Finite Group Databasebeta

← C5×#13326 #11936046

Subgroups — C5×#13326 #11936046

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×SD32C211yesmaximal, characteristic, complemented
2C22×C5×D16C211yesmaximal, characteristic
3C2×C5×#495C211yesmaximal, characteristic, complemented
4C2×C5×#358C211yesmaximal, characteristic, complemented
5C2×C5×#357C211yesmaximal, characteristic, complemented
6C5×#1970C211yesmaximal, characteristic, complemented
7C5×#1374C211yesmaximal, characteristic, complemented
8C22×C5×Q16C2211yescharacteristic
9C22×C5×SD16—21no—
10C22×C5×D8C2211yescharacteristic
11C22×C5×D8C2211yescharacteristic
12C2×C5×#124—21no—
13C2×C5×SD32—22no—
14C2×C5×SD32—22no—
15C2×C5×SD32—22no—
16C2×C5×D16—22no—
17C2×C5×D16—22no—
18C2×C5×D16—22no—
19C22×C5×C16C2211yescharacteristic
20C22×C5×C16C2211yescharacteristic
21C5×#495—22no—
22C5×#495—22no—
23C2×C5×#107C2211yescharacteristic
24C2×C5×#102—21no—
25C2×C4×C5×C8C2211yescharacteristic
26C5×#390—21no—
27C5×#358—22no—
28C5×#357—22no—
29#13326C511yesmaximal, characteristic, direct product factor, Sylow
30C22×C5×Q8—21no—
31C22×C5×D4—21no—
32C22×C5×D4—21no—
33C2×C5×Q16—21no—
34C2×C5×Q16—21no—
35C2×C5×Q16—21no—
36C2×C5×Q16—21no—
37C2×C5×Q16—21no—
38C2×C5×Q16D412yes—
39C2×C5×SD16—24no—
40C2×C5×SD16—42no—
41C2×C5×SD16—42no—
42C2×C5×D8—21no—
43C2×C5×D8—21no—
44C2×C5×D8—21no—
45C2×C5×D8—21no—
46C2×C5×D8—21no—
47C2×C5×D8—21no—
48C2×C5×D8—21no—
49C2×C5×D8—21no—
50C2×C5×D8—21no—
51C2×C5×D8—21no—
52C2×C5×D8D412yes—
53C2×C5×D8D412yes—
54C22×C5×C8D411yescharacteristic
55C22×C5×C8C2311yescharacteristic
56C5×#124—24no—
57C5×#124—42no—
58C2×C5×C4⋊C4—21no—
59C2×C5×C22⋊C4—42no—
60C2×C5×C22⋊C4—42no—
61C2×C42×C5D411yescharacteristic
62C5×SD32—28no—
63C5×D16—28no—
64C2×C5×C16—21no—
65C2×C5×C16—21no—
66C2×C5×C16—21no—
67C2×C5×C16—21no—
68C2×C5×C16D412yes—
69C2×C5×C16D412yes—
70C5×#107—21no—
71C5×#107—21no—
72C5×#102—42no—
73C4×C5×C8—21no—
74C4×C5×C8—21no—
75C22×SD32C2×C511yescharacteristic, complemented
76C22×D16C2×C511yescharacteristic
77C2×#495C2×C511yescharacteristic, complemented
78C2×#358C2×C511yescharacteristic, complemented
79C2×#357C2×C511yescharacteristic, complemented
80#1970C2×C511yescharacteristic, complemented
81#1374C2×C511yescharacteristic, complemented
82C24×C5—41no—
83C24×C5—41no—
84C2×C5×Q8—22no—
85C2×C5×Q8—41no—
86C2×C5×Q8—41no—
87C2×C5×Q8—41no—
88C2×C5×Q8—41no—
89C2×C5×Q8—41no—
90C2×C5×D4—22no—
91C2×C5×D4—22no—
92C2×C5×D4—41no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C2×C5×D4—41no—
96C2×C5×D4—41no—
97C2×C5×D4—41no—
98C2×C5×D4—41no—
99C2×C5×D4—41no—
100C2×C5×D4—41no—
101C2×C5×D4—41no—
102C22×C4×C5—41no—
103C22×C4×C5—41no—
104C22×C4×C5D812yes—
105C22×C4×C5C2×D411yescharacteristic
106C5×Q16—22no—
107C5×Q16—22no—
108C5×Q16—41no—
109C5×Q16—41no—
110C5×SD16—48no—
111C5×D8—22no—
112C5×D8—22no—
113C5×D8—22no—
114C5×D8—22no—
115C5×D8—41no—
116C5×D8—41no—
117C5×D8—41no—
118C5×D8—41no—
119C2×C5×C8—21no—
120C2×C5×C8—21no—
121C2×C5×C8D812yes—
122C2×C5×C8C2×D411yescharacteristic
123C2×C5×C8C2×D411yescharacteristic
124C2×C5×C8C2×D411yescharacteristic
125C2×C5×C8C2×D411yescharacteristic
126C2×C5×C8C2×D411yescharacteristic
127C2×C5×C8#4111yescharacteristic
128C5×C4⋊C4—41no—
129C5×C4⋊C4—41no—
130C5×C22⋊C4—48no—
131C5×C22⋊C4—84no—
132C42×C5—21no—
133C42×C5—21no—
134C5×C16—22no—
135C5×C16—22no—
136C22×Q16C22×C511yescharacteristic
137C22×SD16—21no—
138C22×D8C22×C511yescharacteristic
139C22×D8C22×C511yescharacteristic
140C2×#124—21no—
141C2×SD32—22no—
142C2×SD32—22no—
143C2×SD32—22no—
144C2×D16—22no—
145C2×D16—22no—
146C2×D16—22no—
147C22×C16C22×C511yescharacteristic
148C22×C16C22×C511yescharacteristic
149#495—22no—
150#495—22no—
151C2×#107C22×C511yescharacteristic
152C2×#102—21no—
153C2×C4×C8C22×C511yescharacteristic
154#390—21no—
155#358—22no—
156#357—22no—
157C23×C5—42no—
158C23×C5—42no—
159C23×C5—81no—
160C23×C5—81no—
161C23×C5—81no—
162C23×C5—81no—
163C23×C5—82no—
164C23×C5—82no—
165C23×C5—82no—
166C23×C5—82no—
167C23×C5C2×D811yescharacteristic
168C5×Q8—42no—
169C5×Q8—42no—
170C5×Q8—81no—
171C5×Q8—81no—
172C5×D4—42no—
173C5×D4—42no—
174C5×D4—42no—
175C5×D4—42no—
176C5×D4—81no—
177C5×D4—81no—
178C5×D4—81no—
179C5×D4—81no—
180C2×C4×C5—22no—
181C2×C4×C5—22no—
182C2×C4×C5—22no—
183C2×C4×C5—42no—
184C2×C4×C5—81no—
185C2×C4×C5—81no—
186C2×C4×C5—81no—
187C2×C4×C5—82no—
188C2×C4×C5#12011yescharacteristic
189C2×C4×C5#12111yescharacteristic
190C2×C4×C5#12711yescharacteristic
191C2×C4×C5C2×D811yescharacteristic
192C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
194C5×C8—22no—
195C5×C8#12011yescharacteristic
196C5×C8#12011yescharacteristic
197C5×C8#12111yescharacteristic
198C5×C8#12111yescharacteristic
199C22×Q8—21no—
200C22×D4—21no—
201C22×D4—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16C5×D412yes—
208C2×SD16—24no—
209C2×SD16—42no—
210C2×SD16—42no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—21no—
221C2×D8C5×D412yes—
222C2×D8C5×D412yes—
223C22×C8C5×D411yescharacteristic
224C22×C8C23×C511yescharacteristic
225#124—24no—
226#124—42no—
227C2×C4⋊C4—21no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C42C5×D411yescharacteristic
231SD32—28no—
232D16—28no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16C5×D412yes—
238C2×C16C5×D412yes—
239#107—21no—
240#107—21no—
241#102—42no—
242C4×C8—21no—
243C4×C8—21no—
244C22×C5—84no—
245C22×C5—84no—
246C22×C5—82no—
247C22×C5—82no—
248C22×C5—162no—
249C22×C5—162no—
250C22×C5—161no—
251C22×C5—161no—
252C22×C5—161no—
253C22×C5—161no—
254C22×C5#44711yescharacteristic
255C22×C5#45911yescharacteristic
256C22×C5#49311yescharacteristic
257C22×C5#50812yes—
258C22×C5#50912yes—
259C4×C5—42no—
260C4×C5—82no—
261C4×C5—161no—
262C4×C5#39211yescharacteristic
263C4×C5#44711yescharacteristic
264C4×C5#45911yescharacteristic
265C4×C5#49611yescharacteristic
266C24—41no—
267C24—41no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C5×D812yes—
289C22×C4C2×C5×D411yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—41no—
293Q16—41no—
294SD16—48no—
295D8—22no—
296D8—22no—
297D8—22no—
298D8—22no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8C5×D812yes—
306C2×C8C2×C5×D411yescharacteristic
307C2×C8C2×C5×D411yescharacteristic
308C2×C8C2×C5×D411yescharacteristic
309C2×C8C2×C5×D411yescharacteristic
310C2×C8C2×C5×D411yescharacteristic
311C2×C8C5×#4111yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—48no—
315C22⋊C4—84no—
316C42—21no—
317C42—21no—
318C16—22no—
319C16—22no—
320C2×C5—162no—
321C2×C5—162no—
322C2×C5#201111yescharacteristic
323C2×C5#218612yes—
324C2×C5#221012yes—
325C2×C5#224911yescharacteristic
326C2×C5#225011yescharacteristic
327C23—42no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23C2×C5×D811yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4C5×#12011yescharacteristic
359C2×C4C5×#12111yescharacteristic
360C2×C4C5×#12711yescharacteristic
361C2×C4C2×C5×D811yescharacteristic
362C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
363C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
364C8—22no—
365C8C5×#12011yescharacteristic
366C8C5×#12011yescharacteristic
367C8C5×#12111yescharacteristic
368C8C5×#12111yescharacteristic
369C5#1332611yescharacteristic, direct product factor, Sylow
370C22—84no—
371C22—84no—
372C22—82no—
373C22—82no—
374C22—162no—
375C22—162no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22C5×#44711yescharacteristic
381C22C5×#45911yescharacteristic
382C22C5×#49311yescharacteristic
383C22C5×#50812yes—
384C22C5×#50912yes—
385C4—42no—
386C4—82no—
387C4—161no—
388C4C5×#39211yescharacteristic
389C4C5×#44711yescharacteristic
390C4C5×#45911yescharacteristic
391C4C5×#49611yescharacteristic
392C2—162no—
393C2—162no—
394C2C5×#201111yescharacteristic
395C2C5×#218612yes—
396C2C5×#221012yes—
397C2C5×#224911yescharacteristic
398C2C5×#225011yescharacteristic
subgroupG/H
center, socleC23×C5C2×D8
commutator, Frattini, cosocleC22×C8C23×C5
prime radicalC22×C5×D16C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C5×SD32C2semidirect
3. C2×C5×#495C2semidirect
4. C2×C5×#358C2semidirect
5. C2×C5×#357C2semidirect
6. C5×#1970C2semidirect
7. C5×#1374C2semidirect
29. #13326C5direct (N × H)
75. C22×SD32C2×C5semidirect
77. C2×#495C2×C5semidirect
78. C2×#358C2×C5semidirect
79. C2×#357C2×C5semidirect
80. #1970C2×C5semidirect
81. #1374C2×C5semidirect
369. C5#13326direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×SD322
C22×C5×D162
C2×C5×#4952
C2×C5×#3582
C2×C5×#3572
C5×#19702
C5×#13742
#133265