Finite Group Databasebeta

← #27295142

Subgroups — #27295142

Order 13568.

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

normal subgroups only

This group has 460 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#20272721C214yesmaximal, complemented
4C53×#2486C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133134—21no—
7C2×C22⋊#88322—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×#123×C53—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C53×#398C2212yes—
15C53×#397C412yes—
16C53×#389—21no—
17C53×#381—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
31#123×C53—42no—
32#123×C53—42no—
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53—41no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—22no—
48#95×C53—22no—
49#95×C53C2×C412yes—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C22⋊C411yescharacteristic
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53D812yes—
55C2×D4×C53SD1612yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—42no—
83C42×C53—21no—
84C42×C53—21no—
85#11719917—42no—
86#11719917—42no—
87C2×#88322—42no—
88C2×#88322—42no—
89C2×#88322—82no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53#10212yes—
99C23×C53#10212yes—
100C23×C53C2×C22⋊C411yescharacteristic
101D4×C53—42no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—81no—
122C2×C4×C53—84no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53C2×D811yescharacteristic
126C2×C4×C53C2×SD1611yescharacteristic
127#88322—82no—
128#88322—82no—
129#11881—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—81no—
142C22×C53C2×#9911yescharacteristic
143C22×C53C2×#10211yescharacteristic
144C22×C53#42111yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—82no—
152C4×C53—82no—
153C4×C53#35212yes—
154#2486D5311yescharacteristic
155#1800—531no—
156#1459—531no—
157#1346—534no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—42no—
161C2×C53—42no—
162C2×C53#151911yescharacteristic
163C2×C53C2×#35211yescharacteristic
164C2×C53#213511yescharacteristic
165C2×#123—21no—
166C2×C4×D4C2×D5311yescharacteristic
167C22×C4⋊C4Dic5311yescharacteristic
168#423—1061no—
169C2×#100—1061no—
170C2×C22⋊C8—1061no—
171#398C2×D5312yes—
172#397Dic5312yes—
173#389—21no—
174#381—21no—
175#380—21no—
176#380Dic5311yescharacteristic
177#343—532no—
178#324—532no—
179#323—1061no—
180C53#1188111yescharacteristic, complemented, Sylow
181C22×D4C22⋊D5311yescharacteristic
182C23×C4—21no—
183C23×C4C2×Dic5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C2×M16—1062no—
186C22×C8—2121no—
187#123—42no—
188#123—42no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4C2×Dic5312yes—
197C2×C4⋊C4C22×D5311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C2×Dic5311yescharacteristic
201C2×C22⋊C4C22⋊D5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#100—2122no—
206C22⋊C8—1062no—
207C22⋊C8—2121no—
208#95—22no—
209#95—22no—
210#95C2×Dic5312yes—
211C24C22⋊Dic5311yescharacteristic
212C24C22⋊Dic5311yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1171992212yes—
216C2×D4#1171992312yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22×Dic5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—4242no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1313313812yes—
260C23#1313313812yes—
261C23C2×C22⋊Dic5311yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—84no—
284C2×C4#1313313211yescharacteristic
285C2×C4#1313313211yescharacteristic
286C2×C4C2×#1171992211yescharacteristic
287C2×C4C2×#1171992311yescharacteristic
288C8—4242no—
289C8—4242no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1313313211yescharacteristic
303C22C2×#1313313811yescharacteristic
304C22#1681064011yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1681041312yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2027328311yescharacteristic
319C2C2×#1681041311yescharacteristic
320C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2486C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#2486D53
hypocenterC53#11881
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
180. C53#11881semidirect

Maximal subgroups

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

subgroupindex
#202747072
#202730332
#202727212
C53×#24862
#1188153