Finite Group Databasebeta

← #27388244

Subgroups — #27388244

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810681C212yesmaximal, complemented
2#20277180C211yesmaximal, characteristic
3#20275514C211yesmaximal, characteristic
4#20275438C211yesmaximal, characteristic, complemented
5#20275436C211yesmaximal, characteristic, complemented
6#20275324C212yesmaximal, complemented
7#20275318C212yesmaximal, complemented
8C4⋊C8×Dic53C211yesmaximal, characteristic, complemented
9#20274483C212yesmaximal, complemented
10#97×Dic53C211yesmaximal, characteristic, complemented
11C53×#1955C211yesmaximal, characteristic
12C2×#13133176C2212yes—
13#16810681—28no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133146C2212yes—
16C2×#13133146C2212yes—
17C2×C22⋊#88322C2212yes—
18C2×C22⋊#88322C2212yes—
19C2×#13133128C2211yescharacteristic
20C2×#13133127C2211yescharacteristic
21C2×#13133127C2211yescharacteristic
22C2×C8×Dic53C2212yes—
23C2×C8×Dic53C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25#16810526C412yes—
26#16810522C412yes—
27#16810516C414yes—
28C4×C4⋊Dic53C2211yescharacteristic
29C4×C22⋊Dic53C2212yes—
30C42×Dic53C2211yescharacteristic
31#16810377C2212yes—
32#16810377C2212yes—
33#16810377C2212yes—
34C2×C4×D4×C53C2211yescharacteristic
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×C22⋊C8×C53C2212yes—
37C2×C22⋊C8×C53C2212yes—
38C2×#97×C53C2211yescharacteristic
39C22×#88322C2312yes—
40C22×#88322C2311yescharacteristic
41C22×#88322C2311yescharacteristic
42#13133176—24no—
43#13133176—24no—
44C2×C4⋊Dic53C2×C414yes—
45C2×C4⋊Dic53C2×C412yes—
46C2×C22⋊Dic53C2×C414yes—
47C2×C22⋊Dic53C2×C414yes—
48C2×C4×Dic53C2×C412yes—
49C2×C4×Dic53C2×C412yes—
50C2×C4×Dic53C2312yes—
51C2×C4×Dic53C2311yescharacteristic
52C2×C4×Dic53C2311yescharacteristic
53C4⋊#88322—22no—
54#13133146—24no—
55#13133146—24no—
56C22⋊#88322—24no—
57C22⋊#88322—24no—
58#13133128—22no—
59#13133127—22no—
60#13133127D414yescomplemented
61C8×Dic53—22no—
62C8×Dic53—24no—
63C8×Dic53—22no—
64#13133124C2×C412yes—
65#13133124C2×C414yes—
66C22×D4×C53C2×C411yescharacteristic
67C23×C4×C53C2312yes—
68C22×C8×C53C2312yes—
69C22×C8×C53C2311yescharacteristic
70C22×C8×C53C2311yescharacteristic
71C4×D4×C53—22no—
72C4×D4×C53—22no—
73C2×C4⋊C4×C53C2×C411yescharacteristic
74C2×C22⋊C4×C53C2×C412yes—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—22no—
77C22⋊C8×C53—24no—
78C22⋊C8×C53—24no—
79#97×C53—22no—
80C24×C53C22×C412yes—
81C2×D4×C53—22no—
82C2×D4×C53—42no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53C22×C412yes—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C22×Dic53C22×C414yes—
94C22×Dic53C22×C412yes—
95C22×Dic53C22×C412yes—
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53C2×D412yes—
104C2×C8×C53#4112yes—
105C4⋊C4×C53—22no—
106C22⋊C4×C53—24no—
107C42×C53#4112yes—
108C42×C53#4112yes—
109C22⋊Dic53—28no—
110C22⋊Dic53—28no—
111C4⋊Dic53—28no—
112C4⋊Dic53—42no—
113C4×Dic53—22no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53—22no—
117C4×Dic53—24no—
118C4×Dic53—42no—
119C4×Dic53C2×D412yes—
120C4×Dic53#4112yes—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322C2×D412yes—
129C2×#88322#4112yes—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—24no—
134C23×C53—24no—
135C23×C53C23×C411yescharacteristic
136D4×C53—44no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53#11712yes—
147C2×C4×C53#11712yes—
148C2×C4×C53C22×D411yescharacteristic
149C2×C4×C53C2×#4111yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C2×C4×C53C2×#4111yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—24no—
159C2×Dic53—24no—
160C2×Dic53—24no—
161C2×Dic53C4×D414yes—
162C2×Dic53#13114yes—
163C8×C53—22no—
164C8×C53—22no—
165C8×C53—42no—
166#88322—22no—
167#88322—22no—
168#88322—42no—
169#21269—531nomaximal, Sylow
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—44no—
173C22×C53—44no—
174C22×C53—42no—
175C22×C53C2×#11711yescharacteristic
176C22×C53C2×C4×D411yescharacteristic
177C22×C53C4×#4111yescharacteristic
178C22×C53C2×#13112yes—
179C22×C53#56812yes—
180C4×C53—22no—
181C4×C53—22no—
182C4×C53—42no—
183C4×C53#52911yescharacteristic
184C4×C53#52911yescharacteristic
185C4×C53#53211yescharacteristic
186C4×C53#53211yescharacteristic
187Dic53—24no—
188Dic53—24no—
189Dic53—44no—
190C2×#433—532no—
191#2293—531no—
192#1955—531no—
193#1955D5311yescharacteristic
194#1940—531no—
195#1940—531no—
196#1923—532no—
197#1923—532no—
198C4×C4⋊C8—531no—
199#1750—532no—
200C4×#97—531no—
201C2×C53—82no—
202C2×C53#229311yescharacteristic
203C2×C53#292111yescharacteristic
204C2×C53#292211yescharacteristic
205C2×C53#293512yes—
206C2×C53#296812yes—
207C2×C4×D4C2×D5311yescharacteristic
208C2×#117—532no—
209#433—1068no—
210C2×C4⋊C8—532no—
211C2×C4⋊C8—532no—
212C2×C4⋊C8—531no—
213C2×C4⋊C8C2×D5311yescharacteristic
214C2×C22⋊C8—532no—
215C2×C22⋊C8—532no—
216C2×C22⋊C8C2×D5312yes—
217C2×C22⋊C8C2×D5312yes—
218C2×#97—531no—
219C2×#97—531no—
220C2×#97—531no—
221C2×#97C2×D5311yescharacteristic
222C2×C4×C8—532no—
223C2×C4×C8—531no—
224C2×C4×C8—531no—
225#390—532no—
226#389—532no—
227#388—534no—
228C4×C4⋊C4—531no—
229C4×C22⋊C4—532no—
230C43—531no—
231#336—532no—
232#336—532no—
233#336—532no—
234C53#2126911yescharacteristic, complemented, Sylow
235C22×D4C4×D5311yescharacteristic
236C23×C4C22×D5312yes—
237C22×C8—531no—
238C22×C8—531no—
239C22×C8—532no—
240C22×C8C22×D5312yes—
241C22×C8C22×D5311yescharacteristic
242C22×C8C22×D5311yescharacteristic
243C4×D4—22no—
244C4×D4—22no—
245#117—1064no—
246#117—1064no—
247C2×C4⋊C4—534no—
248C2×C4⋊C4—532no—
249C2×C4⋊C4C4×D5311yescharacteristic
250C2×C22⋊C4—534no—
251C2×C22⋊C4—534no—
252C2×C22⋊C4C4×D5312yes—
253C2×C42—531no—
254C2×C42—532no—
255C2×C42—532no—
256C2×C42—531no—
257C2×C42—532no—
258C2×C42C22×D5311yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—1064no—
261C4⋊C8—1064no—
262C4⋊C8—1062no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—1064no—
266C22⋊C8—1064no—
267#97—22no—
268#97—534no—
269#97—1062no—
270#97—1062no—
271C4×C8—1064no—
272C4×C8—1062no—
273C4×C8—1062no—
274#95—534no—
275#95—532no—
276C24C2×C4×D5312yes—
277C2×D4—22no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—532no—
286C22×C4—532no—
287C22×C4—534no—
288C22×C4C2×C4×D5312yes—
289C22×C4C2×C4×D5311yescharacteristic
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4C23×D5311yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—532no—
300C2×C8—532no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—1061no—
304C2×C8—1061no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8D4×D5312yes—
309C2×C8#1171994712yes—
310C4⋊C4—22no—
311C4⋊C4—1068no—
312C4⋊C4—2122no—
313C22⋊C4—24no—
314C22⋊C4—1068no—
315C22⋊C4—1068no—
316C42—532no—
317C42—532no—
318C42—1064no—
319C42—1062no—
320C42—1061no—
321C42—1062no—
322C42—1061no—
323C42—2122no—
324C42#1171994712yes—
325C42#1171994712yes—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23C22×C4×D5311yescharacteristic
332D4—44no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—534no—
343C2×C4—534no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1064no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4#1313317312yes—
352C2×C4#1313317312yes—
353C2×C4C2×D4×D5311yescharacteristic
354C2×C4#41×D5311yescharacteristic
355C2×C4#41×D5311yescharacteristic
356C2×C4C2×#1171994711yescharacteristic
357C2×C4C2×#1171994711yescharacteristic
358C2×C4C2×#1171994711yescharacteristic
359C8—22no—
360C8—22no—
361C8—42no—
362C8—1062no—
363C8—1062no—
364C8—2122no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22C2×#1313317311yescharacteristic
371C22C4×D4×D5311yescharacteristic
372C22C4×#1171994711yescharacteristic
373C22#131×D5312yes—
374C22#1681138912yes—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—1064no—
379C4—1064no—
380C4—2124no—
381C4#1681115211yescharacteristic
382C4#1681115411yescharacteristic
383C4#1681116811yescharacteristic
384C4#1681117111yescharacteristic
385C2—82no—
386C2#2027717711yescharacteristic
387C2#2028335011yescharacteristic
388C2#2028335511yescharacteristic
389C2#2028345612yes—
390C2#2028370612yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1955C2
prime radicalC22×D4×C53C2×C4
cosocleC22×C4×C53C24
hypercenter#1955D53
hypocenterC53#21269
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#16810681C2semidirect
4. #20275438C2semidirect
5. #20275436C2semidirect
6. #20275324C2semidirect
7. #20275318C2semidirect
8. C4⋊C8×Dic53C2semidirect
9. #20274483C2semidirect
10. #97×Dic53C2semidirect
60. #13133127D4semidirect
234. C53#21269semidirect

Maximal subgroups

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

subgroupindex
C2×#168106812
#202771802
#202755142
#202754382
#202754362
#202753242
#202753182
C4⋊C8×Dic532
#202744832
#97×Dic532
C53×#19552
#2126953