Finite Group Databasebeta

← #27273313

Subgroups — #27273313

Order 13568.

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

normal subgroups only

This group has 362 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#20276272C211yesmaximal, characteristic
3#20276124C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic
5#20272540C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#2072C211yesmaximal, characteristic
8C2×#13133140—21no—
9#16810556—21no—
10#16810555—21no—
11#16810544—21no—
12#16810530—21no—
13#16810529C2211yescharacteristic
14#16810521C2211yescharacteristic
15#16810520—21no—
16#16810506—21no—
17#16810385—21no—
18#16810385—21no—
19#16810385—21no—
20#16810378C2211yescharacteristic
21#16810378C2211yescharacteristic
22C2×#103×C53—21no—
23C53×#391C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#340—21no—
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#13133140—42no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53D411yescharacteristic
47C22×C8×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53#103×C53—42no—
54#95×C53—21no—
55#95×C53—21no—
56C2×Q8×C53—21no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53SD1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—41no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C2×Dic106—21no—
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×C8×C53—41no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—41no—
84C4⋊Dic53—21no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—81no—
88C4⋊Dic53—81no—
89C4⋊Dic53Q1612yes—
90#11719919—41no—
91#11719919—41no—
92#11719919—81no—
93#11719919—81no—
94C4×Dic53—81no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C2×#88322—41no—
100C2×#88322—41no—
101C23×C53#12011yescharacteristic
102Q8×C53—42no—
103Q8×C53—81no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—81no—
117C2×C4×C53—81no—
118C2×C4×C53—81no—
119C2×C4×C53C2×SD1611yescharacteristic
120C2×C4×C53C2×Q1611yescharacteristic
121C2×C4×C53#13511yescharacteristic
122C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137Dic106—42no—
138Dic106—81no—
139C8×C53—81no—
140#88322—81no—
141#88322—81no—
142#7816—531nomaximal, Sylow
143C22×C53#44811yescharacteristic
144C22×C53#44911yescharacteristic
145C22×C53#45111yescharacteristic
146C22×C53#45411yescharacteristic
147C22×C53#45511yescharacteristic
148C22×C53#45511yescharacteristic
149C22×C53#45811yescharacteristic
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—42no—
157C4×C53—81no—
158C4×C53—161no—
159Dic53—42no—
160Dic53—81no—
161Dic53—81no—
162Dic53—81no—
163#2806—531no—
164#2099—531no—
165#2072—531no—
166#2072D5311yescharacteristic
167#1272—531no—
168#1272—531no—
169#1271—531no—
170C2×C53#160811yescharacteristic
171C2×C53#161211yescharacteristic
172C2×C53#161711yescharacteristic
173C2×C53#162611yescharacteristic
174C2×C53#164411yescharacteristic
175C2×C53#164511yescharacteristic
176C2×C53#166211yescharacteristic
177C2×#103—21no—
178C2×#103—1061no—
179#398—1061no—
180#398—1061no—
181#395—1061no—
182#391—531no—
183#391—1061no—
184#391C2×D5311yescharacteristic
185#389—531no—
186#389—1061no—
187#387—1061no—
188#384C2×D5311yescharacteristic
189#340—21no—
190#340—1061no—
191#340—1061no—
192#340—1061no—
193#336—531no—
194#336—531no—
195#336C2×D5311yescharacteristic
196C53#781611yescharacteristic, complemented, Sylow
197C22×Q8—531no—
198C22×Q8C22⋊D5311yescharacteristic
199C22×C8—21no—
200C22×C8—1061no—
201C22×C8—1061no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4C22⋊D5311yescharacteristic
214C2×C42—1061no—
215C2×C42C22×D5311yescharacteristic
216#103—42no—
217#103—2122no—
218#95—21no—
219#95—21no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226C2×Q8—21no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—1061no—
230C2×Q8—1061no—
231C2×Q8—2121no—
232C2×Q8—2121no—
233C2×Q8#1171992312yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—2121no—
248C2×C8—2121no—
249C2×C8—2121no—
250C2×C8—2121no—
251C2×C8—2121no—
252C2×C8—2121no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—532no—
260C4⋊C4—1061no—
261C4⋊C4—4241no—
262C4⋊C4—4241no—
263C4⋊C4—4241no—
264C4⋊C4—4241no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C42—41no—
270C42—4241no—
271C23#1313318511yescharacteristic
272Q8—42no—
273Q8—81no—
274Q8—4241no—
275Q8—2122no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—1062no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4C2×#1171992311yescharacteristic
304C2×C4Q16×D5311yescharacteristic
305C2×C4#1313323811yescharacteristic
306C2×C4#1313324411yescharacteristic
307C2×C4#1313324711yescharacteristic
308C2×C4#1313324911yescharacteristic
309C8—81no—
310C8—4241no—
311C8—4241no—
312C22#1681076111yescharacteristic
313C22#1681076711yescharacteristic
314C22#1681077711yescharacteristic
315C22#1681079411yescharacteristic
316C22#1681079911yescharacteristic
317C22#1681080011yescharacteristic
318C22#1681081211yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—81no—
327C4—161no—
328C4—4241no—
329C4—4241no—
330C4—4241no—
331C4—2122no—
332C2#2027370511yescharacteristic
333C2#2027372811yescharacteristic
334C2#2027375511yescharacteristic
335C2#2027381011yescharacteristic
336C2#2027392611yescharacteristic
337C2#2027393411yescharacteristic
338C2#2027403711yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2072C2
socle, prime radicalC23×C53#120
hypercenter#2072D53
hypocenterC53#7816
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2072

Complements

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

NH ≅ G/Ntype
196. C53#7816semidirect

Maximal subgroups

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

subgroupindex
#202826792
#202762722
#202761242
#202725412
#202725402
#202725372
C53×#20722
#781653