Finite Group Databasebeta

← #10617356

Subgroups — #10617356

Order 640.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66040C211yesmaximal, characteristic
2#65709C211yesmaximal, characteristic, complemented
3#65708C211yesmaximal, characteristic, complemented
4#65707C211yesmaximal, characteristic, complemented
5#65706C211yesmaximal, characteristic, complemented
6#65705C211yesmaximal, characteristic, complemented
7#65704C211yesmaximal, characteristic, complemented
8#65697C211yesmaximal, characteristic, complemented
9#65696C211yesmaximal, characteristic, complemented
10#65695C211yesmaximal, characteristic, complemented
11#65694C211yesmaximal, characteristic, complemented
12#65693C211yesmaximal, characteristic, complemented
13#65693C211yesmaximal, characteristic, complemented
14#65684C211yesmaximal, characteristic, complemented
15#65681C211yesmaximal, characteristic, complemented
16C2×#704C2211yescharacteristic
17C22×#170C2211yescharacteristic
18C2×#700C2211yescharacteristic
19C2×#700C2211yescharacteristic
20C2×#700C2211yescharacteristic
21C2×D5⋊C8C2211yescharacteristic
22C5×#126C411yescharacteristic
23#4042C411yescharacteristic
24#4040C411yescharacteristic
25#4039C2211yescharacteristic
26#4038C2211yescharacteristic
27#4035C411yescharacteristic
28#4033C2211yescharacteristic
29#4031C411yescharacteristic
30#4030C411yescharacteristic
31#4029C411yescharacteristic
32#4027C2211yescharacteristic
33#4027C2211yescharacteristic
34#4023C2211yescharacteristic
35#4022C411yescharacteristic
36C42×D5C2211yescharacteristic
37C22⋊#170C2211yescharacteristic
38C22⋊#170C2211yescharacteristic
39#4004C2211yescharacteristic
40#4004C2211yescharacteristic
41#4004C2211yescharacteristic
42#4004C2211yescharacteristic
43#4003C2211yescharacteristic
44#4003C2211yescharacteristic
45#4003C2211yescharacteristic
46#4003C2211yescharacteristic
47#4002C2211yescharacteristic
48#4002C2211yescharacteristic
49#4002C2211yescharacteristic
50#4002C2211yescharacteristic
51C4⋊#170C2211yescharacteristic
52C4⋊#170C2211yescharacteristic
53C4⋊#170C2211yescharacteristic
54C4⋊#170C2211yescharacteristic
55C4⋊#170C2211yescharacteristic
56C4⋊#170C2211yescharacteristic
57C4×#170C2211yescharacteristic
58C4×#170C2211yescharacteristic
59#2995—51nomaximal, Sylow
60C2×C22⋊D5C2×C411yescharacteristic
61C2×C22⋊D5C2×C411yescharacteristic
62C22×Dic5C2311yescharacteristic
63C2×D20C2×C411yescharacteristic
64C2×C4×D5C2×C411yescharacteristic
65C2×C4×D5C2×C411yescharacteristic
66C2×C4×D5C2311yescharacteristic
67C2×C4×D5C2311yescharacteristic
68C2×Dic10C2×C411yescharacteristic
69#704—41no—
70C2×#170—41no—
71C2×#170C2311yescharacteristic
72C2×#170C2311yescharacteristic
73C2×#170C2311yescharacteristic
74C2×#170C2311yescharacteristic
75C2×#170C2311yescharacteristic
76C2×#170C2311yescharacteristic
77C2×#170C2311yescharacteristic
78C2×#170C2311yescharacteristic
79#700—41no—
80#700—41no—
81#700—41no—
82D5⋊C8—41no—
83C5×C4⋊C4C2×C411yescharacteristic
84C5×C4⋊C4C2×C411yescharacteristic
85C5×C4⋊C4C2×C411yescharacteristic
86C5×C22⋊C4C2×C411yescharacteristic
87C5×C22⋊C4C2×C411yescharacteristic
88C5×C22⋊C4C2×C411yescharacteristic
89C42×C5C2×C411yescharacteristic
90C22⋊Dic5C2×C411yescharacteristic
91#685C2×C411yescharacteristic
92#685C2×C411yescharacteristic
93#685C2×C411yescharacteristic
94#685C2×C411yescharacteristic
95#685C2×C411yescharacteristic
96#685C2×C411yescharacteristic
97C4⋊Dic5C2×C411yescharacteristic
98#683C2×C411yescharacteristic
99#683C2×C411yescharacteristic
100#683C2×C411yescharacteristic
101#683C2×C411yescharacteristic
102#683C2×C411yescharacteristic
103#683C2×C411yescharacteristic
104C4×Dic5C2×C411yescharacteristic
105C4×Dic5C2311yescharacteristic
106C4×Dic5C2311yescharacteristic
107C4×Dic5C2311yescharacteristic
108C4×Dic5C2311yescharacteristic
109#529—51no—
110#446—51no—
111#446—51no—
112#436—51no—
113#436—51no—
114#435—51no—
115#435—51no—
116#435—51no—
117#435—51no—
118#433—51no—
119#433—51no—
120#432—51no—
121#424—51no—
122#424—51no—
123#423—51no—
124C23×C5C22×C411yescharacteristic
125C22×D5C22×C411yescharacteristic
126C22×D5C22×C411yescharacteristic
127C2×C4×C5C22×C411yescharacteristic
128C2×C4×C5C22×C411yescharacteristic
129C2×C4×C5C22×C411yescharacteristic
130C2×C4×C5C22×C411yescharacteristic
131C2×C4×C5C22×C411yescharacteristic
132C2×C4×C5C22×C411yescharacteristic
133C22⋊D5—41no—
134C22⋊D5—41no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5C22×C411yescharacteristic
138C2×Dic5C22×C411yescharacteristic
139C2×Dic5C22×C411yescharacteristic
140C2×Dic5C22×C411yescharacteristic
141C2×Dic5C22×C411yescharacteristic
142C2×Dic5C2411yescharacteristic
143D20—41no—
144C4×D5—21no—
145C4×D5—21no—
146C4×D5—21no—
147C4×D5—21no—
148C4×D5—41no—
149C4×D5—41no—
150Dic10—41no—
151#170—21no—
152#170—21no—
153#170—21no—
154#170—21no—
155#170—21no—
156#170—21no—
157#170—21no—
158#170—21no—
159C2×M16—51no—
160C2×M16—51no—
161C2×M16—51no—
162C2×M16—51no—
163C22×C8—51no—
164C22×C8—51no—
165#126—51no—
166#126AGL(1,𝔽5)11yescharacteristic
167#125—51no—
168#124—51no—
169#123—51no—
170#123—51no—
171C22⋊Q8—51no—
172#121—51no—
173C4×Q8—51no—
174C4×D4—51no—
175C4×D4—51no—
176C4×D4—51no—
177#117—51no—
178#117—51no—
179C2×C42—51no—
180C4⋊C8—51no—
181C4⋊C8—51no—
182C4⋊C8—51no—
183C4⋊C8—51no—
184C4⋊C8—51no—
185C4⋊C8—51no—
186C4⋊C8—51no—
187C4⋊C8—51no—
188C4⋊C8—51no—
189C4⋊C8—51no—
190C22⋊C8—51no—
191C22⋊C8—51no—
192C22⋊C8—51no—
193C22⋊C8—51no—
194C22⋊C8—51no—
195C22⋊C8—51no—
196#97—51no—
197#97—51no—
198#97—51no—
199#97—51no—
200C4×C8—51no—
201C4×C8—51no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5—21no—
205C22×C5C23×C411yescharacteristic
206C2×D5—21no—
207C2×D5—21no—
208C2×D5—21no—
209C2×D5—21no—
210C2×D5—21no—
211C2×D5—21no—
212C4×C5—21no—
213C4×C5—21no—
214C4×C5—21no—
215C4×C5—21no—
216C4×C5—21no—
217C4×C5—21no—
218Dic5—21no—
219Dic5—21no—
220Dic5—21no—
221Dic5—21no—
222Dic5—21no—
223Dic5H(5,𝔽2)11yescharacteristic
224Dic52−1+411yescharacteristic
225C2×Q8—51no—
226C2×D4—51no—
227C2×D4—51no—
228C2×D4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—51no—
234M16—201no—
235M16—201no—
236M16—201no—
237M16—201no—
238C2×C8—51no—
239C2×C8—51no—
240C2×C8—51no—
241C2×C8—51no—
242C2×C8—51no—
243C2×C8—51no—
244C2×C8—51no—
245C2×C8—51no—
246C2×C8—201no—
247C2×C8—201no—
248C4⋊C4—51no—
249C4⋊C4—51no—
250C4⋊C4—51no—
251C4⋊C4—51no—
252C4⋊C4—51no—
253C4⋊C4—51no—
254C4⋊C4—51no—
255C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
256C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
257C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
258C22⋊C4—51no—
259C22⋊C4—51no—
260C22⋊C4—51no—
261C22⋊C4—51no—
262C22⋊C4—51no—
263C22⋊C4—51no—
264C22⋊C4—51no—
265C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
266C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
267C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
268C42—51no—
269C42—51no—
270C42—51no—
271C42—51no—
272C42—51no—
273C42C2×AGL(1,𝔽5)11yescharacteristic
274C2×C5—41no—
275C2×C5#52011yescharacteristic
276C2×C5#56811yescharacteristic
277C2×C5#56811yescharacteristic
278D5—41no—
279D5—41no—
280C23—51no—
281C23—51no—
282C23C22×AGL(1,𝔽5)11yescharacteristic
283Q8—201no—
284D4—201no—
285D4—201no—
286D4—201no—
287C2×C4—51no—
288C2×C4—51no—
289C2×C4—51no—
290C2×C4—51no—
291C2×C4—51no—
292C2×C4—51no—
293C2×C4—101no—
294C2×C4—101no—
295C2×C4—101no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—201no—
300C2×C4—201no—
301C2×C4C22×AGL(1,𝔽5)11yescharacteristic
302C2×C4C22×AGL(1,𝔽5)11yescharacteristic
303C2×C4C22×AGL(1,𝔽5)11yescharacteristic
304C2×C4C22×AGL(1,𝔽5)11yescharacteristic
305C2×C4C22×AGL(1,𝔽5)11yescharacteristic
306C2×C4C22×AGL(1,𝔽5)11yescharacteristic
307C8—101no—
308C8—101no—
309C8—101no—
310C8—101no—
311C8—101no—
312C8—101no—
313C8—101no—
314C8—101no—
315C5#299511yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—101no—
320C22—101no—
321C22—101no—
322C22—101no—
323C22—101no—
324C22—101no—
325C22C23×AGL(1,𝔽5)11yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—51no—
333C4—51no—
334C4—101no—
335C4—101no—
336C4—101no—
337C4—101no—
338C4—101no—
339C2—41no—
340C2—201no—
341C2—201no—
342C2#6626411yescharacteristic
343C2#6626411yescharacteristic
344C2#6626611yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽5)
commutator, socleC22×C5C23×C4
FittingC5×#126C4
prime radical#4030C4
cosocleC2×Dic5C24
hypercenter#126AGL(1,𝔽5)
hypocenterC5#2995
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
2. #65709C2semidirect
3. #65708C2semidirect
4. #65707C2semidirect
5. #65706C2semidirect
6. #65705C2semidirect
7. #65704C2semidirect
8. #65697C2semidirect
9. #65696C2semidirect
10. #65695C2semidirect
11. #65694C2semidirect
12. #65693C2semidirect
13. #65693C2semidirect
14. #65684C2semidirect
15. #65681C2semidirect
315. C5#2995semidirect

Maximal subgroups

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

subgroupindex
#660402
#657092
#657082
#657072
#657062
#657052
#657042
#656972
#656962
#656952
#656942
#656932
#656842
#656812
#29955