Finite Group Databasebeta

← #10614524

Subgroups — #10614524

Order 640.

592 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4092C211yesmaximal, characteristic, complemented
3C2×#4070C211yesmaximal, characteristic, complemented
4C2×#4042C211yesmaximal, characteristic, complemented
5#65518C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic
7#65283C211yesmaximal, characteristic, complemented
8#65283C211yesmaximal, characteristic, complemented
9#65274C211yesmaximal, characteristic, complemented
10#65268C211yesmaximal, characteristic, complemented
11#65261C211yesmaximal, characteristic, complemented
12#65246C211yesmaximal, characteristic
13#65245C211yesmaximal, characteristic
14#65237C211yesmaximal, characteristic, complemented
15#65237C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C5×C22⋊Q8—24no—
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30#4092—24no—
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×C4⋊Dic5C2211yescharacteristic
38C2×C4⋊Dic5C2211yescharacteristic
39#4070—24no—
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×#683C2211yescharacteristic
43C2×#683C2211yescharacteristic
44C2×#683C2211yescharacteristic
45C2×#683C2211yescharacteristic
46C2×#683C2211yescharacteristic
47C2×C4×Dic5C2211yescharacteristic
48#4042—24no—
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2683—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5C2311yescharacteristic
58C2×C5×Q8—22no—
59C2×C5×Q8—22no—
60C2×C5×Q8—41no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—41no—
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C2×C22⋊D5—41no—
71C2×C22⋊D5—41no—
72C2×C22⋊D5Q814yescomplemented
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C2×C4×D5—22no—
81C2×C4×D5—22no—
82C2×C4×D5—41no—
83C2×Dic10—22no—
84C2×Dic10—22no—
85C2×Dic10—41no—
86C5×C4⋊C4—22no—
87C5×C4⋊C4—22no—
88C5×C4⋊C4D414yescomplemented
89C5×C22⋊C4—22no—
90C5×C22⋊C4—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—41no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—22no—
98#685—41no—
99C4⋊Dic5—22no—
100C4⋊Dic5—22no—
101C4⋊Dic5—41no—
102#683—22no—
103#683—22no—
104#683—22no—
105#683—22no—
106#683—41no—
107#683—41no—
108#683D414yescomplemented
109C4×Dic5—41no—
110C2×C22⋊Q8—51no—
111C2×C22⋊Q8—51no—
112C2×C22⋊Q8—51no—
113C2×C22⋊Q8D511yescharacteristic, complemented
114#396—51no—
115#396—51no—
116#396—51no—
117#396—51no—
118#395—51no—
119#395—51no—
120#390—51no—
121#389—51no—
122#389—51no—
123#386—51no—
124#386—51no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5C2×Q812yes—
132C23×C5C2411yescharacteristic
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5C2×Q812yes—
140C5×Q8—44no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—44no—
156C2×C4×C5C2×D412yes—
157C2×C4×C5C2×D412yes—
158C2×C4×C5C2×D412yes—
159C2×C4×C5C2×D412yes—
160C22⋊D5—44no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5C2×D412yes—
180C2×Dic5C2×D412yes—
181C2×Dic5C2×Q812yes—
182C4×D5—44no—
183Dic10—44no—
184C22×Q8—51no—
185C22×Q8C2×D511yescharacteristic
186C22×D4—51no—
187C23×C4—51no—
188C23×C4C2×D511yescharacteristic
189C22⋊Q8—24no—
190C22⋊Q8—104no—
191C22⋊Q8—104no—
192C22⋊Q8—104no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4—51no—
197C2×C4⋊C4—51no—
198C2×C4⋊C4—51no—
199C2×C4⋊C4—51no—
200C2×C4⋊C4—51no—
201C2×C4⋊C4—51no—
202C2×C4⋊C4—51no—
203C2×C4⋊C4C2×D511yescharacteristic
204C2×C4⋊C4C2×D511yescharacteristic
205C2×C4⋊C4C2×D511yescharacteristic
206C2×C22⋊C4—51no—
207C2×C22⋊C4—51no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4—51no—
210C2×C22⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4C2×D511yescharacteristic
215C2×C22⋊C4C2×D511yescharacteristic
216C2×C42—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95—51no—
223C22×C5—22no—
224C22×C5—22no—
225C22×C5—22no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5—41no—
230C22×C5C22×D411yescharacteristic
231C22×C5C22×D411yescharacteristic
232C22×C5C22×Q811yescharacteristic
233C22×C5H(5,𝔽2)11yescharacteristic
234C22×C5H(5,𝔽2)11yescharacteristic
235C22×C52−1+411yescharacteristic
236C22×C52−1+411yescharacteristic
237C2×D5—22no—
238C2×D5—22no—
239C2×D5—22no—
240C2×D5—41no—
241C2×D5—41no—
242C2×D5—41no—
243C2×D5—41no—
244C4×C5—22no—
245C4×C5—22no—
246C4×C5—22no—
247C4×C5—22no—
248C4×C5—41no—
249C4×C5—41no—
250Dic5—22no—
251Dic5—22no—
252Dic5—22no—
253Dic5—41no—
254Dic5—41no—
255Dic5—41no—
256Dic5—41no—
257C24—51no—
258C24C22×D511yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—102no—
263C2×Q8—102no—
264C2×Q8—201no—
265C2×D4—54no—
266C2×D4—201no—
267C2×D4—201no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—51no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—51no—
276C22×C4—51no—
277C22×C4—51no—
278C22×C4—102no—
279C22×C4—102no—
280C22×C4—201no—
281C22×C4C22×D511yescharacteristic
282C22×C4C22×D511yescharacteristic
283C22×C4C22×D511yescharacteristic
284C22×C4C22×D511yescharacteristic
285C22×C4C22×D511yescharacteristic
286C22×C4C22×D511yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—54no—
290C4⋊C4—102no—
291C4⋊C4—102no—
292C4⋊C4—102no—
293C4⋊C4—102no—
294C4⋊C4—102no—
295C4⋊C4—102no—
296C4⋊C4—201no—
297C4⋊C4—201no—
298C4⋊C4—201no—
299C4⋊C4C22⋊D514yescomplemented
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—102no—
303C22⋊C4—102no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—201no—
309C22⋊C4—201no—
310C42—201no—
311C2×C5—42no—
312C2×C5#53711yescharacteristic
313C2×C5#53711yescharacteristic
314C2×C5#54311yescharacteristic
315C2×C5D4211yescharacteristic
316C2×C5D4×Q811yescharacteristic
317C2×C5D4×Q811yescharacteristic
318C2×C5#56311yescharacteristic
319D5—42no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—52no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23—101no—
331C23—101no—
332C23—101no—
333C23Q8×D512yes—
334C23C23×D511yescharacteristic
335Q8—44no—
336Q8—204no—
337D4—204no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—52no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—204no—
375C2×C4D4×D512yes—
376C2×C4C2×C22⋊D512yes—
377C2×C4C2×C22⋊D512yes—
378C2×C4C2×C22⋊D512yes—
379C5#268311yescharacteristic, complemented, Sylow
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—102no—
388C22—102no—
389C22—102no—
390C22—201no—
391C22—201no—
392C22—201no—
393C22—201no—
394C22C2×D4×D511yescharacteristic
395C22#414411yescharacteristic
396C22C2×Q8×D511yescharacteristic
397C22#414711yescharacteristic
398C22#414911yescharacteristic
399C22#415011yescharacteristic
400C22C22×C22⋊D511yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—102no—
408C4—102no—
409C4—102no—
410C4—201no—
411C4—201no—
412C4—201no—
413C4—201no—
414C2—42no—
415C2—202no—
416C2#6596311yescharacteristic
417C2#6596611yescharacteristic
418C2#6596811yescharacteristic
419C2#6598011yescharacteristic
420C2D4×C22⋊D511yescharacteristic
421C2Q8×C22⋊D511yescharacteristic
422C2#6616711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC22×C22⋊D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2683
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
2. C2×#4092C2semidirect
3. C2×#4070C2semidirect
4. C2×#4042C2semidirect
5. #65518C2semidirect
7. #65283C2semidirect
8. #65283C2semidirect
9. #65274C2semidirect
10. #65268C2semidirect
11. #65261C2semidirect
14. #65237C2semidirect
15. #65237C2semidirect
72. C2×C22⋊D5Q8semidirect
88. C5×C4⋊C4D4semidirect
108. #683D4semidirect
113. C2×C22⋊Q8D5semidirect
299. C4⋊C4C22⋊D5semidirect
379. C5#2683semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40922
C2×#40702
C2×#40422
#655182
#655022
#652832
#652742
#652682
#652612
#652462
#652452
#652372
#26835