Finite Group Databasebeta

← #12009609

Subgroups — #12009609

Order 1280.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#10613914C211yesmaximal, characteristic
2C2×C5×#343C211yesmaximal, characteristic
3C2×#64748C211yesmaximal, characteristic
4#10597987C212yesmaximal, complemented
5#10597985C212yesmaximal, complemented
6C22×C5×M16—21no—
7C22×C5×C4⋊C4C2211yescharacteristic
8C2×#4070C411yescharacteristic
9#65501C411yescharacteristic
10#65500—21no—
11#65499—21no—
12C2×C22⋊#168—21no—
13#65275C411yescharacteristic
14#65275C2211yescharacteristic
15#65273—21no—
16#65273—21no—
17#65268C411yescharacteristic
18#65268C2211yescharacteristic
19C5×#343—21no—
20C5×#343C2212yes—
21#64748—21no—
22#64748C2212yes—
23#11722—51nomaximal, Sylow
24C23×C4×C5C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C22×Dic10C2×C411yescharacteristic
28C2×C5×M16—24no—
29C2×C5×M16—22no—
30C2×C5×M16—41no—
31C2×C5×M16—42no—
32C22×C5×C8—41no—
33C2×C5×C4⋊C4—41no—
34C2×C5×C4⋊C4—41no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C5×C4⋊C4C2311yescharacteristic
39C2×C22⋊Dic5—21no—
40C2×C22⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5C2×C411yescharacteristic
43#4070—41no—
44#4070—41no—
45C2×#683—21no—
46C2×#683—21no—
47C2×#683—21no—
48C22×#168—41no—
49#3963—21no—
50#3963—21no—
51#3963—21no—
52#3963—21no—
53#3963—21no—
54#3963—21no—
55#3963C2×C411yescharacteristic
56#3963C2×C411yescharacteristic
57C22⋊#168—22no—
58C22⋊#168—41no—
59#2627—51no—
60C2×#343D511yescharacteristic
61C2×#324—51no—
62#1354—52no—
63#1353—52no—
64C24×C5C22⋊C411yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—81no—
73C22×C4×C5—81no—
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5SD1612yes—
78C22×C4×C5Q1612yes—
79C22×C4×C5C22×C411yescharacteristic
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×Dic5—21no—
83C22×Dic5—21no—
84C22×Dic5—21no—
85C22×Dic5—21no—
86C2×Dic10—41no—
87C2×Dic10—81no—
88C5×M16—24no—
89C5×M16—44no—
90C5×M16—42no—
91C2×C5×C8—44no—
92C2×C5×C8—42no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—41no—
96C5×C4⋊C4—41no—
97C22⋊Dic5—41no—
98C22⋊Dic5—41no—
99C22⋊Dic5—42no—
100C4⋊Dic5—41no—
101C4⋊Dic5—81no—
102#683—42no—
103#683—81no—
104#683—81no—
105C2×#168—42no—
106C2×#168—82no—
107C22×M16—21no—
108C2×C22⋊Q8—51no—
109C22×C4⋊C4C2×D511yescharacteristic
110C2×C22⋊C8—101no—
111#400—51no—
112#400—51no—
113#398—101no—
114#398—101no—
115#395—51no—
116#395—51no—
117#385—101no—
118#381—101no—
119#380—51no—
120#343—21no—
121#343C2×D512yes—
122#324—52no—
123#324—101no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5#9911yescharacteristic
127C23×C5#9911yescharacteristic
128C23×C5#10312yes—
129C23×C5#10412yes—
130C23×C5C2×C22⋊C411yescharacteristic
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—22no—
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—42no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—81no—
149C2×C4×C5—81no—
150C2×C4×C5—81no—
151C2×C4×C5#10312yes—
152C2×C4×C5C2×SD1611yescharacteristic
153C2×C4×C5C2×Q1611yescharacteristic
154C2×Dic5—41no—
155C2×Dic5—41no—
156C2×Dic5—41no—
157C2×Dic5—41no—
158C2×Dic5—41no—
159C2×Dic5—41no—
160C2×Dic5—42no—
161C2×Dic5—41no—
162C2×Dic5—41no—
163C2×Dic5—41no—
164C2×Dic5—41no—
165Dic10—82no—
166C5×C8—44no—
167#168—82no—
168C22×Q8—51no—
169C23×C4C4×D511yescharacteristic
170C23×C4D2011yescharacteristic
171C23×C4C22⋊D511yescharacteristic
172C2×M16—24no—
173C2×M16—22no—
174C2×M16—41no—
175C2×M16—42no—
176C22×C8—41no—
177C22×C8—201no—
178C22⋊Q8—201no—
179C22⋊Q8—201no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—101no—
184C2×C4⋊C4—101no—
185C2×C4⋊C4—101no—
186C2×C4⋊C4—101no—
187C2×C4⋊C4C4×D511yescharacteristic
188C2×C4⋊C4D2011yescharacteristic
189C2×C4⋊C4C22⋊D511yescharacteristic
190C2×C4⋊C4C22×D511yescharacteristic
191C2×C22⋊C4—101no—
192C2×C22⋊C4—101no—
193C22⋊C8—102no—
194C22⋊C8—201no—
195#95—51no—
196#95—51no—
197#95—101no—
198#95—101no—
199#95—101no—
200#95—101no—
201#95—101no—
202#95—101no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5—42no—
210C22×C5—42no—
211C22×C5#32812yes—
212C22×C5#32812yes—
213C22×C5C2×#9911yescharacteristic
214C22×C5C2×#10311yescharacteristic
215C22×C5C2×#10411yescharacteristic
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—21no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223Dic5—81no—
224Dic5—81no—
225Dic5—81no—
226Dic5—81no—
227C24#68511yescharacteristic
228C2×Q8—201no—
229C2×Q8—401no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—101no—
240C22×C4—101no—
241C22×C4—101no—
242C22×C4—101no—
243C22×C4#68511yescharacteristic
244C22×C4#68511yescharacteristic
245C22×C4#68511yescharacteristic
246C22×C4#68712yes—
247C22×C4#68912yes—
248C22×C4C2×C4×D511yescharacteristic
249C22×C4C2×D2011yescharacteristic
250C22×C4C2×C22⋊D511yescharacteristic
251M16—24no—
252M16—44no—
253M16—42no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—202no—
257C2×C8—402no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—201no—
263C4⋊C4—202no—
264C4⋊C4—401no—
265C4⋊C4—401no—
266C4⋊C4—401no—
267C22⋊C4—202no—
268C22⋊C4—201no—
269C22⋊C4—201no—
270C2×C5—21no—
271C2×C5—22no—
272C2×C5—22no—
273C2×C5—21no—
274C2×C5C2×#32811yescharacteristic
275C2×C5#212111yescharacteristic
276C2×C5#213311yescharacteristic
277C23—41no—
278C23—41no—
279C23#393712yes—
280C23#393811yescharacteristic
281C23#393811yescharacteristic
282C23#394212yes—
283C23C2×#68511yescharacteristic
284Q8—402no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—201no—
306C2×C4—201no—
307C2×C4—201no—
308C2×C4—201no—
309C2×C4—201no—
310C2×C4—201no—
311C2×C4—202no—
312C2×C4—201no—
313C2×C4—201no—
314C2×C4—201no—
315C2×C4—201no—
316C2×C4#394212yes—
317C2×C4C2×#68711yescharacteristic
318C2×C4C2×#68911yescharacteristic
319C8—44no—
320C8—402no—
321C5#1172211yescharacteristic, complemented, Sylow
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#6467212yes—
331C22#6467212yes—
332C22C2×#393711yescharacteristic
333C22C2×#393811yescharacteristic
334C22C2×#394211yescharacteristic
335C4—21no—
336C4—22no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—401no—
343C4—401no—
344C4—401no—
345C4—401no—
346C2—21no—
347C2—22no—
348C2—22no—
349C2—21no—
350C2C2×#6467211yescharacteristic
351C2#1060088711yescharacteristic
352C2#1060092111yescharacteristic
subgroupG/H
centerC22C2×#3938
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC2×C5×#343C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenterC2×#343D5
hypocenterC5#11722
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. #10597987C2semidirect
5. #10597985C2semidirect
321. C5#11722semidirect

Maximal subgroups

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

subgroupindex
#106139142
C2×C5×#3432
C2×#647482
#105979872
#105979852
#117225