Finite Group Databasebeta

← #27300901

Subgroups — #27300901

Order 13568.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16810418C211yesmaximal, characteristic
2#20275104C211yesmaximal, characteristic
3#20274995C212yesmaximal
4#20272833C212yesmaximal
5C53×#2379C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810632—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810496—22no—
11#16810418C414yescomplemented
12#16810389—22no—
13#16810383—21no—
14C2×#124×C53C2211yescharacteristic
15C53×#391—21no—
16C53×#388—22no—
17C53×#386C2212yes—
18C53×#385—22no—
19C53×#382—21no—
20C2×#11719917—21no—
21D4×Dic53—41no—
22D4×Dic53—41no—
23#13133176—21no—
24C2×#11719919—42no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27#13133141—41no—
28#13133141—41no—
29#13133134—22no—
30#13133132—22no—
31#13133132C2×C414yes—
32#13133124—42no—
33C22×Q8×C53D411yescharacteristic
34C22×D4×C53C2311yescharacteristic
35C23×C4×C53D412yes—
36#124×C53—22no—
37#124×C53C2×C412yes—
38#124×C53Q812yes—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53D412yes—
45C2×C42×C53—21no—
46C2×C42×C53D411yescharacteristic
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—22no—
50C24×C53C2×D412yes—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53C4⋊C412yescomplemented
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C22×C411yescharacteristic
58C2×D4×C53C2×Q811yescharacteristic
59C2×D4×C53#4112yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—42no—
65C22×C4×C53—82no—
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—42no—
70C4⋊C4×C53—81no—
71C4⋊C4×C53—81no—
72C4⋊C4×C53—81no—
73C22⋊C4×C53—24no—
74C22⋊C4×C53—24no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—82no—
77C42×C53—21no—
78C42×C53—81no—
79C42×C53C4⋊C412yescomplemented
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—41no—
84C4⋊Dic53—21no—
85C4⋊Dic53—21no—
86#11719919—84no—
87C4×Dic53—41no—
88#11719917—41no—
89#11719917—41no—
90C2×#88322—41no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C4×D412yes—
96C23×C53#12011yescharacteristic
97C23×C53C22⋊Q812yes—
98Q8×C53—42no—
99Q8×C53—81no—
100D4×C53—41no—
101D4×C53—41no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—44no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—82no—
119C2×C4×C53C2×C4⋊C411yescharacteristic
120C2×C4×C53#12311yescharacteristic
121C2×Dic53—44no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—82no—
125C2×Dic53—82no—
126C2×Dic53—82no—
127C2×Dic53—82no—
128#88322—81no—
129#12967—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—82no—
139C22×C53#38511yescharacteristic
140C22×C53B(4,𝔽2)11yescharacteristic
141C22×C53#45811yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—82no—
147C4×C53—82no—
148Dic53—84no—
149Dic53—81no—
150Dic53—81no—
151#2379D5311yescharacteristic
152C2×#354—531no—
153#1885—531no—
154#1867—532no—
155#1390—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—42no—
160C2×C53#186711yescharacteristic
161C2×C53#219911yescharacteristic
162C2×C53#220111yescharacteristic
163C2×#124C2×D5311yescharacteristic
164#518—531no—
165#419—1061no—
166C2×#100—531no—
167C2×#99—532no—
168#391—21no—
169#388—22no—
170#386C2×D5312yes—
171#385—22no—
172#385—1062no—
173#382—21no—
174#354—534no—
175#342—1062no—
176#339—1061no—
177C53#1296711yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5311yescharacteristic
179C22×D4C22×D5311yescharacteristic
180C23×C4C22⋊D5312yes—
181C2×M16—1061no—
182#124—22no—
183#124Dic10612yes—
184#124C4×D5312yes—
185C4×D4—2121no—
186C4×D4—2121no—
187#117—1061no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—2122no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4C22⋊D5312yes—
197C2×C42—21no—
198C2×C42C22⋊D5311yescharacteristic
199#103—2121no—
200#103—2121no—
201#100—1062no—
202#99—534no—
203#99—1062no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—2122no—
208C24C2×C22⋊D5312yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#1171991912yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic10611yescharacteristic
216C2×D4C2×C4×D5311yescharacteristic
217C2×D4#1171994512yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—2122no—
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—2121no—
229M16—2121no—
230C2×C8—2121no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—1061no—
235C4⋊C4—1061no—
236C4⋊C4—4244no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—1064no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C22⋊C4—2121no—
245C42—21no—
246C42—81no—
247C42—2121no—
248C42#1171991912yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23C4×C22⋊D5312yes—
254C23#1313318611yescharacteristic
255C23#1313319612yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—4242no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—4242no—
281C2×C4—2124no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4C2×#1171991911yescharacteristic
285C2×C4#1313320011yescharacteristic
286C8—4241no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1681049611yescharacteristic
297C22#1681081011yescharacteristic
298C22#1681081311yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—82no—
305C4—4244no—
306C4—4241no—
307C4—4241no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027499611yescharacteristic
313C2#2027671611yescharacteristic
314C2#2027672811yescharacteristic
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2379C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenter#2379D53
hypocenterC53#12967
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2379

Complements

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

NH ≅ G/Ntype
11. #16810418C4semidirect
54. C2×Q8×C53C4⋊C4semidirect
79. C42×C53C4⋊C4semidirect
177. C53#12967semidirect
212. C2×Q8#11719919semidirect
248. C42#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202751042
#202749952
#202728332
C53×#23792
#1296753