Finite Group Databasebeta

← #28486027

Subgroups — #28486027

Order 15104.

532 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 103 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#21432515C211yesmaximal, characteristic
2#21429794C211yesmaximal, characteristic
3#21422699C212yesmaximal, complemented
4#21421988C212yesmaximal, complemented
5C59×#1464C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273719C2211yescharacteristic
8#16837358—24no—
9C4×#11745664—21no—
10C4×#11745664C2212yes—
11#16837341C411yescharacteristic
12#16837338C411yescharacteristic
13#16837332C411yescharacteristic
14#16837327C411yescharacteristic
15#16837297—21no—
16#16837296—21no—
17#16837286—21no—
18#16837275—21no—
19#16837248C2211yescharacteristic
20#16837143—24no—
21C2×#117×C59C2211yescharacteristic
22C59×#432C2212yes—
23C59×#324—21no—
24C59×#323—21no—
25C2×#11745664—24no—
26C2×#11745664—24no—
27C2×#11745664—22no—
28C2×#11745664—22no—
29C22×C59⋊C8—22no—
30#14273719—22no—
31#14273719—22no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C4×Dic59—21no—
41#14273638—22no—
42#14273638D414yes—
43C4×C59⋊C8—22no—
44C4×C59⋊C8—22no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95C2×C411yescharacteristic
51C59⋊#95C2×C411yescharacteristic
52C23×C4×C59D411yescharacteristic
53#117×C59—22no—
54#117×C59—22no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C2×C42×C59C2311yescharacteristic
61C4⋊C8×C59—22no—
62C22⋊C8×C59—22no—
63#97×C59—22no—
64C24×C59C22⋊C411yescharacteristic
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C2×C8×C59—42no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—22no—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59C2×D412yes—
87C42×C59#4112yes—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—22no—
91C4⋊Dic59—22no—
92Dic59⋊C4—42no—
93Dic59⋊C4—42no—
94Dic59⋊C4—41no—
95Dic59⋊C4—42no—
96Dic59⋊C4—41no—
97C4×Dic59—42no—
98#11745664—28no—
99#11745664—24no—
100#11745664—24no—
101C2×C59⋊C8—28no—
102C2×C59⋊C8—24no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#10412yes—
108C23×C59C2×C22⋊C411yescharacteristic
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—24no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59#10412yes—
123C2×C4×C59C4×D412yes—
124C2×C4×C59#12011yescharacteristic
125C2×C4×C59#12112yes—
126C2×C4×C59#12311yescharacteristic
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C8×C59—82no—
140C59⋊C8—28no—
141#9388—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—81no—
149C22×C59#38611yescharacteristic
150C22×C59C2×#10411yescharacteristic
151C22×C59C2×#10411yescharacteristic
152C4×C59—22no—
153C4×C59—21no—
154C4×C59—22no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—41no—
158C4×C59—41no—
159Dic59—42no—
160Dic59—81no—
161Dic59—81no—
162Dic59—81no—
163#2873—591no—
164#2611—591no—
165#1528—592no—
166#1464D5911yescharacteristic
167#1300—592no—
168C2×C59—21no—
169C2×C59—21no—
170C2×C59—42no—
171C2×C59#188411yescharacteristic
172C2×C59#189611yescharacteristic
173C2×C59#191511yescharacteristic
174C22×M16—1181no—
175C2×#125—591no—
176C2×#117C2×D5911yescharacteristic
177#432C2×D5912yes—
178#405—1184no—
179C4×M16—592no—
180C4×M16—1181no—
181#400—591no—
182#399—591no—
183#398—591no—
184#397—591no—
185#389—1181no—
186#388—1181no—
187#387—1181no—
188#385—1181no—
189#376—591no—
190#330—1184no—
191#324—21no—
192#323—21no—
193C59#938811yescharacteristic, complemented, Sylow
194C23×C4D23611yescharacteristic
195C2×M16—1182no—
196C2×M16—1184no—
197C2×M16—1184no—
198C2×M16—1182no—
199C22×C8—1182no—
200#125—1182no—
201#125—1182no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—591no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4C4×D5911yescharacteristic
211C2×C4⋊C4C22⋊D5911yescharacteristic
212C2×C22⋊C4—1181no—
213C2×C22⋊C4—1181no—
214C2×C22⋊C4C4×D5911yescharacteristic
215C2×C22⋊C4C22⋊D5911yescharacteristic
216C2×C42—1181no—
217C2×C42D23611yescharacteristic
218C2×C42C22×D5911yescharacteristic
219C4⋊C8—22no—
220C22⋊C8—22no—
221#97—22no—
222#97—594no—
223#97—1182no—
224C4×C8—1182no—
225C4×C8—1182no—
226#95—591no—
227#95—591no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233C24(C2×D59)⋊C411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4C2×D23611yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248M16—1188no—
249M16—1184no—
250M16—1184no—
251C2×C8—42no—
252C2×C8—1188no—
253C2×C8—1184no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—1182no—
257C4⋊C4—1182no—
258C4⋊C4—2362no—
259C4⋊C4—2362no—
260C4⋊C4—2361no—
261C4⋊C4—2362no—
262C4⋊C4—2361no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C42—21no—
268C42—21no—
269C42—2362no—
270C42D4×D5912yes—
271C42#1174568712yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1427365312yes—
277C23C2×(C2×D59)⋊C411yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—1182no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4#1427365312yes—
304C2×C4(C22⋊D59)⋊C412yes—
305C2×C4#1427369711yescharacteristic
306C2×C4#1427369912yes—
307C2×C4#1427371111yescharacteristic
308C8—82no—
309C8—1188no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1683727911yescharacteristic
318C22C2×#1427365311yescharacteristic
319C22C2×#1427365311yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—4721no—
328C4—4721no—
329C4—4721no—
330C4—2362no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2142447711yescharacteristic
335C2#2142453711yescharacteristic
336C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1464C2
socleC22×C59#386
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1464D59
hypocenterC59#9388
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #21422699C2semidirect
4. #21421988C2semidirect
193. C59#9388semidirect

Maximal subgroups

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

subgroupindex
#214325152
#214297942
#214226992
#214219882
C59×#14642
#938859