Finite Group Databasebeta

← #27272622

Subgroups — #27272622

Order 13568.

378 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 95 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#20283101C211yesmaximal, characteristic
2#20274830C211yesmaximal, characteristic
3#20274567C211yesmaximal, characteristic
4#20272539C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133146—21no—
9C2×#13133146—21no—
10C2×#13133127—21no—
11#16810522C411yescharacteristic
12#16810521C2211yescharacteristic
13#16810519C411yescharacteristic
14#16810509—21no—
15#16810508—21no—
16#16810505—21no—
17#16810504—21no—
18#16810488C411yescharacteristic
19#16810488C411yescharacteristic
20C42×Dic53C2211yescharacteristic
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40#13133146—24no—
41#13133146—42no—
42#13133127—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×C8×C53—21no—
51C22×C8×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56#95×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C2×C8×C53—22no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C42×C53—41no—
80C4⋊Dic53—21no—
81C4⋊Dic53—21no—
82C4⋊Dic53—42no—
83C4⋊Dic53—41no—
84#11719919—41no—
85#11719919—41no—
86#11719919—42no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C4×Dic53—22no—
94C4×Dic53—22no—
95C4×Dic53—41no—
96C2×#88322—22no—
97C2×#88322—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
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#13111yescharacteristic
116C2×C4×C53#13111yescharacteristic
117C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C53#13711yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53#10112yes—
138C2×Dic53#10412yes—
139C8×C53—42no—
140C8×C53—81no—
141#88322—42no—
142#88322—81no—
143#7697—531nomaximal, Sylow
144C22×C53#40811yescharacteristic
145C22×C53#41011yescharacteristic
146C22×C53C2×#10111yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#41711yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—81no—
158C4×C53—81no—
159Dic53—22no—
160Dic53—22no—
161Dic53—41no—
162Dic53—41no—
163Dic53—81no—
164Dic53—81no—
165#2864—531no—
166#1831—531no—
167#1767—531no—
168#1271—531no—
169#1271—531no—
170#1271—531no—
171#1271D5311yescharacteristic
172C2×C53#147011yescharacteristic
173C2×C53#148311yescharacteristic
174C2×C53#149511yescharacteristic
175C2×C53#150911yescharacteristic
176C2×C53#151211yescharacteristic
177C2×C53#153111yescharacteristic
178C2×C53#156111yescharacteristic
179C2×C4⋊C8—1061no—
180C2×C4⋊C8—1061no—
181C2×#97—1061no—
182#389—531no—
183#389—531no—
184#389—531no—
185#389C2×D5311yescharacteristic
186#387—1061no—
187#387—1061no—
188#387—1061no—
189#387—1061no—
190#382—531no—
191#382—531no—
192C43—531no—
193#336—531no—
194#336—531no—
195#336—1061no—
196#336C2×D5311yescharacteristic
197#336C2×D5311yescharacteristic
198C53#769711yescharacteristic, complemented, Sylow
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—1061no—
202C22×C8—1061no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C4×D5311yescharacteristic
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C42—531no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42C22×D5311yescharacteristic
218C4⋊C8—1064no—
219C4⋊C8—2122no—
220#97—2122no—
221#95—21no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4C2×C4×D5311yescharacteristic
238C22×C4D4×D5311yescharacteristic
239C22×C4D4×D5311yescharacteristic
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—1062no—
247C2×C8—2121no—
248C2×C8—2121no—
249C2×C8—2121no—
250C2×C8—2121no—
251C2×C8—2121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—1061no—
258C4⋊C4—1061no—
259C4⋊C4—2121no—
260C4⋊C4—2122no—
261C4⋊C4—2122no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C42—41no—
265C42—1062no—
266C42—1062no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C42—1061no—
271C42—1061no—
272C42—1061no—
273C42—2121no—
274C23C22⋊C4×D5311yescharacteristic
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—532no—
288C2×C4—532no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4#1313322311yescharacteristic
306C2×C4#1313322511yescharacteristic
307C2×C4#1313324111yescharacteristic
308C2×C4#1313324411yescharacteristic
309C2×C4#1313324611yescharacteristic
310C2×C4#1313324911yescharacteristic
311C8—42no—
312C8—81no—
313C8—4241no—
314C8—2122no—
315C22#1681058811yescharacteristic
316C22#1681059511yescharacteristic
317C22#101×D5311yescharacteristic
318C22#1681062111yescharacteristic
319C22#1681062311yescharacteristic
320C22#104×D5311yescharacteristic
321C22#1681063811yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—81no—
329C4—81no—
330C4—1062no—
331C4—1062no—
332C4—4241no—
333C4—4241no—
334C4—2121no—
335C4—2121no—
336C2#2027306911yescharacteristic
337C2#2027312511yescharacteristic
338C2#2027317911yescharacteristic
339C2#2027323911yescharacteristic
340C2#2027325011yescharacteristic
341C2#2027332811yescharacteristic
342C2#2027345111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7697
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
198. C53#7697semidirect

Maximal subgroups

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

subgroupindex
#202831012
#202748302
#202745672
#202725392
#202725382
#202725372
C53×#12712
#769753