Finite Group Databasebeta

← #12017973

Subgroups — #12017973

Order 1280.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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#10614115C211yesmaximal, characteristic
2C5×#1799C211yesmaximal, characteristic
3C4×#3966C211yesmaximal, characteristic
4#10598186C214yesmaximal
5C2×C4×C5×D4C2211yescharacteristic
6C2×#4015C411yescharacteristic, complemented
7C2×C5×#99C2212yes—
8C5×#380—21no—
9#65510C2212yes—
10C2×#3966C2212yes—
11#65501—21no—
12#65501—21no—
13C4×C22⋊Dic5—21no—
14#65266C412yescomplemented
15#65236—21no—
16#65236—21no—
17C4×C4⋊Dic5C411yescharacteristic, complemented
18C5×#342—21no—
19#64746—21no—
20#13299—51nomaximal, Sylow
21C22×C5×D4C2311yescharacteristic
22C23×C4×C5D411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C22×Dic10—21no—
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—41no—
28C2×C5×C4⋊C4C2×C411yescharacteristic
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4D411yescharacteristic
31C2×C5×C22⋊C4D411yescharacteristic
32C2×C42×C5C2×C411yescharacteristic
33C2×C22⋊Dic5—22no—
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—22no—
36C2×C4⋊Dic5—21no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5C2×C412yes—
39C2×C4×Dic5—41no—
40C2×C4×Dic5C2×C412yes—
41#4015—41no—
42#4015—41no—
43C5×#99—24no—
44C5×#95—41no—
45#3966—22no—
46#3966Q814yescomplemented
47#3963—21no—
48#3963—22no—
49#3963—22no—
50#3963—41no—
51#2563—51no—
52#1799D511yescharacteristic
53C4×#99—51no—
54#1393—54no—
55C24×C5C2×D411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C2×C5×D4C2×Q812yes—
60C2×C5×D4#4112yes—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22×C411yescharacteristic
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—42no—
76C2×Dic10—42no—
77C2×Dic10—82no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—82no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84C42×C5—21no—
85C42×C5—21no—
86C22⋊Dic5—24no—
87C22⋊Dic5—42no—
88C22⋊Dic5—82no—
89C22⋊Dic5—82no—
90C4⋊Dic5—42no—
91C4⋊Dic5—41no—
92C4⋊Dic5—41no—
93C4⋊Dic5—42no—
94C4×Dic5—42no—
95C4×Dic5—82no—
96C2×C4⋊Q8—51no—
97C2×C4×D4C2×D511yescharacteristic
98C2×#99—52no—
99C2×#99C2×D512yes—
100#393—101no—
101#393—101no—
102#390—52no—
103#382—52no—
104#380—21no—
105#380—101no—
106#380—101no—
107C4×C4⋊C4—51no—
108C4×C22⋊C4—101no—
109#342—21no—
110#342—101no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5—41no—
119C23×C5C2×C22⋊C411yescharacteristic
120C23×C5C22⋊Q812yes—
121C23×C5#12411yescharacteristic
122C23×C5C4⋊Q811yescharacteristic
123C5×D4—42no—
124C5×D4—42no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—22no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—81no—
137C2×C4×C5—81no—
138C2×C4×C5—82no—
139C2×C4×C5—82no—
140C2×C4×C5—82no—
141C2×C4×C5#9911yescharacteristic
142C2×C4×C5#9911yescharacteristic
143C2×C4×C5C4×Q812yes—
144C2×Dic5—42no—
145C2×Dic5—44no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151C2×Dic5—82no—
152C2×Dic5—82no—
153Dic10—84no—
154C22×Q8—101no—
155C22×D4C22×D511yescharacteristic
156C23×C4D2011yescharacteristic
157C23×C4C22⋊D511yescharacteristic
158C4⋊Q8—201no—
159C4⋊Q8—201no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—52no—
164C2×C4⋊C4—101no—
165C2×C4⋊C4—101no—
166C2×C4⋊C4C4×D511yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—102no—
169C2×C22⋊C4—102no—
170C2×C22⋊C4—102no—
171C2×C22⋊C4D2011yescharacteristic
172C2×C22⋊C4C22⋊D511yescharacteristic
173C2×C42—52no—
174C2×C42—201no—
175C2×C42C4×D511yescharacteristic
176#99—24no—
177#99—54no—
178#99—102no—
179#95—41no—
180#95—101no—
181#95—102no—
182#95—102no—
183#95—201no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—22no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—42no—
190C22×C5—41no—
191C22×C5—41no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—81no—
197C22×C5—81no—
198C22×C5#39111yescharacteristic
199C22×C5C2×#9911yescharacteristic
200C22×C5#41011yescharacteristic
201C4×C5—21no—
202C4×C5—22no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—162no—
207C4×C5#35412yes—
208Dic5—82no—
209Dic5—84no—
210Dic5—82no—
211C24C2×D2011yescharacteristic
212C24C2×C22⋊D511yescharacteristic
213C2×Q8—202no—
214C2×Q8—402no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4#71112yes—
218C2×D4Q8×D512yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—102no—
227C22×C4—102no—
228C22×C4—202no—
229C22×C4#68511yescharacteristic
230C22×C4#68511yescharacteristic
231C22×C4#68511yescharacteristic
232C22×C4#68511yescharacteristic
233C22×C4C2×C4×D511yescharacteristic
234C4⋊C4—41no—
235C4⋊C4—82no—
236C4⋊C4—202no—
237C4⋊C4—202no—
238C4⋊C4—201no—
239C4⋊C4—201no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—104no—
245C22⋊C4—202no—
246C22⋊C4—402no—
247C22⋊C4—402no—
248C42—21no—
249C42—21no—
250C42—202no—
251C42—402no—
252C2×C5—21no—
253C2×C5—21no—
254C2×C5—42no—
255C2×C5—41no—
256C2×C5—41no—
257C2×C5#199611yescharacteristic
258C2×C5C2×#35411yescharacteristic
259C2×C5#212811yescharacteristic
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#404312yes—
269C23C2×#68511yescharacteristic
270C23#408211yescharacteristic
271C23#409011yescharacteristic
272Q8—404no—
273D4—42no—
274D4—42no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—204no—
292C2×C4—202no—
293C2×C4—202no—
294C2×C4—202no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—402no—
299C2×C4—402no—
300C2×C4#393811yescharacteristic
301C2×C4#393811yescharacteristic
302C2×C4#403312yes—
303C5#1329911yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22C2×#393811yescharacteristic
319C22#6526311yescharacteristic
320C22#6529511yescharacteristic
321C4—21no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—162no—
327C4—402no—
328C4—404no—
329C4—402no—
330C4#6469312yes—
331C2—21no—
332C2—21no—
333C2—41no—
334C2—42no—
335C2—41no—
336C2#1060032211yescharacteristic
337C2C2×#6469311yescharacteristic
338C2#1060085411yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#1799C2
socleC22×C5C2×#99
prime radical, cosocleC22×C5×D4C23
hypercenter#1799D5
hypocenterC5#13299
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C2×#4015C4semidirect
14. #65266C4semidirect
17. C4×C4⋊Dic5C4semidirect
46. #3966Q8semidirect
303. C5#13299semidirect

Maximal subgroups

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

subgroupindex
#106141152
C5×#17992
C4×#39662
#105981862
#132995