Finite Group Databasebeta

← #28505528

Subgroups — #28505528

Order 15104.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16837201C211yesmaximal, characteristic
2#21424490C211yesmaximal, characteristic
3#21424310C212yesmaximal
4#21422223C212yesmaximal
5C59×#2369C211yesmaximal, characteristic
6#16837880C2211yescharacteristic
7#16837442—22no—
8#16837411—21no—
9C2×#14273646C2211yescharacteristic
10C2×#14273646C2212yes—
11#16837201C414yescomplemented
12#16837162—21no—
13#16837162—22no—
14C2×C4⋊Q8×C59C2211yescharacteristic
15C59×#391—21no—
16C59×#391C2212yes—
17C59×#389—22no—
18C4×C4⋊C4×C59—22no—
19C4×C4⋊C4×C59—21no—
20C2×#11745664—22no—
21C2×#11745664—21no—
22Q8×Dic59—41no—
23Q8×Dic59—41no—
24#14273686—21no—
25#14273686—22no—
26#14273660—42no—
27#14273658—42no—
28#14273651—41no—
29#14273651—41no—
30#14273646—22no—
31#14273646—22no—
32#14273646C2×C414yes—
33C22×Q8×C59D411yescharacteristic
34C22×Q8×C59C2311yescharacteristic
35C4⋊Q8×C59—22no—
36C4⋊Q8×C59C2×C412yes—
37C4⋊Q8×C59Q812yes—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59D412yes—
43C2×C42×C59—22no—
44C2×C42×C59—21no—
45C2×C42×C59D411yescharacteristic
46C2×C42×C59D412yes—
47#95×C59—22no—
48#95×C59—22no—
49#95×C59—22no—
50C2×Q8×C59—21no—
51C2×Q8×C59—41no—
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59—41no—
55C2×Q8×C59C4⋊C412yes—
56C2×Q8×C59C22×C411yescharacteristic
57C2×Q8×C59C2×Q811yescharacteristic
58C2×Q8×C59#4112yes—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65C4⋊C4×C59—24no—
66C4⋊C4×C59—24no—
67C4⋊C4×C59—24no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—81no—
71C42×C59—21no—
72C42×C59—44no—
73C42×C59—42no—
74C42×C59—81no—
75C42×C59C4⋊C412yes—
76C22⋊Dic59—41no—
77C4⋊Dic59—21no—
78C4⋊Dic59—21no—
79C4⋊Dic59—22no—
80C4⋊Dic59—22no—
81C4×Dic59—42no—
82C4×Dic59—41no—
83#11745664—24no—
84#11745664—42no—
85#11745664—41no—
86#11745664—41no—
87C2×C59⋊C8—42no—
88C2×C59⋊C8—41no—
89C23×C59#12011yescharacteristic
90Q8×C59—41no—
91Q8×C59—41no—
92Q8×C59—42no—
93Q8×C59—42no—
94Q8×C59—81no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—24no—
103C2×C4×C59—22no—
104C2×C4×C59—42no—
105C2×C4×C59—41no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59C2×C4⋊C411yescharacteristic
112C2×C4×C59C4×D412yes—
113C2×C4×C59C22⋊Q812yes—
114C2×C4×C59#12311yescharacteristic
115C2×Dic59—41no—
116C2×Dic59—41no—
117C59⋊C8—44no—
118C59⋊C8—81no—
119#12973—591nomaximal, Sylow
120C22×C59—21no—
121C22×C59—21no—
122C22×C59#38511yescharacteristic
123C22×C59B(4,𝔽2)11yescharacteristic
124C22×C59#45811yescharacteristic
125C4×C59—24no—
126C4×C59—22no—
127C4×C59—21no—
128C4×C59—21no—
129C4×C59—22no—
130C4×C59—44no—
131C4×C59—42no—
132C4×C59—82no—
133Dic59—81no—
134Dic59—81no—
135#2369D5911yescharacteristic
136C2×#356—591no—
137#1886—591no—
138#1857—592no—
139#1394—592no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59#186711yescharacteristic
143C2×C59#220411yescharacteristic
144C2×C59#220511yescharacteristic
145C2×C4⋊Q8C2×D5911yescharacteristic
146#519—591no—
147#428—1182no—
148#419—1181no—
149C2×#101—592no—
150C2×#101—591no—
151#391—21no—
152#391C2×D5912yes—
153#389—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—21no—
156#356—594no—
157#339—1182no—
158#339—1181no—
159C59#1297311yescharacteristic, complemented, Sylow
160C22×Q8C22⋊D5911yescharacteristic
161C22×Q8C22×D5911yescharacteristic
162C2×M16—1182no—
163C2×M16—1181no—
164C4⋊Q8—22no—
165C4⋊Q8Dic11812yes—
166C4⋊Q8C4×D5912yes—
167C4×Q8—2361no—
168C4×Q8—2361no—
169#117—1181no—
170#117—1182no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C22⋊D5912yes—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42C22⋊D5911yescharacteristic
179C2×C42C22⋊D5912yes—
180#107—2362no—
181#106—2362no—
182#103—2361no—
183#103—2361no—
184#101—594no—
185#101—1182no—
186#101—1182no—
187#95—22no—
188#95—22no—
189#95—22no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8Dic59⋊C412yes—
196C2×Q8C2×Dic11811yescharacteristic
197C2×Q8C2×C4×D5911yescharacteristic
198C2×Q8#1174568512yes—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4C2×C22⋊D5911yescharacteristic
204C22×C4C2×C22⋊D5912yes—
205M16—1184no—
206M16—2361no—
207M16—2361no—
208M16—2362no—
209C2×C8—2361no—
210C2×C8—2362no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—1181no—
218C4⋊C4—1181no—
219C4⋊C4—1182no—
220C4⋊C4—1182no—
221C22⋊C4—2361no—
222C42—21no—
223C42—42no—
224C42—44no—
225C42—81no—
226C42—2362no—
227C42—2361no—
228C42Dic59⋊C412yes—
229C23#1427369611yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—42no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—2361no—
252C2×C4—2361no—
253C2×C4C2×Dic59⋊C411yescharacteristic
254C2×C4C4×C22⋊D5912yes—
255C2×C4#1427370612yes—
256C2×C4#1427371011yescharacteristic
257C8—4721no—
258C8—2364no—
259C22—21no—
260C22—21no—
261C22#1683727511yescharacteristic
262C22#1683758911yescharacteristic
263C22#1683759211yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—4721no—
273C4—4721no—
274C2—21no—
275C2—21no—
276C2#2142437711yescharacteristic
277C2#2142612211yescharacteristic
278C2#2142612611yescharacteristic
subgroupG/H
centerC22#16837589
commutatorC2×Q8×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#2369C2
socleC22×C59B(4,𝔽2)
prime radicalC23×C59#120
cosocleC22×Q8×C59C23
hypercenter#2369D59
hypocenterC59#12973
LCSG ▷ C2×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2369

Complements

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

NH ≅ G/Ntype
11. #16837201C4semidirect
159. C59#12973semidirect

Maximal subgroups

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

subgroupindex
C2×#168372012
#214244902
#214243102
#214222232
C59×#23692
#1297359