Finite Group Databasebeta

← C5×#13122 #11935842

Subgroups — C5×#13122 #11935842

Order 1280.

326 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#364C211yesmaximal, characteristic, complemented
2C2×C5×#350C211yesmaximal, characteristic
3C5×#2113C211yesmaximal, characteristic
4C5×#2108C211yesmaximal, characteristic, complemented
5C5×#1764C211yesmaximal, characteristic
6C5×#1379C211yesmaximal, characteristic, complemented
7C5×#1374C211yesmaximal, characteristic, complemented
8C2×C5×#131C2211yescharacteristic
9C2×C5×M32C2211yescharacteristic
10C2×C5×M32C2211yescharacteristic
11C22×C5×C16C2211yescharacteristic
12C22×C5×C16C2211yescharacteristic
13C2×C5×#104C411yescharacteristic
14C5×#416C411yescharacteristic
15C5×#405C2211yescharacteristic
16C2×C4×C5×C8C2211yescharacteristic
17C5×#364—22no—
18C5×#350—22no—
19C5×C22⋊C16—21no—
20C5×C22⋊C16—21no—
21C5×#346—21no—
22C4×C5×C16—21no—
23C5×#341C411yescharacteristic
24C5×#339C411yescharacteristic
25#13122C511yesmaximal, characteristic, direct product factor, Sylow
26C2×C5×#41C2×C411yescharacteristic
27C5×#131—21no—
28C5×#131—21no—
29C5×#131—21no—
30C5×#131—21no—
31C2×C5×M16—21no—
32C2×C5×M16C2×C411yescharacteristic
33C2×C5×M16C2×C411yescharacteristic
34C22×C5×C8—21no—
35C22×C5×C8C2×C411yescharacteristic
36C22×C5×C8C2311yescharacteristic
37C5×#117C2×C411yescharacteristic
38C2×C42×C5C2×C411yescharacteristic
39C5×M32—21no—
40C5×M32—21no—
41C5×M32—21no—
42C5×M32D412yes—
43C2×C5×C16—21no—
44C2×C5×C16—21no—
45C2×C5×C16—21no—
46C2×C5×C16—21no—
47C2×C5×C16—21no—
48C2×C5×C16—21no—
49C2×C5×C16—21no—
50C2×C5×C16D411yescharacteristic
51C2×C5×C16D411yescharacteristic
52C5×#104—22no—
53C5×#103C811yescharacteristic
54C5×#103C811yescharacteristic
55C5×#102C811yescharacteristic
56C5×#102C811yescharacteristic
57C5×#97—21no—
58C4×C5×C8—21no—
59C4×C5×C8—21no—
60C4×C5×C8—21no—
61C2×#364C2×C511yescharacteristic, complemented
62C2×#350C2×C511yescharacteristic
63#2113C2×C511yescharacteristic
64#2108C2×C511yescharacteristic, complemented
65#1764C2×C511yescharacteristic
66#1379C2×C511yescharacteristic, complemented
67#1374C2×C511yescharacteristic, complemented
68C5×#41—21no—
69C5×#41—21no—
70C5×#41—21no—
71C5×#41—21no—
72C2×C5×Q8C2×C811yescharacteristic
73C2×C5×D4—21no—
74C2×C5×D4C2×C811yescharacteristic
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5C22×C411yescharacteristic
78C5×M16—21no—
79C5×M16—21no—
80C5×M16—21no—
81C5×M16—41no—
82C5×M16—41no—
83C5×M16M1612yes—
84C2×C5×C8—21no—
85C2×C5×C8—21no—
86C2×C5×C8—21no—
87C2×C5×C8—21no—
88C2×C5×C8—41no—
89C2×C5×C8—41no—
90C2×C5×C8C2×C811yescharacteristic
91C2×C5×C8C2×C811yescharacteristic
92C2×C5×C8C2×D411yescharacteristic
93C2×C5×C8C2×D411yescharacteristic
94C2×C5×C8#4111yescharacteristic
95C2×C5×C8#4111yescharacteristic
96C2×C5×C8#4111yescharacteristic
97C2×C5×C8#4111yescharacteristic
98C5×C4⋊C4C2×C811yescharacteristic
99C5×C4⋊C4C2×C811yescharacteristic
100C5×C22⋊C4—21no—
101C42×C5—21no—
102C42×C5—21no—
103C42×C5—21no—
104C5×C16—22no—
105C5×C16—21no—
106C5×C16—21no—
107C5×C16—41no—
108C5×C16—41no—
109C2×#131C22×C511yescharacteristic
110C2×M32C22×C511yescharacteristic
111C2×M32C22×C511yescharacteristic
112C22×C16C22×C511yescharacteristic
113C22×C16C22×C511yescharacteristic
114C2×#104C4×C511yescharacteristic
115#416C4×C511yescharacteristic
116#405C22×C511yescharacteristic
117C2×C4×C8C22×C511yescharacteristic
118#364—22no—
119#350—22no—
120C22⋊C16—21no—
121C22⋊C16—21no—
122#346—21no—
123C4×C16—21no—
124#341C4×C511yescharacteristic
125#339C4×C511yescharacteristic
126C23×C5—21no—
127C23×C5#13111yescharacteristic
128C5×Q8—21no—
129C5×Q8—21no—
130C5×D4—21no—
131C5×D4—21no—
132C5×D4—41no—
133C5×D4—41no—
134C2×C4×C5—22no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5#11711yescharacteristic
143C2×C4×C5C4×D411yescharacteristic
144C2×C4×C5C4×D411yescharacteristic
145C2×C4×C5C22×C811yescharacteristic
146C2×C4×C5C2×M1611yescharacteristic
147C2×C4×C5#13111yescharacteristic
148C5×C8—21no—
149C5×C8—21no—
150C5×C8—22no—
151C5×C8—41no—
152C5×C8#12111yescharacteristic
153C5×C8#12311yescharacteristic
154C5×C8#12411yescharacteristic
155C5×C8#12611yescharacteristic
156C2×#41C2×C4×C511yescharacteristic
157#131—21no—
158#131—21no—
159#131—21no—
160#131—21no—
161C2×M16—21no—
162C2×M16C2×C4×C511yescharacteristic
163C2×M16C2×C4×C511yescharacteristic
164C22×C8—21no—
165C22×C8C2×C4×C511yescharacteristic
166C22×C8C23×C511yescharacteristic
167#117C2×C4×C511yescharacteristic
168C2×C42C2×C4×C511yescharacteristic
169M32—21no—
170M32—21no—
171M32—21no—
172M32C5×D412yes—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16C5×D411yescharacteristic
181C2×C16C5×D411yescharacteristic
182#104—22no—
183#103C5×C811yescharacteristic
184#103C5×C811yescharacteristic
185#102C5×C811yescharacteristic
186#102C5×C811yescharacteristic
187#97—21no—
188C4×C8—21no—
189C4×C8—21no—
190C4×C8—21no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5#43311yescharacteristic
197C22×C5C8×D411yescharacteristic
198C22×C5#43611yescharacteristic
199C4×C5—42no—
200C4×C5—41no—
201C4×C5—41no—
202C4×C5—41no—
203C4×C5#38811yescharacteristic
204C4×C5#43111yescharacteristic
205C4×C5C8×D411yescharacteristic
206C4×C5#43511yescharacteristic
207#41—21no—
208#41—21no—
209#41—21no—
210#41—21no—
211C2×Q8C2×C5×C811yescharacteristic
212C2×D4—21no—
213C2×D4C2×C5×C811yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4C22×C4×C511yescharacteristic
217M16—21no—
218M16—21no—
219M16—21no—
220M16—41no—
221M16—41no—
222M16C5×M1612yes—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8C2×C5×C811yescharacteristic
230C2×C8C2×C5×C811yescharacteristic
231C2×C8C2×C5×D411yescharacteristic
232C2×C8C2×C5×D411yescharacteristic
233C2×C8C5×#4111yescharacteristic
234C2×C8C5×#4111yescharacteristic
235C2×C8C5×#4111yescharacteristic
236C2×C8C5×#4111yescharacteristic
237C4⋊C4C2×C5×C811yescharacteristic
238C4⋊C4C2×C5×C811yescharacteristic
239C22⋊C4—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C16—22no—
244C16—21no—
245C16—21no—
246C16—41no—
247C16—41no—
248C2×C5—21no—
249C2×C5—21no—
250C2×C5—81no—
251C2×C5#192311yescharacteristic
252C2×C5#217011yescharacteristic
253C2×C5#217111yescharacteristic
254C23—21no—
255C23C5×#13111yescharacteristic
256Q8—21no—
257Q8—21no—
258D4—21no—
259D4—21no—
260D4—41no—
261D4—41no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4C5×#11711yescharacteristic
271C2×C4C4×C5×D411yescharacteristic
272C2×C4C4×C5×D411yescharacteristic
273C2×C4C22×C5×C811yescharacteristic
274C2×C4C2×C5×M1611yescharacteristic
275C2×C4C5×#13111yescharacteristic
276C8—21no—
277C8—21no—
278C8—22no—
279C8—41no—
280C8C5×#12111yescharacteristic
281C8C5×#12311yescharacteristic
282C8C5×#12411yescharacteristic
283C8C5×#12611yescharacteristic
284C5#1312211yescharacteristic, direct product factor, Sylow
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22C5×#43311yescharacteristic
291C22C5×C8×D411yescharacteristic
292C22C5×#43611yescharacteristic
293C4—42no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4C5×#38811yescharacteristic
298C4C5×#43111yescharacteristic
299C4C5×C8×D411yescharacteristic
300C4C5×#43511yescharacteristic
301C2—21no—
302C2—21no—
303C2—81no—
304C2C5×#192311yescharacteristic
305C2C5×#217011yescharacteristic
306C2C5×#217111yescharacteristic
subgroupG/H
centerC2×C5×C8C2×D4
commutatorC2×C4C22×C5×C8
Frattini, cosocleC22×C8C23×C5
socleC22×C5C8×D4
prime radicalC2×C5×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#364C2semidirect
4. C5×#2108C2semidirect
6. C5×#1379C2semidirect
7. C5×#1374C2semidirect
25. #13122C5direct (N × H)
61. C2×#364C2×C5semidirect
64. #2108C2×C5semidirect
66. #1379C2×C5semidirect
67. #1374C2×C5semidirect
284. C5#13122direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3642
C2×C5×#3502
C5×#21132
C5×#21082
C5×#17642
C5×#13792
C5×#13742
#131225