Finite Group Databasebeta

← #27275870

Subgroups — #27275870

Order 13568.

782 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 159 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
1#20274711C211yesmaximal, characteristic, complemented
2#20273400C212yesmaximal, complemented
3#20273140C212yesmaximal
4#20273039C211yesmaximal, characteristic, complemented
5C2×C53×#554C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810392—21no—
10#16810369—24no—
11D4⋊#88322—24no—
12#16810349—22no—
13C53×#554—24no—
14C53×#554—24no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C2×#125×C53C412yes—
17C2×C22⋊Q8×C53—21no—
18C2×C22⋊Q8×C53—22no—
19C2×C4×Q8×C53C2211yescharacteristic
20C2×C4×D4×C53C412yes—
21C2×C4×D4×C53C2211yescharacteristic
22C22×C4⋊C4×C53—21no—
23C2×#11719917—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322—22no—
27#13133136—42no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30C22×Q8×C53—21no—
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53C2×C411yescharacteristic
33C23×C4×C53—21no—
34C23×C4×C53—22no—
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53D414yes—
37#125×C53—24no—
38#125×C53—24no—
39C22⋊Q8×C53—216no—
40C22⋊Q8×C53—28no—
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C4×D4×C53—24no—
44C4×D4×C53—22no—
45C4×D4×C53—24no—
46C4×D4×C53D414yes—
47C2×C4⋊C4×C53—28no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53C2×C412yes—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D412yes—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—21no—
57C2×C42×C53C2×C412yes—
58C2×C42×C53C2311yescharacteristic
59C24×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—24no—
62C2×Q8×C53—22no—
63C2×Q8×C53—21no—
64C2×Q8×C53—21no—
65C2×Q8×C53—82no—
66C2×D4×C53—24no—
67C2×D4×C53—21no—
68C2×D4×C53C22⋊C412yes—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—28no—
72C22×C4×C53—24no—
73C22×C4×C53—42no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—84no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22⋊C412yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C4⋊C4×C53—28no—
83C4⋊C4×C53—28no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53C22⋊C414yes—
90C4⋊C4×C53C22⋊C412yes—
91C22⋊C4×C53—28no—
92C22⋊C4×C53—24no—
93C42×C53—22no—
94C42×C53—22no—
95C42×C53C2×D412yes—
96C42×C53C2×D412yes—
97#11719917—42no—
98#11719917—42no—
99C2×#88322—42no—
100C2×#88322—42no—
101C23×C53—24no—
102C23×C53—22no—
103C23×C53—42no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107Q8×C53—42no—
108Q8×C53—42no—
109Q8×C53—84no—
110D4×C53—24no—
111D4×C53—22no—
112D4×C53#10114yescomplemented
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—24no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—44no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—44no—
125C2×C4×C53—44no—
126C2×C4×C53—42no—
127C2×C4×C53—88no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53#12012yes—
131C2×C4×C53#12012yes—
132C2×C4×C53#13512yes—
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135#88322—82no—
136#88322—82no—
137#8398—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—24no—
141C22×C53—44no—
142C22×C53—44no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53#37911yescharacteristic
146C22×C53#41611yescharacteristic
147C22×C53#41711yescharacteristic
148C4×C53—22no—
149C4×C53—21no—
150C4×C53—44no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—42no—
154C4×C53C2×#10112yes—
155C4×C53#44912yes—
156C4×C53#45212yes—
157C2×#554D5311yescharacteristic
158#1801—531no—
159#1549—532no—
160#1486—532no—
161#1461—531no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—44no—
165C2×C53#178411yescharacteristic
166C2×C53#178811yescharacteristic
167C2×C53#179511yescharacteristic
168#554—24no—
169#554—24no—
170C2×C4⋊Q8C2×D5311yescharacteristic
171C2×#125Dic5312yes—
172C2×C22⋊Q8—21no—
173C2×C22⋊Q8—22no—
174C2×C4×Q8C2×D5311yescharacteristic
175C2×C4×D4Dic5312yes—
176C2×C4×D4C2×D5311yescharacteristic
177C22×C4⋊C4—21no—
178#431—532no—
179#423—532no—
180C2×#101—1061no—
181#343—1061no—
182#332—1064no—
183D4⋊C8—1064no—
184#324—1062no—
185C53#839811yescharacteristic, complemented, Sylow
186C22×Q8—21no—
187C22×Q8C22⋊D5311yescharacteristic
188C22×D4C2×Dic5311yescharacteristic
189C23×C4—21no—
190C23×C4—22no—
191C2×M16—1062no—
192C4⋊Q8—22no—
193C4⋊Q8C22⋊D5314yes—
194#125—24no—
195#125—24no—
196C22⋊Q8—28no—
197C22⋊Q8—216no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4C22⋊D5314yes—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C2×Dic5312yes—
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C4⋊C4C22⋊D5312yes—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C42C2×Dic5312yes—
215C2×C42C22×D5311yescharacteristic
216C4⋊C8—1062no—
217C4⋊C8—1062no—
218C4⋊C8—1062no—
219#101—2122no—
220C22⋊C8—1062no—
221C4×C8—1062no—
222C24—21no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—82no—
229C2×D4—24no—
230C2×D4—21no—
231C2×D4C22⋊Dic5312yes—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—28no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—84no—
240C22×C4C22⋊Dic5312yes—
241C22×C4C22⋊Dic5312yes—
242C22×C4C22×Dic5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—2122no—
246M16—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4C22⋊Dic5314yes—
257C4⋊C4C22⋊Dic5312yes—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C42—22no—
261C42—22no—
262C42C2×C22⋊D5312yes—
263C42C2×C22⋊D5312yes—
264C23—24no—
265C23—22no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊Dic5311yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—84no—
273D4—24no—
274D4—22no—
275D4#1313313614yescomplemented
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4C2×C22⋊Dic5311yescharacteristic
292C2×C4C2×C22⋊Dic5311yescharacteristic
293C2×C4#1313318612yes—
294C2×C4#1313318612yes—
295C2×C4#1313323712yes—
296C2×C4#1313324211yescharacteristic
297C2×C4#1313324711yescharacteristic
298C8—4242no—
299C8—4242no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22#1681047911yescharacteristic
308C22#1681061911yescharacteristic
309C22#1681062411yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4C2×#1313313612yes—
317C4#1681076812yes—
318C4#1681078512yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2#2027463911yescharacteristic
323C2#2027465511yescharacteristic
324C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#554C2
socleC22×C53#379
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenterC2×#554D53
hypocenterC53#8398
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #20274711C2semidirect
2. #20273400C2semidirect
4. #20273039C2semidirect
112. D4×C53#101semidirect
185. C53#8398semidirect
275. D4#13133136semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202734002
#202731402
#202730392
C2×C53×#5542
#839853