Finite Group Databasebeta

← #28484710

Subgroups — #28484710

Order 15104.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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#21432660C211yesmaximal, characteristic
2#21429862C211yesmaximal, characteristic
3#21422767C212yesmaximal, complemented
4#21421955C212yesmaximal, complemented
5C59×#1458C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273727C2211yescharacteristic
8C2×#14273706—21no—
9#16837429—24no—
10#16837426—21no—
11#16837426C2212yes—
12#16837332—21no—
13#16837332C412yes—
14#16837327—21no—
15#16837327C412yes—
16#16837308—21no—
17#16837268C2211yescharacteristic
18#16837147—24no—
19C2×#117×C59C2211yescharacteristic
20C59×#431C2212yes—
21C59×#324—21no—
22C59×#323—21no—
23C22×Dic118—21no—
24C2×#11745664—22no—
25C2×#11745664—22no—
26C2×#11745664—42no—
27C2×#11745664—42no—
28C22×C59⋊C8—22no—
29#14273727—22no—
30#14273727—22no—
31#14273706—42no—
32#14273706—42no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59C2×C412yes—
36C2×Dic59⋊C4—22no—
37C2×C22⋊Dic59—22no—
38C4⋊(C59⋊C8)—24no—
39C4⋊(C59⋊C8)—22no—
40C4⋊(C59⋊C8)D414yes—
41C59⋊#95—22no—
42C59⋊#95—22no—
43C59⋊#95—21no—
44C59⋊#95C2×C412yes—
45C23×C4×C59D411yescharacteristic
46#117×C59—22no—
47#117×C59—22no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59C2311yescharacteristic
54C4⋊C8×C59—22no—
55C22⋊C8×C59—22no—
56C4×C8×C59—22no—
57C24×C59C22⋊C411yescharacteristic
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59D812yes—
66C22×C4×C59SD1612yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—22no—
71C22×Dic59—22no—
72C2×Dic118—41no—
73C2×Dic118—41no—
74C2×Dic118—41no—
75C2×Dic118—41no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×C8×C59—42no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—22no—
81C22⋊C4×C59—22no—
82C22⋊C4×C59—22no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59C2×D412yes—
86C42×C59#4112yes—
87C22⋊Dic59—44no—
88C4⋊Dic59—24no—
89C4⋊Dic59—42no—
90C4⋊Dic59—41no—
91C4⋊Dic59—41no—
92Dic59⋊C4—44no—
93#11745664—44no—
94#11745664—42no—
95#11745664—42no—
96C2×C59⋊C8—24no—
97C2×C59⋊C8—44no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59#10212yes—
103C23×C59C2×C22⋊C411yescharacteristic
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59#10212yes—
117C2×C4×C59C4×D412yes—
118C2×C4×C59#12011yescharacteristic
119C2×C4×C59#12112yes—
120C2×C4×C59#12311yescharacteristic
121C2×C4×C59C2×D811yescharacteristic
122C2×C4×C59C2×SD1611yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C59#13711yescharacteristic
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—42no—
131Dic118—81no—
132Dic118—81no—
133Dic118—81no—
134Dic118—81no—
135C8×C59—82no—
136C59⋊C8—44no—
137#9195—591nomaximal, Sylow
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—22no—
141C22×C59—42no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—81no—
145C22×C59#38611yescharacteristic
146C22×C59C2×#10211yescharacteristic
147C22×C59#41711yescharacteristic
148C4×C59—22no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59#44711yescharacteristic
154C4×C59#44811yescharacteristic
155C4×C59#48011yescharacteristic
156C4×C59#48111yescharacteristic
157Dic59—82no—
158Dic59—82no—
159#2905—591no—
160#2621—591no—
161#1546—592no—
162#1458D5911yescharacteristic
163#1286—592no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—42no—
167C2×C59#187411yescharacteristic
168C2×C59#190911yescharacteristic
169C2×C59#191511yescharacteristic
170C22×M16—1181no—
171C2×C4⋊Q8—591no—
172C2×C22⋊Q8—1181no—
173C2×#117C2×D5911yescharacteristic
174#431C2×D5912yes—
175#424—1184no—
176#423—592no—
177#423—1181no—
178#398—592no—
179#398—1181no—
180#397—592no—
181#397—1181no—
182#391—1181no—
183#382—591no—
184#332—1184no—
185#324—21no—
186#323—21no—
187C59#919511yescharacteristic, complemented, Sylow
188C22×Q8—1181no—
189C23×C4D23611yescharacteristic
190C2×M16—1182no—
191C2×M16—1182no—
192C2×M16—2362no—
193C2×M16—2362no—
194C22×C8—1182no—
195C4⋊Q8—1182no—
196C4⋊Q8—1182no—
197C22⋊Q8—2362no—
198C22⋊Q8—2362no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—592no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4C4×D5911yescharacteristic
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C22⋊C4—1182no—
208C2×C22⋊C4C4×D5911yescharacteristic
209C2×C22⋊C4C22⋊D5911yescharacteristic
210C2×C42D23611yescharacteristic
211C2×C42C22×D5911yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—594no—
214C4⋊C8—1184no—
215C4⋊C8—1182no—
216C22⋊C8—22no—
217C4×C8—22no—
218#95—592no—
219#95—1182no—
220#95—1182no—
221#95—1181no—
222C24(C2×D59)⋊C411yescharacteristic
223C2×Q8—2361no—
224C2×Q8—2361no—
225C2×Q8—2361no—
226C2×Q8—2361no—
227C2×Q8—2361no—
228C2×Q8—2361no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—1182no—
234C22×C4—1182no—
235C22×C4#1174566012yes—
236C22×C4D47212yes—
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4C2×C4×D5911yescharacteristic
241C22×C4C2×D23611yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243M16—2364no—
244M16—2362no—
245M16—2362no—
246C2×C8—42no—
247C2×C8—1184no—
248C2×C8—2364no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—1184no—
252C4⋊C4—2364no—
253C4⋊C4—2362no—
254C4⋊C4—2361no—
255C4⋊C4—2361no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—2364no—
259C42—21no—
260C42—21no—
261C42D4×D5912yes—
262C42#1174568712yes—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#1427364912yes—
268C23C2×(C2×D59)⋊C411yescharacteristic
269Q8—4721no—
270Q8—4721no—
271Q8—4721no—
272Q8—4721no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—2362no—
286C2×C4—2362no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4#1427364912yes—
292C2×C4(C22⋊D59)⋊C412yes—
293C2×C4#1427369711yescharacteristic
294C2×C4#1427369912yes—
295C2×C4#1427371111yescharacteristic
296C2×C4C2×D47211yescharacteristic
297C2×C4C2×#1174566011yescharacteristic
298C2×C4#1427375511yescharacteristic
299C2×C4#1427376011yescharacteristic
300C8—82no—
301C8—2364no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#1683727911yescharacteristic
310C22C2×#1427364911yescharacteristic
311C22#1683740411yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—4722no—
318C4—4722no—
319C4#1683753811yescharacteristic
320C4#1683754311yescharacteristic
321C4#1683771611yescharacteristic
322C4#1683772111yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2142441111yescharacteristic
327C2#2142461611yescharacteristic
328C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1458C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1458D59
hypocenterC59#9195
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #21422767C2semidirect
4. #21421955C2semidirect
187. C59#9195semidirect

Maximal subgroups

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

subgroupindex
#214326602
#214298622
#214227672
#214219552
C59×#14582
#919559