Finite Group Databasebeta

← #11997069

Subgroups — #11997069

Order 1280.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2972C211yesmaximal, characteristic
2#10614105C211yesmaximal, characteristic
3#10602412C211yesmaximal, characteristic
4#10600718C212yesmaximal, complemented
5#10598167C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C22×C4⋊Dic5C2211yescharacteristic
8C5×C8×D4—24no—
9C5×#431—21no—
10C2×C5×C22⋊C8—21no—
11C2×C5×C22⋊C8C2212yes—
12#65509—21no—
13C2×#3966—21no—
14#65501—21no—
15#65501C2212yes—
16#65501C2211yescharacteristic
17#65275C412yescomplemented
18#65238C412yescomplemented
19#65235—21no—
20#65217—21no—
21#64771—24no—
22#64746—21no—
23#9469—51nomaximal, Sylow
24C22×C5×D4D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C23×Dic5—21no—
28C22×C5×C8—22no—
29C22×C5×C8D414yes—
30C4×C5×D4—22no—
31C4×C5×D4—22no—
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C22⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4D411yescharacteristic
35C2×C42×C5D411yescharacteristic
36C2×C22⋊Dic5—21no—
37C2×C22⋊Dic5—22no—
38C2×C22⋊Dic5—22no—
39C2×C4⋊Dic5—22no—
40C2×C4⋊Dic5—22no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5—21no—
43C2×C4⋊Dic5—22no—
44C2×C4⋊Dic5C2×C412yes—
45C2×#683—22no—
46C5×C4⋊C8—22no—
47C5×C22⋊C8—22no—
48C5×C22⋊C8—22no—
49C5×C22⋊C8—22no—
50C4×C5×C8—22no—
51#3966—42no—
52#3963—22no—
53#3963—22no—
54#3963—22no—
55#3963C2×C412yes—
56#2972D511yescharacteristic
57#2487—51no—
58#1824—52no—
59#1799—51no—
60#1289—52no—
61C24×C5C22⋊C411yescharacteristic
62C24×C5C2×D411yescharacteristic
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4SD1612yes—
66C2×C5×D4Q1612yes—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22⋊C411yescharacteristic
73C22×C4×C5C22⋊C411yescharacteristic
74C22×C4×C5C22×C411yescharacteristic
75C22×C4×C5C2×D412yes—
76C22×C4×C5C2×D411yescharacteristic
77C22×C4×C5#4112yes—
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—42no—
81C22×Dic5—42no—
82C22×Dic5—82no—
83C2×C5×C8—24no—
84C2×C5×C8—44no—
85C2×C5×C8—82no—
86C5×C4⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4—22no—
89C42×C5—21no—
90C42×C5—21no—
91C22⋊Dic5—42no—
92C22⋊Dic5—42no—
93C22⋊Dic5—42no—
94C22⋊Dic5—82no—
95C4⋊Dic5—24no—
96C4⋊Dic5—42no—
97C4⋊Dic5—41no—
98C4⋊Dic5—41no—
99C4⋊Dic5—44no—
100#683—82no—
101C2×C4×D4C2×D511yescharacteristic
102C22×C4⋊C4—51no—
103C8×D4—24no—
104#431—21no—
105C2×#99—101no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C2×D512yes—
108#400—52no—
109#399—52no—
110#389—101no—
111#386—101no—
112#385—101no—
113#380—51no—
114#380—52no—
115#380—101no—
116#342—101no—
117#340—104no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5#10312yes—
125C23×C5C2×C22⋊C411yescharacteristic
126C23×C5#12112yes—
127C23×C5C2×SD1611yescharacteristic
128C23×C5C2×Q1611yescharacteristic
129C23×C5#13512yes—
130C5×D4—24no—
131C5×D4—24no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—24no—
135C2×C4×C5—22no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—41no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—81no—
144C2×C4×C5#10312yes—
145C2×C4×C5C4×D412yes—
146C2×C4×C5#12011yescharacteristic
147C2×C4×C5#12311yescharacteristic
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154C2×Dic5—84no—
155C2×Dic5—82no—
156C2×Dic5—82no—
157C2×Dic5—82no—
158C5×C8—44no—
159C22×D4D2011yescharacteristic
160C23×C4—101no—
161C23×C4C22⋊D511yescharacteristic
162C23×C4C22×D511yescharacteristic
163C22×C8—22no—
164C22×C8C22⋊D514yes—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—52no—
168C2×C4⋊C4—102no—
169C2×C4⋊C4—102no—
170C2×C4⋊C4—101no—
171C2×C4⋊C4—101no—
172C2×C4⋊C4—102no—
173C2×C4⋊C4—102no—
174C2×C4⋊C4C4×D511yescharacteristic
175C2×C22⋊C4—101no—
176C2×C22⋊C4—102no—
177C2×C22⋊C4—102no—
178C2×C22⋊C4C4×D511yescharacteristic
179C2×C22⋊C4D2011yescharacteristic
180C2×C42C22⋊D511yescharacteristic
181C4⋊C8—22no—
182#99—202no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C4×C8—22no—
187#95—52no—
188#95—102no—
189#95—102no—
190#95—102no—
191C22×C5—21no—
192C22×C5—22no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—22no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—42no—
199C22×C5—42no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—81no—
203C22×C5#38611yescharacteristic
204C22×C5C2×#10311yescharacteristic
205C22×C5#41611yescharacteristic
206C22×C5#46512yes—
207C22×C5#46712yes—
208C4×C5—21no—
209C4×C5—22no—
210C4×C5—22no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5#45812yes—
214Dic5—82no—
215Dic5—82no—
216Dic5—162no—
217C24#68511yescharacteristic
218C24C2×D2011yescharacteristic
219C2×D4—22no—
220C2×D4—22no—
221C2×D4#67712yes—
222C2×D4Dic2012yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—102no—
228C22×C4—102no—
229C22×C4—202no—
230C22×C4—202no—
231C22×C4—402no—
232C22×C4#68511yescharacteristic
233C22×C4#68511yescharacteristic
234C22×C4#68511yescharacteristic
235C22×C4C2×C4×D511yescharacteristic
236C22×C4#70912yes—
237C22×C4C2×C22⋊D512yes—
238C22×C4C2×C22⋊D511yescharacteristic
239C2×C8—24no—
240C2×C8—44no—
241C2×C8—82no—
242C4⋊C4—22no—
243C4⋊C4—104no—
244C4⋊C4—202no—
245C4⋊C4—204no—
246C4⋊C4—201no—
247C4⋊C4—201no—
248C4⋊C4—402no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—202no—
252C22⋊C4—202no—
253C22⋊C4—202no—
254C22⋊C4—402no—
255C42—21no—
256C42—21no—
257C2×C5—21no—
258C2×C5—22no—
259C2×C5—21no—
260C2×C5—22no—
261C2×C5—42no—
262C2×C5#189011yescharacteristic
263C2×C5#189211yescharacteristic
264C2×C5#191011yescharacteristic
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#394812yes—
272C23C2×#67711yescharacteristic
273C23#405012yes—
274C23C2×Dic2011yescharacteristic
275C23C2×#68511yescharacteristic
276C23#407612yes—
277D4—24no—
278D4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—202no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4—202no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—404no—
298C2×C4—402no—
299C2×C4—402no—
300C2×C4—402no—
301C2×C4#394812yes—
302C2×C4C4×C22⋊D512yes—
303C2×C4#407511yescharacteristic
304C2×C4#409911yescharacteristic
305C8—44no—
306C5#946911yescharacteristic, complemented, Sylow
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22C2×#394811yescharacteristic
320C22#6540211yescharacteristic
321C22#6540312yes—
322C22#6540512yes—
323C22#6550211yescharacteristic
324C4—22no—
325C4—21no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—402no—
330C4—402no—
331C4—802no—
332C4#6553212yes—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#1060244811yescharacteristic
339C2#1060276511yescharacteristic
340C2#1060281811yescharacteristic
subgroupG/H
centerC22#65502
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2972C2
socleC22×C5#386
prime radicalC22×C5×D4D4
cosocleC23×C4×C5C23
hypercenter#2972D5
hypocenterC5#9469
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #10600718C2semidirect
5. #10598167C2semidirect
17. #65275C4semidirect
18. #65238C4semidirect
306. C5#9469semidirect

Maximal subgroups

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

subgroupindex
C5×#29722
#106141052
#106024122
#106007182
#105981672
#94695