Finite Group Databasebeta

← #27276530

Subgroups — #27276530

Order 13568.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#20275697C211yesmaximal, characteristic, complemented
2#20273380C211yesmaximal, characteristic, complemented
3#20273201C212yesmaximal, complemented
4#20273180C212yesmaximal, complemented
5C53×#3531C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810647C2211yescharacteristic
8#16810469C2211yescharacteristic
9#16810382—22no—
10#16810369—22no—
11#16810360—22no—
12#16810357—22no—
13Q8⋊#88322—22no—
14D4⋊#88322—22no—
15C53×#564C411yescharacteristic, complemented
16C53×#563—22no—
17C53×#563—22no—
18C53×#563C412yescomplemented
19C53×#562—22no—
20C53×#562C411yescharacteristic, complemented
21C53×#533—22no—
22C53×#533C2211yescharacteristic
23C53×#529—22no—
24C53×#529—22no—
25C53×#529C2212yes—
26C2×#11719917—21no—
27C2×C4×Dic53—21no—
28C4⋊#88322—22no—
29C4⋊#88322—22no—
30C22⋊#88322—22no—
31C4×#88322—22no—
32#13133124—22no—
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53D412yes—
36#126×C53—22no—
37#126×C53—22no—
38#126×C53—22no—
39#126×C53—22no—
40#126×C53—22no—
41#125×C53—22no—
42#125×C53—22no—
43#125×C53—22no—
44#125×C53—22no—
45#125×C53C2×C412yes—
46#125×C53C2×C411yescharacteristic
47#125×C53C2×C411yescharacteristic
48#124×C53—22no—
49#124×C53—22no—
50#124×C53C2×C412yes—
51#123×C53—22no—
52#123×C53—22no—
53#123×C53—22no—
54#123×C53—22no—
55#123×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53—22no—
61C22⋊Q8×C53—22no—
62C22⋊Q8×C53—22no—
63C22⋊Q8×C53D412yes—
64#121×C53—22no—
65#121×C53—22no—
66#121×C53D412yes—
67C4×Q8×C53—22no—
68C4×Q8×C53—22no—
69C4×Q8×C53D412yes—
70C4×D4×C53—22no—
71C4×D4×C53—22no—
72C4×D4×C53D412yes—
73#117×C53—22no—
74#117×C53—22no—
75#117×C53D412yes—
76C2×C42×C53—22no—
77C2×C42×C53C2311yescharacteristic
78C2×Q8×C53—22no—
79C2×Q8×C53—22no—
80C2×Q8×C53C22⋊C412yes—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53C22⋊C412yes—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D412yes—
89C22×Dic53—41no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53C22⋊C412yes—
104C4⋊C4×C53C22⋊C412yes—
105C4⋊C4×C53C22⋊C412yes—
106C4⋊C4×C53C22⋊C412yes—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C42×C53—22no—
118C42×C53—22no—
119C42×C53—22no—
120C42×C53—22no—
121C42×C53C22×C411yescharacteristic
122C42×C53C2×D412yes—
123C42×C53C2×D411yescharacteristic
124C4×Dic53—22no—
125C4×Dic53—41no—
126#11719917—42no—
127C2×#88322—42no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53#12011yescharacteristic
131Q8×C53—42no—
132Q8×C53—82no—
133Q8×C53—82no—
134D4×C53—42no—
135D4×C53—82no—
136D4×C53—82no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—82no—
148C2×C4×C53—82no—
149C2×C4×C53C2×C22⋊C412yes—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53#12012yes—
152C2×C4×C53#12011yescharacteristic
153C2×Dic53—42no—
154C2×Dic53—82no—
155#88322—82no—
156#88322—81no—
157#8505—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53#37911yescharacteristic
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—42no—
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—42no—
171C4×C53—42no—
172C4×C53#45411yescharacteristic
173C4×C53#45611yescharacteristic
174Dic53—82no—
175#3531D5311yescharacteristic
176#1995—531no—
177#1544—531no—
178#1500—532no—
179#1495—532no—
180C2×C53—41no—
181C2×C53—82no—
182C2×C53#179311yescharacteristic
183C2×C53#179412yes—
184#564Dic5311yescharacteristic, complemented
185#563—22no—
186#563—22no—
187#563Dic5312yescomplemented
188#562—22no—
189#562Dic5311yescharacteristic, complemented
190#533—22no—
191#533C2×D5311yescharacteristic
192#529—22no—
193#529—22no—
194#529C2×D5312yes—
195#431—532no—
196#423—531no—
197#376—531no—
198#339—1062no—
199#332—1062no—
200#328—1062no—
201#327—1062no—
202Q8⋊C8—1062no—
203D4⋊C8—1062no—
204C53#850511yescharacteristic, complemented, Sylow
205C2×M16—1061no—
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4⋊Q8C22⋊D5312yes—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#126—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125C2×Dic5312yes—
219#125C2×Dic5311yescharacteristic
220#125C2×Dic5311yescharacteristic
221#124—22no—
222#124—22no—
223#124C2×Dic5312yes—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C22⋊D5312yes—
237#121—22no—
238#121—22no—
239#121C22⋊D5312yes—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8C22⋊D5312yes—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C22⋊D5312yes—
246#117—22no—
247#117—22no—
248#117C22⋊D5312yes—
249C2×C42—22no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252C4⋊C8—1062no—
253C4⋊C8—1062no—
254C22⋊C8—1062no—
255C4×C8—1062no—
256#95—1062no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8C22⋊Dic5312yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊Dic5312yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—2121no—
267C22×C4C2×C22⋊D5311yescharacteristic
268C22×C4C2×C22⋊D5312yes—
269M16—2122no—
270C2×C8—2122no—
271C2×C8—2121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4C22⋊Dic5312yes—
286C4⋊C4C22⋊Dic5312yes—
287C4⋊C4C22⋊Dic5312yes—
288C4⋊C4C22⋊Dic5312yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—1062no—
304C42—2121no—
305C42C22×Dic5311yescharacteristic
306C42C2×C22⋊D5311yescharacteristic
307C42C2×C22⋊D5312yes—
308C23—22no—
309C23#1313318611yescharacteristic
310Q8—42no—
311Q8—82no—
312Q8—82no—
313D4—42no—
314D4—82no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—4242no—
329C2×C4—2122no—
330C2×C4C2×C22⋊Dic5311yescharacteristic
331C2×C4C2×C22⋊Dic5312yes—
332C2×C4#1313318612yes—
333C2×C4#1313318611yescharacteristic
334C8—4242no—
335C8—4241no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#1681047911yescharacteristic
342C4—21no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—4242no—
351C4#1681079611yescharacteristic
352C4#1681080611yescharacteristic
353C2—41no—
354C2—82no—
355C2#2027467411yescharacteristic
356C2#2027467912yes—
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3531C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3531D53
hypocenterC53#8505
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3531

Complements

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

NH ≅ G/Ntype
1. #20275697C2semidirect
2. #20273380C2semidirect
3. #20273201C2semidirect
4. #20273180C2semidirect
15. C53×#564C4semidirect
18. C53×#563C4semidirect
20. C53×#562C4semidirect
184. #564Dic53semidirect
187. #563Dic53semidirect
189. #562Dic53semidirect
204. C53#8505semidirect

Maximal subgroups

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

subgroupindex
#202756972
#202733802
#202732012
#202731802
C53×#35312
#850553