Finite Group Databasebeta

← #10607142

Subgroups — #10607142

Order 640.

416 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#387C211yesmaximal, characteristic, complemented
2#65281C211yesmaximal, characteristic
3#65276C211yesmaximal, characteristic
4#65271C211yesmaximal, characteristic, complemented
5#65268C211yesmaximal, characteristic, complemented
6#65229C211yesmaximal, characteristic
7#65217C211yesmaximal, characteristic, complemented
8#64962C211yesmaximal, characteristic
9#64959C211yesmaximal, characteristic
10#64957C211yesmaximal, characteristic
11#64956C211yesmaximal, characteristic
12#64949C211yesmaximal, characteristic, complemented
13#64945C211yesmaximal, characteristic, complemented
14#64941C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C42×C5C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37C5×#95C2211yescharacteristic
38C5×#95C2211yescharacteristic
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2742—51nomaximal, Sylow
52C23×D5C2311yescharacteristic
53C22×C4×C5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×Dic5C2311yescharacteristic
61C22×Dic5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C2×C4×D5—22no—
68C2×C4×D5—22no—
69C2×C4×D5—41no—
70C2×Dic10—41no—
71C2×Dic10—41no—
72C2×Dic10—41no—
73C5×C4⋊C4—22no—
74C5×C4⋊C4—41no—
75C42×C5—41no—
76#685—41no—
77#685—41no—
78#685—41no—
79#685—41no—
80#685—41no—
81#685Q814yescomplemented
82C4⋊Dic5—22no—
83C4⋊Dic5—41no—
84C4⋊Dic5—41no—
85#683—22no—
86#683—22no—
87#683—41no—
88#683—41no—
89#683—41no—
90#683—41no—
91C4×Dic5—41no—
92#400—51no—
93#400—51no—
94#399—51no—
95#395—51no—
96#395—51no—
97#393—51no—
98#389—51no—
99#389—51no—
100#388—51no—
101#388—51no—
102#387—51no—
103#387D511yescharacteristic, complemented
104#385—51no—
105#380—51no—
106#380—51no—
107C23×C5C2411yescharacteristic
108C22×D5—21no—
109C22×D5—21no—
110C22×D5—21no—
111C22×D5—21no—
112C22×D5—21no—
113C22×D5—21no—
114C22×D5C2×Q812yes—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5C2×Q812yes—
135C2×C4×C5#4112yes—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5C2×Q812yes—
156C2×Dic5#4112yes—
157C4×D5—44no—
158Dic10—44no—
159C22×Q8—51no—
160C23×C4—51no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4—51no—
167C2×C4⋊C4—51no—
168C2×C4⋊C4—51no—
169C2×C4⋊C4—51no—
170C2×C4⋊C4C2×D511yescharacteristic
171C2×C4⋊C4C2×D511yescharacteristic
172C2×C22⋊C4—51no—
173C2×C22⋊C4—51no—
174C2×C22⋊C4—51no—
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C42—51no—
179C2×C42C2×D511yescharacteristic
180#95—51no—
181#95—51no—
182#95—51no—
183#95—51no—
184#95—51no—
185#95—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95C2×D511yescharacteristic
191#95C2×D511yescharacteristic
192#95C2×D511yescharacteristic
193#95C2×D511yescharacteristic
194C22×C5C22×Q811yescharacteristic
195C22×C5C2×#4111yescharacteristic
196C22×C5C2×#4111yescharacteristic
197C22×C5H(5,𝔽2)11yescharacteristic
198C22×C5H(5,𝔽2)11yescharacteristic
199C22×C5H(5,𝔽2)11yescharacteristic
200C22×C52−1+411yescharacteristic
201C2×D5—22no—
202C2×D5—22no—
203C2×D5—22no—
204C2×D5—41no—
205C2×D5—41no—
206C2×D5—41no—
207C2×D5—41no—
208C4×C5—22no—
209C4×C5—22no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215Dic5—22no—
216Dic5—22no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220Dic5—41no—
221Dic5—41no—
222C24—51no—
223C2×Q8—201no—
224C2×Q8—201no—
225C2×Q8—201no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—102no—
234C22×C4—102no—
235C22×C4—201no—
236C22×C4C22×D511yescharacteristic
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C22×C4C22×D511yescharacteristic
240C22×C4C22×D511yescharacteristic
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—102no—
246C4⋊C4—102no—
247C4⋊C4—102no—
248C4⋊C4—201no—
249C4⋊C4—201no—
250C4⋊C4—201no—
251C4⋊C4—201no—
252C4⋊C4—201no—
253C4⋊C4—201no—
254C22⋊C4—54no—
255C22⋊C4—201no—
256C22⋊C4—201no—
257C22⋊C4—201no—
258C22⋊C4—201no—
259C22⋊C4—201no—
260C42—41no—
261C42—201no—
262C2×C5#54211yescharacteristic
263C2×C5#54211yescharacteristic
264C2×C5#54411yescharacteristic
265C2×C5#55111yescharacteristic
266C2×C5#55411yescharacteristic
267C2×C5#55411yescharacteristic
268C2×C5#56011yescharacteristic
269D5—42no—
270C23—52no—
271C23—101no—
272C23—101no—
273C23—101no—
274C23—101no—
275C23—101no—
276C23—101no—
277C23C23×D511yescharacteristic
278Q8—204no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—52no—
299C2×C4—52no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—101no—
306C2×C4—101no—
307C2×C4—101no—
308C2×C4—101no—
309C2×C4—101no—
310C2×C4—101no—
311C2×C4—101no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—204no—
320C2×C4#70912yes—
321C2×C4Q8×D512yes—
322C5#274211yescharacteristic, complemented, Sylow
323C22—102no—
324C22—102no—
325C22—102no—
326C22—201no—
327C22—201no—
328C22—201no—
329C22—201no—
330C22C2×#70911yescharacteristic
331C22#414411yescharacteristic
332C22C2×Q8×D511yescharacteristic
333C22D5×#4111yescharacteristic
334C22#414911yescharacteristic
335C22#414911yescharacteristic
336C22#415011yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—102no—
345C4—102no—
346C4—201no—
347C4—201no—
348C4—201no—
349C4—201no—
350C4—201no—
351C2—202no—
352C2#6586111yescharacteristic
353C2#6589511yescharacteristic
354C2#6591311yescharacteristic
355C2#6597011yescharacteristic
356C2#6600011yescharacteristic
357C2#6602311yescharacteristic
358C2#6604411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#387C2
prime radicalC23×D5C23
hypercenter#387D5
hypocenterC5#2742
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C5×#387C2semidirect
4. #65271C2semidirect
5. #65268C2semidirect
7. #65217C2semidirect
12. #64949C2semidirect
13. #64945C2semidirect
14. #64941C2semidirect
15. #64936C2semidirect
81. #685Q8semidirect
103. #387D5semidirect
322. C5#2742semidirect

Maximal subgroups

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

subgroupindex
C5×#3872
#652812
#652762
#652712
#652682
#652292
#652172
#649622
#649592
#649572
#649562
#649492
#649452
#649412
#649362
#27425