Finite Group Databasebeta

← #27279454

Subgroups — #27279454

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719917C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic
3C2×Q8⋊#88322C212yesmaximal, complemented
4#20272556C212yesmaximal, complemented
5C4×C22⋊Q8×C53C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×C4⋊#88322C2212yes—
8C2×C22⋊#88322—21no—
9C4×#11719917—24no—
10C2×#13133128—21no—
11C2×C4×#88322C2212yes—
12#16810378—21no—
13Q8⋊#88322—24no—
14C2×C22⋊Q8×C53C411yescharacteristic
15C2×C4×Q8×C53C2211yescharacteristic
16C22×C42×C53C2211yescharacteristic
17C53×#391C411yescharacteristic
18C53×#389C411yescharacteristic
19C53×#387—21no—
20C53×#385—21no—
21C53×#380C411yescharacteristic
22C4×C4⋊C4×C53—21no—
23C4×C4⋊C4×C53C2211yescharacteristic
24C4×C22⋊C4×C53—21no—
25C2×#11719917—24no—
26C2×#11719917—24no—
27C2×#11719917—24no—
28C22×#88322—22no—
29C22×#88322—22no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32C22⋊#88322—42no—
33#13133128—24no—
34C4×#88322—22no—
35C4×#88322D414yes—
36C22×Q8×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53—22no—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C42×C53—22no—
53C2×C42×C53—22no—
54C2×C42×C53—21no—
55C2×C42×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53C22⋊C411yescharacteristic
66C2×Q8×C53—21no—
67C2×Q8×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53M1612yescomplemented
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C4⋊C4×C53—24no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53M1612yescomplemented
100C22⋊C4×C53—24no—
101C22⋊C4×C53—42no—
102C42×C53—22no—
103C42×C53—21no—
104C42×C53—22no—
105C42×C53—21no—
106C42×C53—42no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53C2×D412yes—
110C42×C53#4112yes—
111#11719917—216no—
112C2×#88322—24no—
113C2×#88322—24no—
114C2×#88322—24no—
115C2×#88322—42no—
116C2×#88322—42no—
117C2×#88322—42no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53#10412yes—
125C23×C53C2×C22⋊C411yescharacteristic
126Q8×C53—24no—
127Q8×C53—42no—
128C2×C4×C53—24no—
129C2×C4×C53—24no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53#10412yes—
157C2×C4×C53C4×D412yes—
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12112yes—
160C2×C4×C53#12311yescharacteristic
161C2×C4×C53C2×M1611yescharacteristic
162C2×C4×C53C2×M1611yescharacteristic
163C2×C4×C53#13112yes—
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×C4×C53#13712yes—
167#88322—28no—
168#88322—82no—
169#9077—531nomaximal, Sylow
170C22×C53—22no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53#38611yescharacteristic
178C22×C53#40711yescharacteristic
179C22×C53#40811yescharacteristic
180C22×C53#41711yescharacteristic
181C22×C53#41911yescharacteristic
182C22×C53C2×#10411yescharacteristic
183C22×C53C2×#10411yescharacteristic
184C4×C53—21no—
185C4×C53—22no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—42no—
189C4×C53—41no—
190C4×C53—42no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53#43512yes—
194C4×C53#43612yes—
195C4×C53#46812yes—
196C4×C53#47012yes—
197C2×C4×M16—531no—
198C4×C22⋊Q8D5311yescharacteristic
199#1797—531no—
200C2×Q8⋊C8—532no—
201#1279—532no—
202C2×C53—42no—
203C2×C53#148311yescharacteristic
204C2×C53#148511yescharacteristic
205C2×C53#148911yescharacteristic
206C2×C53#149111yescharacteristic
207C2×C53#187011yescharacteristic
208C2×C53#189411yescharacteristic
209C2×C53#189611yescharacteristic
210C22×M16—1061no—
211C2×C22⋊Q8Dic5311yescharacteristic
212C2×C4×Q8C2×D5311yescharacteristic
213C22×C42C2×D5311yescharacteristic
214C2×C4⋊C8—532no—
215C2×C22⋊C8—1061no—
216C4×M16—1064no—
217C2×#97—1061no—
218C2×C4×C8—532no—
219#391Dic5311yescharacteristic
220#389Dic5311yescharacteristic
221#387—21no—
222#385—21no—
223#380Dic5311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2×D5311yescharacteristic
226C4×C22⋊C4—21no—
227#336—1061no—
228Q8⋊C8—1064no—
229C53#907711yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic5311yescharacteristic
231C23×C4C2×Dic5311yescharacteristic
232C23×C4C22⋊D5311yescharacteristic
233C23×C4C22⋊D5311yescharacteristic
234C2×M16—1064no—
235C2×M16—1064no—
236C2×M16—1064no—
237C22×C8—1062no—
238C22×C8—1062no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×Dic5311yescharacteristic
247C2×C4⋊C4C2×Dic5311yescharacteristic
248C2×C4⋊C4C2×Dic5311yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2×Dic5311yescharacteristic
256C2×C42C22⋊D5311yescharacteristic
257C2×C42C22⋊D5311yescharacteristic
258C2×C42C22×D5311yescharacteristic
259C4⋊C8—1062no—
260C4⋊C8—1062no—
261C22⋊C8—2122no—
262#97—1064no—
263C4×C8—534no—
264C4×C8—1062no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24C22⋊Dic5311yescharacteristic
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8#1171991712yescomplemented
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4C22⋊Dic5311yescharacteristic
292C22×C4C22⋊Dic5311yescharacteristic
293C22×C4C22⋊Dic5311yescharacteristic
294C22×C4C22×Dic5311yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297M16—10616no—
298C2×C8—1064no—
299C2×C8—1064no—
300C2×C8—1064no—
301C2×C8—2122no—
302C2×C8—2122no—
303C2×C8—2122no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4#1171991712yescomplemented
312C22⋊C4—24no—
313C22⋊C4—42no—
314C42—22no—
315C42—21no—
316C42—22no—
317C42—21no—
318C42—42no—
319C42—41no—
320C42—41no—
321C42D4×D5312yes—
322C42#1171994712yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#1313314412yes—
330C23C2×C22⋊Dic5311yescharacteristic
331Q8—24no—
332Q8—42no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4#1313314412yes—
362C2×C4D4×Dic5312yes—
363C2×C4#1313318511yescharacteristic
364C2×C4#1313318912yes—
365C2×C4#1313319811yescharacteristic
366C2×C4C2×#1171991711yescharacteristic
367C2×C4C2×#1171991711yescharacteristic
368C2×C4#1313322412yes—
369C2×C4#1313324211yescharacteristic
370C2×C4#1313324711yescharacteristic
371C2×C4#1313324812yes—
372C8—1068no—
373C8—4242no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22#1681050311yescharacteristic
382C22#1681058711yescharacteristic
383C22#1681059011yescharacteristic
384C22#1681062411yescharacteristic
385C22#1681063211yescharacteristic
386C22C2×#1313314411yescharacteristic
387C22C2×#1313314411yescharacteristic
388C4—21no—
389C4—22no—
390C4—21no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4#1681069512yes—
398C4#1681070112yes—
399C4#1681087112yes—
400C4#1681087912yes—
401C2—42no—
402C2#2027312811yescharacteristic
403C2#2027313611yescharacteristic
404C2#2027315411yescharacteristic
405C2#2027316211yescharacteristic
406C2#2027501311yescharacteristic
407C2#2027514911yescharacteristic
408C2#2027515911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×C22⋊Q8D53
hypocenterC53#9077
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#88322C2semidirect
4. #20272556C2semidirect
69. C2×Q8×C53M16semidirect
99. C4⋊C4×C53M16semidirect
229. C53#9077semidirect
274. C2×Q8#11719917semidirect
311. C4⋊C4#11719917semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199172
#202746952
C2×Q8⋊#883222
#202725562
C4×C22⋊Q8×C532
#907753