Finite Group Databasebeta

← #28505523

Subgroups — #28505523

Order 15104.

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

normal subgroups only

This group has 492 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×#16837199C211yesmaximal, characteristic
2#21424500C211yesmaximal, characteristic
3#21424310C212yesmaximal, complemented
4#21422223C212yesmaximal, complemented
5C59×#2372C211yesmaximal, characteristic
6#16837880C2211yescharacteristic
7#16837442—22no—
8#16837407—21no—
9C2×#14273646C2211yescharacteristic
10C2×#14273646C2212yes—
11#16837199C414yescomplemented
12#16837162—21no—
13#16837162—22no—
14C2×#124×C59C2211yescharacteristic
15C59×#391C2212yes—
16C59×#388—22no—
17C59×#386—21no—
18C4×C22⋊C4×C59—22no—
19C59×#376—21no—
20C2×#11745664—22no—
21C2×#11745664—21no—
22Q8×Dic59—42no—
23#14273686—21no—
24#14273686—22no—
25#14273660—42no—
26#14273658—42no—
27#14273648—42no—
28#14273646—22no—
29#14273646—22no—
30#14273646C2×C414yescomplemented
31C22×Q8×C59C2311yescharacteristic
32C22×D4×C59D411yescharacteristic
33C23×C4×C59—21no—
34#124×C59—22no—
35#124×C59C2×C412yes—
36#124×C59Q812yes—
37C2×C4⋊C4×C59D412yes—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—21no—
41C2×C42×C59—22no—
42C2×C42×C59D411yescharacteristic
43C2×C42×C59D412yes—
44#95×C59—21no—
45#95×C59—21no—
46#95×C59—22no—
47#95×C59—22no—
48C24×C59—21no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59C22×C411yescharacteristic
52C2×Q8×C59C2×Q811yescharacteristic
53C2×Q8×C59#4112yes—
54C2×D4×C59—21no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C4⋊C412yes—
58C22×C4×C59—22no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—44no—
62C22×C4×C59—81no—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65C4⋊C4×C59—42no—
66C22⋊C4×C59—24no—
67C22⋊C4×C59—24no—
68C22⋊C4×C59—44no—
69C22⋊C4×C59—81no—
70C42×C59—21no—
71C42×C59—44no—
72C42×C59—42no—
73C42×C59C4⋊C412yes—
74C22⋊Dic59—41no—
75C4⋊Dic59—22no—
76C4⋊Dic59—22no—
77C4⋊Dic59—22no—
78C4×Dic59—42no—
79C4×Dic59—41no—
80#11745664—24no—
81#11745664—42no—
82#11745664—41no—
83#11745664—41no—
84C2×C59⋊C8—42no—
85C2×C59⋊C8—41no—
86C23×C59—22no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59#12011yescharacteristic
92Q8×C59—41no—
93Q8×C59—41no—
94Q8×C59—42no—
95D4×C59—42no—
96D4×C59—81no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—22no—
100C2×C4×C59—21no—
101C2×C4×C59—24no—
102C2×C4×C59—22no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—84no—
110C2×C4×C59C2×C4⋊C411yescharacteristic
111C2×C4×C59C4×D412yes—
112C2×C4×C59C22⋊Q812yes—
113C2×C4×C59#12311yescharacteristic
114C2×Dic59—42no—
115C59⋊C8—44no—
116C59⋊C8—81no—
117#12972—591nomaximal, Sylow
118C22×C59—22no—
119C22×C59—44no—
120C22×C59—42no—
121C22×C59—81no—
122C22×C59—81no—
123C22×C59—82no—
124C22×C59#38511yescharacteristic
125C22×C59B(4,𝔽2)11yescharacteristic
126C22×C59#45811yescharacteristic
127C4×C59—22no—
128C4×C59—21no—
129C4×C59—21no—
130C4×C59—22no—
131C4×C59—42no—
132C4×C59—44no—
133C4×C59—82no—
134Dic59—82no—
135#2372D5911yescharacteristic
136C2×#355—591no—
137#1888—591no—
138#1857—592no—
139#1394—592no—
140C2×C59—21no—
141C2×C59—21no—
142C2×C59—82no—
143C2×C59#186711yescharacteristic
144C2×C59#220412yes—
145C2×#124C2×D5911yescharacteristic
146#519—591no—
147#428—1182no—
148#418—1181no—
149C2×#101—592no—
150C2×#101—591no—
151#391C2×D5912yes—
152#388—22no—
153#386—21no—
154C4×C22⋊C4—22no—
155#376—21no—
156#355—594no—
157#339—1182no—
158#339—1181no—
159C59#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D5911yescharacteristic
161C22×D4C22⋊D5911yescharacteristic
162C23×C4—21no—
163C2×M16—1182no—
164C2×M16—1181no—
165#124—22no—
166#124Dic11812yes—
167#124C4×D5912yes—
168C4×Q8—2362no—
169#117—1181no—
170#117—1182no—
171C2×C4⋊C4C22⋊D5912yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C42—22no—
176C2×C42C22⋊D5911yescharacteristic
177C2×C42C22⋊D5912yes—
178#107—2362no—
179#106—2362no—
180#102—2362no—
181#101—594no—
182#101—1182no—
183#101—1182no—
184#95—21no—
185#95—21no—
186#95—22no—
187#95—22no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic11811yescharacteristic
192C2×Q8C2×C4×D5911yescharacteristic
193C2×Q8#1174568512yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4Dic59⋊C412yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4C2×C22⋊D5911yescharacteristic
204C22×C4C2×C22⋊D5912yes—
205M16—1184no—
206M16—2361no—
207M16—2361no—
208M16—2362no—
209C2×C8—2361no—
210C2×C8—2362no—
211C4⋊C4—42no—
212C4⋊C4—1182no—
213C4⋊C4—1182no—
214C4⋊C4—1182no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—81no—
219C22⋊C4—2361no—
220C42—21no—
221C42—42no—
222C42—44no—
223C42—2362no—
224C42—2361no—
225C42Dic59⋊C412yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1427369611yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—2362no—
251C2×C4C2×Dic59⋊C411yescharacteristic
252C2×C4C4×C22⋊D5912yes—
253C2×C4#1427370612yes—
254C2×C4#1427371011yescharacteristic
255C8—4721no—
256C8—2364no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1683727511yescharacteristic
264C22#1683758911yescharacteristic
265C22#1683759211yescharacteristic
266C4—21no—
267C4—22no—
268C4—21no—
269C4—22no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—4722no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2142437711yescharacteristic
278C2#2142612212yes—
subgroupG/H
centerC22#16837589
commutatorC2×Q8×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC59×#2372C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
cosocleC22×Q8×C59C23
hypercenter#2372D59
hypocenterC59#12972
LCSG ▷ C2×Q8×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #21424310C2semidirect
4. #21422223C2semidirect
11. #16837199C4semidirect
30. #14273646C2×C4semidirect
159. C59#12972semidirect

Maximal subgroups

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

subgroupindex
C2×#168371992
#214245002
#214243102
#214222232
C59×#23722
#1297259