Finite Group Databasebeta

← #28497969

Subgroups — #28497969

Order 15104.

428 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C4×#14273725C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic
3#21422041C212yesmaximal
4#21422029C212yesmaximal
5C59×#1461C211yesmaximal, characteristic
6C2×#14273725C411yescharacteristic
7C2×C4×Dic118C411yescharacteristic
8#16837455—21no—
9#16837309C411yescharacteristic
10#16837307C2211yescharacteristic
11#16837297—21no—
12#16837297—21no—
13#16837271C2211yescharacteristic
14C4×Dic59⋊C4—21no—
15C4×Dic59⋊C4—21no—
16C42×Dic59C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#324C2212yes—
23C22×Dic118C2×C411yescharacteristic
24#14273725—22no—
25#14273725C814yescomplemented
26C4×Dic118—41no—
27C4×Dic118—41no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59C2×C411yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39C4⋊(C59⋊C8)—41no—
40C22⋊(C59⋊C8)—22no—
41C4×C59⋊C8—41no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×Q8×C59C2×C411yescharacteristic
47C4×Q8×C59—41no—
48C4×Q8×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59C2311yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C4⋊C8×C59—41no—
56C22⋊C8×C59—22no—
57C4×C8×C59—41no—
58C2×Q8×C59—41no—
59C2×Q8×C59—41no—
60C2×Q8×C59C2×C812yes—
61C2×Q8×C59M1612yes—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C2×Dic118—21no—
74C2×Dic118—41no—
75C2×Dic118—41no—
76C2×Dic118C2×C812yes—
77C2×C8×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C4⋊Dic59—41no—
88Dic59⋊C4—24no—
89Dic59⋊C4—24no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92Dic59⋊C4—42no—
93C4×Dic59—24no—
94C4×Dic59—21no—
95C4×Dic59—42no—
96C4×Dic59—41no—
97C4×Dic59—41no—
98C4×Dic59—41no—
99C4×Dic59C2×C812yes—
100C2×C59⋊C8—42no—
101C23×C59C2×C22⋊C411yescharacteristic
102Q8×C59—42no—
103Q8×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59C22⋊C812yes—
118C2×C4×C59C22⋊C812yes—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C2×C4×C59C22×C811yescharacteristic
122C2×C4×C59C2×M1611yescharacteristic
123C2×Dic59—22no—
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—22no—
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135Dic118—42no—
136Dic118—42no—
137C8×C59—82no—
138C59⋊C8—82no—
139#11478—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59C2×C22⋊C811yescharacteristic
143C22×C59C2×#9911yescharacteristic
144C22×C59#41311yescharacteristic
145C4×C59—21no—
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—22no—
149C4×C59—22no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59#35612yes—
154Dic59—24no—
155Dic59—42no—
156Dic59—42no—
157Dic59—81no—
158C4×C4⋊Q8—591no—
159#1461—591no—
160#1461D5911yescharacteristic
161#1325—592no—
162#1319—592no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59#145911yescharacteristic
166C2×C59#212611yescharacteristic
167C2×C59C2×#35611yescharacteristic
168C2×C4⋊Q8—591no—
169C2×C4×Q8—591no—
170C2×C4×Q8C2×D5911yescharacteristic
171#431—21no—
172#431—1181no—
173#391—591no—
174#391—591no—
175#389—1181no—
176#389—1181no—
177#384—591no—
178C4×C4⋊C4—1181no—
179C4×C4⋊C4—1181no—
180C43—591no—
181#324—21no—
182#324—592no—
183#324—1181no—
184#324C2×D5912yes—
185C59#1147811yescharacteristic, complemented, Sylow
186C22×Q8—591no—
187C22×Q8C4×D5911yescharacteristic
188C4⋊Q8—594no—
189C4⋊Q8—1182no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—2361no—
193C4×Q8—2361no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4D23611yescharacteristic
201C2×C4⋊C4C22⋊D5911yescharacteristic
202C2×C4⋊C4C22×D5911yescharacteristic
203C2×C42—591no—
204C2×C42—591no—
205C2×C42—1181no—
206C2×C42—1181no—
207C2×C42—1181no—
208C2×C42C4×D5911yescharacteristic
209C2×C42D23611yescharacteristic
210C2×C42C22⋊D5911yescharacteristic
211C4⋊C8—41no—
212C4⋊C8—2361no—
213C22⋊C8—22no—
214C22⋊C8—1182no—
215C4×C8—41no—
216C4×C8—2361no—
217#95—1181no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—592no—
224C2×Q8—1181no—
225C2×Q8—2361no—
226C2×Q8—2361no—
227C2×Q8C8×D5912yes—
228C2×Q8#1174565912yes—
229C22×C4—1181no—
230C22×C4—1181no—
231C22×C4—1181no—
232C22×C4—1181no—
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4C2×C4×D5911yescharacteristic
238C22×C4C2×D23611yescharacteristic
239C22×C4C2×C22⋊D5911yescharacteristic
240C2×C8—42no—
241C2×C8—2362no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—1184no—
246C4⋊C4—1184no—
247C4⋊C4—2361no—
248C4⋊C4—2362no—
249C4⋊C4—2362no—
250C4⋊C4—2362no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—592no—
258C42—1184no—
259C42—1181no—
260C42—2361no—
261C42—2362no—
262C42—2361no—
263C42—2361no—
264C23C2×(C2×D59)⋊C411yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—2362no—
268Q8—2362no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—1182no—
283C2×C4—1182no—
284C2×C4—1182no—
285C2×C4—1182no—
286C2×C4—1182no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4(C2×D59)⋊C812yes—
295C2×C4(C2×D59)⋊C812yes—
296C2×C4#1427364111yescharacteristic
297C2×C4#1427364111yescharacteristic
298C2×C4C2×C8×D5911yescharacteristic
299C2×C4C2×#1174565911yescharacteristic
300C8—82no—
301C8—4722no—
302C22—21no—
303C22—21no—
304C22C2×(C2×D59)⋊C811yescharacteristic
305C22C2×#1427364111yescharacteristic
306C22#1683738411yescharacteristic
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—1184no—
316C4—4721no—
317C4—2362no—
318C4—2362no—
319C4#1683720012yes—
320C2—21no—
321C2—21no—
322C2#2142241211yescharacteristic
323C2#2142575911yescharacteristic
324C2C2×#1683720011yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1461C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1461D59
hypocenterC59#11478
LCSG ▷ C2×C4×C59 ▷ C22×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
25. #14273725C8semidirect
185. C59#11478semidirect

Maximal subgroups

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

subgroupindex
C4×#142737252
#214224202
#214220412
#214220292
C59×#14612
#1147859