422486C2semidirect
4. #21421924C2semidirect
5. C59×#1460C2semidirect
26. #14273723C8semidirect
38. C4×C59⋊C8D4semidirect
74. C2×D236C2×C8semidirect
89. C4×Dic59C2×C8semidirect
160. #1460D59semidirect
184. C59#9101semidirect

Maximal subgroups

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

subgroupindex
#214331202
#214268182
#214224862
#214219242
C59×#14602
#910159