Finite Group Databasebeta

← #12015551

Subgroups — #12015551

Order 1280.

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

normal subgroups only

This group has 628 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
1#10615818C211yesmaximal, characteristic
2C5×#1785C211yesmaximal, characteristic
3#10602932C211yesmaximal, characteristic
4#10598189C214yesmaximal, complemented
5C22×C5×#41C2211yescharacteristic
6C2×#4072C411yescharacteristic, complemented
7C22×C4⋊Dic5C411yescharacteristic, complemented
8C2×C5×#101—21no—
9C2×C5×#100C2212yes—
10C5×#407—21no—
11#65519—21no—
12#65510—21no—
13#65509C412yescomplemented
14#65509C2212yes—
15C2×#3966—21no—
16C2×#3966C2212yes—
17#65501—21no—
18#65501—21no—
19#65500—21no—
20#12855—51nomaximal, Sylow
21C2×C5×#41—24no—
22C2×C5×#41—42no—
23C2×C5×#41—42no—
24C22×C5×Q8D411yescharacteristic
25C22×C5×D4C2×C411yescharacteristic
26C22×C5×D4D411yescharacteristic
27C22×C5×D4C2311yescharacteristic
28C23×C4×C5C2×C411yescharacteristic
29C23×C4×C5D411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C23×Dic5C2×C412yes—
32C2×C5×M16—22no—
33C2×C22⋊Dic5—22no—
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—22no—
36C2×C22⋊Dic5C2×C412yes—
37#4072—42no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—42no—
42C2×C4×Dic5—21no—
43C5×#101—42no—
44C5×#100—24no—
45C5×C22⋊C8—42no—
46#3966—24no—
47#3966—42no—
48#3963—22no—
49#3963—22no—
50#3963—42no—
51#2311—51no—
52#1785D511yescharacteristic
53#1782—51no—
54#1390—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C22×C411yescharacteristic
57C24×C5C2×D411yescharacteristic
58C5×#41—416no—
59C2×C5×Q8—22no—
60C2×C5×Q8—22no—
61C2×C5×Q8—41no—
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—41no—
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C2×C5×D4—41no—
69C2×C5×D4—41no—
70C2×C5×D4#4114yes—
71C22×C4×C5—22no—
72C22×C4×C5—22no—
73C22×C4×C5—22no—
74C22×C4×C5—22no—
75C22×C4×C5—22no—
76C22×C4×C5—22no—
77C22×C4×C5—22no—
78C22×C4×C5—41no—
79C22×C4×C5—41no—
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22⋊C411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C22×Dic5—22no—
85C22×Dic5—22no—
86C22×Dic5—42no—
87C22×Dic5—42no—
88C22×Dic5—82no—
89C5×M16—42no—
90C5×M16—42no—
91C2×C5×C8—42no—
92C22⋊Dic5—42no—
93C22⋊Dic5—42no—
94C22⋊Dic5—42no—
95C22⋊Dic5—82no—
96C22⋊Dic5—82no—
97C4⋊Dic5—42no—
98C4⋊Dic5—42no—
99C4⋊Dic5—42no—
100C4×Dic5—42no—
101C22×#41C2×D511yescharacteristic
102C2×#117—51no—
103C22×C4⋊C4—51no—
104C2×#101—21no—
105C2×#100C2×D512yes—
106C2×#99—52no—
107C2×#99—101no—
108#407—21no—
109#391—101no—
110#390—101no—
111#386—52no—
112#386—52no—
113#381—101no—
114#380—101no—
115#380—101no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5—41no—
129C23×C5C2×C22⋊C411yescharacteristic
130C23×C5#11712yes—
131C23×C5#12312yes—
132C5×Q8—44no—
133C5×D4—44no—
134C5×D4—44no—
135C5×D4—44no—
136C2×C4×C5—28no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—44no—
147C2×C4×C5—44no—
148C2×C4×C5#9912yes—
149C2×C4×C5#9912yes—
150C2×C4×C5#12312yes—
151C2×Dic5—42no—
152C2×Dic5—42no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—42no—
157C2×Dic5—84no—
158C2×Dic5—82no—
159C2×Dic5—82no—
160C2×Dic5—82no—
161C5×C8—82no—
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—42no—
165C22×Q8C22⋊D511yescharacteristic
166C22×D4C4×D511yescharacteristic
167C22×D4D2011yescharacteristic
168C22×D4C22×D511yescharacteristic
169C23×C4—52no—
170C23×C4C4×D511yescharacteristic
171C23×C4D2011yescharacteristic
172C23×C4C22⋊D511yescharacteristic
173C2×M16—22no—
174#117—202no—
175C2×C4⋊C4—101no—
176C2×C4⋊C4—101no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—202no—
179C2×C22⋊C4—52no—
180C2×C22⋊C4—102no—
181C2×C22⋊C4—102no—
182C2×C22⋊C4—102no—
183C2×C42—101no—
184#101—42no—
185#100—24no—
186#99—104no—
187#99—202no—
188C22⋊C8—42no—
189#95—102no—
190#95—102no—
191#95—202no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—22no—
196C22×C5—42no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5—42no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—81no—
208C22×C5—81no—
209C22×C5#38111yescharacteristic
210C22×C5C2×#9911yescharacteristic
211C22×C5C2×#9911yescharacteristic
212C4×C5—22no—
213C4×C5—22no—
214C4×C5—22no—
215C4×C5—41no—
216Dic5—82no—
217Dic5—82no—
218Dic5—162no—
219C24#68511yescharacteristic
220C24C2×C4×D511yescharacteristic
221C24C2×D2011yescharacteristic
222#41—416no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4#71114yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—102no—
245C22×C4—102no—
246C22×C4—202no—
247C22×C4—202no—
248C22×C4—402no—
249C22×C4#68511yescharacteristic
250C22×C4#68511yescharacteristic
251C22×C4#68511yescharacteristic
252C22×C4C2×C22⋊D511yescharacteristic
253M16—42no—
254M16—42no—
255C2×C8—42no—
256C4⋊C4—202no—
257C4⋊C4—202no—
258C4⋊C4—202no—
259C22⋊C4—202no—
260C22⋊C4—202no—
261C22⋊C4—202no—
262C22⋊C4—402no—
263C22⋊C4—402no—
264C42—202no—
265C2×C5—21no—
266C2×C5—21no—
267C2×C5—42no—
268C2×C5—42no—
269C2×C5—81no—
270C2×C5#181911yescharacteristic
271C2×C5#212211yescharacteristic
272C2×C5#212311yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#402312yes—
287C23#403212yes—
288C23C2×#68511yescharacteristic
289Q8—44no—
290D4—44no—
291D4—44no—
292D4—44no—
293C2×C4—28no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—202no—
306C2×C4—202no—
307C2×C4—202no—
308C2×C4—202no—
309C2×C4—202no—
310C2×C4—202no—
311C2×C4—404no—
312C2×C4—402no—
313C2×C4—402no—
314C2×C4—402no—
315C2×C4#393812yes—
316C2×C4#393812yes—
317C2×C4#408112yes—
318C8—82no—
319C5#1285511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#6523211yescharacteristic
338C22C2×#393811yescharacteristic
339C22C2×#393811yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—402no—
345C4—402no—
346C4—802no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2#1060028811yescharacteristic
353C2#1060093911yescharacteristic
354C2#1060094011yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC24×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#1785C2
socleC22×C5C2×#99
prime radicalC22×C5×#41C22
cosocleC22×C5×D4C23
hypercenter#1785D5
hypocenterC5#12855
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
4. #10598189C2semidirect
6. C2×#4072C4semidirect
7. C22×C4⋊Dic5C4semidirect
13. #65509C4semidirect
319. C5#12855semidirect

Maximal subgroups

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

subgroupindex
#106158182
C5×#17852
#106029322
#105981892
#128555