Finite Group Databasebeta

← #12014897

Subgroups — #12014897

Order 1280.

694 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C5×#3510C211yesmaximal, characteristic
2#10602666C211yesmaximal, characteristic, complemented
3#10602664C211yesmaximal, characteristic, complemented
4#10602652C212yesmaximal
5#10602437C212yesmaximal, complemented
6C5×#559—22no—
7C5×#556—22no—
8C5×#551—24no—
9C5×#542—22no—
10C5×#536C2211yescharacteristic
11C5×#535C2211yescharacteristic
12C5×#533C411yescharacteristic, complemented
13C5×#529—22no—
14C2×C5×#124C411yescharacteristic, complemented
15C5×#519C411yescharacteristic, complemented
16C5×#518C411yescharacteristic, complemented
17C4×C5×#41C2211yescharacteristic
18#65529C2212yes—
19#65526C2212yes—
20#64768—22no—
21#64761—22no—
22#64756—24no—
23#64751—22no—
24#64743—22no—
25#12721—51nomaximal, Sylow
26C2×C5×#41C2311yescharacteristic
27C22×C5×Q8C2×C411yescharacteristic
28C22×C5×D4C2×C411yescharacteristic
29C5×C4⋊Q8—21no—
30C5×C4⋊Q8D412yes—
31C5×#127—21no—
32C5×#126—24no—
33C5×#126—24no—
34C5×#125—21no—
35C5×#124—24no—
36C5×#124—22no—
37C5×#124—21no—
38C5×#124—21no—
39C5×#124—21no—
40C5×#124D412yes—
41C5×#123—24no—
42C5×#123—24no—
43C5×#123—22no—
44C5×C22⋊Q8—24no—
45C5×C22⋊Q8—22no—
46C5×C22⋊Q8—22no—
47C5×#121—22no—
48C5×#120D414yes—
49C4×C5×Q8—22no—
50C4×C5×Q8—21no—
51C4×C5×Q8—22no—
52C4×C5×Q8—22no—
53C4×C5×D4—24no—
54C4×C5×D4—22no—
55C4×C5×D4—21no—
56C4×C5×D4—22no—
57C4×C5×D4—22no—
58C5×#117—24no—
59C5×#117—22no—
60C5×#117C2×C411yescharacteristic
61C5×#117C2×C411yescharacteristic
62C5×#117C2×C411yescharacteristic
63C5×#117D412yes—
64C2×C5×C22⋊C4—22no—
65C2×C5×C22⋊C4—22no—
66C2×C42×C5C2×C411yescharacteristic
67C2×C42×C5D412yes—
68#4072—22no—
69C2×#681—22no—
70#3968—22no—
71#3966—22no—
72#3966—22no—
73#3965—22no—
74#3510D511yescharacteristic
75#2135—51no—
76#2129—51no—
77#2122—52no—
78#1827—52no—
79C24×C5C22⋊C412yes—
80C5×#41—42no—
81C2×C5×Q8—22no—
82C2×C5×Q8—21no—
83C2×C5×Q8—42no—
84C2×C5×Q8C22⋊C412yes—
85C2×C5×Q8C2×D411yescharacteristic
86C2×C5×D4—24no—
87C2×C5×D4—21no—
88C2×C5×D4—21no—
89C2×C5×D4—42no—
90C2×C5×D4C2×D411yescharacteristic
91C2×C5×D4C2×D412yes—
92C22×C4×C5—21no—
93C22×C4×C5—24no—
94C22×C4×C5—21no—
95C22×C4×C5—22no—
96C22×C4×C5C22×C411yescharacteristic
97C22×C4×C5C2×D412yes—
98C5×C4⋊C4—22no—
99C5×C4⋊C4—24no—
100C5×C4⋊C4—22no—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—21no—
103C5×C4⋊C4—21no—
104C5×C4⋊C4—21no—
105C5×C4⋊C4—24no—
106C5×C4⋊C4—22no—
107C5×C22⋊C4—24no—
108C5×C22⋊C4—24no—
109C5×C22⋊C4—24no—
110C5×C22⋊C4—21no—
111C5×C22⋊C4—21no—
112C5×C22⋊C4—22no—
113C5×C22⋊C4—24no—
114C5×C22⋊C4—24no—
115C5×C22⋊C4C22⋊C412yes—
116C42×C5—24no—
117C42×C5—21no—
118C42×C5—21no—
119C42×C5—22no—
120C42×C5C22⋊C411yescharacteristic
121C42×C5C22⋊C411yescharacteristic
122C42×C5C22⋊C412yes—
123C42×C5C22⋊C412yes—
124C22⋊Dic5—42no—
125C4⋊Dic5—42no—
126C4×Dic5—42no—
127#681—42no—
128#681—42no—
129C2×#168—42no—
130#559—22no—
131#556—22no—
132#551—24no—
133#542—22no—
134#536C2×D511yescharacteristic
135#535C2×D511yescharacteristic
136#533Dic511yescharacteristic, complemented
137#529—22no—
138C2×#124Dic511yescharacteristic, complemented
139#519Dic511yescharacteristic, complemented
140#518Dic511yescharacteristic, complemented
141C4×#41C2×D511yescharacteristic
142#413—52no—
143#410—52no—
144#356—102no—
145#354—102no—
146AΣL(1,𝔽16)—104no—
147#339—102no—
148H(3,ℤ/4)—102no—
149C23×C5—22no—
150C23×C5—42no—
151C23×C5—82no—
152C23×C5C2×C22⋊C411yescharacteristic
153C23×C5#12012yes—
154C5×Q8—41no—
155C5×Q8—42no—
156C5×Q8—42no—
157C5×Q8—82no—
158C5×D4—42no—
159C5×D4—41no—
160C5×D4—42no—
161C5×D4—42no—
162C5×D4—84no—
163C2×C4×C5—22no—
164C2×C4×C5—22no—
165C2×C4×C5—22no—
166C2×C4×C5—22no—
167C2×C4×C5—21no—
168C2×C4×C5—22no—
169C2×C4×C5—22no—
170C2×C4×C5—21no—
171C2×C4×C5—22no—
172C2×C4×C5—42no—
173C2×C4×C5—82no—
174C2×C4×C5—84no—
175C2×C4×C5—81no—
176C2×C4×C5C2×C22⋊C411yescharacteristic
177C2×C4×C5C2×C22⋊C411yescharacteristic
178C2×C4×C5#12012yes—
179C2×Dic5—82no—
180#168—82no—
181C2×#41C22×D511yescharacteristic
182C22×Q8C2×Dic511yescharacteristic
183C22×D4C2×Dic511yescharacteristic
184C2×M16—102no—
185C4⋊Q8—21no—
186C4⋊Q8C22⋊D512yes—
187#127—21no—
188#126—24no—
189#126—24no—
190#125—21no—
191#124—24no—
192#124—22no—
193#124—21no—
194#124—21no—
195#124—21no—
196#124C22⋊D512yes—
197#123—24no—
198#123—24no—
199#123—22no—
200C22⋊Q8—24no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203#121—22no—
204#120C22⋊D514yes—
205C4×Q8—22no—
206C4×Q8—21no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×D4—24no—
210C4×D4—22no—
211C4×D4—21no—
212C4×D4—22no—
213C4×D4—22no—
214#117—24no—
215#117—22no—
216#117—102no—
217#117C2×Dic511yescharacteristic
218#117C2×Dic511yescharacteristic
219#117C2×Dic511yescharacteristic
220#117C22⋊D512yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42C2×Dic511yescharacteristic
224C2×C42C22⋊D512yes—
225#101—102no—
226#100—102no—
227#99—102no—
228#99—102no—
229C22×C5—22no—
230C22×C5—21no—
231C22×C5—42no—
232C22×C5—41no—
233C22×C5—44no—
234C22×C5—42no—
235C22×C5—162no—
236C22×C5#37911yescharacteristic
237C4×C5—22no—
238C4×C5—21no—
239C4×C5—42no—
240C4×C5—41no—
241C4×C5—42no—
242C4×C5—41no—
243C4×C5—42no—
244C4×C5—42no—
245C4×C5#41012yes—
246Dic5—162no—
247C24C22⋊Dic512yes—
248#41—42no—
249C2×Q8—22no—
250C2×Q8—21no—
251C2×Q8—42no—
252C2×Q8C22⋊Dic512yes—
253C2×Q8C2×C22⋊D511yescharacteristic
254C2×D4—24no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4C2×C22⋊D511yescharacteristic
259C2×D4C2×C22⋊D512yes—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4C22×Dic511yescharacteristic
265C22×C4C2×C22⋊D512yes—
266M16—202no—
267M16—202no—
268C2×C8—202no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—202no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—202no—
288C22⋊C4C22⋊Dic512yes—
289C42—24no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—202no—
294C42C22⋊Dic512yes—
295C42C22⋊Dic511yescharacteristic
296C42C22⋊Dic511yescharacteristic
297C42C22⋊Dic512yes—
298C2×C5—21no—
299C2×C5—42no—
300C2×C5—41no—
301C2×C5—82no—
302C2×C5#178211yescharacteristic
303C23—22no—
304C23—42no—
305C23—82no—
306C23C2×C22⋊Dic511yescharacteristic
307C23#409912yes—
308Q8—41no—
309Q8—42no—
310Q8—42no—
311Q8—82no—
312D4—42no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—84no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—81no—
330C2×C4—402no—
331C2×C4C2×C22⋊Dic511yescharacteristic
332C2×C4C2×C22⋊Dic511yescharacteristic
333C2×C4#409912yes—
334C8—402no—
335C5#1272111yescharacteristic, complemented, Sylow
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—44no—
341C22—42no—
342C22—162no—
343C22#6553611yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—802no—
353C4#6552912yes—
354C2—21no—
355C2—42no—
356C2—41no—
357C2—82no—
358C2#1060293211yescharacteristic
subgroupG/H
centerC2#10602932
commutatorC22×C4×C5C22×C4
FrattiniC2×#41C22×D5
FittingC5×#3510C2
socleC2×C5#1782
prime radicalC5×#535C22
cosocleC2×C5×#41C23
hypercenter#3510D5
hypocenterC5#12721
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. #10602666C2semidirect
3. #10602664C2semidirect
5. #10602437C2semidirect
12. C5×#533C4semidirect
14. C2×C5×#124C4semidirect
15. C5×#519C4semidirect
16. C5×#518C4semidirect
136. #533Dic5semidirect
138. C2×#124Dic5semidirect
139. #519Dic5semidirect
140. #518Dic5semidirect
335. C5#12721semidirect

Maximal subgroups

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

subgroupindex
C5×#35102
#106026662
#106026642
#106026522
#106024372
#127215