Finite Group Databasebeta

← #27272172

Subgroups — #27272172

Order 13568.

566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274783C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3#20272538C214yesmaximal, complemented
4C2×C53×#389C211yesmaximal, characteristic
5C2×C22⋊#88322—21no—
6C2×C22⋊#88322—21no—
7#16810378—21no—
8#16810378—21no—
9#16810378C2212yes—
10#16810378C2212yes—
11C22×C4⋊C4×C53—21no—
12C22×C4⋊C4×C53C411yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#389—21no—
16C53×#389—21no—
17C53×#389C412yes—
18C53×#389C2212yes—
19C2×#95×C53—21no—
20C22×#88322—22no—
21C22×#88322—22no—
22C22⋊#88322—42no—
23C22⋊#88322—42no—
24C23×C4×C53—21no—
25C23×C4×C53—21no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—44no—
35C2×C4⋊C4×C53—42no—
36C2×C4⋊C4×C53C814yescomplemented
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C42×C53—42no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53D411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53C2311yescharacteristic
44#95×C53—22no—
45#95×C53—22no—
46C24×C53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—24no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C2×C812yes—
67C22×C4×C53M1612yes—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C4⋊C4×C53—24no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—44no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—44no—
77C4⋊C4×C53C2×C814yes—
78C42×C53—41no—
79C42×C53—41no—
80C42×C53—41no—
81C42×C53—41no—
82C2×#88322—42no—
83C2×#88322—42no—
84C2×#88322—42no—
85C2×#88322—42no—
86C2×#88322—42no—
87C2×#88322—42no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53C22⋊C812yes—
93C23×C53#9912yes—
94C23×C53#10112yes—
95C23×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53—24no—
97C2×C4×C53—24no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—24no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—44no—
114C2×C4×C53—44no—
115C2×C4×C53—48no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—84no—
122C2×C4×C53C22⋊C812yes—
123C2×C4×C53C22×C811yescharacteristic
124C2×C4×C53C2×M1611yescharacteristic
125C2×C4×C53AGL(1,ℤ/8)12yes—
126C2×C4×C53#13712yes—
127#88322—82no—
128#88322—82no—
129#7593—531nomaximal, Sylow
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53#32414yes—
137C22×C53C2×C22⋊C811yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53C2×#10111yescharacteristic
140C22×C53#41811yescharacteristic
141C22×C53#41911yescharacteristic
142C22×C53#42111yescharacteristic
143C22×C53#42111yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—22no—
148C4×C53—22no—
149C4×C53—44no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—82no—
153C2×#389D5311yescharacteristic
154#1817—531no—
155#1817—531no—
156#1271—534no—
157C2×C53—24no—
158C2×C53C2×#32411yescharacteristic
159C2×C53#147811yescharacteristic
160C2×C53#147911yescharacteristic
161C2×C53#151011yescharacteristic
162C2×C53#151111yescharacteristic
163C2×C53#153411yescharacteristic
164C2×C53#155611yescharacteristic
165C22×C4⋊C4—21no—
166C22×C4⋊C4Dic5311yescharacteristic
167C22×C4⋊C4Dic5311yescharacteristic
168C22×C42C2×D5311yescharacteristic
169C2×C22⋊C8—1061no—
170C2×C22⋊C8—1061no—
171#389—21no—
172#389—21no—
173#389Dic5312yes—
174#389C2×D5312yes—
175C2×#95—21no—
176#336—532no—
177#336—532no—
178#336—1061no—
179#336—1061no—
180C53#759311yescharacteristic, complemented, Sylow
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C2×Dic5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C22×C8—1062no—
187C22×C8—1062no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4#8832214yescomplemented
196C2×C4⋊C4C2×Dic5312yes—
197C2×C4⋊C4C2×Dic5312yes—
198C2×C42—42no—
199C2×C42C2×Dic5311yescharacteristic
200C2×C42C22⋊D5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C2×C42C22×D5311yescharacteristic
203C22⋊C8—2122no—
204C22⋊C8—2122no—
205#95—22no—
206#95—22no—
207C24C22⋊Dic5311yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C2×#8832212yes—
225C22×C4#1171991712yes—
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22×Dic5311yescharacteristic
230C22×C4C2×C22⋊D5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4C2×#8832214yes—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23C22⋊#8832212yes—
254C23#1313313212yes—
255C23#1313313612yes—
256C23C2×C22⋊Dic5311yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4C22⋊#8832212yes—
284C2×C4C22×#8832211yescharacteristic
285C2×C4C2×#1171991711yescharacteristic
286C2×C4#1313324312yes—
287C2×C4#1313324812yes—
288C8—4242no—
289C8—4242no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1681034914yes—
297C22C2×C22⋊#8832211yescharacteristic
298C22C2×#1313313211yescharacteristic
299C22C2×#1313313611yescharacteristic
300C22#1681062811yescharacteristic
301C22#1681063211yescharacteristic
302C22#1681064011yescharacteristic
303C22#1681064011yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—44no—
312C4—82no—
313C2—24no—
314C2C2×#1681034911yescharacteristic
315C2#2027310611yescharacteristic
316C2#2027311011yescharacteristic
317C2#2027324411yescharacteristic
318C2#2027324811yescharacteristic
319C2#2027334211yescharacteristic
320C2#2027343211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×#389C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#389D53
hypocenterC53#7593
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #20272538C2semidirect
36. C2×C4⋊C4×C53C8semidirect
180. C53#7593semidirect
195. C2×C4⋊C4#88322semidirect

Maximal subgroups

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

subgroupindex
#202747832
#202725382
C2×C53×#3892
#759353