Finite Group Databasebeta

← #27278773

Subgroups — #27278773

Order 13568.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#20283800C211yesmaximal, characteristic, complemented
2#20273106C212yesmaximal
3#20273039C211yesmaximal, characteristic, complemented
4#20272548C212yesmaximal, complemented
5C4×#124×C53C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—22no—
8#16810676C2212yes—
9#16810676C2212yes—
10D4⋊#88322—24no—
11#16810349—22no—
12C2×#124×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C53×#391C411yescharacteristic
16C53×#390C411yescharacteristic
17C53×#388—22no—
18C53×#382C411yescharacteristic
19C4×C22⋊C4×C53—22no—
20C43×C53C2211yescharacteristic
21C4⋊#88322—24no—
22C4⋊#88322—22no—
23C4⋊#88322D414yes—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133128—24no—
27C4×#88322—22no—
28C4×#88322—22no—
29C22×Q8×C53C2×C411yescharacteristic
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32#124×C53—22no—
33#124×C53—22no—
34C4×Q8×C53—22no—
35C4×Q8×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—22no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D412yes—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53—21no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D412yes—
49C2×C42×C53C2311yescharacteristic
50#95×C53—22no—
51#95×C53—22no—
52C24×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—22no—
60C2×D4×C53M1612yes—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—44no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C4⋊C4×C53—22no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—22no—
75C22⋊C4×C53—28no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—42no—
78C42×C53—22no—
79C42×C53—22no—
80C42×C53—21no—
81C42×C53—22no—
82C42×C53—42no—
83C42×C53—42no—
84C42×C53M1612yes—
85C42×C53C2×D412yes—
86C42×C53#4112yes—
87C2×#88322—24no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—42no—
97D4×C53—24no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
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—41no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53#12112yes—
119C2×C4×C53#12311yescharacteristic
120C2×C4×C53C2×M1611yescharacteristic
121C2×C4×C53C2×M1611yescharacteristic
122C2×C4×C53#13112yes—
123#88322—44no—
124#88322—82no—
125#8964—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53#38611yescharacteristic
134C22×C53#40711yescharacteristic
135C22×C53#40811yescharacteristic
136C4×C53—22no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53—42no—
141C4×C53—42no—
142C4×C53—81no—
143C4×C53#43512yes—
144C4×C53#43612yes—
145C4×C53#45412yes—
146#2983—531no—
147C4×#124D5311yescharacteristic
148#1478—532no—
149#1461—531no—
150#1276—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#187011yescharacteristic
155C2×C53#188411yescharacteristic
156C2×C53#188811yescharacteristic
157C2×#124Dic5311yescharacteristic
158C2×C4×Q8C2×D5311yescharacteristic
159C2×C4×D4C2×D5311yescharacteristic
160#446—1064no—
161#432—1062no—
162#431—532no—
163#431—532no—
164#391Dic5311yescharacteristic
165#390Dic5311yescharacteristic
166#388—22no—
167#382Dic5311yescharacteristic
168C4×C22⋊C4—22no—
169C43C2×D5311yescharacteristic
170D4⋊C8—1064no—
171#324—1062no—
172C53#896411yescharacteristic, complemented, Sylow
173C22×Q8C2×Dic5311yescharacteristic
174C22×D4C2×Dic5311yescharacteristic
175C23×C4—21no—
176#124—22no—
177#124—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—22no—
181C4×D4—22no—
182C2×C4⋊C4C2×Dic5311yescharacteristic
183C2×C4⋊C4C2×Dic5311yescharacteristic
184C2×C4⋊C4C22⋊D5312yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C2×Dic5311yescharacteristic
191C2×C42C2×Dic5311yescharacteristic
192C2×C42C22⋊D5312yes—
193C2×C42C22×D5311yescharacteristic
194C4⋊C8—534no—
195C4⋊C8—1064no—
196C4⋊C8—1062no—
197C22⋊C8—1062no—
198C22⋊C8—1062no—
199#97—1064no—
200C4×C8—1062no—
201C4×C8—1062no—
202#95—22no—
203#95—22no—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4#1171991712yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—44no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4C22⋊Dic5312yes—
220C22×C4C22⋊Dic5312yes—
221C22×C4C22×Dic5311yescharacteristic
222C22×C4C2×C22⋊D5311yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224C2×C8—1064no—
225C2×C8—2122no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C22⋊C4—28no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C42—22no—
233C42—22no—
234C42—21no—
235C42—22no—
236C42—42no—
237C42—42no—
238C42#1171991712yes—
239C42D4×D5312yes—
240C42#1171994712yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C2×C22⋊Dic5311yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4D4×Dic5312yes—
269C2×C4#1313318511yescharacteristic
270C2×C4#1313318912yes—
271C2×C4#1313319811yescharacteristic
272C2×C4C2×#1171991711yescharacteristic
273C2×C4C2×#1171991711yescharacteristic
274C2×C4#1313322412yes—
275C8—4242no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1681050311yescharacteristic
285C22#1681058711yescharacteristic
286C22#1681059011yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—81no—
294C4#1681069512yes—
295C4#1681070112yes—
296C4#1681079512yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2027501311yescharacteristic
301C2#2027509911yescharacteristic
302C2#2027511911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#124×C53C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenterC4×#124D53
hypocenterC53#8964
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
3. #20273039C2semidirect
4. #20272548C2semidirect
172. C53#8964semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202731062
#202730392
#202725482
C4×#124×C532
#896453