Finite Group Databasebeta

← #28504695

Subgroups — #28504695

Order 15104.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#21426810C211yesmaximal, characteristic
2C2×#16837198C212yesmaximal
3#21424090C212yesmaximal, complemented
4#21422223C211yesmaximal, characteristic, complemented
5C59×#1394C211yesmaximal, characteristic, complemented
6C2×#14273716C2211yescharacteristic
7C2×Q8×Dic59C2211yescharacteristic
8C2×#14273689C411yescharacteristic
9#16837425—22no—
10C2×#14273646C2212yes—
11#16837304C411yescharacteristic
12#16837291—22no—
13#16837276—22no—
14#16837271C411yescharacteristic
15#16837248C411yescharacteristic
16#16837198—24no—
17#16837170—22no—
18C2×#101×C59C2212yes—
19C59×#391C2211yescharacteristic
20C22×D236C2×C411yescharacteristic
21C22×C4×D59—21no—
22C2×#11745664—22no—
23#14273716—22no—
24#14273716D412yes—
25#14273716Q812yes—
26Q8×Dic59—42no—
27#14273689—42no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C4⋊Dic59D412yes—
30C2×Dic59⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—22no—
34C2×C4×Dic59C2×C411yescharacteristic
35C2×C4×Dic59C2×C411yescharacteristic
36C2×C4×Dic59D412yes—
37Dic59⋊C8—44no—
38#14273646—22no—
39#14273646—22no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×Q8×C59C2311yescharacteristic
43C2×M16×C59—22no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C42×C59C2×C411yescharacteristic
46#101×C59—22no—
47#101×C59—22no—
48#95×C59—22no—
49C23×D59—21no—
50C2×Q8×C59—41no—
51C2×Q8×C59—41no—
52C2×Q8×C59C2×D411yescharacteristic
53C2×Q8×C59C2×Q811yescharacteristic
54C2×Q8×C59#4112yes—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D412yes—
59C22×Dic59—21no—
60C22×Dic59C22⋊C411yescharacteristic
61C22×Dic59C22⋊C412yes—
62C22×Dic59C22⋊C411yescharacteristic
63C2×D236—21no—
64C2×D236—41no—
65C2×D236—41no—
66C2×D236C4⋊C412yes—
67C2×C4×D59—42no—
68C2×C4×D59—82no—
69M16×C59—42no—
70M16×C59—42no—
71C2×C8×C59—42no—
72C4⋊C4×C59—41no—
73C42×C59—41no—
74(C2×D59)⋊C4—24no—
75(C2×D59)⋊C4—24no—
76(C2×D59)⋊C4—42no—
77(C2×D59)⋊C4—82no—
78C4⋊Dic59—41no—
79C4⋊Dic59—42no—
80Dic59⋊C4—82no—
81C4×Dic59—21no—
82C4×Dic59—24no—
83C4×Dic59—22no—
84C4×Dic59C4⋊C412yes—
85#11745664—42no—
86#11745664—42no—
87C2×C59⋊C8—42no—
88C23×C59C2×C22⋊C411yescharacteristic
89C22×D59—22no—
90C22×D59—42no—
91C22×D59—42no—
92C22×D59—41no—
93C22×D59—41no—
94Q8×C59—41no—
95Q8×C59—41no—
96Q8×C59—42no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59C2×C4⋊C411yescharacteristic
107C2×C4×C59#11711yescharacteristic
108C2×C4×C59#12112yes—
109C2×C4×C59C22⋊Q812yes—
110C2×Dic59—22no—
111C2×Dic59—21no—
112C2×Dic59—22no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—42no—
117C2×Dic59—41no—
118C2×Dic59#9912yes—
119D236—42no—
120D236—42no—
121C4×D59—84no—
122C8×C59—82no—
123C59⋊C8—82no—
124#12803—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59#38011yescharacteristic
128C22×C59C2×#9911yescharacteristic
129C22×C59#41011yescharacteristic
130C2×D59—44no—
131C2×D59—42no—
132C2×D59—82no—
133C2×D59—81no—
134C2×D59—81no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—22no—
139C4×C59—81no—
140C4×C59—81no—
141Dic59—22no—
142Dic59—42no—
143Dic59—41no—
144Dic59—42no—
145#2326—591no—
146C2×#355—592no—
147#1801—592no—
148#1394—591no—
149#1394D5911yescharacteristic, complemented
150C2×C59—21no—
151C2×C59—21no—
152C2×C59#179911yescharacteristic
153C2×C59#213311yescharacteristic
154C2×C59#213411yescharacteristic
155D59—82no—
156C2×#124—591no—
157C2×C4×Q8—591no—
158C2×C4×D4—591no—
159#423—1182no—
160C2×#101—592no—
161C2×#101C2×D5912yes—
162#391C2×D5911yescharacteristic
163#390—591no—
164#388—1182no—
165#385—1182no—
166#384—591no—
167#376—591no—
168#355—1184no—
169#343—1182no—
170C59#1280311yescharacteristic, complemented, Sylow
171C22×Q8C22×D5911yescharacteristic
172C22×D4—591no—
173C23×C4—1181no—
174C2×M16—22no—
175C2×M16—1182no—
176#124—592no—
177#124—592no—
178#124—1182no—
179C4×Q8—2362no—
180C4×D4—2362no—
181C2×C4⋊C4—591no—
182C2×C4⋊C4—592no—
183C2×C4⋊C4—1182no—
184C2×C4⋊C4C4×D5911yescharacteristic
185C2×C22⋊C4—1182no—
186C2×C22⋊C4—1182no—
187C2×C22⋊C4—1181no—
188C2×C42—591no—
189C2×C42—591no—
190C2×C42—592no—
191C2×C42C4×D5911yescharacteristic
192C4⋊C8—2364no—
193#101—22no—
194#101—22no—
195#101—1182no—
196#101—1182no—
197#95—22no—
198#95—1181no—
199#95—1181no—
200C24—1181no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8#1174568512yes—
204C2×Q8D4×D5911yescharacteristic
205C2×Q8Q8×D5911yescharacteristic
206C2×D4—592no—
207C2×D4—1181no—
208C2×D4—2361no—
209C2×D4—2361no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—591no—
213C22×C4—591no—
214C22×C4—592no—
215C22×C4—1181no—
216C22×C4—4722no—
217C22×C4—2362no—
218C22×C4C2×C4×D5911yescharacteristic
219C22×C4D4×D5912yes—
220M16—42no—
221M16—42no—
222M16—2362no—
223M16—2362no—
224C2×C8—42no—
225C2×C8—2362no—
226C4⋊C4—41no—
227C4⋊C4—4722no—
228C4⋊C4—2361no—
229C4⋊C4—2362no—
230C22⋊C4—1184no—
231C22⋊C4—1184no—
232C22⋊C4—4722no—
233C22⋊C4—2362no—
234C42—41no—
235C42—592no—
236C42—1184no—
237C42—1182no—
238C42—1181no—
239C23—1182no—
240C23—2362no—
241C23—2362no—
242C23—2361no—
243C23—2361no—
244C23C22⋊C4×D5911yescharacteristic
245Q8—42no—
246Q8—41no—
247Q8—41no—
248D4—2362no—
249D4—2362no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—592no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—1181no—
263C2×C4—4724no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4—2362no—
268C2×C4—2361no—
269C2×C4C4⋊C4×D5911yescharacteristic
270C2×C4#1427368411yescharacteristic
271C2×C4#1427370012yes—
272C2×C4#1427370512yes—
273C8—82no—
274C8—4722no—
275C22—21no—
276C22—21no—
277C22—4722no—
278C22—4721no—
279C22—4721no—
280C22—2364no—
281C22—2362no—
282C22#1683726111yescharacteristic
283C22#99×D5911yescharacteristic
284C22#1683737411yescharacteristic
285C4—22no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—1182no—
292C4—2362no—
293C4—2361no—
294C4—2362no—
295C2—21no—
296C2—21no—
297C2—4722no—
298C2#2142408311yescharacteristic
299C2#2142578411yescharacteristic
300C2#2142578611yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#1394C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC22×Q8×C59C23
hypercenter#1394D59
hypocenterC59#12803
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #21424090C2semidirect
4. #21422223C2semidirect
5. C59×#1394C2semidirect
149. #1394D59semidirect
170. C59#12803semidirect

Maximal subgroups

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

subgroupindex
#214268102
C2×#168371982
#214240902
#214222232
C59×#13942
#1280359