Finite Group Databasebeta

← #27282901

Subgroups — #27282901

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133206C211yesmaximal, characteristic
2#20274866C211yesmaximal, characteristic, complemented
3C4×#13133127C211yesmaximal, characteristic, complemented
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6C2×#16810350C211yesmaximal, characteristic
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133206C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133146—22no—
12C2×#13133127—22no—
13C2×C4×#88322C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810524C411yescharacteristic
16#16810513—22no—
17#16810488C411yescharacteristic
18C4×#11719921—22no—
19C42×Dic53C2211yescharacteristic
20Q8⋊#88322—22no—
21#16810352—22no—
22#16810350—22no—
23C2×C4×Q8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212C2×C411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133206—22no—
32#13133206—22no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—21no—
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D412yes—
44C2×C4×Dic53D412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47#13133146—44no—
48#13133127—28no—
49C4×#88322—21no—
50C4×#88322D412yes—
51#13133124—22no—
52#13133124—22no—
53C22×Q8×C53C2×C411yescharacteristic
54C22×C8×C53—21no—
55C22×C8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53D412yes—
68C23×D53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53M1612yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53C22⋊C412yes—
82C22×Dic53C22⋊C412yes—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212M1612yes—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C2×C8×C53—24no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—42no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53—41no—
102C42×C53—41no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107#11719921—28no—
108#11719921—41no—
109#11719921—41no—
110#11719921—44no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53—21no—
117C4×Dic53—22no—
118C4×Dic53—22no—
119C4×Dic53—24no—
120C2×#88322—24no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—42no—
124C23×C53C2×C22⋊C411yescharacteristic
125C22×D53—22no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—42no—
129C22×D53—42no—
130Q8×C53—22no—
131Q8×C53—22no—
132Q8×C53—41no—
133Q8×C53—41no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12411yescharacteristic
145C2×C4×C53#12711yescharacteristic
146C2×C4×C53C2×M1611yescharacteristic
147C2×C4×C53C2×M1611yescharacteristic
148C2×C4×C53#13111yescharacteristic
149C2×C4×C53#13111yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)12yes—
151C2×C4×C53#13712yes—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—41no—
158C2×Dic53—42no—
159C2×Dic53#10414yes—
160D212—22no—
161D212—22no—
162D212—41no—
163D212—41no—
164C4×D53—82no—
165C4×D53—81no—
166C4×D53—81no—
167C8×C53—24no—
168C8×C53—81no—
169#88322—24no—
170#88322—81no—
171#9646—531nomaximal, Sylow
172C22×C53#39011yescharacteristic
173C22×C53#40711yescharacteristic
174C22×C53#40811yescharacteristic
175C22×C53#41712yes—
176C22×C53C2×#10412yes—
177C2×D53—42no—
178C2×D53—44no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—82no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53#43511yescharacteristic
187C4×C53#43511yescharacteristic
188C4×C53#43511yescharacteristic
189C4×C53#43511yescharacteristic
190C4×C53#48912yes—
191C4×C53#49611yescharacteristic
192C4×C53#49711yescharacteristic
193Dic53—24no—
194Dic53—42no—
195Dic53—81no—
196C4×#124—531no—
197#1842—531no—
198C4×#97—531no—
199C2×Q8⋊C8—531no—
200C2×Q8⋊C8D5311yescharacteristic, complemented
201C2×D4⋊C8—531no—
202C2×D4⋊C8—531no—
203C2×C53#148312yes—
204C2×C53#148912yes—
205C2×C53#195511yescharacteristic
206C2×C53#195911yescharacteristic
207C2×C53#196211yescharacteristic
208D53—82no—
209C2×#124—531no—
210C2×C4×Q8C2×D5311yescharacteristic
211C2×C4×D4—531no—
212C2×C4⋊C8—531no—
213C2×C4⋊C8—1062no—
214C2×C4⋊C8C2×D5311yescharacteristic
215C2×#97—1062no—
216C2×C4×C8—531no—
217C2×C4×C8C2×D5311yescharacteristic
218#391—531no—
219#390—531no—
220#388—1062no—
221#382—531no—
222C4×C22⋊C4—1062no—
223C43—531no—
224Q8⋊C8—22no—
225Q8⋊C8—1062no—
226D4⋊C8—1062no—
227D4⋊C8—1062no—
228C53#964611yescharacteristic, complemented, Sylow
229C22×Q8C4×D5311yescharacteristic
230C22×D4—531no—
231C23×C4—1061no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—1061no—
235C22×C8—1061no—
236#124—1062no—
237#124—1062no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245C4×D4—1061no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4C4×D5311yescharacteristic
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1062no—
251C2×C22⋊C4—1061no—
252C2×C42—21no—
253C2×C42—531no—
254C2×C42—531no—
255C2×C42—532no—
256C2×C42—532no—
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—1061no—
261C4⋊C8—1061no—
262C4⋊C8—2124no—
263#97—1068no—
264C4×C8—21no—
265C4×C8—532no—
266C4×C8—1061no—
267C4×C8C22⋊D5312yes—
268#95—1062no—
269#95—1062no—
270C24—1061no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8#1171991212yes—
277C2×D4—532no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—532no—
286C22×C4—532no—
287C22×C4—1061no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4C2×C4×D5311yescharacteristic
293C22×C4D4×D5311yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—1064no—
300C2×C8—1062no—
301C2×C8—2121no—
302C2×C8—2122no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C22⋊C4—1068no—
310C22⋊C4—2124no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C42—41no—
314C42—41no—
315C42—1064no—
316C42—1061no—
317C42—1061no—
318C42—1061no—
319C42—1061no—
320C42—1062no—
321C42—1062no—
322C42#1171994511yescharacteristic
323C42D4×D5311yescharacteristic
324C42#1171994711yescharacteristic
325C42C2×C22⋊D5311yescharacteristic
326C23—1062no—
327C23—2121no—
328C23—2121no—
329C23—2122no—
330C23—2122no—
331C23C22⋊C4×D5311yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336D4—1062no—
337D4—1062no—
338D4—2121no—
339D4—2121no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—534no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—4242no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4#1313317711yescharacteristic
358C2×C4C4×C22⋊D5311yescharacteristic
359C2×C4#1313318911yescharacteristic
360C2×C4#1313319011yescharacteristic
361C2×C4#1313320311yescharacteristic
362C2×C4#1313321311yescharacteristic
363C2×C4M16×D5311yescharacteristic
364C2×C4C2×#1171991211yescharacteristic
365C2×C4#1313322311yescharacteristic
366C2×C4#1313322511yescharacteristic
367C2×C4#1313324412yes—
368C2×C4#1313324912yes—
369C8—24no—
370C8—81no—
371C8—1064no—
372C8—4241no—
373C22—4241no—
374C22—4241no—
375C22—4242no—
376C22—2124no—
377C22—2122no—
378C22#1681052311yescharacteristic
379C22#1681058511yescharacteristic
380C22#1681058811yescharacteristic
381C22#1681062312yes—
382C22#104×D5312yes—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—81no—
387C4—1064no—
388C4—4241no—
389C4—2122no—
390C4#1681069111yescharacteristic
391C4#1681069211yescharacteristic
392C4#1681069411yescharacteristic
393C4#1681069711yescharacteristic
394C4#1681098412yes—
395C4#1681100911yescharacteristic
396C4#1681101611yescharacteristic
397C2—4242no—
398C2#2027312712yes—
399C2#2027315312yes—
400C2#2027551111yescharacteristic
401C2#2027552811yescharacteristic
402C2#2027554111yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#9646
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20274866C2semidirect
3. C4×#13133127C2semidirect
4. C2×Q8⋊#88322C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
200. C2×Q8⋊C8D53semidirect
228. C53#9646semidirect

Maximal subgroups

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

subgroupindex
C4×#131332062
#202748662
C4×#131331272
C2×Q8⋊#883222
C2×#168103522
C2×#168103502
C2×C53×Q8⋊C82
#964653