Finite Group Databasebeta

← #12015555

Subgroups — #12015555

Order 1280.

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

normal subgroups only

This group has 670 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#10615817C211yesmaximal, 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#65500—21no—
18C4×C22⋊Dic5—21no—
19C4×C22⋊Dic5—21no—
20#12857—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⋊Dic5C2×C412yes—
39C2×C4×Dic5—21no—
40C2×C4×Dic5—21no—
41C2×C4×Dic5—42no—
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#2297—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—44no—
88C22×Dic5—42no—
89C22×Dic5—42no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—42no—
92C22⋊Dic5—44no—
93C22⋊Dic5—44no—
94C22⋊Dic5—42no—
95C22⋊Dic5—42no—
96C22⋊Dic5—42no—
97C4⋊Dic5—42no—
98C4×Dic5—42no—
99C4×Dic5—42no—
100C4×Dic5—42no—
101C22×#41C2×D511yescharacteristic
102C2×#117—51no—
103C22×C42—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—
114C4×C22⋊C4—101no—
115C4×C22⋊C4—101no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—42no—
123C23×C5—42no—
124C23×C5—42no—
125C23×C5#9912yes—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5#12312yes—
128C23×C5#12312yes—
129C5×Q8—44no—
130C5×D4—44no—
131C5×D4—44no—
132C5×D4—44no—
133C2×C4×C5—28no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
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—44no—
145C2×C4×C5—44no—
146C2×C4×C5—82no—
147C2×C4×C5—84no—
148C2×C4×C5#9912yes—
149C2×C4×C5#11712yes—
150C2×Dic5—24no—
151C2×Dic5—42no—
152C2×Dic5—48no—
153C2×Dic5—42no—
154C2×Dic5—42no—
155C2×Dic5—42no—
156C2×Dic5—42no—
157C2×Dic5—82no—
158C2×Dic5—84no—
159C2×#41—24no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8C4×D511yescharacteristic
163C22×D4D2011yescharacteristic
164C22×D4C22⋊D511yescharacteristic
165C22×D4C22×D511yescharacteristic
166C23×C4—101no—
167C23×C4C4×D511yescharacteristic
168C23×C4D2011yescharacteristic
169C23×C4C22⋊D511yescharacteristic
170#117—202no—
171C2×C4⋊C4—52no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—101no—
174C2×C22⋊C4—102no—
175C2×C22⋊C4—102no—
176C2×C22⋊C4—102no—
177C2×C42—52no—
178C2×C42—101no—
179C2×C42—101no—
180C2×C42—202no—
181#99—24no—
182#99—42no—
183#99—104no—
184#99—202no—
185#95—42no—
186#95—102no—
187#95—102no—
188#95—202no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—22no—
193C22×C5—42no—
194C22×C5—42no—
195C22×C5—42no—
196C22×C5—42no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5—42no—
200C22×C5—81no—
201C22×C5—81no—
202C22×C5#38111yescharacteristic
203C22×C5C2×#9911yescharacteristic
204C22×C5C2×#9911yescharacteristic
205C4×C5—22no—
206C4×C5—22no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—162no—
210Dic5—44no—
211Dic5—82no—
212Dic5—162no—
213C24#68511yescharacteristic
214C24C2×D2011yescharacteristic
215C24C2×C22⋊D511yescharacteristic
216#41—416no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4#71114yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—102no—
240C22×C4—102no—
241C22×C4—204no—
242C22×C4—202no—
243C22×C4—202no—
244C22×C4#68511yescharacteristic
245C22×C4#68511yescharacteristic
246C22×C4#68511yescharacteristic
247C22×C4C2×C4×D511yescharacteristic
248C4⋊C4—202no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—204no—
252C22⋊C4—204no—
253C22⋊C4—202no—
254C22⋊C4—202no—
255C22⋊C4—202no—
256C42—202no—
257C42—202no—
258C42—202no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5—42no—
262C2×C5—42no—
263C2×C5—42no—
264C2×C5#181911yescharacteristic
265C2×C5#212612yes—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23#393812yes—
276C23#403212yes—
277C23C2×#68511yescharacteristic
278C23#408112yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—104no—
299C2×C4—202no—
300C2×C4—208no—
301C2×C4—202no—
302C2×C4—202no—
303C2×C4—202no—
304C2×C4—202no—
305C2×C4—402no—
306C2×C4—404no—
307C2×C4#393812yes—
308C2×C4#402312yes—
309C5#1285711yescharacteristic, complemented, Sylow
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22#6523211yescharacteristic
324C22C2×#393811yescharacteristic
325C22C2×#393811yescharacteristic
326C4—22no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—162no—
331C4—204no—
332C4—402no—
333C4—802no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2—42no—
339C2#1060028811yescharacteristic
340C2#1060085312yes—
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#12857
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
309. C5#12857semidirect

Maximal subgroups

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

subgroupindex
#106158172
C5×#17822
#106029322
#105981862
#128575