Finite Group Databasebeta

← #12015561

Subgroups — #12015561

Order 1280.

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

normal subgroups only

This group has 642 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×#1782C211yesmaximal, characteristic
3#10602932C211yesmaximal, characteristic
4#10598186C214yesmaximal, complemented
5C22×C5×#41C2211yescharacteristic
6C2×#4072C411yescharacteristic, complemented
7C22×C4⋊Dic5C411yescharacteristic, complemented
8C2×C5×#99—21no—
9C2×C5×#99C2212yes—
10C5×#381—21no—
11#65519C412yescomplemented
12#65510C2212yes—
13#65509—21no—
14#65509—21no—
15C2×#3966—21no—
16C2×#3966C2212yes—
17#65501—21no—
18#65501—21no—
19#65500—21no—
20#12859—51nomaximal, Sylow
21C2×C5×#41—24no—
22C2×C5×#41—42no—
23C2×C5×#41—42no—
24C22×C5×Q8C2×C411yescharacteristic
25C22×C5×D4D411yescharacteristic
26C22×C5×D4D411yescharacteristic
27C22×C5×D4C2311yescharacteristic
28C23×C4×C5C2×C411yescharacteristic
29C23×C4×C5D411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C23×Dic5—21no—
32C2×C5×C22⋊C4—22no—
33C2×C22⋊Dic5—21no—
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—22no—
36C2×C22⋊Dic5—22no—
37#4072—42no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5—21no—
40C2×C4⋊Dic5—42no—
41C2×C4⋊Dic5C2×C412yes—
42C2×C4×Dic5C2×C412yes—
43C5×#99—24no—
44C5×#99—42no—
45C5×#95—42no—
46#3966—24no—
47#3966—42no—
48#3963—22no—
49#3963—22no—
50#3963—42no—
51#2311—51no—
52#1782—51no—
53#1782D511yescharacteristic
54#1393—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C2×D411yescharacteristic
57C24×C5C2×D411yescharacteristic
58C5×#41—416no—
59C2×C5×Q8—22no—
60C2×C5×Q8—41no—
61C2×C5×Q8—41no—
62C2×C5×D4—22no—
63C2×C5×D4—22no—
64C2×C5×D4—22no—
65C2×C5×D4—22no—
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×C5—42no—
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22⋊C411yescharacteristic
83C22×C4×C5C22⋊C411yescharacteristic
84C22×C4×C5C22×C411yescharacteristic
85C22×Dic5—22no—
86C22×Dic5—22no—
87C22×Dic5—42no—
88C22×Dic5—42no—
89C22×Dic5—82no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—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×#99—21no—
105C2×#99—52no—
106C2×#99—101no—
107C2×#99C2×D512yes—
108#391—52no—
109#390—52no—
110#386—101no—
111#386—101no—
112#381—21no—
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—41no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5#9912yes—
129C23×C5C2×C22⋊C411yescharacteristic
130C23×C5#12312yes—
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—21no—
147C2×C4×C5—44no—
148C2×C4×C5—44no—
149C2×C4×C5—82no—
150C2×C4×C5—82no—
151C2×C4×C5—82no—
152C2×C4×C5#9912yes—
153C2×C4×C5#11712yes—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—42no—
157C2×Dic5—42no—
158C2×Dic5—42no—
159C2×Dic5—42no—
160C2×Dic5—84no—
161C2×Dic5—82no—
162C2×Dic5—82no—
163C2×Dic5—82no—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8C4×D511yescharacteristic
168C22×D4D2011yescharacteristic
169C22×D4C22⋊D511yescharacteristic
170C22×D4C22×D511yescharacteristic
171C23×C4—101no—
172C23×C4C4×D511yescharacteristic
173C23×C4D2011yescharacteristic
174C23×C4C22⋊D511yescharacteristic
175#117—202no—
176C2×C4⋊C4—52no—
177C2×C4⋊C4—101no—
178C2×C4⋊C4—101no—
179C2×C4⋊C4—202no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—101no—
182C2×C22⋊C4—102no—
183C2×C22⋊C4—102no—
184C2×C22⋊C4—102no—
185C2×C42—52no—
186#99—24no—
187#99—42no—
188#99—104no—
189#99—202no—
190#95—42no—
191#95—102no—
192#95—102no—
193#95—202no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—22no—
197C22×C5—22no—
198C22×C5—22no—
199C22×C5—42no—
200C22×C5—42no—
201C22×C5—42no—
202C22×C5—42no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—81no—
210C22×C5—81no—
211C22×C5#38111yescharacteristic
212C22×C5C2×#9911yescharacteristic
213C22×C5C2×#9911yescharacteristic
214C4×C5—22no—
215C4×C5—22no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—162no—
219Dic5—82no—
220Dic5—82no—
221Dic5—162no—
222C24#68511yescharacteristic
223C24C2×D2011yescharacteristic
224C24C2×C22⋊D511yescharacteristic
225#41—416no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4#71114yes—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—102no—
249C22×C4—102no—
250C22×C4—202no—
251C22×C4—202no—
252C22×C4—402no—
253C22×C4#68511yescharacteristic
254C22×C4#68511yescharacteristic
255C22×C4#68511yescharacteristic
256C22×C4C2×C4×D511yescharacteristic
257C4⋊C4—202no—
258C4⋊C4—202no—
259C4⋊C4—202no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—202no—
263C22⋊C4—202no—
264C22⋊C4—202no—
265C22⋊C4—402no—
266C22⋊C4—402no—
267C42—202no—
268C2×C5—21no—
269C2×C5—21no—
270C2×C5—42no—
271C2×C5—42no—
272C2×C5—42no—
273C2×C5#181911yescharacteristic
274C2×C5#212811yescharacteristic
275C2×C5#212911yescharacteristic
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#393812yes—
289C23#403212yes—
290C23C2×#68511yescharacteristic
291C23#408112yes—
292Q8—44no—
293D4—44no—
294D4—44no—
295D4—44no—
296C2×C4—28no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—202no—
313C2×C4—202no—
314C2×C4—202no—
315C2×C4—202no—
316C2×C4—202no—
317C2×C4—202no—
318C2×C4—404no—
319C2×C4—402no—
320C2×C4—402no—
321C2×C4—402no—
322C2×C4#393812yes—
323C2×C4#402312yes—
324C5#1285911yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22#6523211yescharacteristic
343C22C2×#393811yescharacteristic
344C22C2×#393811yescharacteristic
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—162no—
350C4—402no—
351C4—402no—
352C4—802no—
353C2—21no—
354C2—21no—
355C2—42no—
356C2—42no—
357C2—42no—
358C2#1060028811yescharacteristic
359C2#1060085211yescharacteristic
360C2#1060085411yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC22×D4C22×D5
FittingC5×#1782C2
socleC22×C5C2×#99
prime radicalC22×C5×#41C22
cosocleC22×C5×D4C23
hypercenter#1782D5
hypocenterC5#12859
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
4. #10598186C2semidirect
6. C2×#4072C4semidirect
7. C22×C4⋊Dic5C4semidirect
11. #65519C4semidirect
324. C5#12859semidirect

Maximal subgroups

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

subgroupindex
#106158182
C5×#17822
#106029322
#105981862
#128595