Finite Group Databasebeta

← #27300896

Subgroups — #27300896

Order 13568.

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

normal subgroups only

This group has 580 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×#16810416C211yesmaximal, characteristic
2#20275114C211yesmaximal, characteristic
3#20274995C212yesmaximal, complemented
4#20272833C212yesmaximal, complemented
5C53×#2380C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810628—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810496—22no—
11#16810416C414yescomplemented
12#16810389—22no—
13#16810383—21no—
14C2×#127×C53C2211yescharacteristic
15C53×#386—21no—
16C53×#386—22no—
17C53×#386—22no—
18C53×#386C2212yes—
19C53×#383—21no—
20C2×#11719917—21no—
21D4×Dic53—42no—
22#13133176—21no—
23C2×#11719919—42no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26#13133138—42no—
27#13133134—22no—
28#13133132—22no—
29#13133132C2×C414yescomplemented
30#13133124—42no—
31C22×D4×C53—22no—
32C22×D4×C53D411yescharacteristic
33C22×D4×C53C2311yescharacteristic
34C23×C4×C53—21no—
35C23×C4×C53D412yes—
36#127×C53—22no—
37#127×C53C2×C412yes—
38#127×C53Q812yes—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53D412yes—
43C2×C42×C53D411yescharacteristic
44#95×C53—21no—
45#95×C53—22no—
46#95×C53—21no—
47#95×C53—22no—
48C24×C53—21no—
49C24×C53C2×D412yes—
50C2×D4×C53—24no—
51C2×D4×C53—24no—
52C2×D4×C53—21no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53—82no—
58C2×D4×C53—82no—
59C2×D4×C53C4⋊C412yescomplemented
60C2×D4×C53C22×C411yescharacteristic
61C2×D4×C53C2×Q811yescharacteristic
62C2×D4×C53#4112yes—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—42no—
67C22×C4×C53—82no—
68C22×C4×C53—82no—
69C22×C4×C53—81no—
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—42no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—82no—
76C22⋊C4×C53—82no—
77C22⋊C4×C53—81no—
78C42×C53—21no—
79C42×C53C4⋊C412yescomplemented
80C22⋊Dic53—24no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C22⋊Dic53—41no—
84C4⋊Dic53—22no—
85#11719919—84no—
86C4×Dic53—41no—
87#11719917—41no—
88#11719917—41no—
89C2×#88322—41no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C4×D412yes—
100C23×C53#12011yescharacteristic
101C23×C53C22⋊Q812yes—
102D4×C53—41no—
103D4×C53—41no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53—81no—
107D4×C53—84no—
108D4×C53—84no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—44no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—84no—
121C2×C4×C53—82no—
122C2×C4×C53C2×C4⋊C411yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—82no—
127C2×Dic53—82no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130#88322—81no—
131#12966—531nomaximal, Sylow
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—82no—
145C22×C53#38511yescharacteristic
146C22×C53B(4,𝔽2)11yescharacteristic
147C22×C53#45811yescharacteristic
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—42no—
151C4×C53—82no—
152C4×C53—82no—
153Dic53—84no—
154Dic53—82no—
155#2380D5311yescharacteristic
156C2×#353—531no—
157#1887—531no—
158#1867—532no—
159#1390—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53—42no—
164C2×C53—82no—
165C2×C53#186711yescharacteristic
166C2×C53#220012yes—
167C2×#127C2×D5311yescharacteristic
168#518—531no—
169#418—1061no—
170C2×#100—531no—
171C2×#99—532no—
172#386—21no—
173#386—22no—
174#386—22no—
175#386C2×D5312yes—
176#385—1062no—
177#383—21no—
178#353—534no—
179#342—1062no—
180#339—1061no—
181C53#1296611yescharacteristic, complemented, Sylow
182C22×D4—22no—
183C22×D4C22⋊D5311yescharacteristic
184C22×D4C22×D5311yescharacteristic
185C23×C4—21no—
186C23×C4C22⋊D5312yes—
187C2×M16—1061no—
188#127—22no—
189#127Dic10612yes—
190#127C4×D5312yes—
191C4×D4—2122no—
192#117—1061no—
193C2×C4⋊C4—2122no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—1062no—
198C2×C22⋊C4—1062no—
199C2×C22⋊C4C22⋊D5312yes—
200C2×C42C22⋊D5311yescharacteristic
201#102—2122no—
202#100—1062no—
203#99—534no—
204#99—1062no—
205#95—21no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—2122no—
210C24—21no—
211C24C2×C22⋊D5312yes—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4#1171991912yescomplemented
222C2×D4C2×Dic10611yescharacteristic
223C2×D4C2×C4×D5311yescharacteristic
224C2×D4#1171994512yes—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—81no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—2122no—
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—2121no—
237M16—2121no—
238C2×C8—2121no—
239C4⋊C4—1062no—
240C4⋊C4—4244no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C22⋊C4—81no—
245C22⋊C4—1064no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C22⋊C4—2121no—
249C42—21no—
250C42—2121no—
251C42#1171991912yescomplemented
252C23—22no—
253C23—22no—
254C23—22no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23C4×C22⋊D5312yes—
262C23#1313318611yescharacteristic
263C23#1313319612yes—
264D4—41no—
265D4—41no—
266D4—42no—
267D4—42no—
268D4—81no—
269D4—84no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—82no—
284C2×C4—4242no—
285C2×C4—4242no—
286C2×C4—4242no—
287C2×C4—4242no—
288C2×C4—2124no—
289C2×C4—2122no—
290C2×C4C2×#1171991911yescharacteristic
291C2×C4#1313320011yescharacteristic
292C8—4241no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—82no—
306C22#1681049611yescharacteristic
307C22#1681081011yescharacteristic
308C22#1681081311yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—82no—
313C4—82no—
314C4—4244no—
315C4—4242no—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2#2027499611yescharacteristic
322C2#2027672212yes—
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2380C2
socleC22×C53B(4,𝔽2)
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenter#2380D53
hypocenterC53#12966
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2380

Complements

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

NH ≅ G/Ntype
3. #20274995C2semidirect
4. #20272833C2semidirect
11. #16810416C4semidirect
29. #13133132C2×C4semidirect
59. C2×D4×C53C4⋊C4semidirect
79. C42×C53C4⋊C4semidirect
181. C53#12966semidirect
221. C2×D4#11719919semidirect
251. C42#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
#202751142
#202749952
#202728332
C53×#23802
#1296653