Finite Group Databasebeta

← #27279151

Subgroups — #27279151

Order 13568.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#16810553C211yesmaximal, characteristic
2#20274799C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C22×#11719919—21no—
8C2×C22⋊#88322C2212yes—
9#16810553—22no—
10#16810553C414yescomplemented
11C2×#13133124—21no—
12C2×#13133124C2211yescharacteristic
13#16810385—24no—
14#16810349—22no—
15C2×#117×C53—21no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#389—24no—
18C53×#380—21no—
19C53×#380C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C22×#88322—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—24no—
28C2×#11719919—42no—
29C2×#11719919—42no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133124—24no—
33#13133124—21no—
34#13133124C2×C412yes—
35C23×C4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53C2311yescharacteristic
38#117×C53—42no—
39#117×C53—42no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C4⋊C4×C53D414yes—
46C2×C4⋊C4×C53D412yes—
47C2×C22⋊C4×C53—22no—
48C2×C42×C53—22no—
49#95×C53—22no—
50#95×C53—22no—
51C24×C53C2×D411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—24no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53D812yes—
64C22×C4×C53SD1612yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—24no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—42no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—44no—
84C4⋊C4×C53—44no—
85C4⋊C4×C53—82no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—44no—
89C4⋊Dic53—24no—
90C4⋊Dic53—42no—
91#11719919—48no—
92C2×#88322—44no—
93C2×#88322—42no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53#12011yescharacteristic
98C23×C53#12311yescharacteristic
99C23×C53C2×D811yescharacteristic
100C23×C53C2×SD1611yescharacteristic
101C23×C53AGL(1,ℤ/8)11yescharacteristic
102C23×C53#13711yescharacteristic
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—24no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—44no—
117C2×C4×C53—44no—
118C2×C4×C53—44no—
119C2×C4×C53—81no—
120C2×C4×C53#10214yes—
121C2×C4×C53C4×D412yes—
122C2×C4×C53#12112yes—
123C2×Dic53—24no—
124C2×Dic53—44no—
125C2×Dic53—44no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—81no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135#88322—82no—
136#9027—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—21no—
145C22×C53#38611yescharacteristic
146C22×C53C2×#10211yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#44711yescharacteristic
149C22×C53#44811yescharacteristic
150C22×C53#45612yes—
151C22×C53#45812yes—
152C22×C53#48011yescharacteristic
153C22×C53#48111yescharacteristic
154C4×C53—24no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—42no—
158C4×C53—84no—
159Dic53—44no—
160Dic53—82no—
161C2×#398—531no—
162#2316D5311yescharacteristic
163#1823—532no—
164C2×#324—531no—
165#1294—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53#161012yes—
171C2×C53#161112yes—
172C2×C53#187411yescharacteristic
173C2×C53#188611yescharacteristic
174C2×C53#189011yescharacteristic
175C2×#117—21no—
176C22×C4⋊C4—531no—
177C22×C4⋊C4—1061no—
178C22×C4⋊C4C2×D5311yescharacteristic
179C2×C22⋊C8—532no—
180#398—534no—
181#398—1062no—
182#389—24no—
183#380—21no—
184#380C2×D5312yes—
185C2×#95—531no—
186C2×#95—1061no—
187#340—1064no—
188#324—1062no—
189C53#902711yescharacteristic, complemented, Sylow
190C23×C4—1061no—
191C23×C4—1061no—
192C23×C4C22⋊D5311yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C22×C8—1062no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—532no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1064no—
207C2×C4⋊C4—2122no—
208C2×C4⋊C4—2122no—
209C2×C4⋊C4C4×D5312yes—
210C2×C4⋊C4D21212yes—
211C2×C4⋊C4C22⋊D5314yes—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C22⋊C8—1062no—
215C22⋊C8—1062no—
216#95—22no—
217#95—22no—
218#95—532no—
219#95—1061no—
220#95—1064no—
221C24C2×C22⋊D5311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—1064no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—2121no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4—2121no—
240C22×C4—2122no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4#1171992112yes—
244C22×C4#1171992112yes—
245C22×C4#1171992212yes—
246C22×C4#1171992412yes—
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4C2×D21211yescharacteristic
249C22×C4#1171994512yes—
250C22×C4C2×C22⋊D5312yes—
251C2×C8—2124no—
252C2×C8—2122no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—82no—
258C4⋊C4—1064no—
259C4⋊C4—2128no—
260C4⋊C4—2122no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—44no—
264C23—22no—
265C23—22no—
266C23C2×#1171992111yescharacteristic
267C23#1313318611yescharacteristic
268C23#1313320011yescharacteristic
269C23C2×#1171992211yescharacteristic
270C23C2×#1171992411yescharacteristic
271C23#1313324311yescharacteristic
272C23#1313324811yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—1064no—
291C2×C4—4241no—
292C2×C4—4241no—
293C2×C4—4241no—
294C2×C4—4241no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2124no—
299C2×C4—2124no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4#1313313714yes—
303C2×C4C4×C22⋊D5312yes—
304C2×C4#1313319112yes—
305C8—4242no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22#1681050211yescharacteristic
315C22C2×#1313313711yescharacteristic
316C22#1681062211yescharacteristic
317C22#1681075711yescharacteristic
318C22#1681076211yescharacteristic
319C22#1681080612yes—
320C22#1681081312yes—
321C22#1681093611yescharacteristic
322C22#1681094111yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—84no—
328C4—4242no—
329C4—2124no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2027371812yes—
335C2#2027372312yes—
336C2#2027503211yescharacteristic
337C2#2027510811yescharacteristic
338C2#2027512811yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2316C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2316D53
hypocenterC53#9027
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
10. #16810553C4semidirect
189. C53#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#168105532
#202747992
C2×#168103492
#202725932
C53×#23162
#902753