Finite Group Databasebeta

← #11996672

Subgroups — #11996672

Order 1280.

594 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 117 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#10614990C211yesmaximal, characteristic
2#10611017C211yesmaximal, characteristic
3C5×#1549C211yesmaximal, characteristic, complemented
4#10602568C211yesmaximal, characteristic, complemented
5#10601320C211yesmaximal, characteristic
6#10598237C211yesmaximal, characteristic, complemented
7#10598206C211yesmaximal, characteristic, complemented
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×#4092—22no—
10C2×C4×D20C2211yescharacteristic
11C5×#431C2211yescharacteristic
12C5×#423C2211yescharacteristic
13C2×#3955—21no—
14#65281C414yes—
15#65277—21no—
16#65265—21no—
17C4×C4⋊Dic5C2211yescharacteristic
18#65214C2211yescharacteristic
19#65213C2211yescharacteristic
20#65123—22no—
21D5×C22⋊C8—22no—
22C8×D20—22no—
23C5×#332—22no—
24#64774—21no—
25#64767—22no—
26#9395—51nomaximal, Sylow
27C22×C5×Q8—21no—
28C22×D20D411yescharacteristic
29C22×C4×D5D412yes—
30C2×C5×M16—21no—
31C5×C4⋊Q8—21no—
32C5×C4⋊Q8—21no—
33C5×C4⋊Q8—21no—
34C5×C4⋊Q8—21no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4C2×C412yes—
37C2×C42×C5C2311yescharacteristic
38#4092—48no—
39C2×#685—24no—
40C2×#685C2×C412yes—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5D411yescharacteristic
43C2×#683—24no—
44C2×C4×Dic5—21no—
45C2×#681—21no—
46C2×C8×D5—24no—
47C4×D20—21no—
48C4×D20—21no—
49C4×D20—21no—
50C4×D20—21no—
51C5×C4⋊C8—21no—
52C5×C4⋊C8—21no—
53C5×C4⋊C8D412yes—
54C5×C22⋊C8—21no—
55C4×C5×C8—21no—
56#3963C2×C412yes—
57C22⋊#168—21no—
58#3955—42no—
59#3952—24no—
60C4⋊#168—21no—
61C4⋊#168—21no—
62C4⋊#168D412yes—
63C4×#168—21no—
64#2972—51no—
65#2542—51no—
66#1800—51no—
67#1549—51no—
68#1549D511yescharacteristic, complemented
69#1303—51no—
70#1303—51no—
71C23×D5C22⋊C412yes—
72C2×C5×Q8—44no—
73C2×C5×Q8—41no—
74C2×C5×Q8—41no—
75C22×C4×C5—22no—
76C22×C4×C5C22×C411yescharacteristic
77C22×C4×C5C2×D411yescharacteristic
78C22×C4×C5C2×D411yescharacteristic
79C22×Dic5—22no—
80C22×Dic5C22⋊C412yes—
81C2×D20—21no—
82C2×D20—21no—
83C2×D20—21no—
84C2×D20—21no—
85C2×D20SD1612yes—
86C2×D20Q1612yes—
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C2×C4×D5—44no—
90C5×M16—41no—
91C5×M16—41no—
92C2×C5×C8—22no—
93C2×C5×C8—41no—
94C5×C4⋊C4—22no—
95C5×C4⋊C4—22no—
96C5×C4⋊C4—41no—
97C5×C4⋊C4—41no—
98C42×C5C2×D411yescharacteristic
99C42×C5C2×D411yescharacteristic
100C42×C5#4111yescharacteristic
101C42×C5#4111yescharacteristic
102#685—22no—
103#685—22no—
104#685—48no—
105C4⋊Dic5—24no—
106C4⋊Dic5—21no—
107C4⋊Dic5—21no—
108#683—48no—
109C4×Dic5—41no—
110C4×Dic5—41no—
111#681—41no—
112#681—41no—
113C2×#168—22no—
114C2×#168—41no—
115C8×D5—84no—
116C2×C4⋊Q8C2×D511yescharacteristic
117C2×C22⋊Q8—102no—
118C2×C4×D4—51no—
119C8×D4—102no—
120C8×D4—102no—
121#431—51no—
122#431C2×D511yescharacteristic
123#423—51no—
124#423C2×D511yescharacteristic
125C2×#100—101no—
126C2×C22⋊C8—102no—
127#399—54no—
128#390—101no—
129#384—101no—
130C4×C4⋊C4—51no—
131#343—101no—
132#332—22no—
133#332—102no—
134C23×C5C2×C22⋊C411yescharacteristic
135C22×D5—22no—
136C22×D5—22no—
137C22×D5—42no—
138C22×D5—42no—
139C22×D5#10314yes—
140C5×Q8—82no—
141C5×Q8—82no—
142C2×C4×C5—22no—
143C2×C4×C5—41no—
144C2×C4×C5—44no—
145C2×C4×C5—42no—
146C2×C4×C5C4×D411yescharacteristic
147C2×C4×C5C4×D411yescharacteristic
148C2×C4×C5#12011yescharacteristic
149C2×C4×C5#12111yescharacteristic
150C2×C4×C5#12111yescharacteristic
151C2×C4×C5#12311yescharacteristic
152C2×C4×C5C2×SD1611yescharacteristic
153C2×C4×C5C2×Q1611yescharacteristic
154C2×C4×C5#13511yescharacteristic
155C2×C4×C5#13511yescharacteristic
156C2×Dic5—24no—
157C2×Dic5—22no—
158C2×Dic5—42no—
159C2×Dic5—42no—
160C2×Dic5—42no—
161D20—22no—
162D20—22no—
163D20—22no—
164D20—22no—
165C4×D5—44no—
166C4×D5—82no—
167C5×C8—42no—
168C5×C8—81no—
169#168—42no—
170#168—81no—
171C22×Q8—21no—
172C22×D4—51no—
173C23×C4—52no—
174C2×M16—21no—
175C2×M16—101no—
176C22×C8—104no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C22⋊Q8—208no—
182C4×D4—101no—
183C4×D4—101no—
184C4×D4—101no—
185C4×D4—101no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—101no—
189C2×C4⋊C4—104no—
190C2×C4⋊C4C4×D512yes—
191C2×C22⋊C4—52no—
192C2×C22⋊C4—104no—
193C2×C42—101no—
194C2×C42C22×D511yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—52no—
198C4⋊C8—101no—
199C4⋊C8—101no—
200C4⋊C8C22⋊D512yes—
201#100—202no—
202C22⋊C8—21no—
203C22⋊C8—104no—
204C22⋊C8—101no—
205C4×C8—21no—
206C4×C8—101no—
207#95—52no—
208C22×C5—21no—
209C22×C5—21no—
210C22×C5#38611yescharacteristic
211C22×C5C2×#10311yescharacteristic
212C22×C5#41611yescharacteristic
213C2×D5—24no—
214C2×D5—42no—
215C2×D5—42no—
216C2×D5—44no—
217C2×D5—82no—
218C4×C5—21no—
219C4×C5—21no—
220C4×C5—21no—
221C4×C5—82no—
222C4×C5#45312yes—
223C4×C5#46511yescharacteristic
224C4×C5#46511yescharacteristic
225C4×C5#46711yescharacteristic
226C4×C5#46711yescharacteristic
227Dic5—44no—
228Dic5—42no—
229C24—52no—
230C2×Q8—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×D4—52no—
234C2×D4—52no—
235C2×D4—101no—
236C2×D4—101no—
237C2×D4—101no—
238C2×D4—101no—
239C22×C4—22no—
240C22×C4—52no—
241C22×C4—102no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4—204no—
245C22×C4C2×C4×D511yescharacteristic
246C22×C4D4×D511yescharacteristic
247C22×C4D4×D511yescharacteristic
248M16—41no—
249M16—41no—
250M16—201no—
251M16—201no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—102no—
255C2×C8—201no—
256C2×C8—404no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—104no—
262C4⋊C4—101no—
263C4⋊C4—101no—
264C4⋊C4—208no—
265C22⋊C4—102no—
266C22⋊C4—102no—
267C22⋊C4—208no—
268C42—201no—
269C42—201no—
270C42#70911yescharacteristic
271C42D4×D511yescharacteristic
272C42#71111yescharacteristic
273C42C2×C22⋊D511yescharacteristic
274C2×C5—21no—
275C2×C5—21no—
276C2×C5#188711yescharacteristic
277C2×C5#189211yescharacteristic
278C2×C5#191511yescharacteristic
279D5—44no—
280C23—54no—
281C23—102no—
282C23—102no—
283C23—202no—
284C23—202no—
285C23D5×C22⋊C411yescharacteristic
286Q8—82no—
287Q8—82no—
288D4—102no—
289D4—102no—
290D4—102no—
291D4—102no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—104no—
297C2×C4—102no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—204no—
301C2×C4—202no—
302C2×C4—402no—
303C2×C4#402711yescharacteristic
304C2×C4#402911yescharacteristic
305C2×C4#403011yescharacteristic
306C2×C4D5×SD1611yescharacteristic
307C2×C4#406211yescharacteristic
308C2×C4D5×Q1611yescharacteristic
309C2×C4#406511yescharacteristic
310C2×C4C4×C22⋊D511yescharacteristic
311C2×C4#408311yescharacteristic
312C2×C4#408411yescharacteristic
313C8—42no—
314C8—81no—
315C8—202no—
316C8—401no—
317C5#939511yescharacteristic, complemented, Sylow
318C22—21no—
319C22—21no—
320C22—104no—
321C22—204no—
322C22—202no—
323C22—202no—
324C22—402no—
325C22D5×#10311yescharacteristic
326C22#6509311yescharacteristic
327C22#6524511yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—204no—
333C4—202no—
334C4#6509611yescharacteristic
335C4#6510211yescharacteristic
336C4#6546011yescharacteristic
337C4#6547711yescharacteristic
338C4#6548712yes—
339C2—21no—
340C2—21no—
341C2—204no—
342C2#1060091711yescharacteristic
343C2#1060106311yescharacteristic
344C2#1060111611yescharacteristic
subgroupG/H
centerC22#65245
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#1549C2
socleC22×C5#386
prime radicalC22×D20D4
cosocleC2×C42×C5C23
hypercenter#1549D5
hypocenterC5#9395
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C5×#1549C2semidirect
4. #10602568C2semidirect
6. #10598237C2semidirect
7. #10598206C2semidirect
68. #1549D5semidirect
317. C5#9395semidirect

Maximal subgroups

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

subgroupindex
#106149902
#106110172
C5×#15492
#106025682
#106013202
#105982372
#105982062
#93955