Finite Group Databasebeta

← #27276229

Subgroups — #27276229

Order 13568.

1038 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 147 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#20274639C211yesmaximal, characteristic, complemented
2#20273219C214yesmaximal, complemented
3#20272986C211yesmaximal, characteristic, complemented
4C53×#3486C211yesmaximal, characteristic
5C2×#13133136—22no—
6#16810587C2212yes—
7C2×#13133124—22no—
8C2×#13133124C2212yes—
9#16810360—24no—
10#16810357—24no—
11C2×2−1+4×C53C411yescharacteristic, complemented
12C23×Q8×C53C2211yescharacteristic
13D4×Q8×C53—24no—
14C53×#548—24no—
15C53×#536—22no—
16C53×#536C412yescomplemented
17C53×#525—22no—
18C53×#525C2212yes—
19C53×#519C411yescharacteristic, complemented
20C23×Dic53—22no—
21C2×#11719917—22no—
22#13133136—44no—
23C22⋊#88322—24no—
24#13133124—24no—
25#13133124—44no—
262−1+4×C53—44no—
27C2×#41×C53—22no—
28C2×#41×C53—22no—
29C2×#41×C53C2×C412yes—
30C22×Q8×C53—22no—
31C22×Q8×C53—24no—
32C22×Q8×C53—44no—
33C22×Q8×C53C2×C411yescharacteristic
34C22×Q8×C53C2×C411yescharacteristic
35C22×Q8×C53D412yes—
36C23×C4×C53D412yes—
37C23×C4×C53C2311yescharacteristic
38C4⋊Q8×C53—24no—
39C4⋊Q8×C53—22no—
40#124×C53—24no—
41#124×C53—22no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53—24no—
45C22⋊Q8×C53—28no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53D414yes—
48#121×C53—24no—
49#121×C53D414yes—
50#120×C53—22no—
51C4×Q8×C53—24no—
52C4×D4×C53—28no—
53C4×D4×C53—24no—
54#117×C53—22no—
55#117×C53C2×C412yes—
56C24×C53C2×D411yescharacteristic
57#41×C53—44no—
58#41×C53—44no—
59#41×C53—48no—
60#41×C53—44no—
61C2×Q8×C53—22no—
62C2×Q8×C53—24no—
63C2×Q8×C53—24no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—24no—
67C2×Q8×C53—42no—
68C2×Q8×C53—42no—
69C2×Q8×C53—416no—
70C2×Q8×C53—48no—
71C2×Q8×C53C22⋊C412yes—
72C2×D4×C53—22no—
73C2×D4×C53—22no—
74C2×D4×C53—24no—
75C2×D4×C53C22⋊C412yes—
76C22×C4×C53—24no—
77C22×C4×C53—24no—
78C22×C4×C53—24no—
79C22×C4×C53—22no—
80C22×C4×C53—42no—
81C22×C4×C53—42no—
82C22×C4×C53C22⋊C412yes—
83C22×C4×C53C22⋊C412yes—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D414yes—
87C22×Dic53—24no—
88C22×Dic53—44no—
89C22×Dic53—42no—
90C22×Dic53—44no—
91C22×Dic53—42no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53—28no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53C22⋊C414yes—
96C22⋊C4×C53—28no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—24no—
99C22⋊C4×C53—24no—
100C42×C53—24no—
101C42×C53—22no—
102#11719917—44no—
103C2×#88322—42no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12012yes—
110Q8×C53—28no—
111Q8×C53—48no—
112Q8×C53—44no—
113Q8×C53—44no—
114Q8×C53—42no—
115D4×C53—42no—
116D4×C53—44no—
117D4×C53—44no—
118D4×C53—44no—
119D4×C53—44no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—24no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—44no—
130C2×C4×C53—48no—
131C2×C4×C53—48no—
132C2×C4×C53—44no—
133C2×C4×C53#9914yes—
134C2×C4×C53C2×C22⋊C412yes—
135C2×C4×C53#12012yes—
136C2×Dic53—48no—
137C2×Dic53—44no—
138C2×Dic53—84no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141#88322—82no—
142#8455—531nomaximal, Sylow
143C22×C53—24no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—44no—
150C22×C53—42no—
151C22×C53#37911yescharacteristic
152C22×C53C2×#9912yes—
153C22×C53#45412yes—
154C22×C53#45612yes—
155C4×C53—22no—
156C4×C53—24no—
157C4×C53—22no—
158C4×C53—44no—
159C4×C53—42no—
160C4×C53—42no—
161Dic53—84no—
162#3486D5311yescharacteristic
163#1784—531no—
164#1504—534no—
165#1446—531no—
166C2×C53—22no—
167C2×C53—22no—
168C2×C53—22no—
169C2×C53—82no—
170C2×C53#178111yescharacteristic
171C2×C53#179312yes—
172C2×2−1+4Dic5311yescharacteristic, complemented
173C23×Q8C2×D5311yescharacteristic
174D4×Q8—24no—
175#548—24no—
176#536—22no—
177#536Dic5312yescomplemented
178#525—22no—
179#525C2×D5312yes—
180#519Dic5311yescharacteristic, complemented
181C2×#101—1062no—
182#407—532no—
183C2×#95—532no—
184C2×#95—1062no—
185#328—1064no—
186#327—1064no—
187C53#845511yescharacteristic, complemented, Sylow
1882−1+4—44no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41C2×Dic5312yes—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8C2×Dic5311yescharacteristic
196C22×Q8C2×Dic5311yescharacteristic
197C22×Q8C22⋊D5312yes—
198C23×C4—1062no—
199C23×C4C22⋊D5312yes—
200C23×C4C22×D5311yescharacteristic
201C2×M16—1062no—
202C4⋊Q8—24no—
203C4⋊Q8—22no—
204#124—24no—
205#124—22no—
206#123—22no—
207#123—24no—
208C22⋊Q8—24no—
209C22⋊Q8—28no—
210C22⋊Q8—24no—
211C22⋊Q8C22⋊D5314yes—
212#121—24no—
213#121C22⋊D5314yes—
214#120—22no—
215C4×Q8—24no—
216C4×D4—28no—
217C4×D4—24no—
218#117—22no—
219#117C2×Dic5312yes—
220#101—2124no—
221C22⋊C8—1064no—
222#95—1064no—
223#95—2124no—
224C24C2×C22⋊D5311yescharacteristic
225#41—44no—
226#41—44no—
227#41—48no—
228#41—44no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—416no—
238C2×Q8—48no—
239C2×Q8C22⋊Dic5312yes—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4C22⋊Dic5312yes—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1064no—
251C22×C4—2124no—
252C22×C4—2122no—
253C22×C4—2124no—
254C22×C4—2122no—
255C22×C4C22⋊Dic5312yes—
256C22×C4C22⋊Dic5312yes—
257C22×C4C22×Dic5311yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259C22×C4C2×C22⋊D5314yes—
260M16—2124no—
261C2×C8—2122no—
262C4⋊C4—24no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4C22⋊Dic5314yes—
266C22⋊C4—28no—
267C22⋊C4—22no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C42—24no—
271C42—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23C2×C22⋊Dic5311yescharacteristic
277C23#1313318612yes—
278Q8—28no—
279Q8—48no—
280Q8—42no—
281Q8—44no—
282Q8—44no—
283D4—42no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—44no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—4244no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—2128no—
305C2×C4—2124no—
306C2×C4#1313313214yes—
307C2×C4C2×C22⋊Dic5312yes—
308C2×C4#1313318612yes—
309C8—4242no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—44no—
318C22#1681047911yescharacteristic
319C22C2×#1313313212yes—
320C22#1681079612yes—
321C22#1681080612yes—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—44no—
328C4—4244no—
329C2—22no—
330C2—22no—
331C2—22no—
332C2—82no—
333C2#2027463011yescharacteristic
334C2#2027467412yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3486C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3486D53
hypocenterC53#8455
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #20274639C2semidirect
2. #20273219C2semidirect
3. #20272986C2semidirect
11. C2×2−1+4×C53C4semidirect
16. C53×#536C4semidirect
19. C53×#519C4semidirect
172. C2×2−1+4Dic53semidirect
177. #536Dic53semidirect
180. #519Dic53semidirect
187. C53#8455semidirect

Maximal subgroups

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

subgroupindex
#202746392
#202732192
#202729862
C53×#34862
#845553