Finite Group Databasebeta

← #27276147

Subgroups — #27276147

Order 13568.

864 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275544C211yesmaximal, characteristic, complemented
2C4×#13133144C212yesmaximal, complemented
3#20273307C211yesmaximal, characteristic, complemented
4#20273154C212yesmaximal, complemented
5C53×#3483C211yesmaximal, characteristic
6#16810647C2211yescharacteristic
7C2×#13133144—22no—
8C2×#13133144—22no—
9C4×#11719917C2212yes—
10C42×Dic53C2211yescharacteristic
11#16810382—22no—
12#16810363—22no—
13Q8⋊#88322—22no—
14D4⋊#88322—22no—
15Q82×C53C411yescharacteristic, complemented
16C53×#555—22no—
17C53×#555—22no—
18C53×#555C411yescharacteristic, complemented
19C53×#550—22no—
20C53×#550C412yescomplemented
21C53×#532—22no—
22C53×#532—22no—
23C53×#532C2211yescharacteristic
24C4×#41×C53—22no—
25C4×#41×C53C2212yes—
26C2×#11719917—22no—
27C2×#11719917—21no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53—22no—
30C4⋊#88322—22no—
31#13133144—28no—
32#13133144—44no—
33#13133128—22no—
34C4×#88322—22no—
35C2×#41×C53—22no—
36C2×#41×C53—22no—
37C2×#41×C53D412yes—
38C2×#41×C53D412yes—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53—22no—
41C4⋊Q8×C53C2×C411yescharacteristic
42#127×C53—22no—
43#127×C53—22no—
44#127×C53C2×C411yescharacteristic
45#124×C53—24no—
46#124×C53—22no—
47#124×C53—22no—
48#124×C53D412yes—
49#121×C53—24no—
50#121×C53—24no—
51#121×C53—22no—
52#121×C53—22no—
53#121×C53—24no—
54#121×C53—22no—
55C4×Q8×C53—22no—
56C4×Q8×C53C2×C412yes—
57C4×Q8×C53D412yes—
58C4×D4×C53—24no—
59C4×D4×C53—24no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53—22no—
63C4×D4×C53—22no—
64C4×D4×C53C2×C412yes—
65C4×D4×C53D412yes—
66#117×C53—24no—
67#117×C53—22no—
68#117×C53—22no—
69#117×C53D412yes—
70C2×C42×C53—22no—
71C2×C42×C53—22no—
72C2×C42×C53C2311yescharacteristic
73#41×C53—24no—
74#41×C53—42no—
75#41×C53—42no—
76#41×C53—42no—
77#41×C53—44no—
78#41×C53—44no—
79#41×C53—82no—
80#41×C53—82no—
81C2×Q8×C53—22no—
82C2×Q8×C53C22⋊C412yes—
83C2×Q8×C53C22⋊C412yes—
84C2×D4×C53—24no—
85C2×D4×C53—22no—
86C2×D4×C53—24no—
87C2×D4×C53—22no—
88C2×D4×C53—22no—
89C2×D4×C53—22no—
90C2×D4×C53C22⋊C412yes—
91C2×D4×C53C22⋊C412yes—
92C22×C4×C53—24no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D411yescharacteristic
99C22×Dic53—41no—
100C4⋊C4×C53—24no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53C22⋊C412yes—
105C4⋊C4×C53C22⋊C412yes—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—24no—
111C22⋊C4×C53—22no—
112C42×C53—22no—
113C42×C53—22no—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53—22no—
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D412yes—
119C42×C53C2×D411yescharacteristic
120C4×Dic53—24no—
121C4×Dic53—22no—
122C4×Dic53—42no—
123C4×Dic53—41no—
124#11719917—24no—
125#11719917—42no—
126#11719917—42no—
127C2×#88322—42no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53#12011yescharacteristic
132Q8×C53—22no—
133Q8×C53—42no—
134Q8×C53—82no—
135Q8×C53#10414yescomplemented
136D4×C53—22no—
137D4×C53—22no—
138D4×C53—44no—
139D4×C53—42no—
140D4×C53—48no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—82no—
144D4×C53—82no—
145D4×C53—84no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—44no—
155C2×C4×C53—48no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53C2×C22⋊C412yes—
164C2×C4×C53#12012yes—
165C2×C4×C53#12011yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—44no—
168#88322—44no—
169#88322—81no—
170#8441—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—44no—
177C22×C53—42no—
178C22×C53#37911yescharacteristic
179C4×C53—21no—
180C4×C53—22no—
181C4×C53—24no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—42no—
185C4×C53C2×#10412yes—
186C4×C53#42112yes—
187C4×C53#45311yescharacteristic
188C4×C53#45511yescharacteristic
189Dic53—44no—
190#3483D5311yescharacteristic
191#1962—531no—
192C4×#104—532no—
193#1525—531no—
194#1489—532no—
195C2×C53—41no—
196C2×C53—82no—
197C2×C53—82no—
198C2×C53#178912yes—
199C2×C53#179211yescharacteristic
200Q82Dic5311yescharacteristic, complemented
201#555—22no—
202#555—22no—
203#555Dic5311yescharacteristic, complemented
204#550—22no—
205#550Dic5312yescomplemented
206#532—22no—
207#532—22no—
208#532C2×D5311yescharacteristic
209C4×#41—22no—
210C4×#41C2×D5312yes—
211#423—531no—
212C2×#104—1062no—
213C2×#104—1062no—
214C4×M16—532no—
215C43—531no—
216#339—1062no—
217#329—1062no—
218Q8⋊C8—1062no—
219D4⋊C8—1062no—
220C53#844111yescharacteristic, complemented, Sylow
221C2×#41—22no—
222C2×#41—22no—
223C2×#41C22⋊D5312yes—
224C2×#41C22⋊D5312yes—
225C2×M16—1062no—
226C2×M16—1061no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8C2×Dic5311yescharacteristic
230#127—22no—
231#127—22no—
232#127C2×Dic5311yescharacteristic
233#124—22no—
234#124—24no—
235#124—22no—
236#124C22⋊D5312yes—
237#121—24no—
238#121—24no—
239#121—22no—
240#121—24no—
241#121—22no—
242#121—22no—
243C4×Q8—22no—
244C4×Q8C2×Dic5312yes—
245C4×Q8C22⋊D5312yes—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4C2×Dic5312yes—
253C4×D4C22⋊D5312yes—
254#117—24no—
255#117—22no—
256#117—22no—
257#117C22⋊D5312yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—1062no—
261C2×C42—1061no—
262C2×C42C22×D5311yescharacteristic
263C4⋊C8—1062no—
264#104—1068no—
265#104—2124no—
266#97—1062no—
267C4×C8—1062no—
268#41—24no—
269#41—42no—
270#41—42no—
271#41—42no—
272#41—44no—
273#41—44no—
274#41—82no—
275#41—82no—
276C2×Q8—22no—
277C2×Q8C22⋊Dic5312yes—
278C2×Q8C22⋊Dic5312yes—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4C22⋊Dic5312yes—
286C2×D4C22⋊Dic5312yes—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—2121no—
293C22×C4C2×C22⋊D5312yes—
294C22×C4C2×C22⋊D5311yescharacteristic
295M16—1064no—
296M16—2122no—
297M16—2122no—
298C2×C8—2121no—
299C2×C8—2122no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4C22⋊Dic5312yes—
305C4⋊C4C22⋊Dic5312yes—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—24no—
311C22⋊C4—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—1064no—
318C42—1062no—
319C42—2121no—
320C42—2122no—
321C42C22×Dic5311yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C42C2×C22⋊D5312yes—
324C23—22no—
325C23—22no—
326C23#1313318611yescharacteristic
327Q8—22no—
328Q8—42no—
329Q8—82no—
330Q8#1313314414yescomplemented
331D4—22no—
332D4—22no—
333D4—44no—
334D4—42no—
335D4—48no—
336D4—42no—
337D4—42no—
338D4—82no—
339D4—82no—
340D4—84no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—2124no—
358C2×C4—2122no—
359C2×C4C2×C22⋊Dic5311yescharacteristic
360C2×C4C2×C22⋊Dic5312yes—
361C2×C4#1313318612yes—
362C2×C4#1313318611yescharacteristic
363C8—4241no—
364C8—2124no—
365C22—22no—
366C22—21no—
367C22—42no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22#1681047911yescharacteristic
373C4—22no—
374C4—24no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4—2124no—
380C4C2×#1313314412yes—
381C4#1681064012yes—
382C4#1681079111yescharacteristic
383C4#1681080111yescharacteristic
384C2—41no—
385C2—82no—
386C2—82no—
387C2#2027465912yes—
388C2#2027466911yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3483C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3483D53
hypocenterC53#8441
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3483

Complements

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

NH ≅ G/Ntype
1. #20275544C2semidirect
2. C4×#13133144C2semidirect
3. #20273307C2semidirect
4. #20273154C2semidirect
15. Q82×C53C4semidirect
18. C53×#555C4semidirect
20. C53×#550C4semidirect
135. Q8×C53#104semidirect
200. Q82Dic53semidirect
203. #555Dic53semidirect
205. #550Dic53semidirect
220. C53#8441semidirect
330. Q8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202755442
C4×#131331442
#202733072
#202731542
C53×#34832
#844153