Finite Group Databasebeta

← #27282896

Subgroups — #27282896

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133206C211yesmaximal, characteristic
2#20274831C211yesmaximal, characteristic, complemented
3#20274460C211yesmaximal, 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×C4×#88322C2211yescharacteristic
12#16810530C411yescharacteristic
13#16810524C411yescharacteristic
14#16810513—22no—
15#16810488C411yescharacteristic
16C4×#11719921—22no—
17C42×Dic53C2211yescharacteristic
18#16810377—22no—
19#16810377—22no—
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—
47C4×#88322—21no—
48C4×#88322D412yes—
49#13133124—22no—
50#13133124—22no—
51C22×Q8×C53C2×C411yescharacteristic
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4×C8×C53—21no—
65C4×C8×C53D412yes—
66C23×D53—21no—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53M1612yes—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53C22⋊C412yes—
80C22×Dic53C22⋊C412yes—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212M1612yes—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—42no—
90C2×C4×D53—42no—
91C2×C8×C53—22no—
92C2×C8×C53—42no—
93C2×C8×C53—41no—
94C2×C8×C53—42no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C42×C53—41no—
100C42×C53—41no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C42×C53#4111yescharacteristic
105#11719921—28no—
106#11719921—44no—
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—22no—
116C4×Dic53—22no—
117C4×Dic53—42no—
118C2×#88322—22no—
119C2×#88322—42no—
120C2×#88322—41no—
121C2×#88322—42no—
122C23×C53C2×C22⋊C411yescharacteristic
123C22×D53—22no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—42no—
127C22×D53—42no—
128Q8×C53—22no—
129Q8×C53—22no—
130Q8×C53—41no—
131Q8×C53—41no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12411yescharacteristic
143C2×C4×C53#12711yescharacteristic
144C2×C4×C53C2×M1611yescharacteristic
145C2×C4×C53C2×M1611yescharacteristic
146C2×C4×C53#13111yescharacteristic
147C2×C4×C53#13111yescharacteristic
148C2×C4×C53#13512yes—
149C2×C4×C53#13512yes—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—22no—
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—82no—
163C4×D53—81no—
164C4×D53—81no—
165C8×C53—42no—
166C8×C53—81no—
167#88322—42no—
168#88322—81no—
169#9645—531nomaximal, Sylow
170C22×C53#39011yescharacteristic
171C22×C53#40711yescharacteristic
172C22×C53#40811yescharacteristic
173C22×C53#41612yes—
174C22×C53#42112yes—
175C2×D53—42no—
176C2×D53—44no—
177C2×D53—81no—
178C2×D53—81no—
179C2×D53—82no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—81no—
184C4×C53#43511yescharacteristic
185C4×C53#43511yescharacteristic
186C4×C53#43511yescharacteristic
187C4×C53#43511yescharacteristic
188C4×C53#48812yes—
189C4×C53#49511yescharacteristic
190C4×C53#49511yescharacteristic
191Dic53—42no—
192Dic53—42no—
193Dic53—81no—
194C4×#124—531no—
195#1831—531no—
196#1744—531no—
197C2×Q8⋊C8—531no—
198C2×Q8⋊C8D5311yescharacteristic, complemented
199C2×D4⋊C8—531no—
200C2×D4⋊C8—531no—
201C2×C53#148212yes—
202C2×C53#148812yes—
203C2×C53#195511yescharacteristic
204C2×C53#195811yescharacteristic
205C2×C53#196311yescharacteristic
206D53—82no—
207C2×#124—531no—
208C2×C4×Q8C2×D5311yescharacteristic
209C2×C4×D4—531no—
210C2×C4⋊C8—531no—
211C2×C4⋊C8C2×D5311yescharacteristic
212C2×C4×C8—531no—
213C2×C4×C8C2×D5311yescharacteristic
214#391—531no—
215#390—531no—
216#388—1062no—
217#382—531no—
218C4×C22⋊C4—1062no—
219C43—531no—
220#336—1062no—
221#336—1062no—
222Q8⋊C8—22no—
223Q8⋊C8—1062no—
224D4⋊C8—1062no—
225D4⋊C8—1062no—
226C53#964511yescharacteristic, complemented, Sylow
227C22×Q8C4×D5311yescharacteristic
228C22×D4—531no—
229C23×C4—1061no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—1061no—
233C22×C8—1061no—
234#124—1062no—
235#124—1062no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—1061no—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4C4×D5311yescharacteristic
247C2×C22⋊C4—1062no—
248C2×C22⋊C4—1062no—
249C2×C22⋊C4—1061no—
250C2×C42—21no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—532no—
254C2×C42—532no—
255C2×C42C22×D5311yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4×C8—21no—
261C4×C8—532no—
262C4×C8—1061no—
263C4×C8C22⋊D5312yes—
264#95—1062no—
265#95—1062no—
266C24—1061no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8#1171991212yes—
273C2×D4—532no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—1061no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4C2×C4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290C22×C4D4×D5311yescharacteristic
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2122no—
298C2×C8—2122no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C22⋊C4—1068no—
306C22⋊C4—2124no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—41no—
310C42—41no—
311C42—1061no—
312C42—1061no—
313C42—1061no—
314C42—1061no—
315C42—1062no—
316C42—1062no—
317C42—2122no—
318C42#1171994511yescharacteristic
319C42D4×D5311yescharacteristic
320C42#1171994711yescharacteristic
321C42C2×C22⋊D5311yescharacteristic
322C23—1062no—
323C23—2121no—
324C23—2121no—
325C23—2122no—
326C23—2122no—
327C23C22⋊C4×D5311yescharacteristic
328Q8—22no—
329Q8—22no—
330Q8—41no—
331Q8—41no—
332D4—1062no—
333D4—1062no—
334D4—2121no—
335D4—2121no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—4241no—
349C2×C4—4242no—
350C2×C4—4241no—
351C2×C4—2121no—
352C2×C4—2122no—
353C2×C4#1313317711yescharacteristic
354C2×C4C4×C22⋊D5311yescharacteristic
355C2×C4#1313318911yescharacteristic
356C2×C4#1313319011yescharacteristic
357C2×C4#1313320311yescharacteristic
358C2×C4#1313321311yescharacteristic
359C2×C4M16×D5311yescharacteristic
360C2×C4C2×#1171991211yescharacteristic
361C2×C4#1313322311yescharacteristic
362C2×C4#1313322511yescharacteristic
363C2×C4#1313323812yes—
364C2×C4#1313324012yes—
365C8—42no—
366C8—81no—
367C8—4241no—
368C8—2122no—
369C22—4241no—
370C22—4241no—
371C22—4242no—
372C22—2122no—
373C22—2124no—
374C22#1681052311yescharacteristic
375C22#1681058511yescharacteristic
376C22#1681058811yescharacteristic
377C22#1681061812yes—
378C22#1681063812yes—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—4241no—
384C4—2122no—
385C4—2122no—
386C4#1681069111yescharacteristic
387C4#1681069211yescharacteristic
388C4#1681069411yescharacteristic
389C4#1681069711yescharacteristic
390C4#1681097912yes—
391C4#1681100411yescharacteristic
392C4#1681100811yescharacteristic
393C2—4242no—
394C2#2027312212yes—
395C2#2027314812yes—
396C2#2027551111yescharacteristic
397C2#2027552311yescharacteristic
398C2#2027554511yescharacteristic
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#9645
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. #20274831C2semidirect
3. #20274460C2semidirect
4. C2×Q8⋊#88322C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
198. C2×Q8⋊C8D53semidirect
226. C53#9645semidirect

Maximal subgroups

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

subgroupindex
C4×#131332062
#202748312
#202744602
C2×Q8⋊#883222
C2×#168103522
C2×#168103502
C2×C53×Q8⋊C82
#964553