Finite Group Databasebeta

← #13101776

Subgroups — #13101776

Order 1600.

294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 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
1Q8×#1039C211yesmaximal, characteristic, complemented
2C2×C5×#689C211yesmaximal, characteristic, complemented
3C2×C5×#688C211yesmaximal, characteristic
4#11716981C211yesmaximal, characteristic
5#11716613C211yesmaximal, characteristic, complemented
6#11716607C211yesmaximal, characteristic, complemented
7#11716604C211yesmaximal, characteristic
8C2×C52×Q8C2211yescharacteristic
9C2×C5×D20C2211yescharacteristic
10C2×C5×Dic10C2211yescharacteristic
11C4⋊#1039—21no—
12C4⋊#1039C2211yescharacteristic
13C4×#1039—21no—
14C4×#1039C2211yescharacteristic
15C5×#689—22no—
16C5×#688—22no—
17C2×C5×#168C2211yescharacteristic
18C2×C5×#168C2211yescharacteristic
19#87778—21no—
20#87778—21no—
21#65474—51nomaximal
22#65455—51nomaximal
23C22×C5×D5—21no—
24C52×Q8—21no—
25C52×Q8D412yes—
26C2×C4×C52—21no—
27C2×C4×C52C2311yescharacteristic
28C2×#1039—21no—
29C2×#1039D411yescharacteristic
30C2×#1039D411yescharacteristic
31C2×C5×Dic5—21no—
32C5×D20—21no—
33C5×D20—21no—
34C5×Dic10—21no—
35C5×Dic10—21no—
36C5×#168—21no—
37C5×#168—21no—
38C2×C5×Q16—51no—
39C2×C5×SD16—51no—
40#4093—51no—
41Q8×Dic5—51no—
42Q8×Dic5—51no—
43Q8×Dic5—102no—
44C2×#689D511yescharacteristic, complemented
45C2×#688D511yescharacteristic
46#4082—51no—
47#3967—51no—
48#3964—51no—
49#3953—51no—
50#3948—51no—
51#3946—51no—
52#3946—51no—
53C22×C52C2×D411yescharacteristic
54C2×C5×D5—41no—
55C2×C5×D5—41no—
56C2×C5×D5—41no—
57C4×C52—22no—
58C4×C52C2×D411yescharacteristic
59C4×C52#4111yescharacteristic
60#1039—21no—
61#1039—21no—
62#1039—41no—
63C5×Dic5—41no—
64C2×C5×Q8—22no—
65C2×C5×Q8—51no—
66C2×C5×Q8C2×D511yescharacteristic
67C2×C5×Q8C2×D511yescharacteristic
68C2×C5×D4—51no—
69C2×D20C2×D511yescharacteristic
70C2×Dic10C2×D511yescharacteristic
71C5×Q16—102no—
72C5×SD16—102no—
73C2×C5×C8—51no—
74C2×C5×C8—51no—
75C22⋊Dic5—101no—
76C22⋊Dic5—101no—
77#689—22no—
78#688—22no—
79#685—101no—
80#685—101no—
81C4⋊Dic5—51no—
82C4⋊Dic5—51no—
83C4⋊Dic5—102no—
84C4⋊Dic5—101no—
85C4⋊Dic5—101no—
86C4⋊Dic5—104no—
87C4×Dic5—51no—
88C4×Dic5—51no—
89C4×Dic5—102no—
90C4×Dic5—101no—
91C4×Dic5—101no—
92C4×Dic5—104no—
93C2×#168C2×D511yescharacteristic
94C2×#168C2×D511yescharacteristic
95#464—251noSylow
96C2×C52#12111yescharacteristic
97C2×C52#13511yescharacteristic
98C2×C52#13711yescharacteristic
99C5×D5—81no—
100C23×C5—101no—
101C22×D5—21no—
102C5×Q8—24no—
103C5×Q8—21no—
104C5×Q8—21no—
105C5×Q8—24no—
106C5×Q8—101no—
107C5×Q8—101no—
108C5×Q8C22⋊D512yes—
109C5×Q8C22⋊D512yes—
110C5×D4—101no—
111C5×D4—101no—
112C2×C4×C5—24no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—22no—
116C2×C4×C5—101no—
117C2×C4×C5C22×D511yescharacteristic
118C2×C4×C5C22×D511yescharacteristic
119C2×Dic5—21no—
120C2×Dic5—51no—
121C2×Dic5—51no—
122C2×Dic5—51no—
123C2×Dic5—51no—
124C2×Dic5—102no—
125C2×Dic5—102no—
126C2×Dic5—104no—
127C2×Dic5—101no—
128C2×Dic5—101no—
129D20—21no—
130D20—21no—
131Dic10—21no—
132Dic10—21no—
133C5×C8—101no—
134C5×C8—101no—
135#168—21no—
136#168—21no—
137C2×Q16—51no—
138C2×SD16—51no—
139#124—251no—
140C4×Q8—251no—
141C4⋊C8—251no—
142#103—251no—
143#102—251no—
144C52#46411yescharacteristic, complemented, Sylow
145C22×C5—22no—
146C22×C5—201no—
147C22×C5—201no—
148C22×C5—201no—
149C22×C5D4×D511yescharacteristic
150C22×C5D4×D511yescharacteristic
151C2×D5—41no—
152C2×D5—41no—
153C2×D5—41no—
154C4×C5—28no—
155C4×C5—22no—
156C4×C5—22no—
157C4×C5—22no—
158C4×C5—22no—
159C4×C5—201no—
160C4×C5#71111yescharacteristic
161C4×C5#71111yescharacteristic
162C4×C5C2×C22⋊D511yescharacteristic
163C4×C5C2×C22⋊D511yescharacteristic
164Dic5—41no—
165Dic5—104no—
166Dic5—101no—
167Dic5—101no—
168Dic5—101no—
169Dic5—101no—
170Dic5—202no—
171Dic5—204no—
172Dic5—201no—
173Dic5—201no—
174C2×Q8—51no—
175C2×Q8D5211yescharacteristic
176C2×D4—51no—
177Q16—102no—
178SD16—102no—
179C2×C8—51no—
180C2×C8—51no—
181C4⋊C4—251no—
182C4⋊C4—501no—
183C22⋊C4—501no—
184C22⋊C4—501no—
185C42—251no—
186C42—501no—
187C2×C5—22no—
188C2×C5—22no—
189C2×C5—22no—
190C2×C5—401no—
191C2×C5#406111yescharacteristic
192C2×C5#406211yescharacteristic
193C2×C5#406411yescharacteristic
194C2×C5#406511yescharacteristic
195C2×C5#408511yescharacteristic
196C2×C5#408511yescharacteristic
197D5—81no—
198C23—101no—
199Q8—21no—
200Q8—101no—
201Q8—101no—
202Q8#603712yes—
203D4—101no—
204D4—101no—
205C2×C4—21no—
206C2×C4—101no—
207C2×C4—251no—
208C2×C4—251no—
209C2×C4—501no—
210C2×C4C2×D5211yescharacteristic
211C8—101no—
212C8—101no—
213C5—22no—
214C5#6545511yescharacteristic, complemented
215C5#6547411yescharacteristic, complemented
216C22—201no—
217C22—201no—
218C22—201no—
219C22#8788611yescharacteristic
220C4—22no—
221C4—201no—
222C4—501no—
223C4—501no—
224C4—1001no—
225C4#8788011yescharacteristic
226C4C2×#603711yescharacteristic
227C2—401no—
228C2#1171706811yescharacteristic
229C2#1171706911yescharacteristic
230C2#1171709811yescharacteristic
subgroupG/H
centerC22#87886
commutator, cosocleC2×C4×C52C23
FrattiniC2×C4C2×D52
FittingC2×C52×Q8C22
socleC22×C52C2×D4
prime radicalC2×C5×D20C22
hypercenterC2×Q8D52
hypocenterC52#464
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1039C2semidirect
2. C2×C5×#689C2semidirect
5. #11716613C2semidirect
6. #11716607C2semidirect
44. C2×#689D5semidirect
144. C52#464semidirect
214. C5#65455semidirect
215. C5#65474semidirect

Maximal subgroups

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

subgroupindex
Q8×#10392
C2×C5×#6892
C2×C5×#6882
#117169812
#117166132
#117166072
#117166042
#654745
#654555