Finite Group Databasebeta

← C2×#10624048 #13089031

Subgroups — C2×#10624048 #13089031

Order 1440.

710 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#67158C211yesmaximal, characteristic, complemented
2C2×#67152C212yesmaximal, complemented
3C2×C5×#625C211yesmaximal, characteristic, complemented
4C2×#67137C211yesmaximal, characteristic, complemented
5C2×C3×D3×Dic5C212yesmaximal, complemented
6#10624048C218yesmaximal, direct product factor
7C2×#6782—32nomaximal
8C22×C3⋊D15C2211yescharacteristic
9C22×C3×C5×D3C2212yes—
10C2×C5×C3⋊Dic3C2211yescharacteristic
11C2×C3×Dic15C2212yes—
12C2×C32×Dic5C2211yescharacteristic
13#67158C2214yescomplemented
14#67152C2218yescomplemented
15C5×#625C2214yescomplemented
16#67137C2214yescomplemented
17C3×D3×Dic5C2218yescomplemented
18C2×#3798—51nomaximal, Hall
19C2×#1220—32no—
20C2×C5×#67—32no—
21C22×C3×Dic5—32no—
22C2×#1204—32no—
23C2×#1203—32no—
24C2×#1200—32no—
25C2×#1200—61no—
26#6782—316no—
27C2×D3×Dic5D312yescomplemented
28C22×C32×C5C2311yescharacteristic
29C2×C3⋊D15—22no—
30C2×C3⋊D15C2312yes—
31C2×C3×C5×D3—24no—
32C2×C3×C5×D3C2314yes—
33C5×C3⋊Dic3C2312yes—
34C3×Dic15C2314yes—
35C32×Dic5C2312yes—
36C2×#711—91noHall
37C2×C4×C3⋊D3—51no—
38C2×C3×C4×D3—52no—
39C2×#627—51no—
40C2×#626—52no—
41C2×#625D511yescharacteristic, complemented
42#3798—58no—
43C23×C3×C5—32no—
44C22×D15—32no—
45C22×D15—61no—
46C22×C5×D3C2×D312yes—
47#1220—38no—
48C2×Dic15C2×D312yes—
49C5×#67—38no—
50C2×C5×Dic3—32no—
51C2×C5×Dic3—61no—
52C2×C3×Dic5—21no—
53C2×C3×Dic5—32no—
54C2×C3×Dic5—38no—
55C2×C3×Dic5C2×D312yes—
56#1204—38no—
57#1203—38no—
58#1200—38no—
59#1200—64no—
60D3×Dic5C2×D318yescomplemented
61C2×#235—152no—
62C2×C32×C5#4112yescomplemented
63C2×C32×C5C2411yescharacteristic
64C3⋊D15—22no—
65C3×C5×D3—24no—
66C2×C5×D4—91no—
67C2×C22⋊D5—92no—
68C22×Dic5—32no—
69#711—98no—
70C2×C4×D5—91no—
71C2×Dic10—91no—
72C22×C3⋊D3—51no—
73C22×C3×D3C2×D512yes—
74C2×C32×C4—51no—
75C2×C3⋊Dic3C2×D511yescharacteristic
76C4×C3⋊D3—54no—
77C2×C3×Dic3—52no—
78C3×C4×D3—58no—
79#627—54no—
80#626—58no—
81#625C2×D514yescomplemented
82C22×C3×C5—21no—
83C22×C3×C5—34no—
84C22×C3×C5—64no—
85C22×C3×C5C22×D312yes—
86C2×D15—34no—
87C2×D15—62no—
88C2×D15—64no—
89C2×D15—64no—
90C2×C5×D3—24no—
91C2×C5×D3C22×D314yes—
92Dic15C22×D314yes—
93C3×Dic5—22no—
94C3×Dic5—34no—
95C3×Dic5C22×D314yes—
96C5×Dic3—34no—
97C5×Dic3—62no—
98C22×C3×C4—152no—
99C2×#67—32no—
100C2×#67—152no—
101#235—1516no—
102C2×D12—152no—
103C2×C4×D3—52no—
104C2×C4×D3—152no—
105C2×C4×D3—301no—
106C2×Dic6—152no—
107C32×C5C2×#4111yescharacteristic, complemented, Hall
108C23×C5—32no—
109C22×D5—91no—
110C5×D4—94no—
111C2×C4×C5—91no—
112C22⋊D5—98no—
113C2×Dic5—32no—
114C2×Dic5—38no—
115C2×Dic5D3211yescharacteristic
116C4×D5—94no—
117Dic10—94no—
118C22×C32C22×D511yescharacteristic
119C2×C3⋊D3—52no—
120C2×C3⋊D3—102no—
121C2×C3×D3—24no—
122C2×C3×D3C22×D514yes—
123C32×C4—52no—
124C3⋊Dic3C22×D512yes—
125C3×Dic3—54no—
126C2×#41—451noSylow
127C2×C3×C5—21no—
128C2×C3×C5—22no—
129C2×C3×C5—64no—
130C2×C3×C5#23514yescomplemented
131C2×C3×C5C23×D312yes—
132D15—64no—
133D15—64no—
134C5×D3—24no—
135C23×C3—32no—
136C22×D3—152no—
137C22×D3—301no—
138C22×D3D3×D512yes—
139C2×C3×C4—52no—
140C2×C3×C4—101no—
141C2×C3×C4—158no—
142C2×C3×C4—152no—
143#67—38no—
144#67—158no—
145C2×Dic3—32no—
146C2×Dic3—52no—
147C2×Dic3—61no—
148D12—158no—
149C4×D3—58no—
150C4×D3—158no—
151C4×D3—304no—
152Dic6—158no—
153C22×C5—34no—
154C22×C5—64no—
155C22×C5C2×D3211yescharacteristic
156C2×D5—92no—
157C2×D5—182no—
158C4×C5—92no—
159Dic5—34no—
160Dic5C2×D3212yes—
161C2×C32#71112yescomplemented
162C2×C32C23×D511yescharacteristic
163C3⋊D3—102no—
164C3×D3—24no—
165#41—458no—
166C2×Q8—451no—
167C2×D4—91no—
168C2×D4—452no—
169C22×C4—152no—
170C22×C4—451no—
171C3×C5—21no—
172C3×C5C2×#23512yescomplemented
173C22×C3—21no—
174C22×C3—34no—
175C22×C3—64no—
176C22×C3C2×D3×D512yes—
177C2×D3—24no—
178C2×D3—154no—
179C2×D3—302no—
180C2×D3—304no—
181C2×D3—304no—
182C2×D3C2×D3×D514yes—
183C3×C4—54no—
184C3×C4—102no—
185C3×C4—154no—
186Dic3—34no—
187Dic3—54no—
188Dic3—62no—
189C2×C5—64no—
190C2×C5#379812yescomplemented
191C2×C5C22×D3211yescharacteristic
192D5—182no—
193C32C2×#71111yescharacteristic, complemented, Sylow
194C23—32no—
195C23—451no—
196Q8—454no—
197D4—94no—
198D4—458no—
199C2×C4—51no—
200C2×C4—91no—
201C2×C4—158no—
202C2×C4—152no—
203C2×C4—454no—
204C2×C3—22no—
205C2×C3—21no—
206C2×C3—64no—
207C2×C3#678214yescomplemented
208C2×C3C22×D3×D512yes—
209D3—24no—
210D3—304no—
211D3—304no—
212C5C2×#379811yescharacteristic, complemented, Sylow
213C22—34no—
214C22—64no—
215C22—452no—
216C22—902no—
217C22D32×D511yescharacteristic
218C4—52no—
219C4—92no—
220C4—154no—
221C3—21no—
222C3C2×#678212yescomplemented
223C2—64no—
224C2—902no—
225C2#1062404812yesdirect product factor
226C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#67158C2semidirect
2. C2×#67152C2semidirect
3. C2×C5×#625C2semidirect
4. C2×#67137C2semidirect
5. C2×C3×D3×Dic5C2semidirect
6. #10624048C2direct (N × H)
13. #67158C22semidirect
14. #67152C22semidirect
15. C5×#625C22semidirect
16. #67137C22semidirect
17. C3×D3×Dic5C22semidirect
27. C2×D3×Dic5D3semidirect
41. C2×#625D5semidirect
60. D3×Dic5C2×D3semidirect
62. C2×C32×C5#41semidirect
81. #625C2×D5semidirect
107. C32×C5C2×#41semidirect
130. C2×C3×C5#235semidirect
161. C2×C32#711semidirect
172. C3×C5C2×#235semidirect
190. C2×C5#3798semidirect
193. C32C2×#711semidirect
207. C2×C3#6782semidirect
212. C5C2×#3798semidirect
222. C3C2×#6782semidirect
225. C2#10624048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#671582
C2×#671522
C2×C5×#6252
C2×#671372
C2×C3×D3×Dic52
#106240482
C2×#67823
C2×#37985