Finite Group Databasebeta

← #13105430

Subgroups — #13105430

Order 1600.

672 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 168 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×#87872C212yesmaximal
2C4⋊C4×C5⋊D5C211yesmaximal, characteristic
3C5×#4034C212yesmaximal, complemented
4#11717036C212yesmaximal, complemented
5#11717018C211yesmaximal, characteristic
6#11717001C212yesmaximal
7C4×#6036C211yesmaximal, characteristic
8#11716957C214yesmaximal, complemented
9C2×C4×C5⋊D5C2212yes—
10C2×C4×C5⋊D5C2211yescharacteristic
11C2×C5×Dic10C2214yes—
12C2×#6039C2214yes—
13C2×#6036C2212yes—
14C2×#6036C2211yescharacteristic
15C2×#6036C2211yescharacteristic
16#87872—28no—
17C52×C4⋊C4C2211yescharacteristic
18C4⋊#1039C2211yescharacteristic
19#87806C2212yes—
20C5×C4⋊Dic5C2212yes—
21C5×#683C2214yes—
22C4×C5×Dic5C2212yes—
23#87781C2212yes—
24#87780C2214yes—
25Dic52C2212yes—
26D5×C4⋊Q8—52nomaximal
27C22×C5⋊D5C2311yescharacteristic
28C2×C4×C52C2312yes—
29C2×C4×C52C2311yescharacteristic
30C2×#1039C2312yes—
31C2×#1039C2311yescharacteristic
32C4×C5⋊D5—24no—
33C4×C5⋊D5Q814yescomplemented
34C2×C5×Dic5C2314yes—
35C2×C5×Dic5C2312yes—
36C2×C5×Dic5C2312yes—
37C5×Dic10—28no—
38#6039—28no—
39#6036—24no—
40#6036D414yes—
41#6036Q814yescomplemented
42C2×Q8×D5—54no—
43C5×C4⋊Q8—52no—
44#4090—54no—
45D5×C4⋊C4—52no—
46D5×C4⋊C4—52no—
47D5×C4⋊C4—54no—
48D5×C4⋊C4—102no—
49#4034—54no—
50#4034—52no—
51#4034D512yescomplemented
52C42×D5—52no—
53#4015—52no—
54C22×C52C2411yescharacteristic
55C2×C5⋊D5C2×D412yes—
56C2×C5⋊D5C2×Q812yes—
57C2×C5⋊D5C2×Q812yes—
58C4×C52—22no—
59C4×C52C2×Q812yescomplemented
60#1039—22no—
61#1039C2×Q812yescomplemented
62C5×Dic5—24no—
63C5×Dic5C2×D414yes—
64C5×Dic5C2×Q814yescomplemented
65C2×C5×Q8—54no—
66Q8×D5—1016no—
67C2×C4×D5—52no—
68C2×C4×D5—52no—
69C2×C4×D5—52no—
70C2×C4×D5—54no—
71C2×C4×D5—54no—
72C2×C4×D5—102no—
73C2×C4×D5—104no—
74C2×Dic10—54no—
75C2×Dic10—54no—
76C2×Dic10C2×D514yes—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—54no—
79C5×C4⋊C4—52no—
80C5×C4⋊C4C2×D512yes—
81C42×C5—52no—
82C4⋊Dic5—52no—
83C4⋊Dic5—54no—
84C4⋊Dic5—102no—
85C4⋊Dic5C2×D512yes—
86#683—54no—
87#683—54no—
88#683—54no—
89#683—104no—
90#683C2×D514yes—
91C4×Dic5—52no—
92C4×Dic5—52no—
93C4×Dic5C2×D512yes—
94C2×C4⋊Q8—251noSylow
95C2×C52C22×D411yescharacteristic
96C2×C52C22×Q811yescharacteristic
97C2×C52C22×Q811yescharacteristic
98C5⋊D5C4⋊Q814yescomplemented
99C22×D5—52no—
100C22×D5—102no—
101C5×Q8—108no—
102C2×C4×C5—22no—
103C2×C4×C5—24no—
104C2×C4×C5—52no—
105C2×C4×C5—52no—
106C2×C4×C5—54no—
107C2×C4×C5C22×D514yes—
108C2×C4×C5C22×D512yes—
109C2×Dic5—52no—
110C2×Dic5—54no—
111C2×Dic5—102no—
112C2×Dic5—104no—
113C2×Dic5C22×D514yes—
114C2×Dic5C22×D512yes—
115C2×Dic5C22×D512yes—
116C4×D5—58no—
117C4×D5—58no—
118C4×D5—58no—
119C4×D5—108no—
120C4×D5—108no—
121C4×D5—108no—
122C4×D5—208no—
123Dic10—28no—
124Dic10—108no—
125Dic10—108no—
126C22×Q8—252no—
127C4⋊Q8—52no—
128C4⋊Q8—252no—
129C4⋊Q8—254no—
130C2×C4⋊C4—252no—
131C2×C4⋊C4—251no—
132C2×C4⋊C4—251no—
133C2×C42—251no—
134C52C2×C4⋊Q811yescharacteristic, complemented, Sylow
135C22×C5—22no—
136C22×C5C23×D512yes—
137C2×D5—54no—
138C2×D5—54no—
139C2×D5—54no—
140C2×D5—104no—
141C2×D5—104no—
142C2×D5—104no—
143C4×C5—24no—
144C4×C5—24no—
145C4×C5—44no—
146C4×C5—54no—
147C4×C5—54no—
148C4×C5—104no—
149C4×C5Q8×D514yescomplemented
150Dic5—24no—
151Dic5—54no—
152Dic5—104no—
153Dic5—104no—
154Dic5—204no—
155Dic5D4×D514yes—
156Dic5Q8×D514yescomplemented
157C2×Q8—54no—
158C2×Q8—254no—
159C2×Q8—508no—
160C22×C4—251no—
161C22×C4—251no—
162C22×C4—252no—
163C22×C4—251no—
164C22×C4—252no—
165C4⋊C4—54no—
166C4⋊C4—52no—
167C4⋊C4—252no—
168C4⋊C4—254no—
169C4⋊C4—252no—
170C4⋊C4—251no—
171C4⋊C4D5211yescharacteristic
172C42—52no—
173C42—252no—
174C2×C5—22no—
175C2×C5—22no—
176C2×C5—22no—
177C2×C5C2×D4×D512yes—
178C2×C5C2×Q8×D512yes—
179C2×C5C2×Q8×D512yes—
180D5—58no—
181D5—108no—
182C23—251no—
183Q8—108no—
184Q8—508no—
185C2×C4—52no—
186C2×C4—52no—
187C2×C4—54no—
188C2×C4—254no—
189C2×C4—254no—
190C2×C4—251no—
191C2×C4—254no—
192C2×C4—252no—
193C2×C4—504no—
194C2×C4—504no—
195C2×C4C2×D5212yes—
196C2×C4C2×D5211yescharacteristic
197C5—22no—
198C5D5×C4⋊Q812yescomplemented
199C22—252no—
200C22—252no—
201C22—252no—
202C22C22×D5211yescharacteristic
203C4—22no—
204C4—54no—
205C4—54no—
206C4—104no—
207C4—252no—
208C4—502no—
209C4#8787212yescomplemented
210C2—254no—
211C2C2×#8787211yescharacteristic
212C2D4×D5211yescharacteristic
213C2Q8×D5211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC22×C5⋊D5C23
hypercenterC4⋊C4D52
hypocenterC52C2×C4⋊Q8
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C5×#4034C2semidirect
4. #11717036C2semidirect
8. #11716957C2semidirect
33. C4×C5⋊D5Q8semidirect
41. #6036Q8semidirect
51. #4034D5semidirect
59. C4×C52C2×Q8semidirect
61. #1039C2×Q8semidirect
64. C5×Dic5C2×Q8semidirect
98. C5⋊D5C4⋊Q8semidirect
134. C52C2×C4⋊Q8semidirect
149. C4×C5Q8×D5semidirect
156. Dic5Q8×D5semidirect
198. C5D5×C4⋊Q8semidirect
209. C4#87872semidirect

Maximal subgroups

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

subgroupindex
C2×#878722
C4⋊C4×C5⋊D52
C5×#40342
#117170362
#117170182
#117170012
C4×#60362
#117169572
D5×C4⋊Q85