Finite Group Databasebeta

← #13105236

Subgroups — #13105236

Order 1600.

762 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 142 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×#6037C211yesmaximal, characteristic, complemented
2C2×#87873C211yesmaximal, characteristic, complemented
3#11717201C211yesmaximal, characteristic, complemented
4C5×D5×C22⋊C4C212yesmaximal, complemented
5#11717100C211yesmaximal, characteristic, complemented
6#11717095C212yesmaximal, complemented
7#11717033C211yesmaximal, characteristic, complemented
8#11717030C211yesmaximal, characteristic, complemented
9#11717024C212yesmaximal, complemented
10#11717023C212yesmaximal, complemented
11C4×#6037C211yesmaximal, characteristic, complemented
12C23×C5×D5C2212yes—
13C2×C22⋊(C5⋊D5)C2211yescharacteristic
14C22×#1039C2211yescharacteristic
15C2×C4×C5⋊D5C2211yescharacteristic, complemented
16C2×C4×C5×D5C2212yescomplemented
17C2×C4×C5×D5C2212yescomplemented
18C2×#6039C2211yescharacteristic, complemented
19C2×#6038C2212yescomplemented
20C2×#6038C2212yescomplemented
21C2×#6037—24no—
22C2×#6037C2212yes—
23C2×#6037C2211yescharacteristic
24C2×#6037C2211yescharacteristic
25#87873—24no—
26C52×C22⋊C4C2211yescharacteristic
27#87808C2211yescharacteristic, complemented
28#87806C2211yescharacteristic, complemented
29C4×#1039C2211yescharacteristic, complemented
30C5×C22⋊Dic5C2212yes—
31C5×#685C2212yescomplemented
32C5×#685C2212yescomplemented
33#87781C2211yescharacteristic, complemented
34#87780C2212yescomplemented
35#87778C2212yescomplemented
36#87778C2212yescomplemented
37#65824—52nomaximal
38C23×C52C2311yescharacteristic
39C22×C5⋊D5C2311yescharacteristic
40C22×C5×D5—24no—
41C22×C5×D5—24no—
42C22×C5×D5—42no—
43C22×C5×D5C2312yes—
44C22×C5×D5C2312yes—
45C2×C4×C52C2311yescharacteristic
46C2×C4×C52C2311yescharacteristic
47C22⋊(C5⋊D5)—22no—
48C2×#1039—22no—
49C2×#1039C2311yescharacteristic
50C2×#1039C2311yescharacteristic
51C2×#1039C2311yescharacteristic
52C4×C5⋊D5—22no—
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2312yes—
55C4×C5×D5—24no—
56C4×C5×D5—42no—
57#6039—22no—
58#6038—24no—
59#6038—42no—
60#6037—24no—
61#6037—22no—
62#6037D414yescomplemented
63C22×C22⋊D5—52no—
64C2×#709—52no—
65C2×C5×C22⋊C4—52no—
66#4099—52no—
67#4076—52no—
68#4075—52no—
69C4×C22⋊D5—52no—
70#4031—52no—
71#4030—52no—
72#4030—52no—
73#4029—52no—
74#4028—52no—
75#4027—52no—
76#4027—102no—
77D5×C22⋊C4D512yescomplemented
78#4024—52no—
79C22×C52—21no—
80C22×C52—21no—
81C22×C52#4112yescomplemented
82C22×C52C2411yescharacteristic
83C2×C5⋊D5—21no—
84C2×C5⋊D5—21no—
85C2×C5⋊D5—21no—
86C2×C5×D5—22no—
87C2×C5×D5—22no—
88C2×C5×D5—22no—
89C2×C5×D5—22no—
90C2×C5×D5—22no—
91C2×C5×D5—48no—
92C2×C5×D5C2×D414yes—
93C4×C52—21no—
94C4×C52—21no—
95#1039—21no—
96#1039—21no—
97#1039C2×D412yes—
98C5×Dic5—22no—
99C5×Dic5—22no—
100C24×C5—52no—
101C23×D5C2×D512yes—
102C22×C4×C5—52no—
103C22×C4×C5—52no—
104C2×C22⋊D5—52no—
105C2×C22⋊D5—52no—
106C2×C22⋊D5—52no—
107C2×C22⋊D5—52no—
108C2×C22⋊D5—52no—
109C2×C22⋊D5—52no—
110C2×C22⋊D5—52no—
111C2×C22⋊D5—102no—
112C2×C22⋊D5—108no—
113C22×Dic5—52no—
114C22×Dic5—102no—
115#709—108no—
116C2×D20—52no—
117C2×D20—52no—
118C2×C4×D5—52no—
119C2×C4×D5—102no—
120C2×C4×D5C2×D512yescomplemented
121C2×C4×D5C2×D512yescomplemented
122C2×Dic10—52no—
123C5×C22⋊C4—22no—
124C5×C22⋊C4—52no—
125C5×C22⋊C4—52no—
126C5×C22⋊C4—52no—
127C5×C22⋊C4C2×D512yes—
128C22⋊Dic5—52no—
129C22⋊Dic5—52no—
130C22⋊Dic5C2×D512yes—
131#685—52no—
132#685—52no—
133#685—52no—
134#685—102no—
135#685C2×D512yescomplemented
136#685C2×D512yescomplemented
137C4⋊Dic5—52no—
138#683—52no—
139#683—52no—
140#683—52no—
141#683—102no—
142C4×Dic5—52no—
143C4×Dic5—102no—
144#546—251noSylow
145C2×C52—22no—
146C2×C52C22×D411yescharacteristic
147C2×C52C2×#4111yescharacteristic
148C2×C52H(5,𝔽2)11yescharacteristic
149C5⋊D5—41no—
150C5×D5—24no—
151C5×D5—42no—
152C23×C5—22no—
153C23×C5—52no—
154C23×C5—52no—
155C23×C5—104no—
156C23×C5—104no—
157C23×C5—202no—
158C23×C5C22×D512yes—
159C22×D5—24no—
160C22×D5—24no—
161C22×D5—42no—
162C22×D5—52no—
163C22×D5—102no—
164C22×D5C22×D512yes—
165C22×D5C22×D512yes—
166C2×C4×C5—22no—
167C2×C4×C5—22no—
168C2×C4×C5—52no—
169C2×C4×C5—52no—
170C2×C4×C5—104no—
171C2×C4×C5—202no—
172C2×C4×C5C22×D512yes—
173C2×C4×C5C22×D512yes—
174C22⋊D5—58no—
175C22⋊D5—104no—
176C22⋊D5—104no—
177C22⋊D5—104no—
178C22⋊D5—108no—
179C22⋊D5—104no—
180C22⋊D5—104no—
181C22⋊D5—202no—
182C2×Dic5—52no—
183C2×Dic5—52no—
184C2×Dic5—52no—
185C2×Dic5—102no—
186C2×Dic5—102no—
187C2×Dic5—102no—
188C2×Dic5—104no—
189C2×Dic5—204no—
190C2×Dic5C22×D512yes—
191C2×Dic5C22×D512yes—
192D20—104no—
193D20—202no—
194C4×D5—24no—
195C4×D5—42no—
196C4×D5—104no—
197C4×D5—204no—
198Dic10—104no—
199C2×#41—251no—
200C22×D4—251no—
201#124—251no—
202#123—252no—
203C22⋊Q8—251no—
204#121—251no—
205#121—252no—
206#120—252no—
207C4×D4—251no—
208C4×D4—251no—
209C2×C22⋊C4—52no—
210C52#54611yescharacteristic, complemented, Sylow
211C22×C5—22no—
212C22×C5—22no—
213C22×C5—24no—
214C22×C5—22no—
215C22×C5—42no—
216C22×C5—42no—
217C22×C5—54no—
218C22×C5—102no—
219C22×C5—102no—
220C22×C5—102no—
221C22×C5—102no—
222C22×C5—102no—
223C22×C5—208no—
224C22×C5#70914yescomplemented
225C22×C5C23×D512yes—
226C2×D5—22no—
227C2×D5—22no—
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—22no—
231C2×D5—48no—
232C2×D5—104no—
233C2×D5—102no—
234C2×D5—102no—
235C2×D5—102no—
236C2×D5—202no—
237C2×D5—202no—
238C2×D5D4×D514yes—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—42no—
242C4×C5—42no—
243C4×C5—102no—
244C4×C5—102no—
245Dic5—22no—
246Dic5—22no—
247Dic5—54no—
248Dic5—104no—
249Dic5—104no—
250Dic5—102no—
251Dic5—102no—
252Dic5—202no—
253C24—52no—
254#41—504no—
255C2×Q8—251no—
256C2×D4—252no—
257C2×D4—252no—
258C2×D4—251no—
259C2×D4—252no—
260C2×D4—251no—
261C2×D4—251no—
262C2×D4—504no—
263C22×C4—52no—
264C22×C4—52no—
265C22×C4—251no—
266C22×C4—251no—
267C4⋊C4—251no—
268C4⋊C4—252no—
269C4⋊C4—251no—
270C22⋊C4—52no—
271C22⋊C4—52no—
272C22⋊C4—52no—
273C22⋊C4—252no—
274C22⋊C4—252no—
275C22⋊C4—251no—
276C22⋊C4D5211yescharacteristic
277C42—251no—
278C2×C5—24no—
279C2×C5—22no—
280C2×C5—22no—
281C2×C5—22no—
282C2×C5—44no—
283C2×C5—104no—
284C2×C5—202no—
285C2×C5C2×#70912yes—
286C2×C5C2×D4×D512yes—
287C2×C5#414412yes—
288D5—24no—
289D5—42no—
290D5—204no—
291D5—202no—
292C23—52no—
293C23—52no—
294C23—104no—
295C23—104no—
296C23—202no—
297C23—251no—
298C23C2×D5211yescharacteristic
299Q8—502no—
300D4—254no—
301D4—504no—
302D4—502no—
303D4—504no—
304D4—502no—
305D4—1002no—
306C2×C4—52no—
307C2×C4—52no—
308C2×C4—104no—
309C2×C4—202no—
310C2×C4—251no—
311C2×C4—251no—
312C2×C4—251no—
313C2×C4—502no—
314C2×C4—502no—
315C2×C4C2×D5211yescharacteristic
316C2×C4C2×D5211yescharacteristic
317C5—22no—
318C5#6582412yescomplemented
319C22—21no—
320C22—21no—
321C22—54no—
322C22—102no—
323C22—102no—
324C22—102no—
325C22—102no—
326C22—102no—
327C22—208no—
328C22—501no—
329C22—501no—
330C22—501no—
331C22#8787312yescomplemented
332C22C22×D5211yescharacteristic
333C4—21no—
334C4—21no—
335C4—102no—
336C4—102no—
337C4—252no—
338C4—501no—
339C4—501no—
340C2—22no—
341C2—104no—
342C2—202no—
343C2—1001no—
344C2C2×#8787311yescharacteristic
345C2D4×D5211yescharacteristic
346C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6037C2semidirect
2. C2×#87873C2semidirect
3. #11717201C2semidirect
4. C5×D5×C22⋊C4C2semidirect
5. #11717100C2semidirect
6. #11717095C2semidirect
7. #11717033C2semidirect
8. #11717030C2semidirect
9. #11717024C2semidirect
10. #11717023C2semidirect
11. C4×#6037C2semidirect
15. C2×C4×C5⋊D5C22semidirect
16. C2×C4×C5×D5C22semidirect
17. C2×C4×C5×D5C22semidirect
18. C2×#6039C22semidirect
19. C2×#6038C22semidirect
20. C2×#6038C22semidirect
27. #87808C22semidirect
28. #87806C22semidirect
29. C4×#1039C22semidirect
31. C5×#685C22semidirect
32. C5×#685C22semidirect
33. #87781C22semidirect
34. #87780C22semidirect
35. #87778C22semidirect
36. #87778C22semidirect
62. #6037D4semidirect
77. D5×C22⋊C4D5semidirect
81. C22×C52#41semidirect
120. C2×C4×D5C2×D5semidirect
121. C2×C4×D5C2×D5semidirect
135. #685C2×D5semidirect
136. #685C2×D5semidirect
210. C52#546semidirect
224. C22×C5#709semidirect
318. C5#65824semidirect
331. C22#87873semidirect

Maximal subgroups

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

subgroupindex
C22×#60372
C2×#878732
#117172012
C5×D5×C22⋊C42
#117171002
#117170952
#117170332
#117170302
#117170242
#117170232
C4×#60372
#658245