Finite Group Databasebeta

← #27273379

Subgroups — #27273379

Order 13568.

358 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 91 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#20276278C211yesmaximal, characteristic
2#20276278C211yesmaximal, characteristic
3#20276278C211yesmaximal, characteristic
4#20272538C211yesmaximal, characteristic
5#20272538C211yesmaximal, characteristic
6#20272538C211yesmaximal, characteristic
7C53×#2726C211yesmaximal, characteristic
8#16810493C2211yescharacteristic
9#16810385—21no—
10#16810385—21no—
11#16810385—21no—
12#16810385—21no—
13#16810385—21no—
14#16810385—21no—
15#16810378C2211yescharacteristic
16#16810378C2211yescharacteristic
17#16810378C2211yescharacteristic
18C2×#125×C53—21no—
19C53×#398—21no—
20C53×#398—21no—
21C53×#398—21no—
22C53×#389—21no—
23C53×#389C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#389C2211yescharacteristic
26C53×#387—21no—
27C22×#88322—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33#125×C53—42no—
34#125×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2311yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—41no—
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53—21no—
69C4⋊C4×C53—21no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—81no—
83C4⋊C4×C53SD1612yes—
84C4⋊C4×C53Q1612yes—
85C42×C53—41no—
86C42×C53—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4⋊Dic53—42no—
90C2×#88322—41no—
91C2×#88322—41no—
92C2×#88322—41no—
93C2×#88322—41no—
94C2×#88322—41no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C2×#88322—41no—
99C23×C53#12011yescharacteristic
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—22no—
107C2×C4×C53—22no—
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—41no—
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—41no—
123C2×C4×C53—41no—
124C2×C4×C53C2×SD1611yescharacteristic
125C2×C4×C53C2×Q1611yescharacteristic
126C2×C4×C53#13511yescharacteristic
127C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C53#13711yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×Dic53—81no—
131C2×Dic53—81no—
132C2×Dic53—81no—
133#88322—81no—
134#88322—81no—
135#88322—81no—
136#7827—531nomaximal, Sylow
137C22×C53#45011yescharacteristic
138C22×C53#45111yescharacteristic
139C22×C53#45211yescharacteristic
140C22×C53#45411yescharacteristic
141C22×C53#45411yescharacteristic
142C22×C53#45411yescharacteristic
143C22×C53#45811yescharacteristic
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—42no—
151C4×C53—42no—
152C4×C53—81no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—161no—
157#2726D5311yescharacteristic
158#2100—531no—
159#2100—531no—
160#2100—531no—
161#1271—531no—
162#1271—531no—
163#1271—531no—
164C2×C53#161511yescharacteristic
165C2×C53#161611yescharacteristic
166C2×C53#161711yescharacteristic
167C2×C53#163511yescharacteristic
168C2×C53#163611yescharacteristic
169C2×C53#163711yescharacteristic
170C2×C53#165811yescharacteristic
171C2×#125—21no—
172#398—21no—
173#398—21no—
174#398—21no—
175#389—21no—
176#389C2×D5311yescharacteristic
177#389C2×D5311yescharacteristic
178#389C2×D5311yescharacteristic
179#387—21no—
180#384—531no—
181#340—1061no—
182#340—1061no—
183#340—1061no—
184#340—1061no—
185#340—1061no—
186#340—1061no—
187#336—531no—
188#336—531no—
189#336—531no—
190C53#782711yescharacteristic, complemented, Sylow
191C22×C8—1061no—
192C22×C8—1061no—
193C22×C8—1061no—
194#125—42no—
195#125—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4C22⋊D5311yescharacteristic
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C4⋊C4C22⋊D5311yescharacteristic
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C42—21no—
213C2×C42C22×D5311yescharacteristic
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—2121no—
227C22×C4C2×C22⋊D5311yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229C22×C4C2×C22⋊D5311yescharacteristic
230C2×C8—2121no—
231C2×C8—2121no—
232C2×C8—2121no—
233C2×C8—2121no—
234C2×C8—2121no—
235C2×C8—2121no—
236C2×C8—2121no—
237C2×C8—2121no—
238C2×C8—2121no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C4⋊C4#1171992312yes—
260C4⋊C4#1171992512yes—
261C42—41no—
262C42—42no—
263C23#1313318611yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—4241no—
289C2×C4—4241no—
290C2×C4—4241no—
291C2×C4C2×#1171992311yescharacteristic
292C2×C4C2×#1171992511yescharacteristic
293C2×C4#1313323711yescharacteristic
294C2×C4#1313324311yescharacteristic
295C2×C4#1313324711yescharacteristic
296C2×C4#1313324811yescharacteristic
297C8—4241no—
298C8—4241no—
299C8—4241no—
300C22#1681077411yescharacteristic
301C22#1681077911yescharacteristic
302C22#1681078511yescharacteristic
303C22#1681079611yescharacteristic
304C22#1681079611yescharacteristic
305C22#1681079611yescharacteristic
306C22#1681081311yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—8481no—
320C2#2027374611yescharacteristic
321C2#2027375211yescharacteristic
322C2#2027375711yescharacteristic
323C2#2027386611yescharacteristic
324C2#2027387311yescharacteristic
325C2#2027388011yescharacteristic
326C2#2027401711yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2726C2
socle, prime radicalC23×C53#120
hypercenter#2726D53
hypocenterC53#7827
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2726

Complements

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

NH ≅ G/Ntype
190. C53#7827semidirect

Maximal subgroups

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

subgroupindex
#202762782
#202725382
C53×#27262
#782753