Finite Group Databasebeta

← #13034666

Subgroups — #13034666

Order 1280.

954 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65729C212yesmaximal
2C2×D5⋊#108C211yesmaximal, characteristic
3C22×#3995C211yesmaximal, characteristic
4#10617108C212yesmaximal
5D5×#405C211yesmaximal, characteristic
6#10603890C214yesmaximal, complemented
7#10603860C212yesmaximal, complemented
8#10603858C212yesmaximal, complemented
9C2×D5⋊M16C2212yes—
10C22×#700C2212yes—
11C2×D5×M16C2211yescharacteristic
12C22×C8×D5C2211yescharacteristic
13D5×#117C2211yescharacteristic
14#65729—28no—
15D5⋊#108—24no—
16C2×#3996C2212yes—
17C2×#3996C2214yes—
18C2×#3995—24no—
19C2×#3995C2212yes—
20C2×#3995C2212yes—
21C2×#3995C2211yescharacteristic
22C2×#3995C2211yescharacteristic
23C2×#3995C2214yes—
24#65708—24no—
25#65696—24no—
26#65682C2214yes—
27#65681C2214yes—
28C5×#405C411yescharacteristic
29#65411C411yescharacteristic
30#65390C411yescharacteristic
31#65286C411yescharacteristic
32#64997C412yes—
33D5×#97C2212yes—
34#64978C412yes—
35C4×C8×D5C2212yes—
36#20807—51nomaximal, Sylow
37C22×C4×D5C2311yescharacteristic
38C2×#704—22no—
39C22×#170—22no—
40D5⋊M16—24no—
41D5⋊M16—24no—
42C2×#700—24no—
43C2×#700—24no—
44C2×#700—22no—
45C2×#700C2312yes—
46C2×#700C2314yes—
47C2×#700C2312yes—
48C2×D5⋊C8—22no—
49C2×C5×M16C2×C411yescharacteristic
50C22×C5×C8C2×C411yescharacteristic
51C5×#117C2×C411yescharacteristic
52#4072C2×C411yescharacteristic
53C2×#681C2×C411yescharacteristic
54C22×#168C2×C411yescharacteristic
55D5×M16D414yes—
56D5×M16Q814yes—
57C2×#676C2×C412yes—
58C2×C8×D5C2×C411yescharacteristic
59C2×C8×D5C2×C411yescharacteristic
60C2×C8×D5D414yes—
61C2×C8×D5Q814yes—
62C2×C8×D5C2311yescharacteristic
63C2×C8×D5C2311yescharacteristic
64C2×C8×D5C2312yes—
65#4038D412yes—
66D5×C4⋊C4Q812yes—
67D5×C22⋊C4Q812yes—
68#4023D412yes—
69#4018C2×C412yes—
70C42×D5C2312yes—
71#4004—28no—
72C4⋊#170—28no—
73#3996—24no—
74#3996—28no—
75#3995—24no—
76#3995—24no—
77#3995—22no—
78#3995—22no—
79#3995—28no—
80C5×#97C2×C412yes—
81C4×C5×C8C2×C412yes—
82#3947C2×C412yes—
83#3947C2×C412yes—
84C8×Dic5C2×C412yes—
85C8×Dic5C2×C412yes—
86#3935C2×C412yes—
87C4×#168C2×C412yes—
88C2×#430—52no—
89C22×#108—51no—
90C22×#108—51no—
91#2905—52no—
92C2×#405—51no—
93#1849—54no—
94#1848—52no—
95#1848—52no—
96C23×D5C2×Q811yescharacteristic
97C22×C4×C5C22×C411yescharacteristic
98C22×Dic5C2×D411yescharacteristic
99C2×C4×D5C22×C411yescharacteristic
100C2×C4×D5C2×D412yes—
101C2×C4×D5C2×D412yes—
102C2×C4×D5C2×D411yescharacteristic
103C2×C4×D5C2×Q814yes—
104C2×C4×D5C2×Q812yes—
105C2×C4×D5C2×Q812yes—
106C2×C4×D5C2×Q811yescharacteristic
107C2×C4×D5C2411yescharacteristic
108#704—44no—
109C2×#170—22no—
110C2×#170—22no—
111C2×#170—24no—
112C2×#170—44no—
113#700—24no—
114#700—24no—
115#700—22no—
116#700—22no—
117#700—22no—
118#700—22no—
119#700—44no—
120D5⋊C8—44no—
121C5×M16C4⋊C414yes—
122C2×C5×C8C4⋊C414yes—
123C2×C5×C8C22×C411yescharacteristic
124C2×C5×C8C22×C411yescharacteristic
125C2×C5×C8C22×C412yes—
126C5×C4⋊C4C4⋊C412yes—
127C5×C22⋊C4C4⋊C412yes—
128C42×C5C22×C412yes—
129C22⋊Dic5C4⋊C412yes—
130#685C4⋊C414yes—
131C4⋊Dic5C4⋊C412yes—
132#683C4⋊C414yes—
133C4×Dic5C22×C412yes—
134C4×Dic5C2×D414yes—
135#681C4⋊C414yes—
136C2×#168C4⋊C414yes—
137C2×#168C22×C411yescharacteristic
138C2×#168C22×C412yes—
139C2×#168C22×C411yescharacteristic
140#676C4⋊C418yes—
141C8×D5C4⋊C414yes—
142C8×D5C4⋊C414yes—
143C8×D5C2×D412yes—
144C8×D5C2×D412yes—
145C8×D5C2×D414yes—
146C8×D5C2×Q812yes—
147C8×D5C2×Q812yes—
148C8×D5C2×Q814yes—
149C22×M16—51no—
150C22×M16—52no—
151C22×M16—52no—
152C23×C8—51no—
153C2×#117—51no—
154#430—108no—
155C2×#108—51no—
156C2×#108—52no—
157C2×#108—51no—
158C2×#108—54no—
159C2×#108—54no—
160C2×#108—52no—
161C2×#108—52no—
162C2×#108—104no—
163C2×#108—104no—
164#424—104no—
165#424—104no—
166#423—54no—
167#423—54no—
168#405—51no—
169#405—51no—
170#405—51no—
171#405—52no—
172#405—52no—
173#405AGL(1,𝔽5)11yescharacteristic
174C2×#97—52no—
175C2×C4×C8—52no—
176C23×C5C2×C4⋊C411yescharacteristic
177C22×D5—42no—
178C22×D5C2×C4⋊C411yescharacteristic
179C22×D5C4⋊Q812yes—
180C22×D5C4⋊Q812yes—
181C22×D5C22×Q811yescharacteristic
182C2×C4×C5C2×C4⋊C414yes—
183C2×C4×C5C2×C4⋊C412yes—
184C2×C4×C5C2×C4⋊C412yes—
185C2×C4×C5C2×C4⋊C411yescharacteristic
186C2×C4×C5C23×C411yescharacteristic
187C2×Dic5C2×C4⋊C414yes—
188C2×Dic5C2×C4⋊C411yescharacteristic
189C2×Dic5#12712yes—
190C2×Dic5C4⋊Q812yes—
191C2×Dic5C22×D411yescharacteristic
192C4×D5—44no—
193C4×D5C2×C4⋊C412yes—
194C4×D5C2×C4⋊C412yes—
195C4×D5#12712yes—
196C4×D5C4⋊Q812yes—
197C4×D5C4⋊Q812yes—
198C4×D5C4⋊Q812yes—
199C4×D5C22×D411yescharacteristic
200C4×D5C22×D411yescharacteristic
201C4×D5C22×Q811yescharacteristic
202C4×D5C22×Q811yescharacteristic
203#170—42no—
204#170—42no—
205#170—44no—
206C5×C8C2×C4⋊C412yes—
207C5×C8C2×C4⋊C412yes—
208C5×C8C2×C4⋊C414yes—
209#168C2×C4⋊C412yes—
210#168C2×C4⋊C412yes—
211#168C2×C4⋊C414yes—
212C23×C4—51no—
213C2×M16—51no—
214C2×M16—54no—
215C2×M16—54no—
216C2×M16—52no—
217C2×M16—52no—
218C2×M16—54no—
219C2×M16—52no—
220C2×M16—102no—
221C2×M16—102no—
222C2×M16—104no—
223C2×M16—104no—
224C2×M16—104no—
225C2×M16—104no—
226C2×M16C2×AGL(1,𝔽5)11yescharacteristic
227C22×C8—51no—
228C22×C8—54no—
229C22×C8—54no—
230C22×C8—51no—
231C22×C8—51no—
232C22×C8—51no—
233C22×C8—51no—
234C22×C8—52no—
235C22×C8—102no—
236C22×C8—102no—
237C22×C8C2×AGL(1,𝔽5)11yescharacteristic
238#117—51no—
239#117—52no—
240#117—52no—
241#117—52no—
242#117C2×AGL(1,𝔽5)11yescharacteristic
243C2×C4⋊C4—52no—
244C2×C22⋊C4—52no—
245C2×C42—52no—
246#108—104no—
247#108—104no—
248#108—102no—
249#108—102no—
250#108—108no—
251#108—104no—
252#108—108no—
253C4⋊C8—108no—
254C4⋊C8—108no—
255#97—52no—
256#97—52no—
257#97—52no—
258#97C2×AGL(1,𝔽5)12yes—
259C4×C8—52no—
260C4×C8—52no—
261C4×C8—52no—
262C4×C8C2×AGL(1,𝔽5)12yes—
263C22×C5—22no—
264C22×C5#38412yes—
265C22×C5C22×C4⋊C411yescharacteristic
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—42no—
269C2×D5—42no—
270C2×D5#38412yes—
271C2×D5C2×C4⋊Q811yescharacteristic
272C2×D5C2×C4⋊Q811yescharacteristic
273C4×C5—24no—
274C4×C5#38412yes—
275C4×C5C22×C4⋊C411yescharacteristic
276C4×C5C22×C4⋊C411yescharacteristic
277Dic5—24no—
278Dic5#38412yes—
279Dic5C2×#12711yescharacteristic
280Dic5C2×C4⋊Q811yescharacteristic
281C24—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—52no—
286C22×C4—52no—
287C22×C4—52no—
288C22×C4—52no—
289C22×C4—51no—
290C22×C4—51no—
291C22×C4—54no—
292C22×C4C22×AGL(1,𝔽5)11yescharacteristic
293M16—54no—
294M16—58no—
295M16—104no—
296M16—102no—
297M16—102no—
298M16—102no—
299M16—104no—
300M16—102no—
301M16—204no—
302M16—204no—
303M16C4⋊AGL(1,𝔽5)14yes—
304C2×C8—54no—
305C2×C8—51no—
306C2×C8—52no—
307C2×C8—51no—
308C2×C8—52no—
309C2×C8—52no—
310C2×C8—54no—
311C2×C8—54no—
312C2×C8—54no—
313C2×C8—52no—
314C2×C8—52no—
315C2×C8—54no—
316C2×C8—102no—
317C2×C8—104no—
318C2×C8—102no—
319C2×C8—204no—
320C2×C8—204no—
321C2×C8C4⋊AGL(1,𝔽5)14yes—
322C2×C8C22×AGL(1,𝔽5)11yescharacteristic
323C2×C8C22×AGL(1,𝔽5)12yes—
324C2×C8C22×AGL(1,𝔽5)11yescharacteristic
325C4⋊C4—54no—
326C4⋊C4—52no—
327C4⋊C4C4⋊AGL(1,𝔽5)12yes—
328C22⋊C4—52no—
329C22⋊C4—54no—
330C22⋊C4C4⋊AGL(1,𝔽5)12yes—
331C42—54no—
332C42—52no—
333C42C22×AGL(1,𝔽5)12yes—
334C2×C5—22no—
335C2×C5C2×#38411yescharacteristic
336C2×C5#291612yes—
337D5—21no—
338D5—21no—
339D5—22no—
340C23—51no—
341C23—51no—
342C23—52no—
343C23—52no—
344C23—202no—
345C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
346C2×C4—52no—
347C2×C4—52no—
348C2×C4—51no—
349C2×C4—51no—
350C2×C4—52no—
351C2×C4—52no—
352C2×C4—52no—
353C2×C4—52no—
354C2×C4—51no—
355C2×C4—51no—
356C2×C4—52no—
357C2×C4—51no—
358C2×C4—51no—
359C2×C4—52no—
360C2×C4—54no—
361C2×C4—204no—
362C2×C4C2×C4⋊AGL(1,𝔽5)14yes—
363C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
364C2×C4C2×C4⋊AGL(1,𝔽5)12yes—
365C2×C4C2×C4⋊AGL(1,𝔽5)12yes—
366C2×C4C23×AGL(1,𝔽5)11yescharacteristic
367C8—52no—
368C8—52no—
369C8—54no—
370C8—202no—
371C8—202no—
372C8—204no—
373C8C2×C4⋊AGL(1,𝔽5)12yes—
374C8C2×C4⋊AGL(1,𝔽5)12yes—
375C8C2×C4⋊AGL(1,𝔽5)14yes—
376C5#2080711yescharacteristic, complemented, Sylow
377C22—22no—
378C22—51no—
379C22—51no—
380C22—52no—
381C22—102no—
382C22—102no—
383C22—202no—
384C22—202no—
385C22#6568812yes—
386C22C22×C4⋊AGL(1,𝔽5)11yescharacteristic
387C4—24no—
388C4—51no—
389C4—51no—
390C4—52no—
391C4—104no—
392C4#6568812yes—
393C4C22×C4⋊AGL(1,𝔽5)11yescharacteristic
394C4C22×C4⋊AGL(1,𝔽5)11yescharacteristic
395C2—22no—
396C2—101no—
397C2—101no—
398C2—102no—
399C2C2×#6568811yescharacteristic
400C2#1061741812yes—
subgroupG/H
centerC22C22×C4⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC5×#405C4
socleC22×C5C22×C4⋊C4
prime radicalC23×D5C2×Q8
cosocleC2×C4×D5C24
hypercenter#405AGL(1,𝔽5)
hypocenterC5#20807
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
6. #10603890C2semidirect
7. #10603860C2semidirect
8. #10603858C2semidirect
376. C5#20807semidirect

Maximal subgroups

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

subgroupindex
C2×#657292
C2×D5⋊#1082
C22×#39952
#106171082
D5×#4052
#106038902
#106038602
#106038582
#208075