Finite Group Databasebeta

← #12010450

Subgroups — #12010450

Order 1280.

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

normal subgroups only

This group has 568 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#10614080C211yesmaximal, characteristic
2C5×#1459C211yesmaximal, characteristic
3#10602410C211yesmaximal, characteristic
4#10597948C214yesmaximal, complemented
5C23×Dic10C411yescharacteristic
6C2×C4×C5×D4C2211yescharacteristic
7C2×#4070—21no—
8C2×C5×C22⋊C8—21no—
9#65509—21no—
10C2×#3965—21no—
11#65501C411yescharacteristic
12#65268C2212yes—
13#65236C412yes—
14#65235—21no—
15#65218—21no—
16#65213—21no—
17C5×#324C2212yes—
18C5×#323—21no—
19#64750C2212yes—
20#11904—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5C2×C411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×Dic5—21no—
25C22×Dic10—21no—
26C22×Dic10—44no—
27C22×Dic10—41no—
28C22×Dic10—41no—
29C22×Dic10C2×C412yes—
30C22×C5×C8—41no—
31C4×C5×D4—41no—
32C4×C5×D4—41no—
33C2×C5×C4⋊C4C2311yescharacteristic
34C2×C5×C22⋊C4C2×C411yescharacteristic
35C2×C5×C22⋊C4D411yescharacteristic
36C2×C42×C5D411yescharacteristic
37C2×C22⋊Dic5—22no—
38C2×C22⋊Dic5—21no—
39C2×C4⋊Dic5—21no—
40#4070—44no—
41C2×#683—22no—
42C2×#681—22no—
43C5×C22⋊C8—22no—
44C5×C22⋊C8—41no—
45#3965—42no—
46#3963—22no—
47#3963—22no—
48#3963C2×C412yes—
49C4⋊#168—41no—
50C4⋊#168—41no—
51#2409—51no—
52#1800—51no—
53#1459D511yescharacteristic
54#1354—54no—
55C24×C5C22⋊C411yescharacteristic
56C24×C5C22⋊C411yescharacteristic
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C2×C5×D4SD1612yes—
60C2×C5×D4Q1612yes—
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×C5C22⋊C411yescharacteristic
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C22×C4×C5C2×D411yescharacteristic
71C22×C4×C5C2×D411yescharacteristic
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—44no—
75C22×Dic5—42no—
76C2×Dic10—24no—
77C2×Dic10—42no—
78C2×Dic10—48no—
79C2×Dic10—44no—
80C2×Dic10—41no—
81C2×Dic10—41no—
82C2×Dic10—42no—
83C2×Dic10—44no—
84C2×Dic10—44no—
85C2×C5×C8—42no—
86C2×C5×C8—82no—
87C5×C4⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C42×C5—21no—
91C42×C5—21no—
92C22⋊Dic5—44no—
93C22⋊Dic5—42no—
94C4⋊Dic5—42no—
95#683—44no—
96#681—42no—
97#681—42no—
98C2×#168—42no—
99C23×Q8—51no—
100C2×C22⋊Q8—101no—
101C2×C4×D4C2×D511yescharacteristic
102#423—101no—
103C2×#100—101no—
104C2×C22⋊C8—21no—
105#395—52no—
106#393—52no—
107#391—101no—
108#386—101no—
109#385—101no—
110#380—51no—
111#343—52no—
112#324C2×D512yes—
113#323—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—41no—
119C23×C5—41no—
120C23×C5—41no—
121C23×C5—41no—
122C23×C5#10312yes—
123C23×C5#10312yes—
124C23×C5C2×C22⋊C411yescharacteristic
125C5×D4—42no—
126C5×D4—42no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—22no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—81no—
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5#9911yescharacteristic
142C2×C4×C5C2×SD1611yescharacteristic
143C2×C4×C5C2×Q1611yescharacteristic
144C2×Dic5—24no—
145C2×Dic5—42no—
146C2×Dic5—48no—
147C2×Dic5—42no—
148C2×Dic5—42no—
149C2×Dic5—42no—
150C2×Dic5—42no—
151Dic10—44no—
152Dic10—44no—
153Dic10—44no—
154Dic10—44no—
155C5×C8—82no—
156#168—82no—
157C22×Q8—52no—
158C22×Q8—101no—
159C22×Q8—201no—
160C22×Q8—201no—
161C22×Q8—204no—
162C22×D4D2011yescharacteristic
163C23×C4—101no—
164C23×C4C4×D511yescharacteristic
165C23×C4C22⋊D511yescharacteristic
166C2×M16—102no—
167C22×C8—41no—
168C22⋊Q8—204no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—101no—
172C2×C4⋊C4—102no—
173C2×C4⋊C4C22×D511yescharacteristic
174C2×C22⋊C4—102no—
175C2×C22⋊C4—101no—
176C2×C22⋊C4C4×D511yescharacteristic
177C2×C22⋊C4C22⋊D511yescharacteristic
178C2×C42D2011yescharacteristic
179C4⋊C8—201no—
180C4⋊C8—201no—
181#100—202no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184#95—52no—
185#95—102no—
186#95—102no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—42no—
192C22×C5—42no—
193C22×C5—42no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—81no—
199C22×C5C2×#9911yescharacteristic
200C22×C5C2×#10311yescharacteristic
201C22×C5#42111yescharacteristic
202C4×C5—21no—
203C4×C5—21no—
204C4×C5—21no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5AΣL(1,𝔽16)12yes—
209Dic5—44no—
210Dic5—82no—
211C24#68511yescharacteristic
212C24#68511yescharacteristic
213C2×Q8—104no—
214C2×Q8—202no—
215C2×Q8—202no—
216C2×Q8—204no—
217C2×Q8—204no—
218C2×Q8—208no—
219C2×Q8—204no—
220C2×Q8—201no—
221C2×Q8—201no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#67712yes—
225C2×D4Dic2012yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—102no—
233C22×C4—102no—
234C22×C4—204no—
235C22×C4—202no—
236C22×C4#68511yescharacteristic
237C22×C4#68511yescharacteristic
238C22×C4C2×C4×D511yescharacteristic
239C22×C4C2×D2011yescharacteristic
240C22×C4C2×C22⋊D511yescharacteristic
241M16—202no—
242M16—202no—
243C2×C8—42no—
244C2×C8—82no—
245C2×C8—202no—
246C4⋊C4—41no—
247C4⋊C4—202no—
248C4⋊C4—204no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—204no—
252C22⋊C4—202no—
253C42—21no—
254C42—21no—
255C2×C5—21no—
256C2×C5—21no—
257C2×C5—42no—
258C2×C5—42no—
259C2×C5#152011yescharacteristic
260C2×C5C2×AΣL(1,𝔽16)11yescharacteristic
261C2×C5#213511yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#394812yes—
271C23#394812yes—
272C23C2×#68511yescharacteristic
273Q8—204no—
274Q8—204no—
275Q8—204no—
276Q8—204no—
277D4—42no—
278D4—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—104no—
293C2×C4—202no—
294C2×C4—208no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—202no—
299C2×C4#393811yescharacteristic
300C2×C4#393811yescharacteristic
301C2×C4C2×#67711yescharacteristic
302C2×C4C2×Dic2011yescharacteristic
303C8—82no—
304C8—402no—
305C5#1190411yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#393811yescharacteristic
319C22C2×#394811yescharacteristic
320C22#6542811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—204no—
328C4—402no—
329C4#6469112yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#1060073911yescharacteristic
335C2C2×#6469111yescharacteristic
336C2#1060094311yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#1459C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C4⋊C4C23
hypercenter#1459D5
hypocenterC5#11904
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
4. #10597948C2semidirect
305. C5#11904semidirect

Maximal subgroups

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

subgroupindex
#106140802
C5×#14592
#106024102
#105979482
#119045