Finite Group Databasebeta

← #13106405

Subgroups — #13106405

Order 1600.

458 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#11717266C211yesmaximal, characteristic
2C5×Q8×Dic5C211yesmaximal, characteristic, complemented
3C5×#4090C211yesmaximal, characteristic, complemented
4#11716992C211yesmaximal, characteristic
5#11716990C211yesmaximal, characteristic
6#11716988C211yesmaximal, characteristic, complemented
7#11716987C212yesmaximal
8#11716986C212yesmaximal, complemented
9#11716966C212yesmaximal
10#11716965C212yesmaximal, complemented
11Dic5×Dic10C211yesmaximal, characteristic, complemented
12C2×C52×Q8C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C5×Dic10C2211yescharacteristic
15#87808C2212yes—
16C4⋊#1039C2211yescharacteristic
17#87806C2212yes—
18C5×C4⋊Dic5C2212yes—
19C5×C4⋊Dic5C2211yescharacteristic
20C5×#683C2212yes—
21C5×#683C2212yes—
22C4×C5×Dic5C2211yescharacteristic
23C4×C5×Dic5C2211yescharacteristic
24C4×C5×Dic5C2212yes—
25#87781C2211yescharacteristic
26#87781C2212yes—
27#87781C2211yescharacteristic
28#87780C2212yes—
29#87780C2212yes—
30#87780C2212yes—
31#87779C2211yescharacteristic
32#87779C2212yes—
33#87779C2211yescharacteristic
34Dic52C2212yes—
35#66065—51nomaximal
36#65917—51nomaximal
37C22×C5⋊D5C2311yescharacteristic
38C52×Q8—22no—
39C2×C4×C52C2312yes—
40C2×C4×C52C2311yescharacteristic
41C2×#1039C2312yes—
42C2×#1039C2311yescharacteristic
43C4×C5⋊D5—22no—
44C2×C5×Dic5C2312yes—
45C2×C5×Dic5C2311yescharacteristic
46C2×C5×Dic5C2311yescharacteristic
47C2×C5×Dic5C2311yescharacteristic
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2311yescharacteristic
50C5×Dic10—22no—
51C5×C4⋊Q8—51no—
52C4×C5×Q8—51no—
53#4092—51no—
54#4092—51no—
55#4092—51no—
56#4092—102no—
57Q8×Dic5—51no—
58Q8×Dic5D511yescharacteristic, complemented
59#4090D511yescharacteristic, complemented
60#4044—52no—
61#4044—52no—
62#4044—52no—
63#4043—51no—
64#4036—51no—
65#4035—52no—
66#4035—52no—
67#4035—52no—
68#4022—52no—
69#4018—51no—
70#4018—51no—
71#4016—51no—
72#4016—52no—
73C4×Dic10—51no—
74C22×C52C2411yescharacteristic
75C2×C5⋊D5—21no—
76C2×C5⋊D5—21no—
77C2×C5⋊D5—21no—
78C4×C52—22no—
79C4×C52#4112yes—
80#1039—21no—
81#1039—22no—
82C5×Dic5—22no—
83C5×Dic5—21no—
84C5×Dic5—21no—
85C5×Dic5—21no—
86C5×Dic5—21no—
87C5×Dic5—22no—
88C2×C5×Q8—22no—
89C2×C5×Q8—51no—
90C2×C5×Q8C2×D511yescharacteristic
91C2×C5×Q8C2×D511yescharacteristic
92C2×C4×D5—51no—
93C2×C4×D5—51no—
94C2×C4×D5—102no—
95C2×Dic10C2×D511yescharacteristic
96C5×C4⋊C4—52no—
97C5×C4⋊C4—51no—
98C5×C4⋊C4—52no—
99C5×C4⋊C4—52no—
100C42×C5—51no—
101C42×C5—52no—
102C42×C5—51no—
103#685—52no—
104#685—52no—
105#685—51no—
106#685—51no—
107#685—51no—
108#685—51no—
109#685—52no—
110#685—52no—
111#685—104no—
112C4⋊Dic5—51no—
113C4⋊Dic5—51no—
114C4⋊Dic5—52no—
115C4⋊Dic5—52no—
116C4⋊Dic5—52no—
117C4⋊Dic5—102no—
118C4⋊Dic5C2×D511yescharacteristic
119C4⋊Dic5C2×D512yes—
120#683—52no—
121#683—52no—
122#683—52no—
123#683—51no—
124#683—52no—
125#683—52no—
126#683—51no—
127#683—51no—
128#683—51no—
129#683—52no—
130#683—52no—
131#683—104no—
132#683C2×D512yes—
133#683C2×D512yes—
134C4×Dic5—52no—
135C4×Dic5—52no—
136C4×Dic5C2×D511yescharacteristic
137C4×Dic5C2×D511yescharacteristic
138C4×Dic5C2×D512yes—
139#541—251noSylow
140C2×C52C2×#4111yescharacteristic
141C2×C522−1+411yescharacteristic
142C2×C522−1+411yescharacteristic
143C5⋊D5—41no—
144C22×D5—51no—
145C22×D5—51no—
146C22×D5—102no—
147C5×Q8—22no—
148C5×Q8—22no—
149C5×Q8—44no—
150C5×Q8—102no—
151C2×C4×C5—22no—
152C2×C4×C5—24no—
153C2×C4×C5—51no—
154C2×C4×C5—52no—
155C2×C4×C5—51no—
156C2×C4×C5—51no—
157C2×C4×C5—51no—
158C2×C4×C5—52no—
159C2×C4×C5C22×D512yes—
160C2×C4×C5C22×D512yes—
161C2×C4×C5C22×D511yescharacteristic
162C2×C4×C5C22×D511yescharacteristic
163C2×Dic5—52no—
164C2×Dic5—52no—
165C2×Dic5—51no—
166C2×Dic5—51no—
167C2×Dic5—102no—
168C2×Dic5—104no—
169C2×Dic5C22×D511yescharacteristic
170C2×Dic5C22×D512yes—
171C2×Dic5C22×D511yescharacteristic
172C2×Dic5C22×D511yescharacteristic
173C2×Dic5C22×D512yes—
174C2×Dic5C22×D511yescharacteristic
175C4×D5—102no—
176C4×D5—102no—
177C4×D5—204no—
178Dic10—22no—
179C4⋊Q8—51no—
180#126—252no—
181#126—252no—
182#125—252no—
183#125—251no—
184#125—252no—
185C22⋊Q8—251no—
186C22⋊Q8—251no—
187C4×Q8—51no—
188C4×Q8—251no—
189#117—251no—
190C52#54111yescharacteristic, complemented, Sylow
191C22×C5—22no—
192C22×C5C23×D511yescharacteristic
193C22×C5C23×D511yescharacteristic
194C2×D5—104no—
195C2×D5—101no—
196C2×D5—101no—
197C2×D5—101no—
198C2×D5—101no—
199C2×D5—101no—
200C2×D5—101no—
201C2×D5—202no—
202C2×D5—202no—
203C4×C5—22no—
204C4×C5—22no—
205C4×C5—24no—
206C4×C5—44no—
207C4×C5—102no—
208C4×C5—101no—
209C4×C5—102no—
210C4×C5—101no—
211C4×C5—101no—
212C4×C5—101no—
213C4×C5#70912yes—
214C4×C5#71112yes—
215Dic5—22no—
216Dic5—21no—
217Dic5—21no—
218Dic5—21no—
219Dic5—21no—
220Dic5—22no—
221Dic5—101no—
222Dic5—101no—
223Dic5—102no—
224Dic5—102no—
225Dic5—202no—
226Dic5—204no—
227C2×Q8—51no—
228C2×Q8D5211yescharacteristic
229C22×C4—251no—
230C4⋊C4—51no—
231C4⋊C4—52no—
232C4⋊C4—52no—
233C4⋊C4—52no—
234C4⋊C4—251no—
235C4⋊C4—251no—
236C4⋊C4—252no—
237C4⋊C4—252no—
238C4⋊C4—252no—
239C4⋊C4—252no—
240C4⋊C4—251no—
241C4⋊C4—252no—
242C22⋊C4—251no—
243C22⋊C4—251no—
244C22⋊C4—252no—
245C22⋊C4—252no—
246C42—51no—
247C42—51no—
248C42—52no—
249C42—252no—
250C2×C5—22no—
251C2×C5—22no—
252C2×C5—22no—
253C2×C5C2×#70911yescharacteristic
254C2×C5C2×#71111yescharacteristic
255C2×C5#414711yescharacteristic
256C2×C5#414711yescharacteristic
257C2×C5#414711yescharacteristic
258C2×C5#415011yescharacteristic
259D5—204no—
260D5—201no—
261D5—201no—
262C23—251no—
263Q8—22no—
264Q8—102no—
265C2×C4—51no—
266C2×C4—51no—
267C2×C4—51no—
268C2×C4—52no—
269C2×C4—51no—
270C2×C4—52no—
271C2×C4—252no—
272C2×C4—251no—
273C2×C4—502no—
274C2×C4C2×D5212yes—
275C2×C4C2×D5211yescharacteristic
276C5—22no—
277C5#6591711yescharacteristic, complemented
278C5#6606511yescharacteristic, complemented
279C22—501no—
280C22—501no—
281C22—501no—
282C22C22×D5211yescharacteristic
283C4—22no—
284C4—101no—
285C4—101no—
286C4—102no—
287C4—101no—
288C4—101no—
289C4—102no—
290C4—502no—
291C4—501no—
292C4#8787912yes—
293C2—1001no—
294C2#1171748011yescharacteristic
295C2#1171748111yescharacteristic
296C2C2×#8787911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
prime radicalC22×C5⋊D5C23
hypercenterC2×Q8D52
hypocenterC52#541
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C5×Q8×Dic5C2semidirect
3. C5×#4090C2semidirect
6. #11716988C2semidirect
8. #11716986C2semidirect
10. #11716965C2semidirect
11. Dic5×Dic10C2semidirect
58. Q8×Dic5D5semidirect
59. #4090D5semidirect
190. C52#541semidirect
277. C5#65917semidirect
278. C5#66065semidirect

Maximal subgroups

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

subgroupindex
#117172662
C5×Q8×Dic52
C5×#40902
#117169922
#117169902
#117169882
#117169872
#117169862
#117169662
#117169652
Dic5×Dic102
#660655
#659175