Finite Group Databasebeta

← C5×#10174 #11932894

Subgroups — C5×#10174 #11932894

Order 1280.

578 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
1C5×#3226C211yesmaximal, characteristic
2C2×C5×#500C211yesmaximal, characteristic, complemented
3C5×#3152C211yesmaximal, characteristic
4C5×#1966C211yesmaximal, characteristic, complemented
5C5×#1546C211yesmaximal, characteristic, complemented
6C5×#1492C211yesmaximal, characteristic
7C5×#1278C211yesmaximal, characteristic, complemented
8C2×C5×#137—22no—
9C5×#532C2211yescharacteristic
10C2×C5×C4⋊Q8C2211yescharacteristic
11C2×C5×C4⋊Q8C2211yescharacteristic
12C5×#500—22no—
13C5×#500—22no—
14C5×#500—21no—
15C5×#497—22no—
16C5×#494—21no—
17C5×#492—21no—
18C5×#486—21no—
19C5×#480—22no—
20C5×#474—22no—
21C5×#431C2211yescharacteristic
22C2×C5×#107—22no—
23C5×#423C2211yescharacteristic
24C2×C5×#101—22no—
25C4×C5×M16C2211yescharacteristic
26C2×C4×C5×C8C2211yescharacteristic
27C5×#332—22no—
28C5×D4⋊C8—22no—
29#10174C511yesmaximal, characteristic, direct product factor, Sylow
30C2×C5×#41—21no—
31C22×C5×Q8—21no—
32C22×C5×Q8—21no—
33C5×#137—44no—
34C5×#137—44no—
35C2×C5×Q16—24no—
36C2×C5×SD16—24no—
37C2×C5×M16—21no—
38C2×C5×M16D412yes—
39C22×C5×C8—21no—
40C5×C4⋊Q8—21no—
41C5×C4⋊Q8—21no—
42C5×C4⋊Q8—21no—
43C5×C4⋊Q8—21no—
44C5×C4⋊Q8—21no—
45C5×C4⋊Q8—21no—
46C5×C4⋊Q8—21no—
47C5×C4⋊Q8D411yescharacteristic
48C5×C4⋊Q8D412yes—
49C5×#127D411yescharacteristic
50C5×#124—21no—
51C5×#121—21no—
52C5×#121—21no—
53C4×C5×D4—21no—
54C4×C5×D4—21no—
55C2×C5×C4⋊C4—21no—
56C2×C5×C4⋊C4—21no—
57C2×C5×C4⋊C4—21no—
58C2×C5×C4⋊C4D412yes—
59C2×C42×C5C2311yescharacteristic
60C5×#107—28no—
61C5×#107—24no—
62C5×C4⋊C8—21no—
63C5×C4⋊C8—21no—
64C5×C4⋊C8D412yes—
65C5×#102—24no—
66C5×#101—44no—
67C5×C22⋊C8—21no—
68C5×#97—21no—
69C4×C5×C8—21no—
70C4×C5×C8—21no—
71C4×C5×C8—21no—
72C4×C5×C8D412yes—
73#3226C2×C511yescharacteristic
74C2×#500C2×C511yescharacteristic, complemented
75#3152C2×C511yescharacteristic
76#1966C2×C511yescharacteristic, complemented
77#1546C2×C511yescharacteristic, complemented
78#1492C2×C511yescharacteristic
79#1278C2×C511yescharacteristic, complemented
80C5×#41—42no—
81C5×#41—42no—
82C2×C5×Q8—21no—
83C2×C5×Q8—22no—
84C2×C5×Q8—41no—
85C2×C5×Q8—41no—
86C2×C5×Q8—41no—
87C2×C5×Q8—42no—
88C2×C5×Q8—42no—
89C2×C5×Q8—42no—
90C2×C5×Q8—42no—
91C2×C5×D4—21no—
92C2×C5×D4—21no—
93C2×C5×D4—21no—
94C2×C5×D4—41no—
95C22×C4×C5—22no—
96C22×C4×C5—41no—
97C22×C4×C5—41no—
98C22×C4×C5C2×D411yescharacteristic
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5#4111yescharacteristic
101C5×Q16—48no—
102C5×SD16—48no—
103C5×M16—22no—
104C5×M16—42no—
105C5×M16D814yes—
106C2×C5×C8—24no—
107C2×C5×C8—22no—
108C2×C5×C8—22no—
109C2×C5×C8—22no—
110C2×C5×C8—41no—
111C5×C4⋊C4—22no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—21no—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—21no—
117C5×C4⋊C4—41no—
118C5×C4⋊C4—41no—
119C5×C4⋊C4—41no—
120C5×C4⋊C4—41no—
121C5×C22⋊C4—41no—
122C5×C22⋊C4—41no—
123C42×C5C2×D411yescharacteristic
124C42×C5C2×D411yescharacteristic
125C42×C5C2×D411yescharacteristic
126C42×C5C2×D411yescharacteristic
127C2×#137—22no—
128#532C22×C511yescharacteristic
129C2×C4⋊Q8C22×C511yescharacteristic
130C2×C4⋊Q8C22×C511yescharacteristic
131#500—22no—
132#500—21no—
133#500—22no—
134#497—22no—
135#494—21no—
136#492—21no—
137#486—21no—
138#480—22no—
139#474—22no—
140#431C22×C511yescharacteristic
141C2×#107—22no—
142#423C22×C511yescharacteristic
143C2×#101—22no—
144C4×M16C22×C511yescharacteristic
145C2×C4×C8C22×C511yescharacteristic
146#332—22no—
147D4⋊C8—22no—
148C23×C5—41no—
149C23×C5#12111yescharacteristic
150C5×Q8—44no—
151C5×Q8—42no—
152C5×Q8—82no—
153C5×Q8—81no—
154C5×Q8—81no—
155C5×Q8—81no—
156C5×Q8—81no—
157C5×D4—42no—
158C5×D4—42no—
159C5×D4—42no—
160C5×D4—82no—
161C2×C4×C5—22no—
162C2×C4×C5—22no—
163C2×C4×C5—41no—
164C2×C4×C5—42no—
165C2×C4×C5—42no—
166C2×C4×C5—44no—
167C2×C4×C5—82no—
168C2×C4×C5—81no—
169C2×C4×C5—81no—
170C2×C4×C5#12011yescharacteristic
171C2×C4×C5#12011yescharacteristic
172C2×C4×C5#12011yescharacteristic
173C2×C4×C5#12111yescharacteristic
174C2×C4×C5#12111yescharacteristic
175C2×C4×C5#12711yescharacteristic
176C2×C4×C5C2×D812yes—
177C2×C4×C5AGL(1,ℤ/8)12yes—
178C5×C8—24no—
179C5×C8—24no—
180C5×C8—42no—
181C5×C8—81no—
182C2×#41—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185#137—44no—
186#137—44no—
187C2×Q16—24no—
188C2×SD16—24no—
189C2×M16—21no—
190C2×M16C5×D412yes—
191C22×C8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C5×D412yes—
200C4⋊Q8C5×D411yescharacteristic
201#127C5×D411yescharacteristic
202#124—21no—
203#121—21no—
204#121—21no—
205C4×D4—21no—
206C4×D4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C5×D412yes—
211C2×C42C23×C511yescharacteristic
212#107—28no—
213#107—24no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8C5×D412yes—
217#102—24no—
218#101—44no—
219C22⋊C8—21no—
220#97—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8C5×D412yes—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—81no—
228C22×C5—81no—
229C22×C5—81no—
230C22×C5#39211yescharacteristic
231C22×C5#45912yes—
232C4×C5—21no—
233C4×C5—21no—
234C4×C5—82no—
235C4×C5—81no—
236C4×C5—82no—
237C4×C5#44712yes—
238C4×C5#45612yes—
239C4×C5#47312yes—
240C4×C5#49311yescharacteristic
241C4×C5#49611yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×C5×D411yescharacteristic
261C22×C4C2×C5×D411yescharacteristic
262C22×C4C5×#4111yescharacteristic
263Q16—48no—
264SD16—48no—
265M16—22no—
266M16—42no—
267M16C5×D814yes—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42C2×C5×D411yescharacteristic
286C42C2×C5×D411yescharacteristic
287C42C2×C5×D411yescharacteristic
288C42C2×C5×D411yescharacteristic
289C2×C5—22no—
290C2×C5—161no—
291C2×C5#200511yescharacteristic
292C2×C5#201111yescharacteristic
293C2×C5#202011yescharacteristic
294C23—41no—
295C23C5×#12111yescharacteristic
296Q8—44no—
297Q8—42no—
298Q8—82no—
299Q8—81no—
300Q8—81no—
301Q8—81no—
302Q8—81no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C5×#12011yescharacteristic
317C2×C4C5×#12011yescharacteristic
318C2×C4C5×#12011yescharacteristic
319C2×C4C5×#12111yescharacteristic
320C2×C4C5×#12111yescharacteristic
321C2×C4C5×#12711yescharacteristic
322C2×C4C2×C5×D812yes—
323C2×C4C5×AGL(1,ℤ/8)12yes—
324C8—24no—
325C8—24no—
326C8—42no—
327C8—81no—
328C5#1017411yescharacteristic, direct product factor, Sylow
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C5×#39211yescharacteristic
335C22C5×#45912yes—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—81no—
340C4—82no—
341C4C5×#44712yes—
342C4C5×#45612yes—
343C4C5×#47312yes—
344C4C5×#49311yescharacteristic
345C4C5×#49611yescharacteristic
346C2—22no—
347C2—161no—
348C2C5×#200511yescharacteristic
349C2C5×#201111yescharacteristic
350C2C5×#202011yescharacteristic
subgroupG/H
center, socleC22×C5#392
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC5×#532C22
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
2. C2×C5×#500C2semidirect
4. C5×#1966C2semidirect
5. C5×#1546C2semidirect
7. C5×#1278C2semidirect
29. #10174C5direct (N × H)
74. C2×#500C2×C5semidirect
76. #1966C2×C5semidirect
77. #1546C2×C5semidirect
79. #1278C2×C5semidirect
328. C5#10174direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#32262
C2×C5×#5002
C5×#31522
C5×#19662
C5×#15462
C5×#14922
C5×#12782
#101745