Finite Group Databasebeta

← #27295237

Subgroups — #27295237

Order 13568.

440 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274707C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272742C214yesmaximal, complemented
4C53×#2487C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134—21no—
7C2×C22⋊#88322—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C22×C4⋊C4×C53C411yescharacteristic
13C53×#400C2212yes—
14C53×#399C412yes—
15C53×#389—21no—
16C53×#386—21no—
17C53×#385—21no—
18C53×#380—21no—
19C53×#380C411yescharacteristic
20C2×#11719917—22no—
21C22×#88322—41no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133134—42no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C22⋊C411yescharacteristic
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53SD1612yes—
53C2×D4×C53Q1612yes—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—82no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—82no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—82no—
79C42×C53—21no—
80C42×C53—21no—
81#11719917—42no—
82#11719917—42no—
83C2×#88322—42no—
84C2×#88322—42no—
85C2×#88322—82no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#10312yes—
95C23×C53#10312yes—
96C23×C53C2×C22⋊C411yescharacteristic
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—81no—
118C2×C4×C53—84no—
119C2×C4×C53#9911yescharacteristic
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53C2×SD1611yescharacteristic
122C2×C4×C53C2×Q1611yescharacteristic
123#88322—82no—
124#88322—82no—
125#11903—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—22no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—81no—
138C22×C53C2×#9911yescharacteristic
139C22×C53C2×#10311yescharacteristic
140C22×C53#42111yescharacteristic
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—82no—
148C4×C53—82no—
149C4×C53#35212yes—
150#2487D5311yescharacteristic
151#1800—531no—
152#1459—531no—
153#1353—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—42no—
157C2×C53—42no—
158C2×C53#152011yescharacteristic
159C2×C53C2×#35211yescharacteristic
160C2×C53#213411yescharacteristic
161C2×C4×D4C2×D5311yescharacteristic
162C22×C4⋊C4Dic5311yescharacteristic
163#423—1061no—
164C2×#100—1061no—
165C2×C22⋊C8—1061no—
166#400C2×D5312yes—
167#399Dic5312yes—
168#389—21no—
169#386—21no—
170#385—21no—
171#380—21no—
172#380Dic5311yescharacteristic
173#343—532no—
174#324—532no—
175#323—1061no—
176C53#1190311yescharacteristic, complemented, Sylow
177C22×D4C22⋊D5311yescharacteristic
178C23×C4—21no—
179C23×C4C2×Dic5311yescharacteristic
180C23×C4C22⋊D5311yescharacteristic
181C2×M16—1062no—
182C22×C8—2121no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×Dic5312yes—
191C2×C4⋊C4C22×D5311yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C2×Dic5311yescharacteristic
195C2×C22⋊C4C22⋊D5311yescharacteristic
196C2×C42C22⋊D5311yescharacteristic
197C4⋊C8—2121no—
198C4⋊C8—2121no—
199#100—2122no—
200C22⋊C8—1062no—
201C22⋊C8—2121no—
202#95—22no—
203#95—22no—
204#95C2×Dic5312yes—
205C24C22⋊Dic5311yescharacteristic
206C24C22⋊Dic5311yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#1171992412yes—
210C2×D4#1171992512yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4C22⋊Dic5311yescharacteristic
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22×Dic5311yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225C22×C4C2×C22⋊D5311yescharacteristic
226M16—2122no—
227M16—2122no—
228C2×C8—4242no—
229C2×C8—2122no—
230C2×C8—2122no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—82no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1313314112yes—
252C23#1313314112yes—
253C23C2×C22⋊Dic5311yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—84no—
276C2×C4#1313313211yescharacteristic
277C2×C4#1313313211yescharacteristic
278C2×C4C2×#1171992411yescharacteristic
279C2×C4C2×#1171992511yescharacteristic
280C8—4242no—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1313313211yescharacteristic
295C22C2×#1313314111yescharacteristic
296C22#1681064011yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—82no—
304C4—82no—
305C4#1681041312yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#2027328711yescharacteristic
311C2C2×#1681041311yescharacteristic
312C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2487C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#2487D53
hypocenterC53#11903
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #20272742C2semidirect
176. C53#11903semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730332
#202727422
C53×#24872
#1190353