Finite Group Databasebeta

← #12073222

Subgroups — #12073222

Order 1280.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#65834C211yesmaximal, characteristic
2C5×#2627C211yesmaximal, characteristic
3#10614472C212yesmaximal
4C2×#64748C211yesmaximal, characteristic
5#10600512C214yesmaximal, complemented
6#10600500C214yesmaximal, complemented
7#10600303C212yesmaximal
8C2×C5×C22⋊Q8C2211yescharacteristic
9C22×C5×C4⋊C4C2211yescharacteristic
10C2×#4072—21no—
11C2×#4070—21no—
12C2×#4070C2212yes—
13#65834—24no—
14#65834—22no—
15#65834C2214yes—
16C2×#4036—22no—
17C2×#4034—22no—
18C5×#400C2212yes—
19C5×#398—22no—
20C5×#395C2212yes—
21C5×#385—21no—
22C5×#381—21no—
23C5×#380C2211yescharacteristic
24#65517C2214yes—
25C2×C22⋊#168C2212yes—
26#65340—24no—
27#65339—22no—
28#65333—22no—
29#65259C2214yes—
30#64748C2214yes—
31#64672C2218yes—
32#18295—51nomaximal, Sylow
33C22×C5×Q8C2311yescharacteristic
34C23×C4×C5D411yescharacteristic
35C23×C4×C5D411yescharacteristic
36C23×C4×C5C2311yescharacteristic
37C22×Dic10—21no—
38C22×Dic10D412yes—
39C5×C22⋊Q8—22no—
40C5×C22⋊Q8—22no—
41C2×C5×C4⋊C4—22no—
42C2×C5×C4⋊C4—22no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4D412yes—
47C2×C5×C4⋊C4C2311yescharacteristic
48C2×C5×C4⋊C4C2312yes—
49C2×C5×C22⋊C4—21no—
50C2×C5×C22⋊C4—21no—
51C2×C22⋊Dic5—22no—
52C2×#689—24no—
53#4072—24no—
54#4072—24no—
55C2×C4⋊Dic5—21no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5—21no—
58C2×C4⋊Dic5D412yes—
59#4070—24no—
60#4070—22no—
61#4070—22no—
62#4070—24no—
63#4070D414yes—
64#4070C2314yes—
65C2×#683—22no—
66C2×#683—22no—
67C2×C4×Dic5—22no—
68C22×#168—22no—
69#4036—28no—
70#4036—28no—
71#4034—28no—
72#4034—28no—
73C5×#95—22no—
74C5×#95—22no—
75C5×#95—22no—
76C5×#95C2312yes—
77#3967—24no—
78C22⋊#168—22no—
79C22⋊#168C2314yes—
80#3942—24no—
81#3939—24no—
82C2×#544—51no—
83#3189—52no—
84#2627D511yescharacteristic
85#1617—54no—
86#1605—54no—
87#1520—52no—
88C2×#324—51no—
89C24×C5C2×D411yescharacteristic
90C2×C5×Q8—22no—
91C2×C5×Q8—81no—
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5—22no—
95C22×C4×C5—21no—
96C22×C4×C5—21no—
97C22×C4×C5—21no—
98C22×C4×C5—22no—
99C22×C4×C5—21no—
100C22×C4×C5—81no—
101C22×C4×C5—81no—
102C22×C4×C5Q1614yescomplemented
103C22×C4×C5C2×D412yes—
104C22×C4×C5C2×D412yes—
105C22×C4×C5C2×D412yes—
106C22×C4×C5C2×D412yes—
107C22×C4×C5C2×D411yescharacteristic
108C22×C4×C5C2411yescharacteristic
109C22×Dic5—22no—
110C22×Dic5—22no—
111C2×Dic10—24no—
112C2×Dic10—22no—
113C2×Dic10—82no—
114C2×Dic10—82no—
115C2×Dic10C2×D414yes—
116C5×C4⋊C4—24no—
117C5×C4⋊C4—24no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—42no—
120C5×C4⋊C4—81no—
121C5×C4⋊C4—82no—
122C5×C22⋊C4—41no—
123C5×C22⋊C4—41no—
124C5×C22⋊C4—42no—
125C22⋊Dic5—24no—
126C22⋊Dic5—24no—
127#689—84no—
128C4⋊Dic5—24no—
129C4⋊Dic5—24no—
130C4⋊Dic5—24no—
131C4⋊Dic5—22no—
132C4⋊Dic5C2×D414yes—
133#683—28no—
134#683—24no—
135#683—24no—
136C4×Dic5—28no—
137C2×#168—24no—
138C2×#168—82no—
139#544—54no—
140#544—104no—
141#544—102no—
142C2×C4⋊Q8—102no—
143C2×#125—102no—
144C2×C22⋊Q8—52no—
145C2×C22⋊Q8—101no—
146C2×C22⋊Q8C2×D511yescharacteristic
147C2×#117—101no—
148C22×C4⋊C4C2×D511yescharacteristic
149#484—54no—
150#484—102no—
151#467—104no—
152#451—54no—
153#451—102no—
154C2×C22⋊C8—52no—
155#400C2×D512yes—
156#398—22no—
157#395C2×D512yes—
158#385—21no—
159#381—21no—
160#380C2×D511yescharacteristic
161#328—58no—
162#324—54no—
163C23×C5—41no—
164C23×C5—41no—
165C23×C5#12011yescharacteristic
166C23×C5#12011yescharacteristic
167C23×C5C2×Q1612yes—
168C23×C5#13712yes—
169C23×C5C22×D411yescharacteristic
170C5×Q8—44no—
171C2×C4×C5—22no—
172C2×C4×C5—22no—
173C2×C4×C5—24no—
174C2×C4×C5—24no—
175C2×C4×C5—22no—
176C2×C4×C5—42no—
177C2×C4×C5—41no—
178C2×C4×C5—42no—
179C2×C4×C5—41no—
180C2×C4×C5—42no—
181C2×C4×C5—42no—
182C2×C4×C5—42no—
183C2×C4×C5—42no—
184C2×C4×C5—42no—
185C2×C4×C5—42no—
186C2×C4×C5—41no—
187C2×C4×C5—41no—
188C2×C4×C5—42no—
189C2×C4×C5—81no—
190C2×C4×C5#12012yes—
191C2×C4×C5C2×Q1614yes—
192C2×C4×C5C22×D412yes—
193C2×Dic5—24no—
194C2×Dic5—24no—
195C2×Dic5—44no—
196C2×Dic5—44no—
197Dic10—44no—
198Dic10—82no—
199Dic10—84no—
200#168—44no—
201C22×Q8—52no—
202C22×Q8—101no—
203C22×Q8C22×D511yescharacteristic
204C23×C4C22⋊D511yescharacteristic
205C23×C4C22⋊D511yescharacteristic
206C23×C4C22×D511yescharacteristic
207C2×Q16—104no—
208C22×C8—102no—
209C4⋊Q8—108no—
210C4⋊Q8—108no—
211#125—108no—
212#125—108no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—54no—
216C22⋊Q8—54no—
217C22⋊Q8—104no—
218C22⋊Q8—102no—
219C22⋊Q8—102no—
220C22⋊Q8—104no—
221#117—104no—
222#117—104no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—52no—
229C2×C4⋊C4—101no—
230C2×C4⋊C4—101no—
231C2×C4⋊C4—101no—
232C2×C4⋊C4—102no—
233C2×C4⋊C4—102no—
234C2×C4⋊C4C22⋊D512yes—
235C2×C4⋊C4C22×D511yescharacteristic
236C2×C4⋊C4C22×D512yes—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—102no—
240C2×C42—102no—
241#107—104no—
242#103—104no—
243#103—104no—
244C22⋊C8—54no—
245C22⋊C8—102no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95C22×D512yes—
250C22×C5—21no—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5—21no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5—42no—
257C22×C5—42no—
258C22×C5#45112yes—
259C22×C5#45112yes—
260C22×C5C2×#12011yescharacteristic
261C22×C5C22×Q1611yescharacteristic
262C22×C5C2×#13711yescharacteristic
263C4×C5—24no—
264C4×C5—22no—
265C4×C5—22no—
266C4×C5—42no—
267C4×C5—82no—
268C4×C5—81no—
269C4×C5—81no—
270Dic5—44no—
271Dic5—44no—
272C24C2×C22⋊D511yescharacteristic
273C2×Q8—22no—
274C2×Q8—54no—
275C2×Q8—81no—
276C2×Q8—104no—
277C2×Q8—102no—
278C2×Q8—402no—
279C2×Q8—402no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—102no—
291C22×C4—102no—
292C22×C4#68914yescomplemented
293C22×C4D4×D512yes—
294C22×C4D4×D512yes—
295C22×C4C2×C22⋊D512yes—
296C22×C4C2×C22⋊D512yes—
297C22×C4C2×C22⋊D511yescharacteristic
298C22×C4C23×D511yescharacteristic
299Q16—404no—
300C2×C8—104no—
301C2×C8—402no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—54no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—104no—
310C4⋊C4—104no—
311C4⋊C4—104no—
312C4⋊C4—108no—
313C4⋊C4—104no—
314C4⋊C4—102no—
315C4⋊C4—104no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—104no—
320C22⋊C4—104no—
321C42—108no—
322C2×C5—21no—
323C2×C5—21no—
324C2×C5—22no—
325C2×C5—22no—
326C2×C5C2×#45111yescharacteristic
327C2×C5#302011yescharacteristic
328C2×C5#302611yescharacteristic
329C23—41no—
330C23—41no—
331C23#408311yescharacteristic
332C23#408311yescharacteristic
333C23C2×#68912yes—
334C23#409712yes—
335C23C22×C22⋊D511yescharacteristic
336Q8—44no—
337Q8—204no—
338Q8—402no—
339Q8—404no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—104no—
360C2×C4—104no—
361C2×C4—204no—
362C2×C4—204no—
363C2×C4#408312yes—
364C2×C4C2×#68914yes—
365C2×C4C2×D4×D512yes—
366C8—204no—
367C5#1829511yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#6533912yes—
377C22#6533912yes—
378C22C2×#408311yescharacteristic
379C22C22×#68911yescharacteristic
380C22C2×#409711yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—204no—
389C4—204no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#6533911yescharacteristic
395C2#1061510611yescharacteristic
396C2#1061514611yescharacteristic
subgroupG/H
centerC22C2×#4083
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2627C2
socleC22×C5C2×#120
prime radicalC24×C5C2×D4
hypercenter#2627D5
hypocenterC5#18295
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
5. #10600512C2semidirect
6. #10600500C2semidirect
102. C22×C4×C5Q16semidirect
292. C22×C4#689semidirect
367. C5#18295semidirect

Maximal subgroups

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

subgroupindex
C2×#658342
C5×#26272
#106144722
C2×#647482
#106005122
#106005002
#106003032
#182955