Finite Group Databasebeta

← #13014821

Subgroups — #13014821

Order 1280.

550 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 110 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
1Dic5×C22⋊Q8C211yesmaximal, characteristic
2#10604900C211yesmaximal, characteristic
3#10604895C211yesmaximal, characteristic
4#10598828C212yesmaximal, complemented
5#10598713C212yesmaximal, complemented
6C22×#704—21no—
7C2×C5×C22⋊Q8C411yescharacteristic
8C2×Q8×Dic5C411yescharacteristic
9C22×C4×Dic5C2211yescharacteristic
10C2×C22⋊#170—21no—
11C2×#4004—21no—
12#65519C2211yescharacteristic
13#65501C411yescharacteristic
14#65274C2211yescharacteristic
15#65270—21no—
16C4⋊C4×Dic5—21no—
17C4⋊C4×Dic5C411yescharacteristic
18#65239—21no—
19C22⋊C4×Dic5—21no—
20#64918—21no—
21#64918C2212yes—
22#64918C2212yes—
23#7617—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5C2×C411yescharacteristic
26C23×Dic5D411yescharacteristic
27C23×Dic5D411yescharacteristic
28C2×#704—24no—
29C2×#704—42no—
30C2×#704—42no—
31C22×#170—22no—
32C22×#170—22no—
33C5×C22⋊Q8—22no—
34C5×C22⋊Q8—22no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C22⋊Dic5—21no—
39Q8×Dic5—42no—
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5C2×C411yescharacteristic
43C2×C4⋊Dic5C2×C411yescharacteristic
44C2×C4×Dic5—21no—
45C2×C4×Dic5—21no—
46C2×C4×Dic5—42no—
47C2×C4×Dic5C2×C411yescharacteristic
48C2×C4×Dic5D411yescharacteristic
49C2×C4×Dic5D411yescharacteristic
50C2×C4×Dic5C2311yescharacteristic
51C22⋊#170—42no—
52#4004—42no—
53#3963—21no—
54#3963—21no—
55#3963—21no—
56#3963—21no—
57#3963—21no—
58C4×C22⋊Q8—51no—
59#1797—51no—
60#1728—51no—
61#1272—52no—
62#1271—52no—
63C24×C5C22⋊C411yescharacteristic
64C2×C5×Q8—21no—
65C2×C5×Q8—41no—
66C2×C5×Q8—41no—
67C2×C5×Q8M1612yescomplemented
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—41no—
73C22×C4×C5—41no—
74C22×C4×C5C22⋊C411yescharacteristic
75C22×C4×C5C22×C411yescharacteristic
76C22×Dic5—22no—
77C22×Dic5—22no—
78C22×Dic5—21no—
79C22×Dic5—21no—
80C22×Dic5—21no—
81C22×Dic5—21no—
82C22×Dic5—41no—
83C22×Dic5—41no—
84C22×Dic5C22⋊C411yescharacteristic
85C22×Dic5C22⋊C411yescharacteristic
86C22×Dic5SD1612yes—
87C22×Dic5Q1612yes—
88C22×Dic5C2×D411yescharacteristic
89C22×Dic5C2×D411yescharacteristic
90#704—48no—
91C2×#170—24no—
92C2×#170—42no—
93C2×#170—42no—
94C2×#170—42no—
95C2×#170—42no—
96C2×#170—42no—
97C5×C4⋊C4—24no—
98C5×C4⋊C4—21no—
99C5×C4⋊C4M1612yescomplemented
100C5×C22⋊C4—24no—
101C22⋊Dic5—42no—
102C4⋊Dic5—42no—
103C4⋊Dic5—41no—
104C4⋊Dic5—41no—
105C4⋊Dic5—41no—
106C4⋊Dic5—41no—
107C4×Dic5—22no—
108C4×Dic5—22no—
109C4×Dic5—41no—
110C4×Dic5—41no—
111C4×Dic5—42no—
112C4×Dic5—42no—
113C22×M16—101no—
114C2×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
115C2×C4×Q8—51no—
116C22×C42—51no—
117C2×C4⋊C8—101no—
118C2×C22⋊C8—101no—
119#391—51no—
120#389—51no—
121#387—101no—
122#385—101no—
123#380—51no—
124C4×C4⋊C4—51no—
125C4×C4⋊C4—101no—
126C4×C22⋊C4—101no—
127#336—52no—
128#336—52no—
129#336—101no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5#10312yes—
137C23×C5C2×C22⋊C411yescharacteristic
138C5×Q8—42no—
139C5×Q8—42no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—22no—
143C2×C4×C5—22no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5C2×M1611yescharacteristic
154C2×C4×C5C2×M1611yescharacteristic
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—24no—
160C2×Dic5—24no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—42no—
165C2×Dic5—42no—
166C2×Dic5—41no—
167C2×Dic5—41no—
168C2×Dic5—41no—
169C2×Dic5—41no—
170C2×Dic5—41no—
171C2×Dic5—41no—
172C2×Dic5#9912yes—
173C2×Dic5#10312yes—
174C2×Dic5#10412yes—
175C2×Dic5C2×SD1611yescharacteristic
176C2×Dic5C2×Q1611yescharacteristic
177C2×Dic5AGL(1,ℤ/8)11yescharacteristic
178C2×Dic5#13711yescharacteristic
179#170—44no—
180#170—82no—
181C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
182C23×C4—51no—
183C23×C4—51no—
184C23×C4C2×AGL(1,𝔽5)11yescharacteristic
185C2×M16—104no—
186C2×M16—202no—
187C2×M16—202no—
188C22×C8—102no—
189C22×C8—102no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C4×Q8—202no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4—101no—
197C2×C4⋊C4—101no—
198C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—101no—
201C2×C42—51no—
202C2×C42—51no—
203C2×C42—51no—
204C2×C42—51no—
205C2×C42—101no—
206C2×C42—101no—
207C2×C42—202no—
208C4⋊C8—202no—
209C22⋊C8—202no—
210#95—101no—
211#95—101no—
212#95—101no—
213#95—101no—
214#95—101no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5#40711yescharacteristic
223C22×C5C2×#9911yescharacteristic
224C22×C5#41311yescharacteristic
225C22×C5C2×#10311yescharacteristic
226C22×C5#41711yescharacteristic
227C22×C5C2×#10411yescharacteristic
228C22×C5#42111yescharacteristic
229C4×C5—21no—
230C4×C5—21no—
231C4×C5—41no—
232C4×C5—42no—
233C4×C5—42no—
234Dic5—21no—
235Dic5—21no—
236Dic5—21no—
237Dic5—21no—
238Dic5—41no—
239Dic5—81no—
240Dic5—81no—
241Dic5#32814yes—
242C24C22⋊AGL(1,𝔽5)11yescharacteristic
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8#70412yescomplemented
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—52no—
257C22×C4—52no—
258C22×C4—51no—
259C22×C4—102no—
260C22×C4—102no—
261C22×C4—101no—
262C22×C4—101no—
263C22×C4—101no—
264C22×C4—101no—
265C22×C4—201no—
266C22×C4—201no—
267C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
268C22×C4C22×AGL(1,𝔽5)11yescharacteristic
269M16—208no—
270C2×C8—104no—
271C2×C8—202no—
272C2×C8—202no—
273C2×C8—202no—
274C2×C8—202no—
275C2×C8—202no—
276C4⋊C4—24no—
277C4⋊C4—21no—
278C4⋊C4—201no—
279C4⋊C4—201no—
280C4⋊C4—201no—
281C4⋊C4—201no—
282C4⋊C4—202no—
283C4⋊C4#70412yescomplemented
284C22⋊C4—24no—
285C22⋊C4—202no—
286C42—102no—
287C42—102no—
288C42—201no—
289C42—201no—
290C42—202no—
291C42—202no—
292C2×C5—42no—
293C2×C5#146511yescharacteristic
294C2×C5#148711yescharacteristic
295C2×C5#148911yescharacteristic
296C2×C5C2×#32811yescharacteristic
297C2×C5#151311yescharacteristic
298C2×C5#152911yescharacteristic
299C2×C5#155011yescharacteristic
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23#400912yes—
307C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
308Q8—42no—
309Q8—42no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—52no—
324C2×C4—52no—
325C2×C4—52no—
326C2×C4—51no—
327C2×C4—51no—
328C2×C4—51no—
329C2×C4—51no—
330C2×C4—104no—
331C2×C4—104no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—202no—
340C2×C4—202no—
341C2×C4—201no—
342C2×C4—201no—
343C2×C4—201no—
344C2×C4—201no—
345C2×C4—201no—
346C2×C4—201no—
347C2×C4C2×#70411yescharacteristic
348C2×C4C2×#70411yescharacteristic
349C8—204no—
350C8—402no—
351C5#761711yescharacteristic, complemented, Sylow
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22D5⋊#9911yescharacteristic
360C22#6577711yescharacteristic
361C22C2×#400911yescharacteristic
362C22#6578411yescharacteristic
363C22D5⋊#10411yescharacteristic
364C22#6579311yescharacteristic
365C22#6579811yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—54no—
372C4—101no—
373C4—101no—
374C4—101no—
375C4—101no—
376C4—201no—
377C4—401no—
378C4—401no—
379C2—42no—
380C2D5⋊#32811yescharacteristic
381C2#1060450211yescharacteristic
382C2#1060450611yescharacteristic
383C2#1060451811yescharacteristic
384C2#1060493311yescharacteristic
385C2#1060496811yescharacteristic
386C2#1060500311yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×C22⋊Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×C22⋊Q8AGL(1,𝔽5)
hypocenterC5#7617
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #10598828C2semidirect
5. #10598713C2semidirect
67. C2×C5×Q8M16semidirect
99. C5×C4⋊C4M16semidirect
246. C2×Q8#704semidirect
283. C4⋊C4#704semidirect
351. C5#7617semidirect

Maximal subgroups

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

subgroupindex
Dic5×C22⋊Q82
#106049002
#106048952
#105988282
#105987132
#76175