Finite Group Databasebeta

← #28484432

Subgroups — #28484432

Order 15104.

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

normal subgroups only

This group has 756 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#21432427C211yesmaximal, characteristic
2C2×#16837248C211yesmaximal, characteristic, complemented
3C2×#16837137C212yesmaximal
4#21421976C212yesmaximal, complemented
5C59×#1817C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C2211yescharacteristic
8C22×C4×Dic59—21no—
9#16837304C412yescomplemented
10#16837300C412yescomplemented
11#16837293—22no—
12#16837281—21no—
13#16837262C2211yescharacteristic
14#16837248—24no—
15C2×C59⋊#95—21no—
16C2×C59⋊#95C2212yes—
17#16837137—24no—
18C22×C42×C59C2211yescharacteristic
19C2×C22⋊C8×C59C2212yes—
20C59×#336—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23C23×Dic59—22no—
24#14273710—44no—
25#14273701—22no—
26#14273701—22no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—21no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—21no—
34C2×C4×Dic59—24no—
35C2×C4×Dic59—28no—
36C59⋊#95—24no—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C59⋊#95—21no—
40C59⋊#95D414yes—
41C23×C4×C59D411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C23×C4×C59C2311yescharacteristic
44C22×C8×C59—22no—
45C2×C42×C59—22no—
46C2×C42×C59—22no—
47C2×C42×C59C2×C412yes—
48C2×C42×C59D412yes—
49C22⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C24×C59C2×D411yescharacteristic
52C23×D59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—24no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—42no—
61C22×C4×C59C22⋊C412yes—
62C22×C4×C59C22⋊C412yes—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×C4×C59C2×D412yes—
66C22×C4×C59#4112yes—
67C2×C22⋊D59—41no—
68C2×C22⋊D59—41no—
69C2×C22⋊D59—42no—
70C2×C22⋊D59—42no—
71C22×Dic59—28no—
72C22×Dic59—24no—
73C22×Dic59—22no—
74C22×Dic59—21no—
75C22×Dic59—44no—
76C22×Dic59—42no—
77C22×Dic59—42no—
78C2×D236—22no—
79C2×D236—42no—
80C2×C8×C59—44no—
81C2×C8×C59—42no—
82C42×C59—24no—
83C42×C59—24no—
84C22⋊Dic59—42no—
85(C2×D59)⋊C4—44no—
86(C2×D59)⋊C4—44no—
87(C2×D59)⋊C4—42no—
88C4⋊Dic59—22no—
89C4⋊Dic59—42no—
90Dic59⋊C4—44no—
91C4×Dic59—216no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59C2×C22⋊C411yescharacteristic
95C23×C59#12011yescharacteristic
96C23×C59#12311yescharacteristic
97C23×C59#13512yes—
98C23×C59AGL(1,ℤ/8)12yes—
99C22×D59—41no—
100C22×D59—42no—
101C22×D59—41no—
102C22×D59—41no—
103C22×D59—42no—
104C2×C4×C59—24no—
105C2×C4×C59—24no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59#10414yes—
118C2×C4×C59C4×D412yes—
119C2×C4×C59#12112yes—
120C22⋊D59—82no—
121C22⋊D59—82no—
122C2×Dic59—28no—
123C2×Dic59—44no—
124C2×Dic59—48no—
125C2×Dic59—48no—
126C2×Dic59—44no—
127C2×Dic59—41no—
128C2×Dic59—42no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—42no—
132C2×Dic59—42no—
133D236—82no—
134C8×C59—82no—
135#9149—591nomaximal, Sylow
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—24no—
141C22×C59—24no—
142C22×C59#38611yescharacteristic
143C22×C59#41012yes—
144C22×C59#41611yescharacteristic
145C22×C59#41811yescharacteristic
146C22×C59C2×#10412yes—
147C22×C59#44912yes—
148C22×C59#48212yes—
149C2×D59—81no—
150C2×D59—81no—
151C2×D59—81no—
152C2×D59—82no—
153C2×D59—82no—
154C4×C59—24no—
155C4×C59—22no—
156C4×C59—22no—
157C4×C59—42no—
158Dic59—48no—
159Dic59—81no—
160Dic59—82no—
161#2854—591no—
162C2×#376—591no—
163#1817D5911yescharacteristic, complemented
164C2×#327—592no—
165#1294—592no—
166C2×C59—22no—
167C2×C59—22no—
168C2×C59#150012yes—
169C2×C59#150512yes—
170C2×C59#187511yescharacteristic
171C2×C59#189611yescharacteristic
172C2×C59#189911yescharacteristic
173D59—161no—
174C2×#123—1181no—
175C2×#121—591no—
176C22×C42—1181no—
177C22×C42C2×D5911yescharacteristic
178C2×C22⋊C8C2×D5912yes—
179#390—592no—
180#389—592no—
181#388—1182no—
182#386—1181no—
183#380—591no—
184#376—1184no—
185C2×#95—592no—
186C2×#95—1181no—
187#336—22no—
188#327—1184no—
189C59#914911yescharacteristic, complemented, Sylow
190C22×D4—591no—
191C22×D4—1181no—
192C23×C4—1182no—
193C23×C4D23611yescharacteristic
194C23×C4D23611yescharacteristic
195C23×C4C22×D5911yescharacteristic
196C22×C8—22no—
197#123—2364no—
198#121—1182no—
199#121—1182no—
200C2×C4⋊C4—591no—
201C2×C4⋊C4—592no—
202C2×C4⋊C4—1182no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4—1182no—
206C2×C22⋊C4—1181no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—1184no—
210C2×C42—1188no—
211C2×C42C4×D5912yes—
212C2×C42C22⋊D5912yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—594no—
216#95—1184no—
217#95—1182no—
218#95—1182no—
219#95—1181no—
220C24—1181no—
221C24C2×D23611yescharacteristic
222C2×D4—1182no—
223C2×D4—2362no—
224C2×D4—2361no—
225C2×D4—2361no—
226C2×D4—2362no—
227C2×D4—2362no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—1188no—
237C22×C4—1184no—
238C22×C4—1182no—
239C22×C4—1181no—
240C22×C4—2364no—
241C22×C4—2362no—
242C22×C4—2362no—
243C22×C4(C2×D59)⋊C412yes—
244C22×C4(C2×D59)⋊C412yes—
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4D4×D5912yes—
247C22×C4#1174568712yes—
248C22×C4C2×C22⋊D5911yescharacteristic
249C2×C8—44no—
250C2×C8—42no—
251C4⋊C4—1182no—
252C4⋊C4—2362no—
253C4⋊C4—2364no—
254C22⋊C4—2364no—
255C22⋊C4—2362no—
256C22⋊C4—2364no—
257C22⋊C4—2362no—
258C42—24no—
259C42—24no—
260C42—11816no—
261C23—22no—
262C23—22no—
263C23—2361no—
264C23—2361no—
265C23—2362no—
266C23—2361no—
267C23—2362no—
268C23C2×(C2×D59)⋊C411yescharacteristic
269C23#1427369711yescharacteristic
270C23#1427371111yescharacteristic
271C23#1427374912yes—
272C23#1427375512yes—
273D4—4722no—
274D4—4722no—
275D4—4722no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—1188no—
290C2×C4—2361no—
291C2×C4—2362no—
292C2×C4—2361no—
293C2×C4—2362no—
294C2×C4—2361no—
295C2×C4—2362no—
296C2×C4—2368no—
297C2×C4—2364no—
298C2×C4—2368no—
299C2×C4—2364no—
300C2×C4#1427365314yes—
301C2×C4(C22⋊D59)⋊C412yes—
302C2×C4#1427369912yes—
303C8—82no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—4722no—
311C22—4721no—
312C22—4721no—
313C22—4721no—
314C22—4722no—
315C22#1683727911yescharacteristic
316C22#1683737512yes—
317C22#1683739911yescharacteristic
318C22#1683740811yescharacteristic
319C22C2×#1427365312yes—
320C22#1683754912yes—
321C22#1683772712yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—4722no—
327C4—4721no—
328C4—2368no—
329C2—22no—
330C2—22no—
331C2—9441no—
332C2#2142258312yes—
333C2#2142260512yes—
334C2#2142441611yescharacteristic
335C2#2142453711yescharacteristic
336C2#2142455211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1817D59
hypocenterC59#9149
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#16837248C2semidirect
4. #21421976C2semidirect
5. C59×#1817C2semidirect
9. #16837304C4semidirect
10. #16837300C4semidirect
163. #1817D59semidirect
189. C59#9149semidirect

Maximal subgroups

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

subgroupindex
#214324272
C2×#168372482
C2×#168371372
#214219762
C59×#18172
#914959