Finite Group Databasebeta

← #27273270

Subgroups — #27273270

Order 13568.

378 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 85 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#20282679C211yesmaximal, characteristic
2#20276187C211yesmaximal, characteristic
3#20276124C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272540C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2096C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133140—21no—
10#16810556—21no—
11#16810555—21no—
12#16810544—21no—
13#16810530—21no—
14#16810529C2211yescharacteristic
15#16810521C2211yescharacteristic
16#16810520—21no—
17#16810506—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×#106×C53—21no—
23C2×#103×C53—21no—
24C53×#391C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C4×Dic53—21no—
39#13133150—42no—
40#13133140—42no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×Q8×C53D411yescharacteristic
48C22×C8×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C42×C53C2311yescharacteristic
54#106×C53—42no—
55#103×C53—42no—
56#95×C53—21no—
57#95×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53Q1612yes—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×Dic106—21no—
73C2×Dic106—21no—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×C8×C53—22no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C42×C53—41no—
87C4⋊Dic53—21no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—81no—
91C4⋊Dic53—81no—
92C4⋊Dic53SD1612yes—
93#11719919—41no—
94#11719919—41no—
95#11719919—81no—
96#11719919—81no—
97C4×Dic53—81no—
98C2×#88322—22no—
99C2×#88322—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C23×C53#12011yescharacteristic
105Q8×C53—42no—
106Q8×C53—81no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53C2×SD1611yescharacteristic
123C2×C4×C53C2×Q1611yescharacteristic
124C2×C4×C53#13511yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C53#13711yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140Dic106—42no—
141Dic106—81no—
142C8×C53—42no—
143#88322—42no—
144#88322—81no—
145#7809—531nomaximal, Sylow
146C22×C53#44811yescharacteristic
147C22×C53#44911yescharacteristic
148C22×C53#45111yescharacteristic
149C22×C53#45311yescharacteristic
150C22×C53#45511yescharacteristic
151C22×C53#45611yescharacteristic
152C22×C53#45811yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—42no—
160C4×C53—81no—
161C4×C53—161no—
162Dic53—42no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166#2806—531no—
167#2096D5311yescharacteristic
168#2083—531no—
169#2072—531no—
170#1272—531no—
171#1272—531no—
172#1271—531no—
173C2×C53#161111yescharacteristic
174C2×C53#161211yescharacteristic
175C2×C53#161411yescharacteristic
176C2×C53#162211yescharacteristic
177C2×C53#164511yescharacteristic
178C2×C53#164811yescharacteristic
179C2×C53#166111yescharacteristic
180C2×#107—1061no—
181C2×#106—21no—
182C2×#103—21no—
183C2×#103—1061no—
184#398—1061no—
185#398—1061no—
186#395—1061no—
187#391—531no—
188#391—1061no—
189#391C2×D5311yescharacteristic
190#389—531no—
191#389—1061no—
192#387—1061no—
193#384C2×D5311yescharacteristic
194#340—1061no—
195#340—1061no—
196#336—531no—
197#336—531no—
198#336C2×D5311yescharacteristic
199C53#780911yescharacteristic, complemented, Sylow
200C22×Q8—531no—
201C22×Q8C22⋊D5311yescharacteristic
202C22×C8—21no—
203C22×C8—1061no—
204C22×C8—1061no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C42—1061no—
218C2×C42C22×D5311yescharacteristic
219#107—2122no—
220#106—42no—
221#103—42no—
222#103—2122no—
223#95—21no—
224#95—21no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—1061no—
235C2×Q8—1061no—
236C2×Q8—2121no—
237C2×Q8—2121no—
238C2×Q8#1171992512yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4D4×D5311yescharacteristic
247C22×C4D4×D5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—1062no—
253C2×C8—2121no—
254C2×C8—2121no—
255C2×C8—2121no—
256C2×C8—2121no—
257C2×C8—2121no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—532no—
266C4⋊C4—1061no—
267C4⋊C4—4241no—
268C4⋊C4—4241no—
269C4⋊C4—4241no—
270C4⋊C4—4241no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C42—41no—
276C42—4241no—
277C23#1313318511yescharacteristic
278Q8—42no—
279Q8—81no—
280Q8—4241no—
281Q8—2122no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—1062no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4SD16×D5311yescharacteristic
310C2×C4C2×#1171992511yescharacteristic
311C2×C4#1313323811yescharacteristic
312C2×C4#1313324411yescharacteristic
313C2×C4#1313324611yescharacteristic
314C2×C4#1313324711yescharacteristic
315C8—42no—
316C8—4241no—
317C8—2122no—
318C22#1681076011yescharacteristic
319C22#1681076711yescharacteristic
320C22#1681077811yescharacteristic
321C22#1681079011yescharacteristic
322C22#1681079911yescharacteristic
323C22#1681080411yescharacteristic
324C22#1681081211yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—81no—
333C4—161no—
334C4—4241no—
335C4—4241no—
336C4—4241no—
337C4—2122no—
338C2#2027372111yescharacteristic
339C2#2027372811yescharacteristic
340C2#2027373911yescharacteristic
341C2#2027378411yescharacteristic
342C2#2027393511yescharacteristic
343C2#2027395611yescharacteristic
344C2#2027402911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2096C2
socle, prime radicalC23×C53#120
hypercenter#2096D53
hypocenterC53#7809
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2096

Complements

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

NH ≅ G/Ntype
199. C53#7809semidirect

Maximal subgroups

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

subgroupindex
#202826792
#202761872
#202761242
#202725412
#202725402
#202725372
C53×#20962
#780953