Finite Group Databasebeta

← #12969645

Subgroups — #12969645

Order 1280.

1830 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618163C212yesmaximal
2C5×#3160C211yesmaximal, characteristic, complemented
3#10615068C212yesmaximal, complemented
4#10613665C214yesmaximal, complemented
5#10613641C218yesmaximal, complemented
6#10613620C214yesmaximal, complemented
7#10610915C214yesmaximal
8#10610900C214yesmaximal, complemented
9C2×#64988C211yesmaximal, characteristic, complemented
10C4×#4047C211yesmaximal, characteristic
11C5×#533C2212yes—
12C22×Dic20C2214yes—
13C2×#4047C2212yes—
14#66062C2218yes—
15#66031C2218yes—
16#65976—28no—
17#65918—24no—
18#65904—24no—
19#65860C2214yes—
20#65850C2214yes—
21C4×#709C2211yescharacteristic
22C2×#4015C2212yes—
23C5×#500C2212yes—
24C5×#499C2214yes—
25C5×#498C2212yes—
26C5×#427C2214yes—
27C2×C4×C5×C8C2211yescharacteristic
28#65390C2212yes—
29#65382C2218yes—
30#65358C2218yes—
31#65293C2218yes—
32C4×#681C2211yescharacteristic
33#65176C22116yes—
34#65162C22116yes—
35#65157C22116yes—
36#65142C22116yes—
37#64988C2212yes—
38#64988C2214yes—
39#64988C2212yes—
40#64978C2214yes—
41C4×#676C2212yes—
42C4×C8×D5C2212yes—
43#60715—51nomaximal, Sylow
44C2×#709C2311yescharacteristic
45C22×Dic10C2314yes—
46C22×C5×C8C2312yes—
47C5×C4⋊Q8C2314yes—
48C5×#125C2314yes—
49C5×C22⋊Q8—24no—
50C4×C5×Q8—24no—
51C5×#117C2314yes—
52C2×C42×C5C2311yescharacteristic
53#4092—28no—
54Q8×Dic5—28no—
55C4×C22⋊D5D414yes—
56#4072C2311yescharacteristic
57C2×C4⋊Dic5—24no—
58C2×#681C2312yes—
59C2×Dic20—216no—
60C2×Dic20C2318yes—
61C2×Dic20C2318yes—
62#4047—28no—
63C2×#676C2314yes—
64C2×C8×D5C2314yes—
65#4044—216no—
66#4043C23116yes—
67#4038—28no—
68#4036—216no—
69#4036—28no—
70#4033C23116yes—
71#4032—28no—
72#4023—28no—
73#4021—24no—
74C4×D20D412yes—
75#4018C2312yes—
76C42×D5C2312yes—
77#4015—24no—
78#4015—24no—
79#4015C2314yes—
80#4015C2314yes—
81C4×Dic10—28no—
82C4×Dic10D412yes—
83C5×#107C2318yes—
84C5×#106C2318yes—
85C4×C5×C8C2311yescharacteristic
86C4×C5×C8C2312yes—
87C4×C5×C8C2311yescharacteristic
88#3947C2314yes—
89C8×Dic5C2314yes—
90#3942C23132yes—
91#3935C2312yes—
92C4×#168C2312yes—
93#3509—52no—
94#3185—54no—
95#3177—58no—
96#3170—54no—
97#3160D511yescharacteristic, complemented
98C2×#494—51no—
99#3139—52no—
100#3059—54no—
101#2950—54no—
102C4×#131—51no—
103C2×C5×Q8—24no—
104C22×C4×C5C2×D412yes—
105C22×C4×C5C2411yescharacteristic
106C2×C22⋊D5C2×D412yes—
107C22×Dic5—24no—
108#709—24no—
109C2×D20C2×D411yescharacteristic
110C2×C4×D5C2×D414yes—
111C2×C4×D5C2412yes—
112C2×Dic10—216no—
113C2×Dic10—24no—
114C2×Dic10C2×D411yescharacteristic
115C2×Dic10C2418yes—
116C2×C5×C8#4118yes—
117C2×C5×C8C2412yes—
118C2×C5×C8C2412yes—
119C5×C4⋊C4—28no—
120C5×C4⋊C4—24no—
121C5×C4⋊C4C2418yes—
122C5×C22⋊C4—24no—
123C42×C5—24no—
124C42×C5C2×D412yes—
125C42×C5C2411yescharacteristic
126C42×C5C2411yescharacteristic
127C22⋊Dic5C2×D412yes—
128#685—216no—
129#685C2×D414yes—
130C4⋊Dic5—28no—
131C4⋊Dic5—28no—
132C4⋊Dic5—216no—
133C4⋊Dic5C2×D412yes—
134#683—216no—
135#683C2×D414yes—
136C4×Dic5—216no—
137C4×Dic5C2×D414yes—
138C4×Dic5C2412yes—
139#681—24no—
140C2×#168C2414yes—
141Dic20—216no—
142Dic20—216no—
143#676—28no—
144C8×D5—28no—
145C22×Q16—54no—
146C2×#131—52no—
147#557—104no—
148#556—104no—
149#552—108no—
150#541—58no—
151#536—54no—
152#533—58no—
153#533C2×D512yes—
154C2×C4⋊Q8—52no—
155#519—54no—
156C4×#41—51no—
157#500C2×D512yes—
158#499C2×D514yes—
159#498C2×D512yes—
160#494—52no—
161#494—54no—
162#494—52no—
163#491—58no—
164#487—58no—
165#470—516no—
166#467—516no—
167#441—516no—
168C4×Q16—516no—
169#427C2×D514yes—
170#419—58no—
171#405—52no—
172#405—54no—
173C4×M16—51no—
174C4×M16—52no—
175C2×C4×C8—52no—
176C2×C4×C8C2×D511yescharacteristic
177C23×C5C22×D411yescharacteristic
178C22×D5C22×D412yes—
179C5×Q8—48no—
180C2×C4×C5—24no—
181C2×C4×C5—28no—
182C2×C4×C5C22×D412yes—
183C2×C4×C5C22×D411yescharacteristic
184C2×C4×C5C22×D412yes—
185C2×C4×C5C2×#4114yes—
186C2×C4×C5C2511yescharacteristic
187C22⋊D5—24no—
188C2×Dic5—28no—
189C2×Dic5—48no—
190C2×Dic5C22×D414yes—
191C2×Dic5C22×D412yes—
192D20—22no—
193C4×D5—28no—
194C4×D5—24no—
195Dic10—22no—
196Dic10—216no—
197Dic10—48no—
198C5×C8C2×#4114yes—
199C5×C8C2×#4114yes—
200#168—24no—
201C2×#41—51no—
202C22×Q8—54no—
203C2×Q16—58no—
204C2×Q16—58no—
205C2×Q16—1016no—
206#131—108no—
207C2×M16—52no—
208C2×M16—54no—
209C22×C8—54no—
210C22×C8C22×D512yes—
211C4⋊Q8—54no—
212C4⋊Q8—54no—
213C4⋊Q8—104no—
214C4⋊Q8—104no—
215C4⋊Q8C22×D514yes—
216#126—1016no—
217#125—108no—
218#125—1016no—
219#125C22×D514yes—
220#124—104no—
221#123—108no—
222C22⋊Q8—24no—
223C22⋊Q8—516no—
224C22⋊Q8—108no—
225C4×Q8—24no—
226C4×Q8—52no—
227C4×Q8—516no—
228C4×Q8—108no—
229C4×Q8—108no—
230C4×D4—54no—
231C4×D4—52no—
232#117—51no—
233#117—52no—
234#117—108no—
235#117—108no—
236#117C22×D514yes—
237C2×C4⋊C4—104no—
238C2×C42—52no—
239C2×C42C22×D511yescharacteristic
240#107C22×D518yes—
241#106C22×D518yes—
242#103—532no—
243#97—54no—
244#97—52no—
245C4×C8—54no—
246C4×C8—52no—
247C4×C8C22×D511yescharacteristic
248C4×C8C22×D511yescharacteristic
249C4×C8C22×D512yes—
250C22×C5—22no—
251C22×C5#53212yes—
252C22×C5C23×D411yescharacteristic
253C2×D5—22no—
254C2×D5—24no—
255C4×C5—22no—
256C4×C5—48no—
257C4×C5#53212yes—
258C4×C5#57814yes—
259C4×C5C22×#4112yes—
260Dic5—22no—
261Dic5—24no—
262Dic5—48no—
263#41—104no—
264C2×Q8—24no—
265C2×Q8—51no—
266C2×Q8—58no—
267C2×Q8—104no—
268C2×Q8—1016no—
269C2×D4—52no—
270C2×D4—51no—
271C22×C4—54no—
272C22×C4—52no—
273C22×C4—104no—
274C22×C4D4×D512yes—
275C22×C4C23×D511yescharacteristic
276Q16—1016no—
277Q16—1016no—
278M16—104no—
279M16—108no—
280C2×C8—54no—
281C2×C8—108no—
282C2×C8#71318yes—
283C2×C8C23×D512yes—
284C2×C8C23×D512yes—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—52no—
288C4⋊C4—54no—
289C4⋊C4—1016no—
290C4⋊C4—1016no—
291C4⋊C4—108no—
292C4⋊C4—108no—
293C4⋊C4C23×D518yes—
294C22⋊C4—24no—
295C22⋊C4—52no—
296C22⋊C4—54no—
297C22⋊C4—1016no—
298C42—24no—
299C42—54no—
300C42—52no—
301C42—1016no—
302C42D4×D512yes—
303C42C23×D511yescharacteristic
304C42C23×D511yescharacteristic
305C2×C5—22no—
306C2×C5C2×#53211yescharacteristic
307C2×C5C2×#57812yes—
308D5—42no—
309C23—52no—
310C23C2×D4×D511yescharacteristic
311Q8—48no—
312Q8—102no—
313Q8—1016no—
314Q8—208no—
315D4—104no—
316D4—102no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—54no—
320C2×C4—52no—
321C2×C4—108no—
322C2×C4—104no—
323C2×C4—108no—
324C2×C4—208no—
325C2×C4C2×D4×D512yes—
326C2×C4C2×D4×D511yescharacteristic
327C2×C4C2×D4×D512yes—
328C2×C4C2×#71314yes—
329C2×C4C24×D511yescharacteristic
330C8—104no—
331C8C2×#71314yes—
332C8C2×#71314yes—
333C5#6071511yescharacteristic, complemented, Sylow
334C22—22no—
335C22—104no—
336C22—102no—
337C22#6605912yes—
338C22C22×D4×D511yescharacteristic
339C4—22no—
340C4—48no—
341C4—104no—
342C4—102no—
343C4—208no—
344C4#6605912yes—
345C4#6610514yes—
346C4C22×#71312yes—
347C2—22no—
348C2—202no—
349C2C2×#6605911yescharacteristic
350C2C2×#6610512yes—
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×#3160C2
socleC22×C5C23×D4
prime radicalC2×#709C23
hypercenter#3160D5
hypocenterC5#60715
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. C5×#3160C2semidirect
3. #10615068C2semidirect
4. #10613665C2semidirect
5. #10613641C2semidirect
6. #10613620C2semidirect
8. #10610900C2semidirect
9. C2×#64988C2semidirect
97. #3160D5semidirect
333. C5#60715semidirect

Maximal subgroups

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

subgroupindex
#106181632
C5×#31602
#106150682
#106136652
#106136412
#106136202
#106109152
#106109002
C2×#649882
C4×#40472
#607155