Finite Group Databasebeta

← C2×#10619041 #13007136

Subgroups — C2×#10619041 #13007136

Order 1280.

6338 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#564C211yesmaximal, characteristic, complemented
2#10619041C2132yesmaximal, direct product factor
3C2×C5⋊#564C213yesmaximal, complemented
4C2×#66019C2112yesmaximal, complemented
5C2×#66030C2112yesmaximal
6C2×#66022C213yesmaximal
7C5×#564C22116yescomplemented
8C2×C5×#125C2213yes—
9C2×C5×#125C22112yes—
10#66030C221192yes—
11#66022C22148yes—
12C5⋊#564C22148yescomplemented
13#66019C221192yescomplemented
14C2×#4044C22112yes—
15C2×#4042C22112yes—
16C2×#4038C2216yes—
17C2×#4036C22112yes—
18C2×#4035C22112yes—
19C2×#4035C22112yes—
20C2×#4035C22112yes—
21C2×#4034C2216yes—
22C2×#4034C22112yes—
23C2×#4033C22112yes—
24C2×#4033C22112yes—
25C2×#4022C2214yes—
26C2×C42×D5C2211yescharacteristic
27C2×#4016C2213yes—
28C2×C4×Dic10C22112yes—
29C2×#3556—51nomaximal, Sylow
30C22×C4×D5C2313yes—
31C22×Dic10C23112yes—
32C5×#125C23196yes—
33C5×#125C23124yes—
34C2×C5×C4⋊C4C23112yes—
35C2×C5×C4⋊C4C23112yes—
36C2×C5×C4⋊C4C2316yes—
37C2×C42×C5C2314yes—
38C2×C42×C5C2311yescharacteristic
39C2×#685C23112yes—
40C2×C4⋊Dic5C23112yes—
41C2×C4⋊Dic5C2316yes—
42C2×#683C23112yes—
43C2×#683C23112yes—
44C2×#683C23112yes—
45C2×#683C23112yes—
46C2×C4×Dic5C23112yes—
47C2×C4×Dic5C23112yes—
48C2×C4×Dic5C2313yes—
49#4044C23196yes—
50#4042C23196yes—
51#4038C23148yes—
52#4036C23196yes—
53#4035C23196yes—
54#4035C23196yes—
55#4035C23196yes—
56#4034C23148yes—
57#4034C23196yes—
58#4033C23196yes—
59#4033C23196yes—
60#4022C23132yes—
61C42×D5C2318yes—
62#4016C23124yes—
63C4×Dic10C23196yes—
64#3556—532no—
65C2×#564—53no—
66C2×#564D511yescharacteristic, complemented
67C2×#557—512no—
68C2×#541—512no—
69C2×#533—53no—
70C23×D5C2411yescharacteristic
71C22×C4×C5C24112yes—
72C22×C4×C5C2413yes—
73C22×Dic5C24112yes—
74C22×Dic5C2413yes—
75C2×C4×D5—212no—
76C2×C4×D5C24112yes—
77C2×Dic10—248no—
78C2×Dic10C24148yes—
79C5×C4⋊C4C24148yes—
80C5×C4⋊C4C24148yes—
81C5×C4⋊C4C24124yes—
82C42×C5C24116yes—
83C42×C5C2414yes—
84#685C24148yes—
85C4⋊Dic5C24148yes—
86C4⋊Dic5C24124yes—
87#683C24148yes—
88#683C24148yes—
89#683C24148yes—
90#683C24148yes—
91C4×Dic5C24148yes—
92C4×Dic5C24148yes—
93C4×Dic5C24112yes—
94#564—548no—
95#564C2×D5116yescomplemented
96#557—5192no—
97#541—5192no—
98#533—548no—
99C2×C4⋊Q8—56no—
100C2×C4⋊Q8—512no—
101C2×#126—512no—
102C2×#126—54no—
103C2×#125—512no—
104C2×#125—512no—
105C2×#125—512no—
106C2×#125—512no—
107C2×#125—53no—
108C2×#125C2×D513yes—
109C2×#125C2×D5112yes—
110C2×C22⋊Q8—512no—
111C2×C4×Q8—512no—
112C2×C4×Q8—512no—
113C2×C4×Q8—512no—
114C2×#117—56no—
115C22×C42—51no—
116C23×C5C2511yescharacteristic
117C22×D5—26no—
118C22×D5C2512yes—
119C2×C4×C5—224no—
120C2×C4×C5—26no—
121C2×C4×C5C25124yes—
122C2×C4×C5C2516yes—
123C2×Dic5—224no—
124C2×Dic52−1+4112yes—
125C2×Dic5C25124yes—
126C2×Dic5C2516yes—
127C4×D5—224no—
128Dic10—296no—
129C22×Q8—512no—
130C23×C4—53no—
131C4⋊Q8—548no—
132C4⋊Q8—596no—
133#126—596no—
134#126—532no—
135#125—596no—
136#125—596no—
137#125—596no—
138#125—596no—
139#125—524no—
140#125C22×D5124yes—
141#125C22×D5196yes—
142C22⋊Q8—596no—
143C4×Q8—596no—
144C4×Q8—596no—
145C4×Q8—596no—
146#117—548no—
147C2×C4⋊C4—56no—
148C2×C4⋊C4—512no—
149C2×C4⋊C4—512no—
150C2×C4⋊C4—512no—
151C2×C4⋊C4—512no—
152C2×C4⋊C4—512no—
153C2×C4⋊C4C22×D516yes—
154C2×C4⋊C4C22×D5112yes—
155C2×C4⋊C4C22×D5112yes—
156C2×C22⋊C4—512no—
157C2×C42—512no—
158C2×C42—53no—
159C2×C42—512no—
160C2×C42—58no—
161C2×C42C22×D514yes—
162C2×C42C22×D511yescharacteristic
163C22×C5C2×2−1+416yes—
164C22×C5C2611yescharacteristic
165C2×D5—26no—
166C2×D5—44no—
167C4×C5—224no—
168C4×C5—26no—
169Dic5—224no—
170Dic5C2×2−1+4112yes—
171C24—51no—
172C2×Q8—548no—
173C2×Q8—1048no—
174C22×C4—53no—
175C22×C4—512no—
176C22×C4—512no—
177C22×C4—1012no—
178C22×C4C23×D5112yes—
179C22×C4C23×D513yes—
180C4⋊C4—524no—
181C4⋊C4—548no—
182C4⋊C4—548no—
183C4⋊C4—548no—
184C4⋊C4—548no—
185C4⋊C4—548no—
186C4⋊C4C23×D5124yes—
187C4⋊C4C23×D5148yes—
188C4⋊C4C23×D5148yes—
189C22⋊C4—548no—
190C42—512no—
191C42—548no—
192C42—548no—
193C42C23×D514yes—
194C42C23×D5116yes—
195C2×C5#355614yescomplemented
196C2×C5C22×2−1+413yes—
197D5—42no—
198C23—52no—
199C23—106no—
200C23C24×D511yescharacteristic
201Q8—1096no—
202C2×C4—26no—
203C2×C4—224no—
204C2×C4—512no—
205C2×C4—56no—
206C2×C4—524no—
207C2×C4—1024no—
208C2×C4—1024no—
209C2×C4C24×D5124yes—
210C2×C4C24×D516yes—
211C5C2×#355611yescharacteristic, complemented, Sylow
212C22—106no—
213C22—204no—
214C22D5×2−1+416yes—
215C22C25×D511yescharacteristic
216C4—26no—
217C4—224no—
218C4—512no—
219C4—1024no—
220C2—202no—
221C2#1061904114yesdirect product factor
222C2C2×D5×2−1+413yes—
subgroupG/H
centerC23C24×D5
commutator, cosocleC22×C5C26
FrattiniC22C25×D5
FittingC2×C5×#564C2
socleC23×C5C25
prime radicalC23×D5C24
hypercenterC2×#564D5
hypocenterC5C2×#3556
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#564

Complements

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

NH ≅ G/Ntype
1. C2×C5×#564C2semidirect
2. #10619041C2direct (N × H)
3. C2×C5⋊#564C2semidirect
4. C2×#66019C2semidirect
7. C5×#564C22semidirect
12. C5⋊#564C22semidirect
13. #66019C22semidirect
66. C2×#564D5semidirect
95. #564C2×D5semidirect
195. C2×C5#3556semidirect
211. C5C2×#3556semidirect
221. C2#10619041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5642
#106190412
C2×C5⋊#5642
C2×#660192
C2×#660302
C2×#660222
C2×#35565