Finite Group Databasebeta

← #27384629

Subgroups — #27384629

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C8×D53C211yesmaximal, characteristic
2#20277120C212yesmaximal
3#20274875C211yesmaximal, characteristic, complemented
4#20274842C212yesmaximal, complemented
5#20274741C214yesmaximal
6#107×Dic53C214yesmaximal, complemented
7C53×#1845C211yesmaximal, characteristic, complemented
8C22×C8×D53C2212yes—
9C2×#13133175—22no—
10C2×C42×D53C2211yescharacteristic
11C2×#13133151C2214yes—
12C2×C4×#88322C2211yescharacteristic
13C2×C8×Dic53C2212yes—
14C4×C8×D53C418yescomplemented
15#16810493C2212yes—
16#16810488—22no—
17#16810480—22no—
18#16810480C2214yes—
19C4⋊C4×Dic53—22no—
20C4⋊C4×Dic53C2214yes—
21#16810385C2218yes—
22C2×#107×C53C2214yes—
23C2×C4×C8×C53C2211yescharacteristic
24C53×#384C2212yes—
25C22×C4×D53D411yescharacteristic
26C22×C4×D53D411yescharacteristic
27C22×C4×D53C2311yescharacteristic
28C22×#88322C2312yes—
29C2×C8×D53—24no—
30C2×C8×D53—24no—
31C2×C8×D53C2×C418yes—
32#13133175—48no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53C2314yes—
36C2×#11719921—24no—
37C2×C4×Dic53—24no—
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41C42×D53C2×C414yes—
42C42×D53Q814yes—
43#13133151—28no—
44C4×#88322C2×C414yes—
45C8×Dic53C2×C418yes—
46#13133124—24no—
47#13133124—24no—
48C22×C8×C53C2312yes—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53C2314yes—
52C2×C42×C53C2311yescharacteristic
53#107×C53—28no—
54C4×C8×C53C2×C414yes—
55C23×D53C2×D411yescharacteristic
56C22×C4×C53—24no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2411yescharacteristic
60C22×Dic53—24no—
61C22×Dic53C2×D411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53C4⋊C414yes—
69C2×C4×D53C4⋊C414yes—
70C2×C4×D53D814yes—
71C2×C4×D53Q1614yes—
72C2×C4×D53C22×C412yes—
73C2×C4×D53C2×Q812yes—
74C2×C8×C53C22×C414yes—
75C2×C8×C53#4114yes—
76C2×C8×C53#4114yes—
77C4⋊C4×C53—28no—
78C4⋊C4×C53—44no—
79C4⋊C4×C53—44no—
80C42×C53C22×C412yes—
81C42×C53C2×Q812yes—
82#11719921—48no—
83C4⋊Dic53—28no—
84C4⋊Dic53—44no—
85C4⋊Dic53—44no—
86C4×Dic53—48no—
87C4×Dic53C4⋊C414yes—
88C4×Dic53C4⋊C414yes—
89C4×Dic53C22×C412yes—
90C4×Dic53C2×Q812yes—
91C2×#88322—22no—
92C2×#88322—22no—
93C2×#88322C22×C414yes—
94C8×D53—216no—
95C23×C53C22×D411yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53C2×C4⋊C412yes—
101C22×D53C2×D812yes—
102C22×D53C2×Q1612yes—
103C2×C4×C53—44no—
104C2×C4×C53—44no—
105C2×C4×C53—44no—
106C2×C4×C53C2×C4⋊C412yes—
107C2×C4×C53C2×C4⋊C412yes—
108C2×C4×C53C2×D812yes—
109C2×C4×C53C2×Q1612yes—
110C2×C4×C53#13512yes—
111C2×C4×C53#13512yes—
112C2×C4×C53C23×C411yescharacteristic
113C2×C4×C53C22×Q811yescharacteristic
114C2×C4×C53C2×#4112yes—
115C2×C4×C53C2×#4112yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—44no—
121C2×Dic53—44no—
122C2×Dic53—44no—
123C2×Dic53C2×C4⋊C412yes—
124C2×Dic53C2×C4⋊C412yes—
125C2×Dic53C2×C4⋊C412yes—
126C2×Dic53C2×D812yes—
127C2×Dic53C2×Q1612yes—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—28no—
131C4×D53#107116yescomplemented
132C8×C53#11718yescomplemented
133#88322—24no—
134#20817—531nomaximal, Sylow
135C22×C53C22×C4⋊C411yescharacteristic
136C22×C53#53211yescharacteristic
137C22×C53#53211yescharacteristic
138C22×C53C22×D811yescharacteristic
139C22×C53C22×Q1611yescharacteristic
140C22×C53C2×#13511yescharacteristic
141C22×C53C2×#13511yescharacteristic
142C2×D53—22no—
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—22no—
148C2×D53C2×#10714yes—
149C4×C53—84no—
150C4×C53C2×#10714yes—
151C4×C53#42714yes—
152C4×C53C2×#11712yes—
153C4×C53#53312yes—
154Dic53—22no—
155Dic53—22no—
156Dic53—84no—
157Dic53C2×#10714yes—
158C22×C4×C8—531no—
159#2285—532no—
160#1845—531no—
161#1845D5311yescharacteristic, complemented
162#1834—532no—
163#1808—534no—
164C4×#107—534no—
165C2×C53#228511yescharacteristic
166C2×C53C22×#10711yescharacteristic
167C2×C53C2×#42711yescharacteristic
168C2×C53#316911yescharacteristic
169C2×C53#316911yescharacteristic
170C2×C53#317011yescharacteristic
171C2×C53#317011yescharacteristic
172D53—24no—
173C23×C8—532no—
174C2×#117—1062no—
175C22×C42—531no—
176C2×#107—534no—
177C2×#107C2×D5314yes—
178C2×C4×C8—532no—
179C2×C4×C8—531no—
180C2×C4×C8—538no—
181C2×C4×C8C2×D5311yescharacteristic
182#384—532no—
183#384C2×D5312yes—
184#382—1062no—
185#380—534no—
186#380—1062no—
187C4×C4⋊C4—534no—
188C4×C4⋊C4—1062no—
189#340—538no—
190C53#2081711yescharacteristic, complemented, Sylow
191C23×C4—531no—
192C23×C4—531no—
193C23×C4—531no—
194C22×C8—532no—
195C22×C8—538no—
196C22×C8—1064no—
197C22×C8—1064no—
198C22×C8C22×D5312yes—
199#117—2128no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—534no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4C22×D5314yes—
206C2×C22⋊C4—1064no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—534no—
211C2×C42—534no—
212C2×C42—1064no—
213C2×C42C22×D5311yescharacteristic
214#107—28no—
215#107—1068no—
216C4×C8—538no—
217C4×C8—534no—
218C4×C8C4×D5314yes—
219#95—1064no—
220#95—1064no—
221C24—531no—
222C22×C4—24no—
223C22×C4—531no—
224C22×C4—534no—
225C22×C4—534no—
226C22×C4—534no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—534no—
230C22×C4—532no—
231C22×C4—532no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1062no—
235C22×C4—1062no—
236C22×C4—1064no—
237C22×C4D4×D5311yescharacteristic
238C22×C4D4×D5311yescharacteristic
239C22×C4C23×D5311yescharacteristic
240C2×C8—534no—
241C2×C8—1062no—
242C2×C8—1062no—
243C2×C8—10616no—
244C2×C8C2×C4×D5314yes—
245C2×C8#1171994714yes—
246C2×C8#1171994914yes—
247C4⋊C4—28no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—1068no—
251C4⋊C4—2124no—
252C4⋊C4—2124no—
253C22⋊C4—2128no—
254C42—534no—
255C42—534no—
256C42—532no—
257C42—532no—
258C42—2128no—
259C42C2×C4×D5312yes—
260C42Q8×D5312yes—
261C23—532no—
262C23—532no—
263C23—532no—
264C23—1061no—
265C23—1061no—
266C23—1061no—
267C23—1061no—
268C23C2×D4×D5311yescharacteristic
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—532no—
273C2×C4—532no—
274C2×C4—532no—
275C2×C4—5316no—
276C2×C4—532no—
277C2×C4—532no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1064no—
283C2×C4—1068no—
284C2×C4—1064no—
285C2×C4—2124no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4C4⋊C4×D5312yes—
289C2×C4C4⋊C4×D5312yes—
290C2×C4D8×D5312yes—
291C2×C4Q16×D5312yes—
292C2×C4#1313323612yes—
293C2×C4#1313324012yes—
294C2×C4C22×C4×D5311yescharacteristic
295C2×C4C2×Q8×D5311yescharacteristic
296C2×C4C2×#1171994712yes—
297C2×C4C2×#1171994912yes—
298C8—1064no—
299C8#1313317518yescomplemented
300C22—534no—
301C22—1062no—
302C22—1062no—
303C22—1062no—
304C22—1062no—
305C22—1062no—
306C22—1062no—
307C22C2×C4⋊C4×D5311yescharacteristic
308C22#1681117111yescharacteristic
309C22#1681117411yescharacteristic
310C22C2×D8×D5311yescharacteristic
311C22C2×Q16×D5311yescharacteristic
312C22C2×#1313323611yescharacteristic
313C22C2×#1313324011yescharacteristic
314C4—84no—
315C4—534no—
316C4—1062no—
317C4—1062no—
318C4—4244no—
319C4#107×D5314yes—
320C4#1681066014yes—
321C4C2×#1313317512yes—
322C4#1681117812yes—
323C2—1064no—
324C2#2027712011yescharacteristic
325C2C2×#107×D5311yescharacteristic
326C2C2×#1681066011yescharacteristic
327C2#2028529211yescharacteristic
328C2#2028529511yescharacteristic
329C2#2028529911yescharacteristic
330C2#2028530211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1845C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1845D53
hypocenterC53#20817
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
3. #20274875C2semidirect
4. #20274842C2semidirect
6. #107×Dic53C2semidirect
7. C53×#1845C2semidirect
14. C4×C8×D53C4semidirect
131. C4×D53#107semidirect
132. C8×C53#117semidirect
161. #1845D53semidirect
190. C53#20817semidirect
299. C8#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C8×D532
#202771202
#202748752
#202748422
#202747412
#107×Dic532
C53×#18452
#2081753