Finite Group Databasebeta

← #13098958

Subgroups — #13098958

Order 1600.

438 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 127 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×#87748C211yesmaximal, characteristic, complemented
2C2×#87744C211yesmaximal, characteristic, complemented
3C25×C22⋊Q8C211yesmaximal, characteristic, complemented
4#11716506C211yesmaximal, characteristic, complemented
5C4×C22⋊D25C211yesmaximal, characteristic, complemented
6#11716459C211yesmaximal, characteristic, complemented
7#11716456C212yesmaximal, complemented
8#11716455C211yesmaximal, characteristic, complemented
9C4⋊C4×D25C212yesmaximal, complemented
10#11716446C212yesmaximal, complemented
11#11716445C212yesmaximal, complemented
12C2×Q8×C25C2211yescharacteristic
13C22×C4×C25C2211yescharacteristic
14C2×C22⋊D25C2212yes—
15C2×C22⋊D25C2211yescharacteristic
16#87748—24no—
17#87744—24no—
18C2×D100C2212yes—
19C2×D100C2211yescharacteristic
20C2×C4×D25C2212yes—
21C2×C4×D25C2212yes—
22C2×C4×D25C2212yes—
23C2×C4×D25C2211yescharacteristic
24C2×Dic50C2211yescharacteristic
25C4⋊C4×C25C2212yes—
26C4⋊C4×C25C2211yescharacteristic
27C22⋊C4×C25C2212yes—
28C22⋊Dic25C2211yescharacteristic
29#87720C2212yes—
30#87720C2212yes—
31#87720C2211yescharacteristic
32C4⋊Dic25C2212yes—
33#87718C2212yes—
34#87718C2212yes—
35#87718C2211yescharacteristic
36C4×Dic25C2211yescharacteristic
37#65965—51nomaximal
38C23×C25C2311yescharacteristic
39C22×D25C2311yescharacteristic
40C22×D25C2312yes—
41Q8×C25—22no—
42C2×C4×C25—22no—
43C2×C4×C25C2312yes—
44C2×C4×C25C2311yescharacteristic
45C2×C4×C25C2312yes—
46C2×C4×C25C2311yescharacteristic
47C22⋊D25—24no—
48C22⋊D25—22no—
49C2×Dic25C2311yescharacteristic
50C2×Dic25C2312yes—
51C2×Dic25C2311yescharacteristic
52C2×Dic25C2311yescharacteristic
53D100—24no—
54D100D414yescomplemented
55C4×D25—24no—
56C4×D25—24no—
57C4×D25—22no—
58C4×D25—24no—
59Dic50—22no—
60C2×#713—51no—
61C2×#709—51no—
62C5×C22⋊Q8D511yescharacteristic
63#4090—51no—
64C4×C22⋊D5—51no—
65#4043—51no—
66#4040—52no—
67#4039—51no—
68D5×C4⋊C4—52no—
69#4030—52no—
70#4029—52no—
71C22×C25—21no—
72C22×C25—21no—
73C22×C25—21no—
74C22×C25C2411yescharacteristic
75C2×D25—22no—
76C2×D25—22no—
77C2×D25—21no—
78C2×D25—21no—
79C2×D25—21no—
80C2×D25C2×D414yes—
81C4×C25—21no—
82C4×C25—22no—
83C4×C25—22no—
84C4×C25C2×D412yes—
85Dic25—22no—
86Dic25—21no—
87Dic25—21no—
88Dic25#4112yescomplemented
89C2×C5×Q8C2×D511yescharacteristic
90C22×C4×C5C2×D511yescharacteristic
91C2×C22⋊D5—52no—
92C2×C22⋊D5—51no—
93#713—104no—
94#709—104no—
95C2×D20—51no—
96C2×D20—52no—
97C2×C4×D5—52no—
98C2×C4×D5—52no—
99C2×C4×D5—51no—
100C2×C4×D5—52no—
101C2×Dic10—51no—
102C5×C4⋊C4C2×D512yes—
103C5×C4⋊C4C2×D511yescharacteristic
104C5×C22⋊C4C2×D512yes—
105C22⋊Dic5—51no—
106#685—52no—
107#685—52no—
108#685—51no—
109C4⋊Dic5—52no—
110#683—52no—
111#683—52no—
112#683—51no—
113C4×Dic5—51no—
114#547—251noSylow
115C2×C25—41no—
116C2×C25C22×D411yescharacteristic
117C2×C25C2×#4111yescharacteristic
118C2×C252−1+411yescharacteristic
119D25—24no—
120D25—41no—
121C23×C5C22×D511yescharacteristic
122C22×D5—52no—
123C22×D5—51no—
124C5×Q8—22no—
125C2×C4×C5—22no—
126C2×C4×C5C22×D512yes—
127C2×C4×C5C22×D511yescharacteristic
128C2×C4×C5C22×D512yes—
129C2×C4×C5C22×D511yescharacteristic
130C22⋊D5—102no—
131C22⋊D5—104no—
132C2×Dic5—51no—
133C2×Dic5—51no—
134C2×Dic5—52no—
135C2×Dic5—51no—
136D20—54no—
137D20—104no—
138C4×D5—104no—
139C4×D5—104no—
140C4×D5—104no—
141C4×D5—102no—
142Dic10—102no—
143C2×#41—251no—
144C2×#41—251no—
145C4⋊Q8—251no—
146#123—252no—
147#123—252no—
148C22⋊Q8—251no—
149C22⋊Q8D2511yescharacteristic, complemented
150#121—252no—
151C4×D4—251no—
152C4×D4—251no—
153C2×C4⋊C4—252no—
154C25#54711yescharacteristic, complemented, Sylow
155C22×C5—21no—
156C22×C5—21no—
157C22×C5—21no—
158C22×C5C23×D511yescharacteristic
159C2×D5—54no—
160C2×D5—102no—
161C2×D5—102no—
162C2×D5—101no—
163C2×D5—101no—
164C2×D5—101no—
165C4×C5—21no—
166C4×C5—22no—
167C4×C5—22no—
168C4×C5D4×D512yes—
169Dic5—52no—
170Dic5—102no—
171Dic5—101no—
172Dic5—101no—
173#41—504no—
174#41—504no—
175C2×Q8—251no—
176C2×Q8C2×D2511yescharacteristic
177C2×D4—251no—
178C2×D4—252no—
179C2×D4—251no—
180C2×D4—252no—
181C22×C4—252no—
182C22×C4—252no—
183C22×C4—252no—
184C22×C4—251no—
185C22×C4C2×D2511yescharacteristic
186C4⋊C4—251no—
187C4⋊C4—252no—
188C4⋊C4—252no—
189C4⋊C4—252no—
190C4⋊C4C2×D2512yes—
191C4⋊C4C2×D2511yescharacteristic
192C22⋊C4—251no—
193C22⋊C4—252no—
194C22⋊C4—251no—
195C22⋊C4—252no—
196C22⋊C4C2×D2512yes—
197C42—251no—
198C2×C5—41no—
199C2×C5C2×D4×D511yescharacteristic
200C2×C5#414711yescharacteristic
201C2×C5D5×#4111yescharacteristic
202D5—104no—
203D5—201no—
204C23—252no—
205C23—251no—
206C23C22×D2511yescharacteristic
207Q8—22no—
208Q8—502no—
209D4—254no—
210D4—504no—
211D4—502no—
212D4—504no—
213C2×C4—22no—
214C2×C4—251no—
215C2×C4—251no—
216C2×C4—251no—
217C2×C4—252no—
218C2×C4—504no—
219C2×C4—504no—
220C2×C4—504no—
221C2×C4—502no—
222C2×C4C22×D2512yes—
223C2×C4C22×D2511yescharacteristic
224C2×C4C22×D2512yes—
225C2×C4C22×D2511yescharacteristic
226C5#6596511yescharacteristic
227C22—21no—
228C22—21no—
229C22—21no—
230C22—254no—
231C22—502no—
232C22—502no—
233C22—501no—
234C22—501no—
235C22—501no—
236C22C23×D2511yescharacteristic
237C4—21no—
238C4—22no—
239C4—22no—
240C4—252no—
241C4—501no—
242C4—502no—
243C4—501no—
244C4D4×D2512yes—
245C2—41no—
246C2—504no—
247C2—1001no—
248C2C2×D4×D2511yescharacteristic
249C2#1171656611yescharacteristic
250C2#41×D2511yescharacteristic
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×C22⋊Q8C2
hypercenterC22⋊Q8D25
hypocenterC25#547
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#87748C2semidirect
2. C2×#87744C2semidirect
3. C25×C22⋊Q8C2semidirect
4. #11716506C2semidirect
5. C4×C22⋊D25C2semidirect
6. #11716459C2semidirect
7. #11716456C2semidirect
8. #11716455C2semidirect
9. C4⋊C4×D25C2semidirect
10. #11716446C2semidirect
11. #11716445C2semidirect
54. D100D4semidirect
88. Dic25#41semidirect
149. C22⋊Q8D25semidirect
154. C25#547semidirect

Maximal subgroups

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

subgroupindex
C2×#877482
C2×#877442
C25×C22⋊Q82
#117165062
C4×C22⋊D252
#117164592
#117164562
#117164552
C4⋊C4×D252
#117164462
#117164452
#659655