Finite Group Databasebeta

← #27279131

Subgroups — #27279131

Order 13568.

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

normal subgroups only

This group has 616 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×#16810562C211yesmaximal, characteristic
2#20274803C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8#16810562—22no—
9#16810562C414yescomplemented
10C2×#13133124—21no—
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—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—24no—
30#13133124—24no—
31#13133124—21no—
32#13133124C2×C412yes—
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36#117×C53—42no—
37#117×C53—42no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D414yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46C2×C42×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—24no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53SD1612yes—
62C22×C4×C53Q1612yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4112yes—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C22×Dic53—44no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C4⋊C4×C53—24no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—44no—
79C4⋊C4×C53—44no—
80C4⋊C4×C53—82no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C42×C53—44no—
84C4⋊Dic53—24no—
85C4⋊Dic53—42no—
86C2×#88322—44no—
87C2×#88322—42no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53C2×C22⋊C411yescharacteristic
91C23×C53#12011yescharacteristic
92C23×C53#12311yescharacteristic
93C23×C53C2×SD1611yescharacteristic
94C23×C53C2×Q1611yescharacteristic
95C23×C53AGL(1,ℤ/8)11yescharacteristic
96C23×C53#13711yescharacteristic
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—24no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—44no—
111C2×C4×C53—44no—
112C2×C4×C53—44no—
113C2×C4×C53—81no—
114C2×C4×C53#10314yes—
115C2×C4×C53C4×D412yes—
116C2×C4×C53#12112yes—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—44no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—82no—
123C2×Dic53—82no—
124C2×Dic53—82no—
125C2×Dic53—82no—
126#88322—82no—
127#9023—531nomaximal, Sylow
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53#38611yescharacteristic
137C22×C53C2×#10311yescharacteristic
138C22×C53#41711yescharacteristic
139C22×C53#45011yescharacteristic
140C22×C53#45111yescharacteristic
141C22×C53#45412yes—
142C22×C53#45812yes—
143C22×C53#48311yescharacteristic
144C22×C53#48411yescharacteristic
145C4×C53—24no—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—42no—
149C4×C53—84no—
150Dic53—82no—
151Dic53—82no—
152C2×#400—531no—
153#2316D5311yescharacteristic
154#1824—532no—
155C2×#324—531no—
156#1294—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53#161612yes—
162C2×C53#161712yes—
163C2×C53#187211yescharacteristic
164C2×C53#188811yescharacteristic
165C2×C53#189011yescharacteristic
166C2×#117—21no—
167C22×C4⋊C4—531no—
168C22×C4⋊C4C2×D5311yescharacteristic
169C2×C22⋊C8—532no—
170#400—534no—
171#400—1062no—
172#389—24no—
173#380—21no—
174#380C2×D5312yes—
175C2×#95—531no—
176C2×#95—1061no—
177C2×#95—1061no—
178#340—1064no—
179#324—1062no—
180C53#902311yescharacteristic, complemented, Sylow
181C23×C4—1061no—
182C23×C4—1061no—
183C23×C4C22⋊D5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22×D5311yescharacteristic
186C22×C8—1062no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—532no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C4×D5312yes—
198C2×C4⋊C4D21212yes—
199C2×C4⋊C4C22⋊D5314yes—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C22⋊C8—1062no—
203C22⋊C8—1062no—
204#95—22no—
205#95—22no—
206#95—532no—
207#95—1061no—
208#95—1064no—
209#95—1064no—
210C24C2×C22⋊D5311yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—2122no—
223C22×C4—2122no—
224C22×C4—2124no—
225C22×C4—2121no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4#1171992112yes—
230C22×C4#1171992112yes—
231C22×C4#1171992312yes—
232C22×C4#1171992512yes—
233C22×C4C2×C4×D5311yescharacteristic
234C22×C4C2×D21211yescharacteristic
235C22×C4#1171994512yes—
236C22×C4C2×C22⋊D5312yes—
237C2×C8—2124no—
238C2×C8—2122no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—82no—
244C4⋊C4—1064no—
245C4⋊C4—2122no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—44no—
249C23—22no—
250C23—22no—
251C23C2×#1171992111yescharacteristic
252C23#1313318611yescharacteristic
253C23#1313320011yescharacteristic
254C23C2×#1171992311yescharacteristic
255C23C2×#1171992511yescharacteristic
256C23#1313324311yescharacteristic
257C23#1313324811yescharacteristic
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—81no—
275C2×C4—4242no—
276C2×C4—4242no—
277C2×C4—4242no—
278C2×C4—4242no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2124no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4#1313314014yes—
285C2×C4C4×C22⋊D5312yes—
286C2×C4#1313319112yes—
287C8—4242no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22#1681050211yescharacteristic
297C22C2×#1313314011yescharacteristic
298C22#1681062211yescharacteristic
299C22#1681077411yescharacteristic
300C22#1681077911yescharacteristic
301C22#1681079612yes—
302C22#1681081312yes—
303C22#1681095311yescharacteristic
304C22#1681095811yescharacteristic
305C4—24no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—84no—
310C4—4242no—
311C4—4242no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#2027375212yes—
317C2#2027375712yes—
318C2#2027502211yescharacteristic
319C2#2027511811yescharacteristic
320C2#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#9023
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
9. #16810562C4semidirect
180. C53#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#168105622
#202748032
C2×#168103492
#202725932
C53×#23162
#902353