Finite Group Databasebeta

← #11996654

Subgroups — #11996654

Order 1280.

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

normal subgroups only

This group has 538 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#10611017C211yesmaximal, characteristic
2#10605538C211yesmaximal, characteristic
3#10601572C211yesmaximal, characteristic, complemented
4#10601320C211yesmaximal, characteristic
5#10601300C211yesmaximal, characteristic, complemented
6C5×#1303C211yesmaximal, characteristic, complemented
7#10598111C211yesmaximal, characteristic, complemented
8C2×C4×D20C2211yescharacteristic
9C2×#4015C2211yescharacteristic
10C5×#431C2211yescharacteristic
11C5×#423C2211yescharacteristic
12C4×C5×C4⋊C4C2211yescharacteristic
13C2×#3955—21no—
14#65276—22no—
15#65228—21no—
16#65225—21no—
17#65214C2211yescharacteristic
18#65213C2211yescharacteristic
19#65123—22no—
20D5×C22⋊C8—22no—
21C8×D20—22no—
22#64962C414yes—
23#64774—21no—
24#64701—22no—
25#64701—22no—
26#9391—51nomaximal, Sylow
27C22×D20D411yescharacteristic
28C22×C4×D5D412yes—
29C22×Dic10C2×C412yes—
30C2×C5×M16—21no—
31C2×C5×C4⋊C4—21no—
32C2×C42×C5—21no—
33C2×C42×C5C2311yescharacteristic
34C2×#685—24no—
35C2×#685C2×C412yes—
36C2×C4⋊Dic5—21no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5D411yescharacteristic
39C2×#681—21no—
40C2×C8×D5—24no—
41C4×D20—21no—
42C4×D20—21no—
43C4×D20—21no—
44C4×D20—21no—
45#4015—21no—
46#4015—21no—
47#4015—21no—
48#4015—21no—
49C5×C4⋊C8—21no—
50C5×C4⋊C8—21no—
51C5×C4⋊C8D412yes—
52C5×C22⋊C8—21no—
53C4×C5×C8—21no—
54C5×#95C2×C412yes—
55#3963—24no—
56C22⋊#168—21no—
57#3955—42no—
58#3952—24no—
59C4⋊#168—21no—
60C4⋊#168—21no—
61C4⋊#168D412yes—
62C4×#168—21no—
63#2972—51no—
64#2563—51no—
65#1800—51no—
66#1548—51no—
67#1548—51no—
68#1303—51no—
69#1303D511yescharacteristic, complemented
70C23×D5C22⋊C412yes—
71C22×C4×C5—22no—
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C22×C4×C5C2×D411yescharacteristic
75C22×Dic5—22no—
76C22×Dic5C22⋊C412yes—
77C2×D20—21no—
78C2×D20—21no—
79C2×D20—21no—
80C2×D20—21no—
81C2×D20SD1612yes—
82C2×D20Q1612yes—
83C2×C4×D5—22no—
84C2×C4×D5—22no—
85C2×C4×D5—44no—
86C2×Dic10—22no—
87C2×Dic10—22no—
88C2×Dic10—82no—
89C5×M16—41no—
90C5×M16—41no—
91C2×C5×C8—22no—
92C2×C5×C8—41no—
93C5×C4⋊C4—42no—
94C42×C5—41no—
95C42×C5—41no—
96C42×C5C2×D411yescharacteristic
97C42×C5C2×D411yescharacteristic
98C42×C5#4111yescharacteristic
99C42×C5#4111yescharacteristic
100#685—22no—
101#685—22no—
102#685—84no—
103C4⋊Dic5—21no—
104C4⋊Dic5—21no—
105C4⋊Dic5—41no—
106C4⋊Dic5—41no—
107C4⋊Dic5—41no—
108C4⋊Dic5—41no—
109#681—41no—
110#681—41no—
111C2×#168—22no—
112C2×#168—41no—
113C8×D5—84no—
114C2×C4⋊Q8—51no—
115C2×C4×D4—51no—
116C8×D4—102no—
117C8×D4—102no—
118#431—51no—
119#431C2×D511yescharacteristic
120#423—51no—
121#423C2×D511yescharacteristic
122C2×#100—101no—
123C2×C22⋊C8—102no—
124#393—54no—
125#390—101no—
126#382—101no—
127#380—102no—
128C4×C4⋊C4C2×D511yescharacteristic
129#343—101no—
130#333—102no—
131#333—102no—
132C23×C5C2×C22⋊C411yescharacteristic
133C22×D5—22no—
134C22×D5—22no—
135C22×D5—42no—
136C22×D5—42no—
137C22×D5#10314yes—
138C2×C4×C5—22no—
139C2×C4×C5—41no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5C4×D411yescharacteristic
144C2×C4×C5C4×D411yescharacteristic
145C2×C4×C5#12011yescharacteristic
146C2×C4×C5#12111yescharacteristic
147C2×C4×C5#12111yescharacteristic
148C2×C4×C5#12311yescharacteristic
149C2×C4×C5C2×SD1611yescharacteristic
150C2×C4×C5C2×Q1611yescharacteristic
151C2×C4×C5#13511yescharacteristic
152C2×C4×C5#13511yescharacteristic
153C2×Dic5—22no—
154C2×Dic5—42no—
155C2×Dic5—44no—
156C2×Dic5—42no—
157D20—22no—
158D20—22no—
159D20—22no—
160D20—22no—
161C4×D5—44no—
162C4×D5—82no—
163Dic10—82no—
164Dic10—82no—
165C5×C8—42no—
166C5×C8—81no—
167#168—42no—
168#168—81no—
169C22×Q8—52no—
170C22×D4—51no—
171C23×C4—52no—
172C2×M16—21no—
173C2×M16—101no—
174C22×C8—104no—
175C4⋊Q8—101no—
176C4⋊Q8—101no—
177C4⋊Q8—101no—
178C4⋊Q8—101no—
179C4×D4—101no—
180C4×D4—101no—
181C4×D4—101no—
182C4×D4—101no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—101no—
186C2×C4⋊C4—101no—
187C2×C22⋊C4—52no—
188C2×C22⋊C4—104no—
189C2×C42—21no—
190C2×C42C22×D511yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—52no—
194C4⋊C8—101no—
195C4⋊C8—101no—
196C4⋊C8D2012yes—
197#100—202no—
198C22⋊C8—21no—
199C22⋊C8—104no—
200C22⋊C8—101no—
201C4×C8—21no—
202C4×C8—101no—
203#95—104no—
204#95C4×D512yes—
205C22×C5—21no—
206C22×C5—21no—
207C22×C5#38611yescharacteristic
208C22×C5C2×#10311yescharacteristic
209C22×C5#41611yescharacteristic
210C2×D5—24no—
211C2×D5—42no—
212C2×D5—42no—
213C2×D5—44no—
214C2×D5—82no—
215C4×C5—21no—
216C4×C5—21no—
217C4×C5—21no—
218C4×C5—82no—
219C4×C5#45512yes—
220C4×C5#46511yescharacteristic
221C4×C5#46511yescharacteristic
222C4×C5#46711yescharacteristic
223C4×C5#46711yescharacteristic
224Dic5—42no—
225Dic5—82no—
226C24—52no—
227C2×Q8—102no—
228C2×Q8—102no—
229C2×Q8—402no—
230C2×D4—52no—
231C2×D4—52no—
232C2×D4—101no—
233C2×D4—101no—
234C2×D4—101no—
235C2×D4—101no—
236C22×C4—22no—
237C22×C4—52no—
238C22×C4—102no—
239C22×C4—102no—
240C22×C4—102no—
241C22×C4—204no—
242C22×C4C2×C4×D511yescharacteristic
243C22×C4D4×D511yescharacteristic
244C22×C4D4×D511yescharacteristic
245M16—41no—
246M16—41no—
247M16—201no—
248M16—201no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—102no—
252C2×C8—201no—
253C2×C8—404no—
254C4⋊C4—42no—
255C4⋊C4—101no—
256C4⋊C4—101no—
257C4⋊C4—201no—
258C4⋊C4—201no—
259C4⋊C4—201no—
260C4⋊C4—201no—
261C22⋊C4—102no—
262C22⋊C4—102no—
263C22⋊C4—404no—
264C42—41no—
265C42—41no—
266C42C2×D2011yescharacteristic
267C42#70911yescharacteristic
268C42D4×D511yescharacteristic
269C42#71311yescharacteristic
270C2×C5—21no—
271C2×C5—21no—
272C2×C5#188511yescharacteristic
273C2×C5#189211yescharacteristic
274C2×C5#191411yescharacteristic
275D5—44no—
276C23—54no—
277C23—102no—
278C23—102no—
279C23—202no—
280C23—202no—
281C23D5×C22⋊C411yescharacteristic
282Q8—402no—
283Q8—402no—
284D4—102no—
285D4—102no—
286D4—102no—
287D4—102no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—102no—
294C2×C4—204no—
295C2×C4—202no—
296C2×C4—204no—
297C2×C4—202no—
298C2×C4—402no—
299C2×C4C4×D2011yescharacteristic
300C2×C4#402811yescharacteristic
301C2×C4#403011yescharacteristic
302C2×C4#403911yescharacteristic
303C2×C4#404011yescharacteristic
304C2×C4#404111yescharacteristic
305C2×C4#405811yescharacteristic
306C2×C4D5×SD1611yescharacteristic
307C2×C4#406211yescharacteristic
308C2×C4D5×Q1611yescharacteristic
309C8—42no—
310C8—81no—
311C8—202no—
312C8—401no—
313C5#939111yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22—104no—
317C22—204no—
318C22—202no—
319C22—202no—
320C22—402no—
321C22#6495411yescharacteristic
322C22#6506111yescharacteristic
323C22D5×#10311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—202no—
329C4—402no—
330C4#6506711yescharacteristic
331C4#6510211yescharacteristic
332C4#6511412yes—
333C4#6515311yescharacteristic
334C4#6517611yescharacteristic
335C2—21no—
336C2—21no—
337C2—204no—
338C2#1059955211yescharacteristic
339C2#1059967511yescharacteristic
340C2#1059983311yescharacteristic
subgroupG/H
centerC22#64954
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#1303C2
socleC22×C5#386
prime radicalC22×D20D4
cosocleC2×C42×C5C23
hypercenter#1303D5
hypocenterC5#9391
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #10601572C2semidirect
5. #10601300C2semidirect
6. C5×#1303C2semidirect
7. #10598111C2semidirect
69. #1303D5semidirect
313. C5#9391semidirect

Maximal subgroups

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

subgroupindex
#106110172
#106055382
#106015722
#106013202
#106013002
C5×#13032
#105981112
#93915