Finite Group Databasebeta

← C2×#11717089 #13106601

Subgroups — C2×#11717089 #13106601

Order 1600.

1088 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 184 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×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C22×C5×C22⋊D5C211yesmaximal, characteristic, complemented
3C22×#6038C211yesmaximal, characteristic, complemented
4C22×D5×Dic5C211yesmaximal, characteristic, complemented
5C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
6#11717089C218yesmaximal, direct product factor
7C2×#87781C211yesmaximal, characteristic, complemented
8C2×#87779C211yesmaximal, characteristic, complemented
9C24×C52C2211yescharacteristic
10C23×C5⋊D5C2211yescharacteristic
11C23×C5×D5C2211yescharacteristic
12C2×C22⋊(C5⋊D5)—24no—
13C2×C22⋊(C5⋊D5)C2214yescomplemented
14C22×#1039C2211yescharacteristic
15C2×C5×C22⋊D5—24no—
16C2×C5×C22⋊D5C2214yescomplemented
17C22×C5×Dic5C2211yescharacteristic
18C22×C5×Dic5C2211yescharacteristic
19C22×C5×Dic5C2211yescharacteristic
20C2×#6038—24no—
21C2×#6038C2214yescomplemented
22C2×D5×Dic5—24no—
23C2×D5×Dic5C2214yescomplemented
24C5×C22⋊Dic5C2214yescomplemented
25#87781C2214yescomplemented
26#87779C2214yescomplemented
27C2×#4085—51nomaximal
28C2×#4030—51nomaximal
29C23×C52—22no—
30C23×C52—22no—
31C23×C52—22no—
32C23×C52C2312yes—
33C23×C52C2311yescharacteristic
34C22×C5⋊D5—22no—
35C22×C5⋊D5—22no—
36C22×C5⋊D5—22no—
37C22×C5⋊D5C2312yes—
38C22×C5×D5—22no—
39C22×C5×D5—22no—
40C22×C5×D5D414yescomplemented
41C22×C5×D5C2312yes—
42C22⋊(C5⋊D5)—28no—
43C2×#1039D414yescomplemented
44C2×#1039C2312yes—
45C5×C22⋊D5—28no—
46C2×C5×Dic5—22no—
47C2×C5×Dic5—22no—
48C2×C5×Dic5—22no—
49C2×C5×Dic5C2312yes—
50C2×C5×Dic5C2312yes—
51C2×C5×Dic5C2312yes—
52#6038—28no—
53D5×Dic5—28no—
54C22×C5×D4—51no—
55C22×C22⋊D5—51no—
56C22×C22⋊D5—51no—
57C22×C22⋊D5—51no—
58C22×C22⋊D5—102no—
59C22×C22⋊D5D511yescharacteristic, complemented
60C23×Dic5—51no—
61C22×D20—51no—
62C22×C4×D5—51no—
63C2×C5×C22⋊C4—51no—
64C2×C22⋊Dic5D511yescharacteristic, complemented
65#4085—58no—
66C2×#685—51no—
67C2×#685—51no—
68C2×#683—51no—
69C2×#683—51no—
70#4030—58no—
71C22×C52—22no—
72C22×C52—22no—
73C22×C52—22no—
74C22×C52—44no—
75C22×C52C2×D412yes—
76C22×C52C2×D412yes—
77C22×C52#4112yes—
78C22×C52C2411yescharacteristic
79C2×C5⋊D5—22no—
80C2×C5⋊D5—22no—
81C2×C5⋊D5—22no—
82C2×C5⋊D5—44no—
83C2×C5×D5—28no—
84C2×C5×D5—22no—
85C2×C5×D5—22no—
86C2×C5×D5C2×D414yescomplemented
87#1039C2×D414yescomplemented
88C5×Dic5—22no—
89C5×Dic5—22no—
90C5×Dic5—22no—
91C24×C5—22no—
92C24×C5—51no—
93C24×C5C2×D511yescharacteristic
94C24×C5C2×D511yescharacteristic
95C23×D5—51no—
96C23×D5—51no—
97C23×D5—102no—
98C23×D5C2×D511yescharacteristic
99C2×C5×D4—54no—
100C2×C5×D4—104no—
101C22×C4×C5—51no—
102C22×C4×C5—51no—
103C22×C4×C5—51no—
104C2×C22⋊D5—24no—
105C2×C22⋊D5—54no—
106C2×C22⋊D5—54no—
107C2×C22⋊D5—54no—
108C2×C22⋊D5—108no—
109C2×C22⋊D5—104no—
110C2×C22⋊D5—104no—
111C2×C22⋊D5—104no—
112C2×C22⋊D5—1016no—
113C2×C22⋊D5C2×D514yescomplemented
114C22×Dic5—51no—
115C22×Dic5—51no—
116C22×Dic5—54no—
117C22×Dic5—102no—
118C22×Dic5—104no—
119C22×Dic5C2×D511yescharacteristic
120C22×Dic5C2×D511yescharacteristic
121C22×Dic5C2×D511yescharacteristic
122C2×D20—54no—
123C2×D20—104no—
124C2×C4×D5—54no—
125C2×C4×D5—104no—
126C5×C22⋊C4—54no—
127C22⋊Dic5C2×D514yescomplemented
128#685—54no—
129#685—54no—
130#683—54no—
131#683—54no—
132C2×#121—251noSylow
133C2×C52—42no—
134C2×C52#12114yescomplemented
135C2×C52C22×D411yescharacteristic
136C2×C52C22×D411yescharacteristic
137C2×C52C2×#4111yescharacteristic
138C5⋊D5—42no—
139C5×D5—24no—
140C23×C5—28no—
141C23×C5—22no—
142C23×C5—22no—
143C23×C5—22no—
144C23×C5—22no—
145C23×C5—22no—
146C23×C5—22no—
147C23×C5—24no—
148C23×C5—22no—
149C23×C5—44no—
150C23×C5—44no—
151C23×C5—54no—
152C23×C5—52no—
153C23×C5—102no—
154C23×C5—102no—
155C23×C5C22×D512yes—
156C23×C5C22×D512yes—
157C23×C5C22×D511yescharacteristic
158C23×C5C22×D511yescharacteristic
159C22×D5—22no—
160C22×D5—22no—
161C22×D5—52no—
162C22×D5—52no—
163C22×D5—104no—
164C22×D5—108no—
165C22×D5—102no—
166C22×D5—102no—
167C22×D5—102no—
168C22×D5—102no—
169C22×D5—102no—
170C22×D5—102no—
171C22×D5—204no—
172C22×D5—204no—
173C22×D5C22⋊D514yescomplemented
174C22×D5C22×D512yes—
175C5×D4—108no—
176C2×C4×C5—52no—
177C2×C4×C5—52no—
178C2×C4×C5—52no—
179C2×C4×C5—102no—
180C2×C4×C5—102no—
181C2×C4×C5—102no—
182C22⋊D5—28no—
183C22⋊D5—108no—
184C22⋊D5—108no—
185C22⋊D5—108no—
186C22⋊D5—1032no—
187C2×Dic5—22no—
188C2×Dic5—22no—
189C2×Dic5—22no—
190C2×Dic5—54no—
191C2×Dic5—54no—
192C2×Dic5—52no—
193C2×Dic5—52no—
194C2×Dic5—108no—
195C2×Dic5—104no—
196C2×Dic5—108no—
197C2×Dic5C22×D512yes—
198C2×Dic5C22×D512yes—
199C2×Dic5C22×D512yes—
200D20—108no—
201C4×D5—108no—
202C22×D4—51no—
203C22×D4—251no—
204C22×D4—251no—
205C23×C4—251no—
206#121—258no—
207C2×C4⋊C4—251no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4—251no—
210C52C2×#12111yescharacteristic, complemented, Sylow
211C22×C5—28no—
212C22×C5—22no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—24no—
219C22×C5—24no—
220C22×C5—24no—
221C22×C5—22no—
222C22×C5—44no—
223C22×C5—44no—
224C22×C5—416no—
225C22×C5—44no—
226C22×C5—44no—
227C22×C5—54no—
228C22×C5—102no—
229C22×C5—102no—
230C22×C5—108no—
231C22×C5#70912yes—
232C22×C5D4×D512yes—
233C22×C5D4×D512yes—
234C22×C5D4×D512yes—
235C22×C5#71112yes—
236C22×C5C2×C22⋊D512yes—
237C22×C5C23×D511yescharacteristic
238C22×C5C23×D511yescharacteristic
239C2×D5—22no—
240C2×D5—22no—
241C2×D5—28no—
242C2×D5—108no—
243C2×D5—102no—
244C2×D5—102no—
245C2×D5—102no—
246C2×D5—102no—
247C2×D5—102no—
248C2×D5—102no—
249C2×D5—204no—
250C2×D5—204no—
251C2×D5—2016no—
252C2×D5—204no—
253C2×D5—204no—
254C2×D5C2×C22⋊D514yescomplemented
255C4×C5—102no—
256C4×C5—102no—
257C4×C5—102no—
258Dic5—22no—
259Dic5—22no—
260Dic5—22no—
261Dic5—54no—
262Dic5—54no—
263Dic5—108no—
264C24—51no—
265C24—251no—
266C24D5211yescharacteristic
267C2×D4—54no—
268C2×D4—104no—
269C2×D4—254no—
270C2×D4—254no—
271C2×D4—504no—
272C2×D4—504no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—51no—
276C22×C4—254no—
277C22×C4—251no—
278C22×C4—504no—
279C4⋊C4—254no—
280C22⋊C4—54no—
281C22⋊C4—254no—
282C2×C5—22no—
283C2×C5—22no—
284C2×C5—22no—
285C2×C5—28no—
286C2×C5—48no—
287C2×C5—42no—
288C2×C5—42no—
289C2×C5—104no—
290C2×C5#403014yescomplemented
291C2×C5#408514yescomplemented
292C2×C5C2×#70911yescharacteristic
293C2×C5C2×D4×D511yescharacteristic
294C2×C5C2×D4×D511yescharacteristic
295C2×C5C2×D4×D511yescharacteristic
296C2×C5C2×#71111yescharacteristic
297C2×C5C22×C22⋊D511yescharacteristic
298D5—24no—
299D5—208no—
300D5—202no—
301D5—202no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—54no—
306C23—52no—
307C23—102no—
308C23—102no—
309C23—252no—
310C23—502no—
311C23—502no—
312C23—502no—
313C23C2×D5212yes—
314C23C2×D5211yescharacteristic
315D4—108no—
316D4—508no—
317D4—508no—
318C2×C4—52no—
319C2×C4—52no—
320C2×C4—52no—
321C2×C4—102no—
322C2×C4—102no—
323C2×C4—102no—
324C2×C4—254no—
325C2×C4—252no—
326C2×C4—508no—
327C5—22no—
328C5C2×#403011yescharacteristic, complemented
329C5C2×#408511yescharacteristic, complemented
330C22—22no—
331C22—22no—
332C22—22no—
333C22—44no—
334C22—54no—
335C22—102no—
336C22—102no—
337C22—108no—
338C22—502no—
339C22—502no—
340C22—502no—
341C22—1004no—
342C22#8787912yes—
343C22D5×C22⋊D512yes—
344C22#8788612yes—
345C22C22×D5211yescharacteristic
346C4—102no—
347C4—102no—
348C4—102no—
349C4—254no—
350C2—42no—
351C2—104no—
352C2—1002no—
353C2#1171708914yesdirect product factor
354C2C2×#8787911yescharacteristic
355C2C2×D5×C22⋊D511yescharacteristic
356C2C2×#8788611yescharacteristic
subgroupG/H
centerC23C2×D52
commutator, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC24×C52C22
socleC23×C52C23
hypercenterC24D52
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C22×C5×C22⋊D5C2semidirect
3. C22×#6038C2semidirect
4. C22×D5×Dic5C2semidirect
5. C2×C5×C22⋊Dic5C2semidirect
6. #11717089C2direct (N × H)
7. C2×#87781C2semidirect
8. C2×#87779C2semidirect
13. C2×C22⋊(C5⋊D5)C22semidirect
16. C2×C5×C22⋊D5C22semidirect
21. C2×#6038C22semidirect
23. C2×D5×Dic5C22semidirect
24. C5×C22⋊Dic5C22semidirect
25. #87781C22semidirect
26. #87779C22semidirect
40. C22×C5×D5D4semidirect
43. C2×#1039D4semidirect
59. C22×C22⋊D5D5semidirect
64. C2×C22⋊Dic5D5semidirect
86. C2×C5×D5C2×D4semidirect
87. #1039C2×D4semidirect
113. C2×C22⋊D5C2×D5semidirect
127. C22⋊Dic5C2×D5semidirect
134. C2×C52#121semidirect
173. C22×D5C22⋊D5semidirect
210. C52C2×#121semidirect
254. C2×D5C2×C22⋊D5semidirect
290. C2×C5#4030semidirect
291. C2×C5#4085semidirect
328. C5C2×#4030semidirect
329. C5C2×#4085semidirect
353. C2#11717089direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊(C5⋊D5)2
C22×C5×C22⋊D52
C22×#60382
C22×D5×Dic52
C2×C5×C22⋊Dic52
#117170892
C2×#877812
C2×#877792
C2×#40855
C2×#40305