Finite Group Databasebeta

← #28499069

Subgroups — #28499069

Order 15104.

402 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427387C211yesmaximal, characteristic
2C4×#14273646C211yesmaximal, characteristic
3#21422127C212yesmaximal
4#21422103C212yesmaximal
5C59×#1461C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic
7C2×#14273646—21no—
8C4×#11745664—21no—
9#16837333—21no—
10#16837332C411yescharacteristic
11#16837332C2211yescharacteristic
12#16837322C411yescharacteristic
13#16837322C2211yescharacteristic
14#16837321—21no—
15C4×Dic59⋊C4—21no—
16C4×Dic59⋊C4—21no—
17#16837248C411yescharacteristic
18#16837171C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C2×#11745664—22no—
25C4×Dic118—41no—
26C4×Dic118—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35#14273646—24no—
36#14273638—41no—
37C4×C59⋊C8—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×Q8×C59D411yescharacteristic
46C4×Q8×C59—41no—
47C4×Q8×C59—41no—
48C2×C4⋊C4×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C22⋊C8×C59—22no—
56C4×C8×C59—41no—
57C2×Q8×C59—22no—
58C2×Q8×C59—22no—
59C2×Q8×C59—41no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×Dic118—41no—
72C2×Dic118—42no—
73C2×C8×C59—42no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C4⋊Dic59—41no—
84Dic59⋊C4—42no—
85Dic59⋊C4—42no—
86Dic59⋊C4—42no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89C4×Dic59—24no—
90C4×Dic59—41no—
91C4×Dic59—41no—
92#11745664—24no—
93#11745664—42no—
94C2×C59⋊C8—42no—
95C23×C59C2×C22⋊C411yescharacteristic
96Q8×C59—44no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59#9911yescharacteristic
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#10412yes—
114C2×C4×C59#13512yes—
115C2×Dic59—22no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127Dic118—42no—
128Dic118—42no—
129C8×C59—82no—
130C59⋊C8—44no—
131#11736—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59C2×#9911yescharacteristic
135C22×C59#41611yescharacteristic
136C22×C59C2×#10411yescharacteristic
137C4×C59—21no—
138C4×C59—22no—
139C4×C59—22no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59#35612yes—
145Dic59—42no—
146Dic59—42no—
147Dic59—42no—
148Dic59—81no—
149#2399—591no—
150C4×#101—591no—
151#1461D5911yescharacteristic
152#1354—592no—
153#1346—592no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59#150111yescharacteristic
157C2×C59#212011yescharacteristic
158C2×C59C2×#35611yescharacteristic
159C2×C4×Q8—591no—
160C2×C4×Q8C2×D5911yescharacteristic
161#431—21no—
162C2×#101—1181no—
163C4×M16—1181no—
164#398—591no—
165#398—591no—
166#398—1181no—
167#395—591no—
168#395—591no—
169#395—1181no—
170C4×C4⋊C4—1181no—
171C4×C4⋊C4—1181no—
172#376—591no—
173#343—592no—
174#324—21no—
175#324C2×D5912yes—
176C59#1173611yescharacteristic, complemented, Sylow
177C22×Q8—591no—
178C22×Q8D23611yescharacteristic
179C2×M16—1182no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—2361no—
183C4×Q8—2361no—
184C2×C4⋊C4—591no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4C4×D5911yescharacteristic
191C2×C4⋊C4C22⋊D5911yescharacteristic
192C2×C4⋊C4C22×D5911yescharacteristic
193C2×C42—1181no—
194C2×C42—1181no—
195C2×C42C4×D5911yescharacteristic
196C2×C42D23611yescharacteristic
197C2×C42C22⋊D5911yescharacteristic
198C4⋊C8—41no—
199#101—1184no—
200C22⋊C8—22no—
201#97—2361no—
202C4×C8—41no—
203C4×C8—2361no—
204#95—591no—
205#95—591no—
206#95—1181no—
207#95—1181no—
208#95—1181no—
209#95—1181no—
210#95—1181no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×Q8—2361no—
215C2×Q8—2362no—
216C22×C4—1181no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4—1181no—
220C22×C4(C2×D59)⋊C411yescharacteristic
221C22×C4(C2×D59)⋊C411yescharacteristic
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4C2×C4×D5911yescharacteristic
225C22×C4C2×D23611yescharacteristic
226C22×C4C2×C22⋊D5911yescharacteristic
227M16—1184no—
228M16—2362no—
229C2×C8—42no—
230C2×C8—2362no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—2361no—
235C4⋊C4—2362no—
236C4⋊C4—2362no—
237C4⋊C4—2362no—
238C4⋊C4—2362no—
239C4⋊C4—2362no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—1184no—
247C42—2361no—
248C42—2361no—
249C23C2×(C2×D59)⋊C411yescharacteristic
250Q8—44no—
251Q8—2362no—
252Q8—2362no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—1182no—
268C2×C4—1182no—
269C2×C4—1182no—
270C2×C4—2361no—
271C2×C4—2361no—
272C2×C4—2361no—
273C2×C4—2361no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4#1427364111yescharacteristic
280C2×C4#1427364111yescharacteristic
281C2×C4#1427365312yes—
282C2×C4#1427374912yes—
283C8—82no—
284C8—2364no—
285C22—21no—
286C22—21no—
287C22C2×#1427364111yescharacteristic
288C22#1683739911yescharacteristic
289C22C2×#1427365311yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—4721no—
298C4—2362no—
299C4—2362no—
300C4—2362no—
301C4#1683720012yes—
302C2—21no—
303C2—21no—
304C2#2142258811yescharacteristic
305C2#2142573511yescharacteristic
306C2C2×#1683720011yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1461C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1461D59
hypocenterC59#11736
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
176. C59#11736semidirect

Maximal subgroups

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

subgroupindex
#214273872
C4×#142736462
#214221272
#214221032
C59×#14612
#1173659