Finite Group Databasebeta

← #27301164

Subgroups — #27301164

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133162C211yesmaximal, characteristic
2C2×#16810705C211yesmaximal, characteristic
3C2×#16810426C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic
5#20275574C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×Dic212C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×#13133179—21no—
11C2×#13133162—22no—
12C2×#13133162—22no—
13C2×#13133162C414yescomplemented
14C22×#11719908C2211yescharacteristic
15#16810705—22no—
16#16810705—22no—
17C2×#13133140—21no—
18C2×#13133137—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810528—21no—
21#16810426—22no—
22#16810423—22no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×Dic106—21no—
27C22×D212—21no—
28C22×C4×D53—41no—
29C2×Dic212—21no—
30C2×Dic212—21no—
31C2×Dic212—21no—
32C2×Dic212—21no—
33C2×Dic212—21no—
34C2×Dic212C2×C412yescomplemented
35C2×D424—21no—
36C2×D424—21no—
37C2×D424—21no—
38C2×D424C2×C412yes—
39C2×D424D412yes—
40C2×D424D412yes—
41C22×#88322D411yescharacteristic
42#13133179—24no—
43#13133179—42no—
44C2×#11719921—41no—
45C2×C4×Dic53D411yescharacteristic
46#13133162—28no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908—21no—
50#13133140—42no—
51#13133137—24no—
52C8×Dic53—21no—
53C8×Dic53—21no—
54#13133124—41no—
55#13133124—41no—
56C22×C8×C53C2311yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53C2×C412yescomplemented
62#107×C53—21no—
63#107×C53—21no—
64#107×C53—21no—
65#107×C53D412yes—
66C23×D53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—41no—
71C22×Dic53D811yescharacteristic
72C22×Dic53D811yescharacteristic
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×C4×D53—44no—
80C2×C4×D53—82no—
81C2×C4×D53—82no—
82C2×Dic106—22no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×Dic106—41no—
86C2×Dic106—41no—
87C2×Dic106—41no—
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98#11719921—44no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322—21no—
104Dic212—22no—
105Dic212—22no—
106Dic212—41no—
107Dic212—41no—
108D424—22no—
109D424—22no—
110D424—22no—
111D424C22⋊C414yes—
112C16×C53—22no—
113#11719908—41no—
114C23×C53C2×D811yescharacteristic
115C22×D53—42no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—81no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—81no—
122C2×C4×C53—42no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53C4×D411yescharacteristic
129C2×C4×C53#12011yescharacteristic
130C2×C4×C53#12111yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53SD3212yescomplemented
143D212—24no—
144D212—42no—
145D212—42no—
146D212—42no—
147C4×D53—88no—
148Dic106—42no—
149Dic106—42no—
150Dic106—81no—
151Dic106—81no—
152C8×C53C2×C22⋊C411yescharacteristic
153C8×C53C4×D411yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#88322—41no—
157#13014—531nomaximal, Sylow
158C22×C53C4×D811yescharacteristic
159C22×C53#44711yescharacteristic
160C22×C53#45911yescharacteristic
161C22×C53C2×SD3211yescharacteristic
162C22×C53#50811yescharacteristic
163C22×C53#50911yescharacteristic
164C22×C53#51011yescharacteristic
165C2×D53—44no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—82no—
171C2×D53—82no—
172C4×C53—82no—
173C4×C53—161no—
174C4×C53#38611yescharacteristic
175C4×C53#44211yescharacteristic
176C4×C53#44911yescharacteristic
177C4×C53#46311yescharacteristic
178Dic53—22no—
179Dic53—41no—
180Dic53—82no—
181C22×SD32—531no—
182C2×C4×D8—531no—
183C2×#358—531no—
184C2×#357—531no—
185#1969—531no—
186#1380D5311yescharacteristic, complemented
187#1374—531no—
188C2×C53#187911yescharacteristic
189C2×C53C4×SD3211yescharacteristic
190C2×C53#217511yescharacteristic
191C2×C53#218711yescharacteristic
192C2×C53#218911yescharacteristic
193C2×C53#220811yescharacteristic
194C2×C53#221011yescharacteristic
195D53—84no—
196C22×Q16—531no—
197C22×D8—531no—
198C2×C4×D4—1061no—
199C2×SD32—534no—
200C2×SD32—1062no—
201C2×SD32—1062no—
202C22×C16—531no—
203C22×C16C2×D5311yescharacteristic
204C4×D8—1062no—
205C4×D8—1062no—
206C2×#107C2×D5311yescharacteristic
207C2×#107C2×D5311yescharacteristic
208C2×#103—1061no—
209C2×#102—1061no—
210C2×C4×C8—531no—
211#391—1061no—
212#358—1062no—
213#357—1062no—
214C53#1301411yescharacteristic, complemented, Sylow
215C22×Q8—1061no—
216C22×D4—1061no—
217C23×C4—2121no—
218C2×Q16—532no—
219C2×Q16—1061no—
220C2×Q16—1061no—
221C2×Q16—1061no—
222C2×Q16—1061no—
223C2×Q16—1061no—
224C2×D8—532no—
225C2×D8—532no—
226C2×D8—532no—
227C2×D8—1061no—
228C2×D8—1061no—
229C2×D8—1061no—
230C22×C8—531no—
231C22×C8C22×D5311yescharacteristic
232C4×D4—1064no—
233C4×D4—2122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C22⋊C4—2121no—
237C2×C42—531no—
238SD32—1068no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16C4×D5312yescomplemented
245#107—21no—
246#107—21no—
247#107—21no—
248#107D21212yes—
249#103—2122no—
250#102—1064no—
251C4×C8—1061no—
252C4×C8—1061no—
253#95—2121no—
254#95—2121no—
255C24—2121no—
256C2×Q8—1062no—
257C2×Q8—2121no—
258C2×Q8—2121no—
259C2×Q8—2121no—
260C2×Q8—2121no—
261C2×Q8—2121no—
262C2×D4—1062no—
263C2×D4—1062no—
264C2×D4—1062no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—4242no—
273C22×C4—4242no—
274C22×C4—2124no—
275C22×C4—2121no—
276C22×C4D4×D5311yescharacteristic
277Q16—1062no—
278Q16—1062no—
279Q16—2121no—
280Q16—2121no—
281D8—534no—
282D8—1062no—
283D8—1062no—
284D8—1062no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8C2×C4×D5311yescharacteristic
289C2×C8C2×D21211yescharacteristic
290C2×C8#1171994511yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8#1171994911yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—2124no—
299C42—1061no—
300C42—1061no—
301C16—22no—
302C16—2121no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—4241no—
307C23—2122no—
308C23—2122no—
309C23—2122no—
310C23D8×D5311yescharacteristic
311Q8—4241no—
312Q8—4241no—
313Q8—2122no—
314Q8—2122no—
315D4—1064no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—532no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4248no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—2122no—
335C2×C4#1313317911yescharacteristic
336C2×C4#1313318711yescharacteristic
337C2×C4#1313319011yescharacteristic
338C2×C4#1313324011yescharacteristic
339C2×C4#1313324411yescharacteristic
340C2×C4#1313324411yescharacteristic
341C8—2121no—
342C8C22⋊C4×D5311yescharacteristic
343C8C4×D21211yescharacteristic
344C8#1313319211yescharacteristic
345C8#1313320211yescharacteristic
346C22—4242no—
347C22—4242no—
348C22—4242no—
349C22—4242no—
350C22—4242no—
351C22—4242no—
352C22—2124no—
353C22#1681070511yescharacteristic
354C22#1681075811yescharacteristic
355C22#1681081711yescharacteristic
356C22SD32×D5311yescharacteristic
357C22#1681105411yescharacteristic
358C22#1681105711yescharacteristic
359C22#1681106511yescharacteristic
360C4—82no—
361C4—161no—
362C4—1062no—
363C4—4242no—
364C4—2121no—
365C4#1681050111yescharacteristic
366C4#1681073411yescharacteristic
367C4#1681077111yescharacteristic
368C4#1681084111yescharacteristic
369C2—4244no—
370C2#2027506411yescharacteristic
371C2#2027656611yescharacteristic
372C2#2027657811yescharacteristic
373C2#2027664511yescharacteristic
374C2#2027665811yescharacteristic
375C2#2027675911yescharacteristic
376C2#2027676911yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC22×D424C22
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#13014
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
3. C2×#16810426C2semidirect
5. #20275574C2semidirect
6. #20272792C2semidirect
7. C53×#1380C2semidirect
13. C2×#13133162C4semidirect
34. C2×Dic212C2×C4semidirect
61. C2×C16×C53C2×C4semidirect
142. C2×Dic53SD32semidirect
186. #1380D53semidirect
214. C53#13014semidirect
244. C2×C16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331622
C2×#168107052
C2×#168104262
C2×#168104232
#202755742
#202727922
C53×#13802
#1301453