Finite Group Databasebeta

← #27276736

Subgroups — #27276736

Order 13568.

778 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#13133208C211yesmaximal, characteristic
2#20274793C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20273044C212yesmaximal
5C53×#1463C211yesmaximal, characteristic
6C2×#13133208—24no—
7C2×#13133208—24no—
8C2×#13133208—22no—
9C2×#13133208—22no—
10C22×C4⋊Dic53C411yescharacteristic
11C22×C4⋊Dic53C411yescharacteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×C4×Dic53C411yescharacteristic
16#16810587C2212yes—
17#16810391—24no—
18#16810349—22no—
19#16810348—24no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C53×#407C2212yes—
22C53×#324—22no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C2×#11719917—22no—
28#13133208—216no—
29#13133208—216no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53D414yes—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919—22no—
45C2×#11719919—22no—
46C2×#11719919D414yes—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—22no—
50C22⋊#88322—22no—
51C22⋊#88322—22no—
52C23×C4×C53C2×C411yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53D412yes—
59C2×C4⋊C4×C53D412yes—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C24×C53C22×C411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—24no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D412yes—
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—42no—
85C22×Dic53—42no—
86C22×Dic53—42no—
87C22×Dic53—42no—
88C22×Dic53C22⋊C412yes—
89C22×Dic53C22⋊C412yes—
90C22×Dic53C22⋊C412yes—
91C22×Dic53C22⋊C412yes—
92M16×C53—44no—
93C2×C8×C53—42no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—24no—
96C4⋊Dic53—28no—
97C4⋊Dic53—28no—
98C4⋊Dic53—24no—
99C4⋊Dic53—24no—
100#11719919—28no—
101#11719919—24no—
102#11719919—24no—
103C4×Dic53—28no—
104#11719917—44no—
105C2×#88322—42no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53C2×C22⋊C411yescharacteristic
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—22no—
115C2×C4×C53—24no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53#10114yes—
125C2×C4×C53#12012yes—
126C2×C4×C53#12012yes—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—24no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C8×C53—82no—
144#88322—82no—
145#8554—531nomaximal, Sylow
146C22×C53—28no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53#37911yescharacteristic
154C22×C53#41012yes—
155C22×C53C2×#10112yes—
156C22×C53#41312yes—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—24no—
160C4×C53—42no—
161Dic53—42no—
162Dic53—42no—
163Dic53—42no—
164Dic53—42no—
165C22×#125—531no—
166#1821—532no—
167#1463—532no—
168#1463—531no—
169#1463D5311yescharacteristic
170C2×C53—22no—
171C2×C53—22no—
172C2×C53—22no—
173C2×C53#178211yescharacteristic
174C2×C53#178411yescharacteristic
175C2×C53#178511yescharacteristic
176C2×#125—1064no—
177C2×#125—1064no—
178C2×#125—1062no—
179C2×#125—1062no—
180C22×C4⋊C4—531no—
181C22×C4⋊C4—531no—
182C22×C4⋊C4—531no—
183C22×C4⋊C4—531no—
184C22×C4⋊C4—531no—
185C22×C4⋊C4C2×D5311yescharacteristic
186C22×C42—531no—
187#407—532no—
188#407C2×D5312yes—
189#343—1064no—
190#324—22no—
191#324—1064no—
192#324—1062no—
193C53#855411yescharacteristic, complemented, Sylow
194C23×C4—531no—
195C23×C4—531no—
196C23×C4—531no—
197C23×C4—531no—
198C23×C4C4×D5311yescharacteristic
199C23×C4C4×D5311yescharacteristic
200C23×C4C22×D5311yescharacteristic
201C2×M16—22no—
202C2×M16—1062no—
203#125—10616no—
204#125—10616no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—534no—
208C2×C4⋊C4—534no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4—1062no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1062no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1062no—
218C2×C4⋊C4—1062no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1062no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C4⋊C4D21212yes—
225C2×C4⋊C4C22⋊D5312yes—
226C2×C42—1062no—
227C2×C42—1062no—
228C2×C42—1062no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—1062no—
232C22⋊C8—1062no—
233C24C2×C4×D5311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—532no—
242C22×C4—532no—
243C22×C4—532no—
244C22×C4—532no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—2122no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4#1171992112yes—
258C22×C4#1171992112yes—
259C22×C4C2×D21211yescharacteristic
260C22×C4D4×D5312yes—
261C22×C4D4×D5312yes—
262C22×C4C2×C22⋊D5311yescharacteristic
263M16—44no—
264M16—2124no—
265C2×C8—42no—
266C2×C8—2122no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—1068no—
270C4⋊C4—1068no—
271C4⋊C4—1068no—
272C4⋊C4—1064no—
273C4⋊C4—1064no—
274C4⋊C4—1064no—
275C4⋊C4—1064no—
276C42—1068no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23C22⋊C4×D5311yescharacteristic
281C23C22⋊C4×D5311yescharacteristic
282C23C2×#1171992111yescharacteristic
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—1064no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—1062no—
299C2×C4—1062no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4#1313313514yes—
312C2×C4#1313318512yes—
313C2×C4#1313318712yes—
314C8—82no—
315C8—4242no—
316C22—28no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#1681047811yescharacteristic
324C22#1681059512yes—
325C22C2×#1313313512yes—
326C22#1681060612yes—
327C4—22no—
328C4—22no—
329C4—24no—
330C4—42no—
331C4—2122no—
332C4—2122no—
333C4—2122no—
334C4—2122no—
335C2—22no—
336C2—22no—
337C2—22no—
338C2#2027463111yescharacteristic
339C2#2027463811yescharacteristic
340C2#2027464211yescharacteristic
subgroupG/H
centerC22#16810478
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1463C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#1463D53
hypocenterC53#8554
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
193. C53#8554semidirect

Maximal subgroups

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

subgroupindex
C22×#131332082
#202747932
#202730452
#202730442
C53×#14632
#855453