Finite Group Databasebeta

← #12002292

Subgroups — #12002292

Order 1280.

640 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C5×#2646C211yesmaximal, characteristic
2#10600318C212yesmaximal
3#10600217C211yesmaximal, characteristic
4#10598115C212yesmaximal
5#10598104C211yesmaximal, characteristic
6C23×Dic10C2211yescharacteristic
7C5×#400—22no—
8C5×#398—22no—
9C5×#385—22no—
10C5×#380C2212yes—
11C2×C5×#95C2211yescharacteristic
12C2×#3963—22no—
13C2×#3963C2211yescharacteristic
14C2×C22⋊#168C2212yes—
15C2×#3942—22no—
16C2×#3942—22no—
17#10345—51nomaximal, Sylow
18C23×C4×C5D412yes—
19C23×C4×C5C2311yescharacteristic
20C23×Dic5—21no—
21C23×Dic5—21no—
22C22×Dic10—22no—
23C22×Dic10—22no—
24C22×Dic10—21no—
25C22×Dic10—44no—
26C22×Dic10Q812yes—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4D412yes—
30C2×C5×C4⋊C4D412yes—
31C2×C5×C22⋊C4—22no—
32C22×#168—22no—
33C5×#95—22no—
34C5×#95—22no—
35C5×#95—22no—
36C5×#95—22no—
37#3963—22no—
38#3963—44no—
39C22⋊#168—22no—
40C22⋊#168—22no—
41#3942—44no—
42#3942—44no—
43#2646D511yescharacteristic
44#1787—52no—
45#1446—51no—
46#1294—52no—
47#1280—51no—
48C24×C5C2×D411yescharacteristic
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—22no—
53C22×C4×C5—22no—
54C22×C4×C5—42no—
55C22×C4×C5—42no—
56C22×C4×C5—42no—
57C22×C4×C5C2×D412yes—
58C22×C4×C5C2×Q811yescharacteristic
59C22×C4×C5#4112yes—
60C22×C4×C5#4111yescharacteristic
61C22×Dic5—24no—
62C22×Dic5—22no—
63C22×Dic5—22no—
64C22×Dic5—42no—
65C22×Dic5—41no—
66C22×Dic5—41no—
67C22×Dic5—41no—
68C22×Dic5—42no—
69C22×Dic5—41no—
70C22×Dic5—42no—
71C2×Dic10—24no—
72C2×Dic10—24no—
73C2×Dic10—24no—
74C2×Dic10—44no—
75C2×Dic10—48no—
76C2×Dic10—44no—
77C2×Dic10—42no—
78C2×Dic10—42no—
79C2×Dic10—42no—
80C5×C4⋊C4—42no—
81C5×C4⋊C4—42no—
82C5×C4⋊C4—82no—
83C5×C4⋊C4—82no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—42no—
86C2×#168—42no—
87C2×#168—42no—
88C2×#168—42no—
89C23×Q8—51no—
90C2×#103—102no—
91C2×#103—102no—
92C2×C22⋊C8—52no—
93#400—22no—
94#398—22no—
95#385—22no—
96#380C2×D512yes—
97C2×#95—51no—
98C2×#95—102no—
99C2×#95C2×D511yescharacteristic
100C23×C5—21no—
101C23×C5—21no—
102C23×C5—22no—
103C23×C5#12011yescharacteristic
104C23×C5C22⋊Q811yescharacteristic
105C23×C5#12411yescharacteristic
106C23×C5#13512yes—
107C23×C5#13712yes—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—42no—
113C2×C4×C5—44no—
114C2×C4×C5—42no—
115C2×C4×C5—42no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5—42no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—82no—
124C2×C4×C5C22⋊Q812yes—
125C2×C4×C5#12412yes—
126C2×Dic5—24no—
127C2×Dic5—42no—
128C2×Dic5—44no—
129C2×Dic5—42no—
130C2×Dic5—44no—
131C2×Dic5—44no—
132C2×Dic5—42no—
133C2×Dic5—82no—
134C2×Dic5—81no—
135C2×Dic5—81no—
136Dic10—28no—
137Dic10—44no—
138Dic10—44no—
139Dic10—44no—
140#168—82no—
141C22×Q8—52no—
142C22×Q8—101no—
143C22×Q8—102no—
144C22×Q8—102no—
145C22×Q8—204no—
146C23×C4—101no—
147C23×C4—101no—
148C23×C4C22⋊D512yes—
149C23×C4C22×D511yescharacteristic
150C22×C8—102no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D512yes—
154C2×C4⋊C4C22⋊D512yes—
155C2×C22⋊C4—22no—
156#103—204no—
157#103—204no—
158C22⋊C8—102no—
159C22⋊C8—102no—
160#95—22no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—102no—
165#95—204no—
166C22×C5—22no—
167C22×C5—22no—
168C22×C5—22no—
169C22×C5—22no—
170C22×C5—21no—
171C22×C5—21no—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5#39311yescharacteristic
175C22×C5#45212yes—
176C22×C5#45511yescharacteristic
177C22×C5#45611yescharacteristic
178C22×C5B(4,𝔽2)12yes—
179C22×C5#47912yes—
180C22×C5#48811yescharacteristic
181C22×C5#49111yescharacteristic
182C4×C5—22no—
183C4×C5—22no—
184C4×C5—82no—
185C4×C5—82no—
186C4×C5—82no—
187Dic5—44no—
188Dic5—82no—
189C24C2×C22⋊D511yescharacteristic
190C2×Q8—104no—
191C2×Q8—104no—
192C2×Q8—104no—
193C2×Q8—202no—
194C2×Q8—202no—
195C2×Q8—204no—
196C2×Q8—204no—
197C2×Q8—208no—
198C2×Q8—202no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—104no—
208C22×C4—102no—
209C22×C4—102no—
210C22×C4—202no—
211C22×C4—201no—
212C22×C4—201no—
213C22×C4—202no—
214C22×C4—201no—
215C22×C4—202no—
216C22×C4—201no—
217C22×C4Q8×D511yescharacteristic
218C22×C4#71312yes—
219C22×C4#71311yescharacteristic
220C22×C4C2×C22⋊D512yes—
221C2×C8—202no—
222C2×C8—202no—
223C2×C8—202no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C2×C5—21no—
231C2×C5—22no—
232C2×C5—21no—
233C2×C5#160312yes—
234C2×C5#160612yes—
235C2×C5#202611yescharacteristic
236C2×C5#202811yescharacteristic
237C2×C5#202911yescharacteristic
238C23—21no—
239C23—22no—
240C23—21no—
241C23#409211yescharacteristic
242C23#409311yescharacteristic
243C23#409612yes—
244C23#409712yes—
245C23#409911yescharacteristic
246Q8—108no—
247Q8—204no—
248Q8—204no—
249Q8—204no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—104no—
267C2×C4—202no—
268C2×C4—204no—
269C2×C4—202no—
270C2×C4—204no—
271C2×C4—204no—
272C2×C4—202no—
273C2×C4—402no—
274C2×C4—401no—
275C2×C4—401no—
276C2×C4#409212yes—
277C2×C4#409312yes—
278C8—402no—
279C5#1034511yescharacteristic, complemented, Sylow
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22#6535212yes—
289C22#6535711yescharacteristic
290C22#6535811yescharacteristic
291C22#6552011yescharacteristic
292C22#6552812yes—
293C22#6553111yescharacteristic
294C22#6553312yes—
295C22#6553511yescharacteristic
296C4—22no—
297C4—22no—
298C4—82no—
299C4—82no—
300C4—82no—
301C4—204no—
302C4—402no—
303C2—21no—
304C2—22no—
305C2—21no—
306C2#1060253711yescharacteristic
307C2#1060267812yes—
308C2#1060268212yes—
309C2#1060269011yescharacteristic
310C2#1060269811yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC23×C4×C5C23
FrattiniC23×C4C22×D5
FittingC5×#2646C2
socleC23×C5#120
prime radicalC24×C5C2×D4
hypercenter#2646D5
hypocenterC5#10345
LCSG ▷ C23×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
279. C5#10345semidirect

Maximal subgroups

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

subgroupindex
C5×#26462
#106003182
#106002172
#105981152
#105981042
#103455