Finite Group Databasebeta

← #13106207

Subgroups — #13106207

Order 1600.

662 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#711C211yesmaximal, characteristic, complemented
2C2×#87880C211yesmaximal, characteristic, complemented
3#11717255C211yesmaximal, characteristic, complemented
4C5×#4085C211yesmaximal, characteristic, complemented
5Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
6Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
7#11717088C211yesmaximal, characteristic, complemented
8#11717087C211yesmaximal, characteristic, complemented
9C2×#87780C211yesmaximal, characteristic, complemented
10#11717082C211yesmaximal, characteristic, complemented
11#11717079C211yesmaximal, characteristic, complemented
12#11717023C211yesmaximal, characteristic, complemented
13#11717006C211yesmaximal, characteristic, complemented
14D5×#683C211yesmaximal, characteristic, complemented
15#11716957C211yesmaximal, characteristic, complemented
16C2×C52×D4C2211yescharacteristic
17C22×#1039C2211yescharacteristic
18C2×C5×C22⋊D5C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C22×C5×Dic5C2211yescharacteristic
23C22×C5×Dic5C2211yescharacteristic
24C22×C5×Dic5C2211yescharacteristic
25C5×#711—24no—
26C2×C4×C5×D5C2211yescharacteristic
27C2×C5×Dic10C2211yescharacteristic
28C2×#6039C2211yescharacteristic
29C2×#6037C2211yescharacteristic
30#87880—24no—
31C2×D5×Dic5C2211yescharacteristic
32C2×D5×Dic5C2211yescharacteristic
33C2×D5×Dic5C2211yescharacteristic
34C22⋊#1039C2211yescharacteristic
35C22⋊#1039C2211yescharacteristic
36C22⋊#1039C2211yescharacteristic
37#87806C2211yescharacteristic
38#87806C2211yescharacteristic
39C5×C22⋊Dic5C2211yescharacteristic
40C5×#685C2211yescharacteristic
41C5×#683C2211yescharacteristic
42#87781C2211yescharacteristic
43#87781C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87780C2211yescharacteristic
48#87780C2211yescharacteristic
49#87778C2211yescharacteristic
50#87778C2211yescharacteristic
51#87778C2211yescharacteristic
52Dic52C2211yescharacteristic
53#66158—51nomaximal
54#65945—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5×D5C2311yescharacteristic
58C22×C5×D5C2311yescharacteristic
59C52×D4—22no—
60C2×C4×C52C2311yescharacteristic
61C2×#1039—22no—
62C2×#1039C2311yescharacteristic
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C5×C22⋊D5—22no—
67C5×C22⋊D5—22no—
68C5×C22⋊D5—22no—
69C5×C22⋊D5D414yescomplemented
70C2×C5×Dic5—22no—
71C2×C5×Dic5—22no—
72C2×C5×Dic5—22no—
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C4×C5×D5—22no—
80C5×Dic10—22no—
81#6039—22no—
82#6037—22no—
83D5×Dic5—22no—
84D5×Dic5—22no—
85D5×Dic5—22no—
86C2×C5×#41—51no—
87C2×#711—51no—
88C2×#711—51no—
89C2×#711D511yescharacteristic, complemented
90C5×#121—51no—
91#4092—51no—
92#4090—51no—
93#4085—51no—
94#4085D511yescharacteristic, complemented
95#4084—51no—
96#4081—51no—
97#4081—51no—
98#4081—51no—
99#4081—51no—
100#4081—102no—
101D4×Dic5—51no—
102D4×Dic5—51no—
103#4075—51no—
104#4075—51no—
105C4×C22⋊D5—51no—
106C4×C22⋊D5—51no—
107C2×C4⋊Dic5—51no—
108#4070—51no—
109C2×#683—51no—
110C2×#683—51no—
111D5×C4⋊C4—51no—
112#4034—51no—
113#4029—51no—
114#4029—51no—
115#4024—51no—
116#4024—51no—
117C22×C52—21no—
118C22×C52—21no—
119C22×C52—21no—
120C22×C52—21no—
121C22×C52—21no—
122C22×C52C2×D412yes—
123C22×C52C2411yescharacteristic
124C2×C5×D5—21no—
125C2×C5×D5—21no—
126C2×C5×D5—21no—
127C2×C5×D5—21no—
128C2×C5×D5—21no—
129C2×C5×D5C2×D412yes—
130C4×C52—21no—
131#1039—21no—
132#1039—21no—
133#1039—21no—
134#1039—21no—
135C5×Dic5—21no—
136C5×Dic5—21no—
137C5×Dic5—21no—
138C5×Dic5—21no—
139C5×Dic5C2×D412yes—
140C5×Dic5#4112yescomplemented
141C5×#41—104no—
142C2×C5×Q8—51no—
143C2×C5×D4—22no—
144C2×C5×D4—51no—
145C2×C5×D4—51no—
146C2×C5×D4—51no—
147C2×C5×D4—51no—
148C2×C5×D4C2×D511yescharacteristic
149C2×C5×D4C2×D511yescharacteristic
150C22×C4×C5—51no—
151C22×C4×C5—51no—
152C22×C4×C5—51no—
153C22×C4×C5—51no—
154C2×C22⋊D5—51no—
155C2×C22⋊D5—51no—
156C2×C22⋊D5C2×D511yescharacteristic
157C2×C22⋊D5C2×D511yescharacteristic
158C2×C22⋊D5C2×D511yescharacteristic
159C2×C22⋊D5C2×D511yescharacteristic
160C22×Dic5—51no—
161C22×Dic5—51no—
162C22×Dic5—51no—
163C22×Dic5—51no—
164C22×Dic5—51no—
165C22×Dic5—102no—
166C22×Dic5C2×D511yescharacteristic
167C22×Dic5C2×D511yescharacteristic
168C22×Dic5C2×D511yescharacteristic
169#711—24no—
170#711—104no—
171#711—104no—
172C2×C4×D5—51no—
173C2×C4×D5—51no—
174C2×C4×D5—51no—
175C2×C4×D5C2×D511yescharacteristic
176C2×Dic10—51no—
177C2×Dic10—51no—
178C2×Dic10C2×D511yescharacteristic
179C5×C4⋊C4—51no—
180C5×C22⋊C4—51no—
181C5×C22⋊C4—51no—
182C22⋊Dic5—51no—
183C22⋊Dic5—51no—
184C22⋊Dic5—51no—
185C22⋊Dic5—51no—
186C22⋊Dic5—51no—
187C22⋊Dic5—51no—
188C22⋊Dic5—51no—
189C22⋊Dic5—51no—
190C22⋊Dic5—51no—
191C22⋊Dic5—102no—
192C22⋊Dic5—102no—
193C22⋊Dic5—102no—
194C22⋊Dic5C2×D511yescharacteristic
195#685—51no—
196#685—51no—
197#685—51no—
198#685C2×D511yescharacteristic
199C4⋊Dic5—51no—
200C4⋊Dic5—51no—
201C4⋊Dic5—51no—
202C4⋊Dic5—51no—
203C4⋊Dic5—51no—
204#683—51no—
205#683—51no—
206#683—51no—
207#683—51no—
208#683—51no—
209#683—51no—
210#683—51no—
211#683—51no—
212#683—51no—
213#683—51no—
214#683—51no—
215#683—51no—
216#683—51no—
217#683—102no—
218#683—102no—
219#683C2×D511yescharacteristic
220C4×Dic5—51no—
221C4×Dic5—51no—
222#547—251noSylow
223C2×C52—22no—
224C2×C52—41no—
225C2×C52C22×D411yescharacteristic
226C2×C52C2×#4111yescharacteristic
227C2×C522−1+411yescharacteristic
228C5×D5—22no—
229C5×D5—41no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—51no—
233C23×C5—51no—
234C23×C5C22×D511yescharacteristic
235C23×C5C22×D511yescharacteristic
236C23×C5C22×D511yescharacteristic
237C23×C5C22×D511yescharacteristic
238C22×D5C22×D511yescharacteristic
239C22×D5C22×D511yescharacteristic
240C5×Q8—102no—
241C5×D4—22no—
242C5×D4—22no—
243C5×D4—44no—
244C5×D4—54no—
245C5×D4—102no—
246C5×D4—102no—
247C5×D4—102no—
248C2×C4×C5—22no—
249C2×C4×C5—51no—
250C2×C4×C5—51no—
251C2×C4×C5—51no—
252C2×C4×C5—51no—
253C2×C4×C5—51no—
254C2×C4×C5—51no—
255C2×C4×C5—102no—
256C2×C4×C5—102no—
257C2×C4×C5—102no—
258C2×C4×C5—102no—
259C2×C4×C5C22×D511yescharacteristic
260C2×C4×C5C22×D511yescharacteristic
261C22⋊D5—22no—
262C22⋊D5—22no—
263C22⋊D5—22no—
264C22⋊D5—102no—
265C22⋊D5—102no—
266C22⋊D5C22⋊D514yescomplemented
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—22no—
270C2×Dic5—51no—
271C2×Dic5—51no—
272C2×Dic5—51no—
273C2×Dic5—51no—
274C2×Dic5—51no—
275C2×Dic5—51no—
276C2×Dic5—51no—
277C2×Dic5—51no—
278C2×Dic5—102no—
279C2×Dic5—102no—
280C2×Dic5—102no—
281C2×Dic5—102no—
282C2×Dic5—102no—
283C2×Dic5—102no—
284C2×Dic5—102no—
285C2×Dic5—102no—
286C2×Dic5—102no—
287C2×Dic5—204no—
288C2×Dic5C22×D511yescharacteristic
289C2×Dic5C22×D511yescharacteristic
290C2×Dic5C22×D511yescharacteristic
291C2×Dic5C22×D511yescharacteristic
292C2×Dic5C22×D511yescharacteristic
293C2×Dic5C22×D511yescharacteristic
294C4×D5—22no—
295C4×D5—102no—
296C4×D5—102no—
297C4×D5—102no—
298Dic10—22no—
299Dic10—102no—
300Dic10—102no—
301C2×#41—51no—
302C2×#41—251no—
303C4⋊Q8—251no—
304#123—251no—
305#123—251no—
306#123—251no—
307#123—251no—
308C22⋊Q8—251no—
309C22⋊Q8—251no—
310#121—51no—
311#121—251no—
312C4×D4—251no—
313C4×D4—251no—
314C2×C4⋊C4—251no—
315C2×C4⋊C4—251no—
316C52#54711yescharacteristic, complemented, Sylow
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—24no—
328C22×C5—22no—
329C22×C5—42no—
330C22×C5—42no—
331C22×C5—42no—
332C22×C5—42no—
333C22×C5—42no—
334C22×C5—52no—
335C22×C5—101no—
336C22×C5—101no—
337C22×C5—101no—
338C22×C5—101no—
339C22×C5—101no—
340C22×C5D4×D512yes—
341C22×C5C2×C22⋊D512yes—
342C22×C5C23×D511yescharacteristic
343C22×C5C23×D511yescharacteristic
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—21no—
347C2×D5—21no—
348C2×D5—21no—
349C2×D5C2×C22⋊D512yes—
350C4×C5—21no—
351C4×C5—21no—
352C4×C5—42no—
353C4×C5—52no—
354C4×C5—52no—
355C4×C5—101no—
356C4×C5—101no—
357C4×C5—101no—
358C4×C5—101no—
359Dic5—21no—
360Dic5—21no—
361Dic5—21no—
362Dic5—21no—
363Dic5—101no—
364Dic5—101no—
365Dic5—101no—
366Dic5—101no—
367Dic5—101no—
368Dic5—101no—
369Dic5—101no—
370Dic5—101no—
371Dic5—202no—
372Dic5—202no—
373Dic5—202no—
374Dic5—202no—
375Dic5#71112yescomplemented
376Dic5C2×C22⋊D512yes—
377#41—104no—
378#41—504no—
379C2×Q8—51no—
380C2×Q8—251no—
381C2×D4—51no—
382C2×D4—51no—
383C2×D4—51no—
384C2×D4—51no—
385C2×D4—251no—
386C2×D4D5211yescharacteristic
387C22×C4—51no—
388C22×C4—51no—
389C22×C4—51no—
390C22×C4—51no—
391C22×C4—251no—
392C22×C4—251no—
393C22×C4—251no—
394C22×C4—251no—
395C4⋊C4—51no—
396C4⋊C4—251no—
397C4⋊C4—251no—
398C4⋊C4—251no—
399C4⋊C4—251no—
400C4⋊C4—251no—
401C4⋊C4—251no—
402C4⋊C4—251no—
403C4⋊C4—251no—
404C4⋊C4—251no—
405C22⋊C4—51no—
406C22⋊C4—51no—
407C22⋊C4—251no—
408C22⋊C4—251no—
409C22⋊C4—251no—
410C22⋊C4—251no—
411C22⋊C4—251no—
412C22⋊C4—251no—
413C42—251no—
414C2×C5—22no—
415C2×C5—22no—
416C2×C5—22no—
417C2×C5—22no—
418C2×C5—22no—
419C2×C5—44no—
420C2×C5—41no—
421C2×C5—41no—
422C2×C5—82no—
423C2×C5—102no—
424C2×C5—201no—
425C2×C5C2×D4×D511yescharacteristic
426C2×C5C2×#71111yescharacteristic
427C2×C5D5×#4111yescharacteristic
428C2×C5#415011yescharacteristic
429C2×C5#415011yescharacteristic
430C2×C5C22×C22⋊D511yescharacteristic
431D5—22no—
432D5—41no—
433C23—51no—
434C23—51no—
435C23C2×D5211yescharacteristic
436C23C2×D5211yescharacteristic
437Q8—102no—
438Q8—502no—
439D4—22no—
440D4—54no—
441D4—102no—
442D4—102no—
443D4—102no—
444D4—502no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—51no—
449C2×C4—51no—
450C2×C4—51no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—102no—
455C2×C4—251no—
456C2×C4—251no—
457C2×C4—251no—
458C2×C4—251no—
459C2×C4—502no—
460C2×C4—502no—
461C2×C4—502no—
462C2×C4—502no—
463C2×C4C2×D5211yescharacteristic
464C5—22no—
465C5#6594511yescharacteristic, complemented
466C5#6615811yescharacteristic, complemented
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—52no—
473C22—101no—
474C22—101no—
475C22—101no—
476C22—101no—
477C22—101no—
478C22D5×C22⋊D512yes—
479C22C22×D5211yescharacteristic
480C4—21no—
481C4—52no—
482C4—52no—
483C4—101no—
484C4—101no—
485C4—101no—
486C4—101no—
487C4—501no—
488C4—501no—
489C4—501no—
490C4—501no—
491C2—22no—
492C2—41no—
493C2—102no—
494C2—201no—
495C2#1171747411yescharacteristic
496C2D5×#71111yescharacteristic
497C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#547
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#711C2semidirect
2. C2×#87880C2semidirect
3. #11717255C2semidirect
4. C5×#4085C2semidirect
5. Dic5×C22⋊D5C2semidirect
6. Dic5×C22⋊D5C2semidirect
7. #11717088C2semidirect
8. #11717087C2semidirect
9. C2×#87780C2semidirect
10. #11717082C2semidirect
11. #11717079C2semidirect
12. #11717023C2semidirect
13. #11717006C2semidirect
14. D5×#683C2semidirect
15. #11716957C2semidirect
69. C5×C22⋊D5D4semidirect
89. C2×#711D5semidirect
94. #4085D5semidirect
140. C5×Dic5#41semidirect
266. C22⋊D5C22⋊D5semidirect
316. C52#547semidirect
375. Dic5#711semidirect
465. C5#65945semidirect
466. C5#66158semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7112
C2×#878802
#117172552
C5×#40852
Dic5×C22⋊D52
#117170882
#117170872
C2×#877802
#117170822
#117170792
#117170232
#117170062
D5×#6832
#117169572
#661585
#659455