Finite Group Databasebeta

← #27301028

Subgroups — #27301028

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810859C211yesmaximal, characteristic
2C22×#13133151C211yesmaximal, characteristic, complemented
3C2×#16810425C212yesmaximal, complemented
4#20272807C212yesmaximal, complemented
5C2×C53×C22⋊C16C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic
7C2×#13133191—21no—
8C22×C4⋊Dic53—21no—
9#16810859—22no—
10#16810859C414yescomplemented
11C2×#13133151—24no—
12C2×#13133151—21no—
13C2×#13133151C2211yescharacteristic
14C2×#13133151C2212yes—
15C2×#13133139—21no—
16#16810425—24no—
17C23×C8×C53C2211yescharacteristic
18C22×C16×C53C2212yes—
19C53×C22⋊C16—22no—
20C22×D212—21no—
21C22×C22⋊D53—41no—
22C23×Dic53—41no—
23C2×D424—21no—
24C2×D424—22no—
25C2×D424—22no—
26C2×D424C2×C412yescomplemented
27#13133191—24no—
28#13133191—42no—
29C2×C4⋊Dic53—24no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—42no—
33C2×C4⋊Dic53—41no—
34C2×#11719921—41no—
35#13133151—24no—
36#13133151—21no—
37#13133151—22no—
38#13133151C2×C412yescomplemented
39#13133151D414yescomplemented
40#13133139—24no—
41C23×C4×C53D411yescharacteristic
42C22×C8×C53—22no—
43C22×C8×C53—22no—
44C22×C8×C53C2×C412yes—
45C22×C8×C53D412yes—
46C22×C8×C53D411yescharacteristic
47C22×C8×C53C2311yescharacteristic
48C2×C16×C53—22no—
49C2×C16×C53—22no—
50C2×C16×C53—22no—
51C24×C53D811yescharacteristic
52C23×D53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53D811yescharacteristic
59C22×C4×C53SD1612yes—
60C22×C4×C53C2×D411yescharacteristic
61C2×C22⋊D53—44no—
62C2×C22⋊D53—82no—
63C2×C22⋊D53—82no—
64C22×Dic53—44no—
65C22×Dic53—41no—
66C22×Dic53—42no—
67C22×Dic53—82no—
68C22×Dic53—82no—
69C2×D212—22no—
70C2×D212—41no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212—41no—
74C2×D212—41no—
75C2×C8×C53—21no—
76C2×C8×C53—21no—
77C2×C8×C53—24no—
78C2×C8×C53—24no—
79C2×C8×C53C22⋊C412yes—
80C2×C8×C53C22×C411yescharacteristic
81C2×C8×C53C2×D412yes—
82C2×C8×C53C2×D411yescharacteristic
83C2×C8×C53#4112yes—
84#11719921—44no—
85C4⋊Dic53—22no—
86C4⋊Dic53—24no—
87C4⋊Dic53—44no—
88C4⋊Dic53—41no—
89C4⋊Dic53—42no—
90D424—24no—
91D424—42no—
92C16×C53—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53#10212yes—
98C23×C53D1612yes—
99C23×C53SD3212yes—
100C23×C53C2×D811yescharacteristic
101C22×D53—42no—
102C22×D53—81no—
103C22×D53—81no—
104C22×D53—81no—
105C22×D53—81no—
106C22×D53—81no—
107C22×D53—81no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53#10212yes—
115C2×C4×C53C2×C22⋊C411yescharacteristic
116C2×C4×C53#12011yescharacteristic
117C2×C4×C53#12311yescharacteristic
118C2×C4×C53C2×SD1611yescharacteristic
119C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C53#13711yescharacteristic
121C22⋊D53—88no—
122C2×Dic53—42no—
123C2×Dic53—44no—
124C2×Dic53—88no—
125C2×Dic53—82no—
126C2×Dic53—81no—
127C2×Dic53—81no—
128C2×Dic53—82no—
129D212—42no—
130D212—42no—
131D212—81no—
132D212—81no—
133C8×C53—22no—
134C8×C53C4×D412yes—
135C8×C53#12112yes—
136#12991—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53#35714yescomplemented
144C22×C53C2×#10211yescharacteristic
145C22×C53#44711yescharacteristic
146C22×C53#48011yescharacteristic
147C22×C53C2×D1611yescharacteristic
148C22×C53C2×SD3211yescharacteristic
149C22×C53#50911yescharacteristic
150C22×C53#51011yescharacteristic
151C2×D53—82no—
152C2×D53—82no—
153C2×D53—82no—
154C2×D53—161no—
155C2×D53—161no—
156C2×D53—161no—
157C2×D53—161no—
158C4×C53—22no—
159C4×C53#38611yescharacteristic
160C4×C53#41711yescharacteristic
161C4×C53#44811yescharacteristic
162C4×C53#48111yescharacteristic
163Dic53—82no—
164Dic53—84no—
165C2×#466—531no—
166C22×#107—531no—
167C2×#357—532no—
168C2×C22⋊C16D5311yescharacteristic, complemented
169#1380—532no—
170C2×C53—24no—
171C2×C53#187411yescharacteristic
172C2×C53C2×#35711yescharacteristic
173C2×C53#213911yescharacteristic
174C2×C53#218411yescharacteristic
175C2×C53#218511yescharacteristic
176C2×C53#223211yescharacteristic
177C2×C53#223311yescharacteristic
178D53—162no—
179C22×D8—531no—
180C23×C8C2×D5311yescharacteristic
181C2×#121—1061no—
182C22×C4⋊C4—1061no—
183C22×C16C2×D5312yes—
184#466—534no—
185#466—1062no—
186C2×#107—532no—
187C2×#107—531no—
188C2×#107—1064no—
189C2×#107—1061no—
190C2×#102—1061no—
191#357—1064no—
192C22⋊C16—22no—
193C53#1299111yescharacteristic, complemented, Sylow
194C22×D4—1061no—
195C22×D4—2121no—
196C23×C4—2121no—
197C23×C4D21211yescharacteristic
198C2×D8—532no—
199C2×D8—1062no—
200C2×D8—1062no—
201C2×D8—1061no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C4×D5312yes—
205C22×C8D21211yescharacteristic
206C22×C8C22⋊D5312yes—
207C22×C8C22×D5311yescharacteristic
208#121—1064no—
209#121—2122no—
210C2×C4⋊C4—1064no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—2122no—
214C2×C4⋊C4—2121no—
215C2×C22⋊C4—2121no—
216C2×C16—22no—
217C2×C16—22no—
218C2×C16—22no—
219#107—532no—
220#107—534no—
221#107—1064no—
222#107—1062no—
223#107—1061no—
224#102—1064no—
225C24—2121no—
226C24D42411yescharacteristic
227C2×D4—1062no—
228C2×D4—4242no—
229C2×D4—4242no—
230C2×D4—2124no—
231C2×D4—2121no—
232C2×D4—2121no—
233C2×D4—2121no—
234C2×D4—2121no—
235C2×D4—2121no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—4242no—
241C22×C4—4242no—
242C22×C4—2124no—
243C22×C4—2121no—
244C22×C4—2122no—
245C22×C4#1171991312yes—
246C22×C4D42411yescharacteristic
247C22×C4#1171992112yes—
248C22×C4C2×D21211yescharacteristic
249D8—1064no—
250D8—2122no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8#1171992112yes—
256C2×C8C2×C4×D5311yescharacteristic
257C2×C8D4×D5312yes—
258C2×C8#1171994712yes—
259C2×C8C2×C22⋊D5311yescharacteristic
260C4⋊C4—1062no—
261C4⋊C4—1064no—
262C4⋊C4—2124no—
263C4⋊C4—2121no—
264C4⋊C4—2122no—
265C22⋊C4—2124no—
266C16—42no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—4241no—
272C23—4241no—
273C23—4241no—
274C23—4241no—
275C23—4241no—
276C23—4241no—
277C23—2122no—
278C23#1313313912yes—
279C23D84812yes—
280C23#1313316212yes—
281C23C2×D42411yescharacteristic
282D4—4248no—
283D4—4241no—
284D4—4241no—
285D4—2122no—
286D4—2122no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—4248no—
294C2×C4—4242no—
295C2×C4—4241no—
296C2×C4—4241no—
297C2×C4—4242no—
298C2×C4—2124no—
299C2×C4—2122no—
300C2×C4#1313313912yes—
301C2×C4C2×#1171992111yescharacteristic
302C2×C4#1313318711yescharacteristic
303C2×C4#1313320111yescharacteristic
304C2×C4C2×#1171991311yescharacteristic
305C2×C4#1313324511yescharacteristic
306C2×C4#1313325011yescharacteristic
307C8—22no—
308C8#1313317712yes—
309C8#1313318912yes—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—8481no—
317C22—8481no—
318C22—8481no—
319C22—8481no—
320C22—4242no—
321C22—4242no—
322C22—4242no—
323C22#1681042514yescomplemented
324C22C2×#1313313911yescharacteristic
325C22#1681075911yescharacteristic
326C22#1681093711yescharacteristic
327C22C2×D84811yescharacteristic
328C22C2×#1313316211yescharacteristic
329C22#1681106111yescharacteristic
330C22#1681106611yescharacteristic
331C4—22no—
332C4—4244no—
333C4—4242no—
334C4#1681050011yescharacteristic
335C4#1681062511yescharacteristic
336C4#1681076411yescharacteristic
337C4#1681094211yescharacteristic
338C2—24no—
339C2—8482no—
340C2#2027503011yescharacteristic
341C2C2×#1681042511yescharacteristic
342C2#2027642811yescharacteristic
343C2#2027662911yescharacteristic
344C2#2027663411yescharacteristic
345C2#2027689611yescharacteristic
346C2#2027690111yescharacteristic
subgroupG/H
centerC23C2×D424
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×C22⋊C16C2
socleC23×C53C2×D8
prime radicalC2×#16810859C2
cosocleC22×C8×C53C23
hypercenterC2×C22⋊C16D53
hypocenterC53#12991
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C22×#13133151C2semidirect
3. C2×#16810425C2semidirect
4. #20272807C2semidirect
5. C2×C53×C22⋊C16C2semidirect
10. #16810859C4semidirect
26. C2×D424C2×C4semidirect
38. #13133151C2×C4semidirect
39. #13133151D4semidirect
143. C22×C53#357semidirect
168. C2×C22⋊C16D53semidirect
193. C53#12991semidirect
323. C22#16810425semidirect

Maximal subgroups

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

subgroupindex
C2×#168108592
C22×#131331512
C2×#168104252
#202728072
C2×C53×C22⋊C162
#1299153