Finite Group Databasebeta

← #27294491

Subgroups — #27294491

Order 13568.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#13133136C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272742C212yesmaximal
4#20272721C212yesmaximal
5C53×#2634C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×#125×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#400—21no—
15C53×#400C411yescharacteristic
16C53×#400C2211yescharacteristic
17C53×#398C411yescharacteristic
18C53×#398C2211yescharacteristic
19C53×#395—21no—
20C53×#387—21no—
21C53×#387—21no—
22C53×#383C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28C4×#88322—41no—
29C4×#88322—41no—
30C22×Q8×C53D411yescharacteristic
31#125×C53—41no—
32#125×C53—41no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—21no—
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87#11719917—24no—
88#11719917—42no—
89C2×#88322—42no—
90C2×#88322—42no—
91C23×C53C2×C22⋊C411yescharacteristic
92Q8×C53—44no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
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—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53#9911yescharacteristic
120C2×C4×C53#10412yes—
121C2×C4×C53#13512yes—
122#88322—44no—
123#88322—82no—
124#11740—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53C2×#9911yescharacteristic
128C22×C53#41611yescharacteristic
129C22×C53C2×#10411yescharacteristic
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—81no—
139C4×C53—81no—
140C4×C53—81no—
141C4×C53—81no—
142#2634D5311yescharacteristic
143C4×#101—531no—
144#1461—531no—
145#1353—532no—
146#1346—532no—
147C2×C53—21no—
148C2×C53—21no—
149C2×C53#150111yescharacteristic
150C2×C53#212311yescharacteristic
151C2×C53#213311yescharacteristic
152C2×#125Dic5311yescharacteristic
153C2×C4×Q8C2×D5311yescharacteristic
154#431—1061no—
155C2×#101—1061no—
156C4×M16—1061no—
157#400—21no—
158#400Dic5311yescharacteristic
159#400C2×D5311yescharacteristic
160#398Dic5311yescharacteristic
161#398C2×D5311yescharacteristic
162#395—21no—
163#387—21no—
164#387—21no—
165#383Dic5311yescharacteristic
166#343—532no—
167#324—532no—
168#324—1061no—
169C53#1174011yescharacteristic, complemented, Sylow
170C22×Q8C22⋊D5311yescharacteristic
171C2×M16—1062no—
172#125—41no—
173#125—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C2×Dic5311yescharacteristic
181C2×C4⋊C4C2×Dic5311yescharacteristic
182C2×C4⋊C4C2×Dic5311yescharacteristic
183C2×C4⋊C4C22⋊D5311yescharacteristic
184C2×C4⋊C4C22×D5311yescharacteristic
185C2×C42C2×Dic5311yescharacteristic
186C2×C42C22⋊D5311yescharacteristic
187C2×C42C22⋊D5311yescharacteristic
188C4⋊C8—2121no—
189#101—1064no—
190C22⋊C8—1062no—
191#97—2121no—
192C4×C8—2121no—
193C4×C8—2121no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2×Dic5311yescharacteristic
203#95C2×Dic5311yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C22⋊Dic5311yescharacteristic
212C22×C4C22⋊Dic5311yescharacteristic
213C22×C4C22⋊Dic5311yescharacteristic
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22×Dic5311yescharacteristic
216C22×C4C2×C22⋊D5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218M16—1064no—
219M16—2122no—
220C2×C8—2122no—
221C2×C8—2122no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23C2×C22⋊Dic5311yescharacteristic
239Q8—44no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4#1313313211yescharacteristic
266C2×C4#1313313211yescharacteristic
267C2×C4#1313314412yes—
268C2×C4#1313323712yes—
269C8—4242no—
270C8—2124no—
271C22—21no—
272C22—21no—
273C22C2×#1313313211yescharacteristic
274C22#1681061911yescharacteristic
275C22C2×#1313314411yescharacteristic
276C4—22no—
277C4—22no—
278C4—21no—
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C2—21no—
289C2—21no—
290C2#2027320611yescharacteristic
291C2#2027636911yescharacteristic
292C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2634C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2634D53
hypocenterC53#11740
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2634

Complements

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

NH ≅ G/Ntype
169. C53#11740semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730392
#202727422
#202727212
C53×#26342
#1174053