Finite Group Databasebeta

← #27294928

Subgroups — #27294928

Order 13568.

340 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274711C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272742C212yesmaximal
4#20272721C212yesmaximal
5C53×#2634C211yesmaximal, characteristic
6#16810680—21no—
7#16810647—21no—
8C2×#13133136—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—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—42no—
28C22⋊#88322—22no—
29#13133128—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—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89#11719917—42no—
90#11719917—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
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—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C53#13711yescharacteristic
126#88322—82no—
127#88322—82no—
128#11835—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41711yescharacteristic
133C22×C53#42111yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—81no—
144C4×C53—81no—
145C4×C53—81no—
146C4×C53—81no—
147#2634D5311yescharacteristic
148#1801—531no—
149#1467—531no—
150#1353—532no—
151#1346—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53#151311yescharacteristic
155C2×C53#212311yescharacteristic
156C2×C53#213311yescharacteristic
157C2×#125Dic5311yescharacteristic
158C2×C4×Q8C2×D5311yescharacteristic
159#432—1061no—
160#423—1061no—
161C2×#101—1061no—
162#400—21no—
163#400Dic5311yescharacteristic
164#400C2×D5311yescharacteristic
165#398Dic5311yescharacteristic
166#398C2×D5311yescharacteristic
167#395—21no—
168#387—21no—
169#387—21no—
170#383Dic5311yescharacteristic
171#343—532no—
172#324—532no—
173#324—1061no—
174C53#1183511yescharacteristic, complemented, Sylow
175C22×Q8C22⋊D5311yescharacteristic
176C2×M16—1062no—
177#125—41no—
178#125—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic5311yescharacteristic
186C2×C4⋊C4C2×Dic5311yescharacteristic
187C2×C4⋊C4C2×Dic5311yescharacteristic
188C2×C4⋊C4C22⋊D5311yescharacteristic
189C2×C4⋊C4C22×D5311yescharacteristic
190C2×C42C2×Dic5311yescharacteristic
191C2×C42C22⋊D5311yescharacteristic
192C2×C42C22⋊D5311yescharacteristic
193C4⋊C8—2121no—
194C4⋊C8—2121no—
195C4⋊C8—2121no—
196#101—2122no—
197C22⋊C8—1062no—
198#97—2121no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95C2×Dic5311yescharacteristic
208#95C2×Dic5311yescharacteristic
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22⋊Dic5311yescharacteristic
219C22×C4C22⋊Dic5311yescharacteristic
220C22×C4C22⋊Dic5311yescharacteristic
221C22×C4C22×Dic5311yescharacteristic
222C22×C4C2×C22⋊D5311yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224M16—2122no—
225M16—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C23C2×C22⋊Dic5311yescharacteristic
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4#1313313211yescharacteristic
275C2×C4#1313313211yescharacteristic
276C2×C4#1313324211yescharacteristic
277C2×C4#1313324711yescharacteristic
278C8—4242no—
279C8—4242no—
280C22—21no—
281C22—21no—
282C22C2×#1313313211yescharacteristic
283C22#1681062411yescharacteristic
284C22#1681064011yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C2—21no—
299C2—21no—
300C2#2027325811yescharacteristic
301C2#2027636911yescharacteristic
302C2#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#11835
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
174. C53#11835semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730592
#202727422
#202727212
C53×#26342
#1183553