Finite Group Databasebeta

← C5×#13314 #11936034

Subgroups — C5×#13314 #11936034

Order 1280.

536 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×SD32C211yesmaximal, characteristic, complemented
2C2×C5×#473C211yesmaximal, characteristic, complemented
3C2×C5×#472C211yesmaximal, characteristic
4C2×C5×#362C211yesmaximal, characteristic, complemented
5C2×C5×#361C211yesmaximal, characteristic, complemented
6C5×#2113C211yesmaximal, characteristic, complemented
7C5×#1381C211yesmaximal, characteristic, complemented
8C2×C5×#137—21no—
9C2×C5×AGL(1,ℤ/8)—21no—
10C22×C5×Q16C2211yescharacteristic
11C22×C5×D8C2211yescharacteristic
12C2×C5×#131C2211yescharacteristic
13C2×C5×SD32—22no—
14C2×C5×SD32—22no—
15C2×C5×SD32—22no—
16C2×C5×M32C2211yescharacteristic
17C22×C5×C16C2211yescharacteristic
18C5×#473—22no—
19C5×#473—22no—
20C5×#472—22no—
21C5×#472—22no—
22C2×C5×#108C2211yescharacteristic
23C2×C5×#108C2211yescharacteristic
24C2×C5×#101—21no—
25C2×C5×#100—21no—
26C5×#362—22no—
27C5×#361—22no—
28C5×C22⋊C16—22no—
29#13314C511yesmaximal, characteristic, direct product factor, Sylow
30C2×C5×#41D411yescharacteristic
31C22×C5×Q8—21no—
32C22×C5×D4—21no—
33C5×#137—24no—
34C5×#137—42no—
35C5×AGL(1,ℤ/8)—24no—
36C5×AGL(1,ℤ/8)—42no—
37C2×C5×Q16—21no—
38C2×C5×Q16—21no—
39C2×C5×Q16—21no—
40C2×C5×Q16—21no—
41C2×C5×Q16—21no—
42C2×C5×Q16—41no—
43C2×C5×Q16D412yes—
44C2×C5×SD16—41no—
45C2×C5×SD16—41no—
46C2×C5×D8—21no—
47C2×C5×D8—21no—
48C2×C5×D8—21no—
49C2×C5×D8—21no—
50C2×C5×D8—21no—
51C2×C5×D8—41no—
52C2×C5×D8D412yes—
53C5×#131—21no—
54C5×#131—21no—
55C5×#131—21no—
56C5×#131—21no—
57C2×C5×M16—21no—
58C2×C5×M16—21no—
59C2×C5×M16—21no—
60C2×C5×M16D411yescharacteristic
61C22×C5×C8—21no—
62C22×C5×C8C2311yescharacteristic
63C5×SD32—28no—
64C5×M32—21no—
65C5×M32—21no—
66C2×C5×C16—21no—
67C2×C5×C16—21no—
68C2×C5×C16—21no—
69C2×C5×C16D412yes—
70C5×#108—21no—
71C5×#108—21no—
72C5×#108D412yes—
73C5×#108D412yes—
74C5×#101—24no—
75C5×#100—24no—
76C22×SD32C2×C511yescharacteristic, complemented
77C2×#473C2×C511yescharacteristic, complemented
78C2×#472C2×C511yescharacteristic
79C2×#362C2×C511yescharacteristic, complemented
80C2×#361C2×C511yescharacteristic, complemented
81#2113C2×C511yescharacteristic, complemented
82#1381C2×C511yescharacteristic, complemented
83C24×C5—41no—
84C5×#41—21no—
85C5×#41—21no—
86C5×#41—21no—
87C5×#41—21no—
88C2×C5×Q8—22no—
89C2×C5×Q8—41no—
90C2×C5×Q8—41no—
91C2×C5×Q8—41no—
92C2×C5×Q8—41no—
93C2×C5×Q8—41no—
94C2×C5×Q8D811yescharacteristic
95C2×C5×D4—21no—
96C2×C5×D4—22no—
97C2×C5×D4—41no—
98C2×C5×D4—41no—
99C2×C5×D4—41no—
100C2×C5×D4—41no—
101C2×C5×D4—41no—
102C2×C5×D4D811yescharacteristic
103C22×C4×C5—21no—
104C22×C4×C5—41no—
105C22×C4×C5C2×D411yescharacteristic
106C5×Q16—22no—
107C5×Q16—22no—
108C5×Q16—44no—
109C5×Q16—41no—
110C5×Q16—41no—
111C5×SD16—44no—
112C5×SD16—44no—
113C5×D8—22no—
114C5×D8—22no—
115C5×D8—44no—
116C5×D8—41no—
117C5×D8—41no—
118C5×M16—22no—
119C5×M16—22no—
120C5×M16—21no—
121C5×M16—21no—
122C5×M16—41no—
123C5×M16—41no—
124C5×M16—41no—
125C5×M16—41no—
126C2×C5×C8—21no—
127C2×C5×C8—41no—
128C2×C5×C8—41no—
129C2×C5×C8—41no—
130C2×C5×C8—41no—
131C2×C5×C8C2×D411yescharacteristic
132C2×C5×C8C2×D411yescharacteristic
133C2×C5×C8C2×D411yescharacteristic
134C2×C5×C8C2×D411yescharacteristic
135C2×C5×C8C2×D411yescharacteristic
136C2×C5×C8#4111yescharacteristic
137C5×C16—22no—
138C5×C16—41no—
139C2×#137—21no—
140C2×AGL(1,ℤ/8)—21no—
141C22×Q16C22×C511yescharacteristic
142C22×D8C22×C511yescharacteristic
143C2×#131C22×C511yescharacteristic
144C2×SD32—22no—
145C2×SD32—22no—
146C2×SD32—22no—
147C2×M32C22×C511yescharacteristic
148C22×C16C22×C511yescharacteristic
149#473—22no—
150#473—22no—
151#472—22no—
152#472—22no—
153C2×#108C22×C511yescharacteristic
154C2×#108C22×C511yescharacteristic
155C2×#101—21no—
156C2×#100—21no—
157#362—22no—
158#361—22no—
159C22⋊C16—22no—
160C23×C5—21no—
161C23×C5—42no—
162C23×C5—81no—
163C23×C5—81no—
164C23×C5—162no—
165C23×C5#13511yescharacteristic
166C5×Q8—21no—
167C5×Q8—21no—
168C5×Q8—42no—
169C5×Q8—42no—
170C5×Q8—81no—
171C5×Q8—81no—
172C5×D4—21no—
173C5×D4—21no—
174C5×D4—41no—
175C5×D4—41no—
176C5×D4—42no—
177C5×D4—42no—
178C5×D4—81no—
179C5×D4—81no—
180C2×C4×C5—21no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—42no—
184C2×C4×C5—81no—
185C2×C4×C5—81no—
186C2×C4×C5#12011yescharacteristic
187C2×C4×C5#12011yescharacteristic
188C2×C4×C5#12111yescharacteristic
189C2×C4×C5C2×D811yescharacteristic
190C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
192C5×C8—41no—
193C5×C8—42no—
194C5×C8—42no—
195C5×C8#12011yescharacteristic
196C5×C8#12111yescharacteristic
197C5×C8#12111yescharacteristic
198C5×C8#12711yescharacteristic
199C2×#41C5×D411yescharacteristic
200C22×Q8—21no—
201C22×D4—21no—
202#137—24no—
203#137—42no—
204AGL(1,ℤ/8)—24no—
205AGL(1,ℤ/8)—42no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16—21no—
209C2×Q16—21no—
210C2×Q16—21no—
211C2×Q16—41no—
212C2×Q16C5×D412yes—
213C2×SD16—41no—
214C2×SD16—41no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—41no—
221C2×D8C5×D412yes—
222#131—21no—
223#131—21no—
224#131—21no—
225#131—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16C5×D411yescharacteristic
230C22×C8—21no—
231C22×C8C23×C511yescharacteristic
232SD32—28no—
233M32—21no—
234M32—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16C5×D412yes—
239#108—21no—
240#108—21no—
241#108C5×D412yes—
242#108C5×D412yes—
243#101—24no—
244#100—24no—
245C22×C5—22no—
246C22×C5—42no—
247C22×C5—41no—
248C22×C5—82no—
249C22×C5—162no—
250C22×C5—162no—
251C22×C5—162no—
252C22×C5#44911yescharacteristic
253C22×C5#44911yescharacteristic
254C22×C5#46611yescharacteristic
255C4×C5—41no—
256C4×C5—82no—
257C4×C5#39211yescharacteristic
258C4×C5#44711yescharacteristic
259C4×C5#44711yescharacteristic
260C4×C5#46911yescharacteristic
261C24—41no—
262#41—21no—
263#41—21no—
264#41—21no—
265#41—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8C5×D811yescharacteristic
273C2×D4—21no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4C5×D811yescharacteristic
281C22×C4—21no—
282C22×C4—41no—
283C22×C4C2×C5×D411yescharacteristic
284Q16—22no—
285Q16—22no—
286Q16—44no—
287Q16—41no—
288Q16—41no—
289SD16—44no—
290SD16—44no—
291D8—22no—
292D8—22no—
293D8—44no—
294D8—41no—
295D8—41no—
296M16—21no—
297M16—22no—
298M16—21no—
299M16—22no—
300M16—41no—
301M16—41no—
302M16—41no—
303M16—41no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8C2×C5×D411yescharacteristic
310C2×C8C2×C5×D411yescharacteristic
311C2×C8C2×C5×D411yescharacteristic
312C2×C8C2×C5×D411yescharacteristic
313C2×C8C2×C5×D411yescharacteristic
314C2×C8C5×#4111yescharacteristic
315C16—22no—
316C16—41no—
317C2×C5—21no—
318C2×C5—21no—
319C2×C5—81no—
320C2×C5—162no—
321C2×C5#199911yescharacteristic
322C2×C5#222312yes—
323C23—21no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—162no—
328C23C5×#13511yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—42no—
332Q8—42no—
333Q8—81no—
334Q8—81no—
335D4—21no—
336D4—21no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4C5×#12011yescharacteristic
350C2×C4C5×#12011yescharacteristic
351C2×C4C5×#12111yescharacteristic
352C2×C4C2×C5×D811yescharacteristic
353C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
355C8—41no—
356C8—42no—
357C8—42no—
358C8C5×#12011yescharacteristic
359C8C5×#12111yescharacteristic
360C8C5×#12111yescharacteristic
361C8C5×#12711yescharacteristic
362C5#1331411yescharacteristic, direct product factor, Sylow
363C22—22no—
364C22—42no—
365C22—41no—
366C22—82no—
367C22—162no—
368C22—162no—
369C22—162no—
370C22C5×#44911yescharacteristic
371C22C5×#44911yescharacteristic
372C22C5×#46611yescharacteristic
373C4—41no—
374C4—82no—
375C4C5×#39211yescharacteristic
376C4C5×#44711yescharacteristic
377C4C5×#44711yescharacteristic
378C4C5×#46911yescharacteristic
379C2—21no—
380C2—21no—
381C2—81no—
382C2—162no—
383C2C5×#199911yescharacteristic
384C2C5×#222312yes—
subgroupG/H
center, socleC22×C5#466
commutator, Frattini, cosocleC22×C8C23×C5
prime radicalC2×C5×#472C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C5×SD32C2semidirect
2. C2×C5×#473C2semidirect
4. C2×C5×#362C2semidirect
5. C2×C5×#361C2semidirect
6. C5×#2113C2semidirect
7. C5×#1381C2semidirect
29. #13314C5direct (N × H)
76. C22×SD32C2×C5semidirect
77. C2×#473C2×C5semidirect
79. C2×#362C2×C5semidirect
80. C2×#361C2×C5semidirect
81. #2113C2×C5semidirect
82. #1381C2×C5semidirect
362. C5#13314direct (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
C2×C5×#4732
C2×C5×#4722
C2×C5×#3622
C2×C5×#3612
C5×#21132
C5×#13812
#133145