Finite Group Databasebeta

← #27273259

Subgroups — #27273259

Order 13568.

366 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 83 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#20282668C211yesmaximal, characteristic
2#20276257C211yesmaximal, characteristic
3#20276193C211yesmaximal, characteristic
4#20276186C211yesmaximal, characteristic
5#20272540C211yesmaximal, characteristic
6#20272539C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×#13133149—21no—
10C2×#13133149—21no—
11C2×#13133142—21no—
12#16810562—21no—
13#16810562—21no—
14#16810544—21no—
15#16810529C2211yescharacteristic
16#16810522—21no—
17#16810521C2211yescharacteristic
18#16810510—21no—
19#16810509—21no—
20#16810493C2211yescharacteristic
21#16810386—21no—
22#16810385—21no—
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106D411yescharacteristic
28C22×#88322—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—21no—
38C2×C4×Dic53—21no—
39#13133150—42no—
40#13133149—42no—
41#13133149—42no—
42#13133142—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C2×Dic106—21no—
69C2×Dic106—21no—
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×C8×C53—22no—
73C2×C8×C53—22no—
74C2×C8×C53—41no—
75C2×C8×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C4⋊C4×C53—21no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53SD1612yes—
83C42×C53—41no—
84C4⋊Dic53—21no—
85C4⋊Dic53—21no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94C4⋊Dic53—81no—
95C4⋊Dic53—81no—
96#11719919—41no—
97#11719919—41no—
98C4×Dic53—81no—
99C2×#88322—22no—
100C2×#88322—41no—
101C2×#88322—41no—
102C23×C53#12011yescharacteristic
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—22no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53C2×SD1611yescharacteristic
116C2×C4×C53#13511yescharacteristic
117C2×C4×C53#13511yescharacteristic
118C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C53#13711yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136Dic106—42no—
137Dic106—81no—
138C8×C53—42no—
139C8×C53—42no—
140#88322—42no—
141#7807—531nomaximal, Sylow
142C22×C53#44811yescharacteristic
143C22×C53#44911yescharacteristic
144C22×C53#45211yescharacteristic
145C22×C53#45311yescharacteristic
146C22×C53#45611yescharacteristic
147C22×C53#45611yescharacteristic
148C22×C53#45811yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—42no—
156C4×C53—81no—
157Dic53—81no—
158Dic53—81no—
159Dic53—81no—
160Dic53—81no—
161Dic53—161no—
162#2805—531no—
163#2096—531no—
164#2084—531no—
165#2083—531no—
166#1272—531no—
167#1271—531no—
168#1271D5311yescharacteristic
169C2×C53#160911yescharacteristic
170C2×C53#161111yescharacteristic
171C2×C53#161811yescharacteristic
172C2×C53#162211yescharacteristic
173C2×C53#165011yescharacteristic
174C2×C53#165311yescharacteristic
175C2×C53#166011yescharacteristic
176C2×#107—1061no—
177C2×#106—1061no—
178C2×#106—1061no—
179C2×#103—1061no—
180#400—1061no—
181#400—1061no—
182#395—1061no—
183#391—531no—
184#389—531no—
185#389—1061no—
186#389C2×D5311yescharacteristic
187#387—1061no—
188#387—1061no—
189#384—531no—
190#340—1061no—
191#340—1061no—
192#336—531no—
193#336C2×D5311yescharacteristic
194#336C2×D5311yescharacteristic
195C53#780711yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—1061no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4D21211yescharacteristic
211C2×C4⋊C4D21211yescharacteristic
212C2×C42—1061no—
213C2×C42C22×D5311yescharacteristic
214#107—2122no—
215#106—2122no—
216#106—2122no—
217#103—2122no—
218#95—21no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227C2×Q8—1061no—
228C2×Q8—1061no—
229C2×Q8—2121no—
230C2×Q8—2121no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—2121no—
238C22×C4C2×D21211yescharacteristic
239C22×C4D4×D5311yescharacteristic
240C22×C4D4×D5311yescharacteristic
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—1062no—
248C2×C8—2121no—
249C2×C8—2121no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—4241no—
257C4⋊C4—4241no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4#1171991312yes—
269C42—41no—
270C42—4241no—
271C23#1313318711yescharacteristic
272Q8—4241no—
273Q8—2122no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—4241no—
287C2×C4—4241no—
288C2×C4—4241no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
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×C4C2×#1171991311yescharacteristic
302C2×C4#1313323811yescharacteristic
303C2×C4#1313323811yescharacteristic
304C2×C4#1313324111yescharacteristic
305C2×C4#1313324911yescharacteristic
306C2×C4#1313325011yescharacteristic
307C8—42no—
308C8—42no—
309C8—2122no—
310C22#1681076411yescharacteristic
311C22#1681076911yescharacteristic
312C22#1681078811yescharacteristic
313C22#1681079211yescharacteristic
314C22#1681080711yescharacteristic
315C22#1681080811yescharacteristic
316C22#1681081411yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C4—8481no—
326C4—4241no—
327C4—4241no—
328C4—4241no—
329C4—4241no—
330C2#2027371311yescharacteristic
331C2#2027372511yescharacteristic
332C2#2027376611yescharacteristic
333C2#2027378911yescharacteristic
334C2#2027397211yescharacteristic
335C2#2027399411yescharacteristic
336C2#2027402811yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53#120
hypercenter#1271D53
hypocenterC53#7807
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
195. C53#7807semidirect

Maximal subgroups

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

subgroupindex
#202826682
#202762572
#202761932
#202761862
#202725402
#202725392
C53×#12712
#780753