Finite Group Databasebeta

← C2×#11717005 #13106072

Subgroups — C2×#11717005 #13106072

Order 1600.

984 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 192 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×C5×D20C211yesmaximal, characteristic, complemented
2C22×#6038C211yesmaximal, characteristic, complemented
3C22×#6038C211yesmaximal, characteristic, complemented
4C22×D5×Dic5C211yesmaximal, characteristic, complemented
5C2×#87808C211yesmaximal, characteristic, complemented
6C2×C5×#683C211yesmaximal, characteristic, complemented
7C2×#87778C211yesmaximal, characteristic, complemented
8#11717005C218yesmaximal, direct product factor
9C23×C5⋊D5C2211yescharacteristic
10C23×C5×D5C2211yescharacteristic
11C23×C5×D5C2211yescharacteristic
12C22×C4×C52C2211yescharacteristic
13C22×#1039C2211yescharacteristic
14C22×C5×Dic5C2211yescharacteristic
15C22×C5×Dic5C2211yescharacteristic
16C2×C5×D20—24no—
17C2×C5×D20C2214yescomplemented
18C2×#6038—24no—
19C2×#6038—24no—
20C2×#6038C2214yescomplemented
21C2×#6038C2214yescomplemented
22C2×D5×Dic5—24no—
23C2×D5×Dic5C2214yescomplemented
24#87808C2214yescomplemented
25C5×#683C2214yescomplemented
26#87778C2214yescomplemented
27C2×#4085—51nomaximal
28C2×#4041—51nomaximal
29C23×C52C2311yescharacteristic
30C22×C5⋊D5—22no—
31C22×C5⋊D5—22no—
32C22×C5⋊D5—22no—
33C22×C5⋊D5C2312yes—
34C22×C5×D5—22no—
35C22×C5×D5—22no—
36C22×C5×D5—22no—
37C22×C5×D5—22no—
38C22×C5×D5—22no—
39C22×C5×D5D414yescomplemented
40C22×C5×D5C2312yes—
41C22×C5×D5C2312yes—
42C2×C4×C52—22no—
43C2×C4×C52C2312yes—
44C2×#1039—22no—
45C2×#1039C2312yes—
46C2×C5×Dic5—22no—
47C2×C5×Dic5D414yescomplemented
48C2×C5×Dic5C2312yes—
49C2×C5×Dic5C2312yes—
50C5×D20—28no—
51#6038—28no—
52#6038—28no—
53D5×Dic5—28no—
54C22×C5×D4—51no—
55C22×C22⋊D5—51no—
56C22×C22⋊D5—51no—
57C23×Dic5—51no—
58C22×D20—51no—
59C22×D20—51no—
60C22×D20D511yescharacteristic, complemented
61C22×C4×D5—51no—
62C2×C5×C4⋊C4—51no—
63C2×C22⋊Dic5—51no—
64#4085—58no—
65C2×#685—51no—
66C2×#685—51no—
67C2×#685—51no—
68C2×#685—102no—
69C2×#683D511yescharacteristic, complemented
70#4041—58no—
71C22×C52C2×D412yes—
72C22×C52C2×D412yes—
73C22×C52#4112yes—
74C22×C52C2411yescharacteristic
75C2×C5⋊D5—22no—
76C2×C5⋊D5—22no—
77C2×C5⋊D5—22no—
78C2×C5⋊D5—44no—
79C2×C5×D5—28no—
80C2×C5×D5—22no—
81C2×C5×D5—22no—
82C2×C5×D5—22no—
83C2×C5×D5—22no—
84C2×C5×D5—22no—
85C2×C5×D5—44no—
86C2×C5×D5C2×D414yescomplemented
87C4×C52—22no—
88#1039—22no—
89C5×Dic5—22no—
90C5×Dic5C2×D414yescomplemented
91C24×C5—51no—
92C24×C5—51no—
93C23×D5—51no—
94C23×D5—51no—
95C23×D5—102no—
96C23×D5C2×D511yescharacteristic
97C23×D5C2×D511yescharacteristic
98C2×C5×D4—54no—
99C2×C5×D4—104no—
100C22×C4×C5—22no—
101C22×C4×C5—51no—
102C22×C4×C5—51no—
103C22×C4×C5C2×D511yescharacteristic
104C22×C4×C5C2×D511yescharacteristic
105C2×C22⋊D5—54no—
106C2×C22⋊D5—54no—
107C2×C22⋊D5—104no—
108C2×C22⋊D5—104no—
109C22×Dic5—51no—
110C22×Dic5—51no—
111C22×Dic5—54no—
112C22×Dic5—102no—
113C22×Dic5—104no—
114C22×Dic5C2×D511yescharacteristic
115C22×Dic5C2×D511yescharacteristic
116C2×D20—24no—
117C2×D20—54no—
118C2×D20—54no—
119C2×D20—104no—
120C2×D20—104no—
121C2×D20C2×D514yescomplemented
122C2×C4×D5—54no—
123C2×C4×D5—104no—
124C5×C4⋊C4—54no—
125C22⋊Dic5—54no—
126#685—54no—
127#685—54no—
128#685—54no—
129#685—108no—
130#683C2×D514yescomplemented
131C2×#121—251noSylow
132C2×C52#12114yescomplemented
133C2×C52C22×D411yescharacteristic
134C2×C52C22×D411yescharacteristic
135C2×C52C2×#4111yescharacteristic
136C5⋊D5—42no—
137C5×D5—24no—
138C5×D5—42no—
139C23×C5—22no—
140C23×C5—54no—
141C23×C5—52no—
142C23×C5—52no—
143C23×C5—102no—
144C23×C5—102no—
145C23×C5—102no—
146C23×C5—102no—
147C23×C5—102no—
148C23×C5C22×D511yescharacteristic
149C23×C5C22×D511yescharacteristic
150C22×D5—22no—
151C22×D5—22no—
152C22×D5—22no—
153C22×D5—22no—
154C22×D5—22no—
155C22×D5—52no—
156C22×D5—52no—
157C22×D5—104no—
158C22×D5—108no—
159C22×D5—102no—
160C22×D5—102no—
161C22×D5—102no—
162C22×D5—102no—
163C22×D5—102no—
164C22×D5—102no—
165C22×D5—204no—
166C22×D5—204no—
167C22×D5C22⋊D514yescomplemented
168C22×D5C22×D512yes—
169C22×D5C22×D512yes—
170C5×D4—108no—
171C2×C4×C5—22no—
172C2×C4×C5—22no—
173C2×C4×C5—24no—
174C2×C4×C5—44no—
175C2×C4×C5—54no—
176C2×C4×C5—52no—
177C2×C4×C5—52no—
178C2×C4×C5—102no—
179C2×C4×C5C22×D512yes—
180C2×C4×C5C22×D512yes—
181C22⋊D5—108no—
182C22⋊D5—108no—
183C2×Dic5—22no—
184C2×Dic5—52no—
185C2×Dic5—52no—
186C2×Dic5—104no—
187C2×Dic5—102no—
188C2×Dic5—102no—
189C2×Dic5—108no—
190C2×Dic5—204no—
191C2×Dic5D2014yescomplemented
192C2×Dic5C22×D512yes—
193C2×Dic5C22×D512yes—
194D20—28no—
195D20—108no—
196D20—108no—
197C4×D5—108no—
198C22×D4—51no—
199C22×D4—251no—
200C22×D4—251no—
201C23×C4—251no—
202#121—258no—
203C2×C4⋊C4—51no—
204C2×C22⋊C4—251no—
205C2×C22⋊C4—251no—
206C52C2×#12111yescharacteristic, complemented, Sylow
207C22×C5—24no—
208C22×C5—24no—
209C22×C5—24no—
210C22×C5—22no—
211C22×C5—54no—
212C22×C5—102no—
213C22×C5—102no—
214C22×C5—108no—
215C22×C5—102no—
216C22×C5—102no—
217C22×C5—102no—
218C22×C5—204no—
219C22×C5C2×D2012yes—
220C22×C5D4×D512yes—
221C22×C5D4×D512yes—
222C22×C5#71112yes—
223C22×C5#71312yes—
224C22×C5C2×C22⋊D512yes—
225C22×C5C23×D511yescharacteristic
226C22×C5C23×D511yescharacteristic
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—28no—
230C2×D5—22no—
231C2×D5—22no—
232C2×D5—22no—
233C2×D5—44no—
234C2×D5—108no—
235C2×D5—102no—
236C2×D5—102no—
237C2×D5—102no—
238C2×D5—102no—
239C2×D5—102no—
240C2×D5—102no—
241C2×D5—204no—
242C2×D5—204no—
243C2×D5—2016no—
244C2×D5—204no—
245C2×D5—204no—
246C2×D5C2×C22⋊D514yescomplemented
247C4×C5—22no—
248C4×C5—22no—
249C4×C5—44no—
250C4×C5—54no—
251C4×C5—102no—
252Dic5—22no—
253Dic5—102no—
254Dic5—102no—
255Dic5—204no—
256Dic5C2×D2014yescomplemented
257C24—51no—
258C24—51no—
259C24—251no—
260C2×D4—54no—
261C2×D4—104no—
262C2×D4—254no—
263C2×D4—254no—
264C2×D4—504no—
265C2×D4—504no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—254no—
269C22×C4—251no—
270C22×C4—504no—
271C22×C4D5211yescharacteristic
272C4⋊C4—54no—
273C22⋊C4—254no—
274C22⋊C4—254no—
275C2×C5—22no—
276C2×C5—22no—
277C2×C5—22no—
278C2×C5—28no—
279C2×C5—104no—
280C2×C5—202no—
281C2×C5#404114yescomplemented
282C2×C5#408514yescomplemented
283C2×C5C22×D2011yescharacteristic
284C2×C5C2×D4×D511yescharacteristic
285C2×C5C2×D4×D511yescharacteristic
286C2×C5C2×#71111yescharacteristic
287C2×C5C2×#71311yescharacteristic
288C2×C5C22×C22⋊D511yescharacteristic
289D5—24no—
290D5—42no—
291D5—208no—
292D5—202no—
293D5—202no—
294C23—54no—
295C23—52no—
296C23—52no—
297C23—102no—
298C23—102no—
299C23—102no—
300C23—102no—
301C23—102no—
302C23—252no—
303C23—502no—
304C23—502no—
305C23—502no—
306C23C2×D5211yescharacteristic
307D4—108no—
308D4—508no—
309D4—508no—
310C2×C4—22no—
311C2×C4—54no—
312C2×C4—52no—
313C2×C4—52no—
314C2×C4—102no—
315C2×C4—252no—
316C2×C4—508no—
317C2×C4—502no—
318C2×C4C2×D5212yes—
319C5—22no—
320C5C2×#404111yescharacteristic, complemented
321C5C2×#408511yescharacteristic, complemented
322C22—54no—
323C22—102no—
324C22—102no—
325C22—108no—
326C22—102no—
327C22—102no—
328C22—102no—
329C22—204no—
330C22—502no—
331C22—502no—
332C22—502no—
333C22—1004no—
334C22#8787112yes—
335C22D5×D2012yes—
336C22D5×C22⋊D512yes—
337C22C22×D5211yescharacteristic
338C4—22no—
339C4—54no—
340C4—102no—
341C4—502no—
342C2—104no—
343C2—202no—
344C2—1002no—
345C2#1171700514yesdirect product factor
346C2C2×#8787111yescharacteristic
347C2C2×D5×D2011yescharacteristic
348C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
centerC23C2×D52
commutator, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
socleC23×C52C23
hypercenterC22×C4D52
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×D20C2semidirect
2. C22×#6038C2semidirect
3. C22×#6038C2semidirect
4. C22×D5×Dic5C2semidirect
5. C2×#87808C2semidirect
6. C2×C5×#683C2semidirect
7. C2×#87778C2semidirect
8. #11717005C2direct (N × H)
17. C2×C5×D20C22semidirect
20. C2×#6038C22semidirect
21. C2×#6038C22semidirect
23. C2×D5×Dic5C22semidirect
24. #87808C22semidirect
25. C5×#683C22semidirect
26. #87778C22semidirect
39. C22×C5×D5D4semidirect
47. C2×C5×Dic5D4semidirect
60. C22×D20D5semidirect
69. C2×#683D5semidirect
86. C2×C5×D5C2×D4semidirect
90. C5×Dic5C2×D4semidirect
121. C2×D20C2×D5semidirect
130. #683C2×D5semidirect
132. C2×C52#121semidirect
167. C22×D5C22⋊D5semidirect
191. C2×Dic5D20semidirect
206. C52C2×#121semidirect
246. C2×D5C2×C22⋊D5semidirect
256. Dic5C2×D20semidirect
281. C2×C5#4041semidirect
282. C2×C5#4085semidirect
320. C5C2×#4041semidirect
321. C5C2×#4085semidirect
345. C2#11717005direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D202
C22×#60382
C22×D5×Dic52
C2×#878082
C2×C5×#6832
C2×#877782
#117170052
C2×#40855
C2×#40415