Finite Group Databasebeta

← C5×#10178 #11932898

Subgroups — C5×#10178 #11932898

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3208C211yesmaximal, characteristic, complemented
2C5×#3077C211yesmaximal, characteristic, complemented
3C2×C5×#459C211yesmaximal, characteristic
4C5×#1545C211yesmaximal, characteristic, complemented
5C5×#1481C211yesmaximal, characteristic, complemented
6C5×#1460C211yesmaximal, characteristic, complemented
7C5×#1301C211yesmaximal, characteristic, complemented
8C22×C5×D8—21no—
9C5×#533C2211yescharacteristic
10C2×C5×#127C2211yescharacteristic
11C2×C4×C5×D4C2211yescharacteristic
12C5×#492—21no—
13C5×#486—21no—
14C5×#478—22no—
15C5×#476—21no—
16C5×#474—21no—
17C5×#460—22no—
18C5×#459—22no—
19C5×#459—22no—
20C5×#450—22no—
21C5×#431C2211yescharacteristic
22C5×#431C2211yescharacteristic
23C5×#428—21no—
24C5×#423C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C5×#418—21no—
27C2×C5×#102—21no—
28C5×#331—22no—
29C5×Q8⋊C8—22no—
30C5×#323—22no—
31#10178C511yesmaximal, characteristic, direct product factor, Sylow
32C22×C5×D4—21no—
33C22×C5×D4—21no—
34C22×C5×D4—21no—
35C22×C5×D4D411yescharacteristic
36C23×C4×C5—21no—
37C2×C5×SD16—24no—
38C2×C5×D8—24no—
39C2×C5×D8—42no—
40C2×C5×D8—42no—
41C2×C5×M16—21no—
42C22×C5×C8—21no—
43C5×C4⋊Q8D411yescharacteristic
44C5×C4⋊Q8D411yescharacteristic
45C5×#127—21no—
46C5×#127—21no—
47C5×#127—21no—
48C5×#127D412yes—
49C5×#125—21no—
50C5×C22⋊Q8—21no—
51C5×C22⋊Q8—21no—
52C4×C5×Q8—21no—
53C4×C5×Q8—21no—
54C4×C5×D4—21no—
55C4×C5×D4—21no—
56C4×C5×D4—21no—
57C4×C5×D4—21no—
58C5×#117—21no—
59C2×C5×C4⋊C4D411yescharacteristic
60C2×C5×C22⋊C4—21no—
61C2×C42×C5C2311yescharacteristic
62C5×#107—22no—
63C5×#106—22no—
64C5×C4⋊C8—21no—
65C5×C4⋊C8—21no—
66C5×C4⋊C8—21no—
67C5×C4⋊C8—21no—
68C5×C4⋊C8D412yes—
69C5×C4⋊C8D412yes—
70C5×#102—24no—
71C5×#102—22no—
72C5×#102—22no—
73C5×#102—24no—
74C5×C22⋊C8—21no—
75C5×C22⋊C8D412yes—
76C4×C5×C8—21no—
77C4×C5×C8—21no—
78#3208C2×C511yescharacteristic, complemented
79#3077C2×C511yescharacteristic, complemented
80C2×#459C2×C511yescharacteristic
81#1545C2×C511yescharacteristic, complemented
82#1481C2×C511yescharacteristic, complemented
83#1460C2×C511yescharacteristic, complemented
84#1301C2×C511yescharacteristic, complemented
85C24×C5—21no—
86C24×C5—41no—
87C2×C5×Q8—21no—
88C2×C5×Q8—21no—
89C2×C5×D4—21no—
90C2×C5×D4—21no—
91C2×C5×D4—22no—
92C2×C5×D4—22no—
93C2×C5×D4—22no—
94C2×C5×D4—21no—
95C2×C5×D4—21no—
96C2×C5×D4—21no—
97C2×C5×D4—41no—
98C2×C5×D4—41no—
99C2×C5×D4—41no—
100C2×C5×D4—41no—
101C2×C5×D4—41no—
102C2×C5×D4—41no—
103C2×C5×D4—41no—
104C2×C5×D4—82no—
105C2×C5×D4—82no—
106C2×C5×D4SD1612yescomplemented
107C22×C4×C5—21no—
108C22×C4×C5—41no—
109C22×C4×C5—41no—
110C22×C4×C5—82no—
111C22×C4×C5C2×D411yescharacteristic
112C22×C4×C5C2×D411yescharacteristic
113C22×C4×C5#4111yescharacteristic
114C5×SD16—84no—
115C5×D8—48no—
116C5×M16—41no—
117C5×M16—41no—
118C2×C5×C8—22no—
119C2×C5×C8—22no—
120C2×C5×C8—22no—
121C2×C5×C8—41no—
122C2×C5×C8—41no—
123C2×C5×C8—41no—
124C5×C4⋊C4—21no—
125C5×C4⋊C4—21no—
126C5×C4⋊C4—21no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—41no—
131C5×C4⋊C4—41no—
132C5×C22⋊C4—41no—
133C5×C22⋊C4—41no—
134C5×C22⋊C4—41no—
135C42×C5—41no—
136C42×C5C2×D411yescharacteristic
137C42×C5C2×D411yescharacteristic
138C42×C5C2×D411yescharacteristic
139C42×C5C2×D411yescharacteristic
140C22×D8—21no—
141#533C22×C511yescharacteristic
142C2×#127C22×C511yescharacteristic
143C2×C4×D4C22×C511yescharacteristic
144#492—21no—
145#486—21no—
146#478—22no—
147#476—21no—
148#474—21no—
149#460—22no—
150#459—22no—
151#459—22no—
152#450—22no—
153#431C22×C511yescharacteristic
154#431C22×C511yescharacteristic
155#428—21no—
156#423C22×C511yescharacteristic
157C2×C4⋊C8C22×C511yescharacteristic
158#418—21no—
159C2×#102—21no—
160#331—22no—
161Q8⋊C8—22no—
162#323—22no—
163C23×C5—22no—
164C23×C5—41no—
165C23×C5—41no—
166C23×C5—42no—
167C23×C5—81no—
168C23×C5—81no—
169C23×C5—82no—
170C23×C5—82no—
171C23×C5—81no—
172C23×C5—81no—
173C23×C5#12111yescharacteristic
174C5×Q8—42no—
175C5×Q8—42no—
176C5×D4—22no—
177C5×D4—22no—
178C5×D4—44no—
179C5×D4—41no—
180C5×D4—42no—
181C5×D4—41no—
182C5×D4—42no—
183C5×D4—82no—
184C5×D4—82no—
185C5×D4—81no—
186C5×D4—81no—
187C5×D4—82no—
188C2×C4×C5—22no—
189C2×C4×C5—41no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—41no—
193C2×C4×C5—82no—
194C2×C4×C5—82no—
195C2×C4×C5—81no—
196C2×C4×C5#12011yescharacteristic
197C2×C4×C5#12011yescharacteristic
198C2×C4×C5#12011yescharacteristic
199C2×C4×C5#12111yescharacteristic
200C2×C4×C5#12111yescharacteristic
201C2×C4×C5#12711yescharacteristic
202C2×C4×C5C2×SD1611yescharacteristic
203C2×C4×C5#13511yescharacteristic
204C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C5#13711yescharacteristic
206C5×C8—42no—
207C5×C8—42no—
208C5×C8—42no—
209C5×C8—81no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C5×D411yescharacteristic
214C23×C4—21no—
215C2×SD16—24no—
216C2×D8—24no—
217C2×D8—42no—
218C2×D8—42no—
219C2×M16—21no—
220C22×C8—21no—
221C4⋊Q8C5×D411yescharacteristic
222C4⋊Q8C5×D411yescharacteristic
223#127—21no—
224#127—21no—
225#127—21no—
226#127C5×D412yes—
227#125—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—21no—
237C2×C4⋊C4C5×D411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C23×C511yescharacteristic
240#107—22no—
241#106—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8C5×D412yes—
247C4⋊C8C5×D412yes—
248#102—24no—
249#102—22no—
250#102—22no—
251#102—24no—
252C22⋊C8—21no—
253C22⋊C8C5×D412yes—
254C4×C8—21no—
255C4×C8—21no—
256C22×C5—21no—
257C22×C5—21no—
258C22×C5—42no—
259C22×C5—82no—
260C22×C5—82no—
261C22×C5—81no—
262C22×C5—81no—
263C22×C5—82no—
264C22×C5—82no—
265C22×C5—82no—
266C22×C5—162no—
267C22×C5—162no—
268C22×C5#39211yescharacteristic
269C22×C5#46511yescharacteristic
270C22×C5#46811yescharacteristic
271C4×C5—21no—
272C4×C5—21no—
273C4×C5—21no—
274C4×C5—81no—
275C4×C5—81no—
276C4×C5—81no—
277C4×C5#44811yescharacteristic
278C4×C5#44911yescharacteristic
279C4×C5#45011yescharacteristic
280C4×C5#45211yescharacteristic
281C4×C5#47212yes—
282C24—21no—
283C24—41no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4C5×SD1612yescomplemented
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4C2×C5×D411yescharacteristic
309C22×C4C2×C5×D411yescharacteristic
310C22×C4C5×#4111yescharacteristic
311SD16—84no—
312D8—48no—
313M16—41no—
314M16—41no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42—41no—
333C42C2×C5×D411yescharacteristic
334C42C2×C5×D411yescharacteristic
335C42C2×C5×D411yescharacteristic
336C42C2×C5×D411yescharacteristic
337C2×C5—21no—
338C2×C5—21no—
339C2×C5—82no—
340C2×C5—162no—
341C2×C5#200011yescharacteristic
342C2×C5#200811yescharacteristic
343C2×C5#201811yescharacteristic
344C23—22no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—82no—
351C23—82no—
352C23—81no—
353C23—81no—
354C23C5×#12111yescharacteristic
355Q8—42no—
356Q8—42no—
357D4—22no—
358D4—22no—
359D4—44no—
360D4—41no—
361D4—42no—
362D4—41no—
363D4—42no—
364D4—82no—
365D4—82no—
366D4—81no—
367D4—81no—
368D4—82no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4C5×#12011yescharacteristic
378C2×C4C5×#12011yescharacteristic
379C2×C4C5×#12011yescharacteristic
380C2×C4C5×#12111yescharacteristic
381C2×C4C5×#12111yescharacteristic
382C2×C4C5×#12711yescharacteristic
383C2×C4C2×C5×SD1611yescharacteristic
384C2×C4C5×#13511yescharacteristic
385C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
386C2×C4C5×#13711yescharacteristic
387C8—42no—
388C8—42no—
389C8—42no—
390C8—81no—
391C5#1017811yescharacteristic, direct product factor, Sylow
392C22—21no—
393C22—21no—
394C22—42no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—162no—
403C22—162no—
404C22C5×#39211yescharacteristic
405C22C5×#46511yescharacteristic
406C22C5×#46811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4C5×#44811yescharacteristic
414C4C5×#44911yescharacteristic
415C4C5×#45011yescharacteristic
416C4C5×#45211yescharacteristic
417C4C5×#47212yes—
418C2—21no—
419C2—21no—
420C2—82no—
421C2—162no—
422C2C5×#200011yescharacteristic
423C2C5×#200811yescharacteristic
424C2C5×#201811yescharacteristic
subgroupG/H
center, socleC22×C5#392
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC2×C5×#459C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3208C2semidirect
2. C5×#3077C2semidirect
4. C5×#1545C2semidirect
5. C5×#1481C2semidirect
6. C5×#1460C2semidirect
7. C5×#1301C2semidirect
31. #10178C5direct (N × H)
78. #3208C2×C5semidirect
79. #3077C2×C5semidirect
81. #1545C2×C5semidirect
82. #1481C2×C5semidirect
83. #1460C2×C5semidirect
84. #1301C2×C5semidirect
106. C2×C5×D4SD16semidirect
303. C2×D4C5×SD16semidirect
391. C5#10178direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#32082
C5×#30772
C2×C5×#4592
C5×#15452
C5×#14812
C5×#14602
C5×#13012
#101785