Finite Group Databasebeta

← C2×#11716984 #13106130

Subgroups — C2×#11716984 #13106130

Order 1600.

880 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C22×C4×C5⋊D5C211yesmaximal, characteristic
2C22×#6039C211yesmaximal, characteristic, complemented
3C2×C5×#683C212yesmaximal, complemented
4C2×#87781C211yesmaximal, characteristic, complemented
5C2×#87779C212yesmaximal
6#11716984C218yesmaximal, direct product factor
7C23×C5⋊D5C2211yescharacteristic
8C22×C4×C52C2211yescharacteristic
9C22×#1039C2211yescharacteristic
10C2×C4×C5⋊D5—24no—
11C2×C4×C5⋊D5C2214yes—
12C22×C5×Dic5C2212yes—
13C22×C5×Dic5C2212yes—
14C2×#6039—24no—
15C2×#6039C2214yescomplemented
16C5×#683C2218yescomplemented
17#87781C2214yescomplemented
18#87779C2218yes—
19C2×#4042—52nomaximal
20C23×C52C2311yescharacteristic
21C22×C5⋊D5—22no—
22C22×C5⋊D5—22no—
23C22×C5⋊D5Q814yescomplemented
24C22×C5⋊D5C2312yes—
25C2×C4×C52—22no—
26C2×C4×C52C2312yes—
27C2×#1039D414yes—
28C2×#1039C2312yes—
29C4×C5⋊D5—28no—
30C2×C5×Dic5—24no—
31C2×C5×Dic5—24no—
32C2×C5×Dic5C2314yes—
33C2×C5×Dic5C2314yes—
34#6039—28no—
35C22×C4×D5—52no—
36C22×C4×D5—102no—
37C22×Dic10—52no—
38C2×C5×C4⋊C4—52no—
39C2×#685—52no—
40C2×#685—52no—
41C2×#683—52no—
42C2×#683D512yescomplemented
43#4042—516no—
44C22×C52C2×D412yes—
45C22×C52C2×Q812yes—
46C22×C52#4112yes—
47C22×C52C2411yescharacteristic
48C2×C5⋊D5—28no—
49C2×C5⋊D5—22no—
50C2×C5⋊D5—22no—
51C2×C5⋊D5C2×Q814yescomplemented
52C4×C52—22no—
53#1039C2×D414yes—
54C5×Dic5—24no—
55C5×Dic5—24no—
56C23×D5—52no—
57C23×D5—102no—
58C22×C4×C5—22no—
59C22×C4×C5—52no—
60C22×C4×C5—52no—
61C22×C4×C5C2×D512yes—
62C22×Dic5—52no—
63C22×Dic5—102no—
64C22×Dic5C2×D512yes—
65C22×Dic5C2×D512yes—
66C2×C4×D5—58no—
67C2×C4×D5—108no—
68C2×C4×D5—108no—
69C2×C4×D5—1016no—
70C2×Dic10—58no—
71C2×Dic10—108no—
72C5×C4⋊C4—58no—
73#685—58no—
74#685—58no—
75#683—58no—
76#683C2×D518yescomplemented
77C2×C22⋊Q8—251noSylow
78C2×C52C22⋊Q814yescomplemented
79C2×C52C22×D411yescharacteristic
80C2×C52C22×Q811yescharacteristic
81C2×C52C2×#4111yescharacteristic
82C5⋊D5—24no—
83C23×C5—22no—
84C23×C5C22×D512yes—
85C22×D5—58no—
86C22×D5—54no—
87C22×D5—108no—
88C22×D5—104no—
89C22×D5—108no—
90C22×D5—108no—
91C22×D5—104no—
92C22×D5—104no—
93C2×C4×C5—28no—
94C2×C4×C5—24no—
95C2×C4×C5—24no—
96C2×C4×C5—54no—
97C2×C4×C5—54no—
98C2×C4×C5—104no—
99C2×C4×C5—104no—
100C2×C4×C5C22×D514yes—
101C2×Dic5—24no—
102C2×Dic5—24no—
103C2×Dic5—58no—
104C2×Dic5—54no—
105C2×Dic5—108no—
106C2×Dic5—104no—
107C2×Dic5C22×D514yes—
108C2×Dic5C22×D514yes—
109C4×D5—1016no—
110C4×D5—1032no—
111Dic10—1016no—
112C22×Q8—251no—
113C23×C4—251no—
114C22⋊Q8—258no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—251no—
117C2×C22⋊C4—252no—
118C52C2×C22⋊Q811yescharacteristic, complemented, Sylow
119C22×C5—24no—
120C22×C5—24no—
121C22×C5—24no—
122C22×C5—22no—
123C22×C5#70914yes—
124C22×C5D4×D514yes—
125C22×C5Q8×D514yes—
126C22×C5C23×D512yes—
127C2×D5—58no—
128C2×D5—108no—
129C2×D5—108no—
130C2×D5—108no—
131C2×D5—1032no—
132C2×D5—104no—
133C2×D5—104no—
134C2×D5—1016no—
135C4×C5—28no—
136C4×C5—24no—
137C4×C5—104no—
138C4×C5—104no—
139Dic5—24no—
140Dic5—24no—
141Dic5—58no—
142Dic5—108no—
143C24—251no—
144C2×Q8—254no—
145C2×Q8—504no—
146C22×C4—52no—
147C22×C4—52no—
148C22×C4—251no—
149C22×C4—254no—
150C22×C4—504no—
151C22×C4D5211yescharacteristic
152C4⋊C4—58no—
153C4⋊C4—254no—
154C22⋊C4—258no—
155C2×C5—22no—
156C2×C5—22no—
157C2×C5—22no—
158C2×C5—28no—
159C2×C5#404218yescomplemented
160C2×C5C2×#70912yes—
161C2×C5C2×D4×D512yes—
162C2×C5C2×Q8×D512yes—
163D5—1016no—
164D5—108no—
165C23—254no—
166C23—252no—
167C23—502no—
168C23—502no—
169C23C2×D5211yescharacteristic
170Q8—508no—
171C2×C4—22no—
172C2×C4—54no—
173C2×C4—54no—
174C2×C4—104no—
175C2×C4—104no—
176C2×C4—254no—
177C2×C4—252no—
178C2×C4—508no—
179C2×C4C2×D5212yes—
180C5—22no—
181C5C2×#404212yescomplemented
182C22—254no—
183C22—502no—
184C22—502no—
185C22—508no—
186C22#8787212yes—
187C22#8787312yes—
188C22#8788612yes—
189C22C22×D5211yescharacteristic
190C4—22no—
191C4—104no—
192C4—104no—
193C4—254no—
194C2—504no—
195C2#1171698414yesdirect product factor
196C2C2×#8787211yescharacteristic
197C2C2×#8787311yescharacteristic
198C2C2×#8788611yescharacteristic
subgroupG/H
centerC23C2×D52
commutator, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
socleC23×C52C23
prime radicalC23×C5⋊D5C22
hypercenterC22×C4D52
hypocenterC52C2×C22⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6039C2semidirect
3. C2×C5×#683C2semidirect
4. C2×#87781C2semidirect
6. #11716984C2direct (N × H)
15. C2×#6039C22semidirect
16. C5×#683C22semidirect
17. #87781C22semidirect
23. C22×C5⋊D5Q8semidirect
42. C2×#683D5semidirect
51. C2×C5⋊D5C2×Q8semidirect
76. #683C2×D5semidirect
78. C2×C52C22⋊Q8semidirect
118. C52C2×C22⋊Q8semidirect
159. C2×C5#4042semidirect
181. C5C2×#4042semidirect
195. C2#11716984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C5⋊D52
C22×#60392
C2×C5×#6832
C2×#877812
C2×#877792
#117169842
C2×#40425