Finite Group Databasebeta

← #27276271

Subgroups — #27276271

Order 13568.

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

normal subgroups only

This group has 718 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#20274792C211yesmaximal, characteristic, complemented
2#20274786C211yesmaximal, characteristic, complemented
3#20273271C212yesmaximal, complemented
4#20273271C212yesmaximal, complemented
5C53×#3493C211yesmaximal, characteristic
6C2×C22⋊#88322C2211yescharacteristic
7C2×C22⋊#88322C2211yescharacteristic
8#16810486C2212yes—
9#16810390—22no—
10#16810378—22no—
11#16810360—22no—
12#16810360—22no—
13#16810357—22no—
14#16810357—22no—
15C53×#556—22no—
16C53×#555—22no—
17C53×#553—22no—
18C53×#552—22no—
19C53×#544C411yescharacteristic, complemented
20C53×#538—22no—
21C53×#537C411yescharacteristic, complemented
22C53×#529—22no—
23C53×#525C2212yes—
24C53×#520C412yescomplemented
25C2×#117×C53C2211yescharacteristic
26C22×#88322—21no—
27C22×#88322—21no—
28C2×C22⋊Dic53—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124—22no—
33C2×#41×C53C2×C412yes—
34C23×C4×C53C2311yescharacteristic
35C4⋊Q8×C53—22no—
36#126×C53—24no—
37#126×C53—24no—
38#125×C53—22no—
39#125×C53—22no—
40#124×C53—22no—
41#124×C53—24no—
42#123×C53—24no—
43#123×C53—22no—
44C22⋊Q8×C53—22no—
45C22⋊Q8×C53—22no—
46C22⋊Q8×C53D412yes—
47C22⋊Q8×C53D412yes—
48#121×C53—24no—
49#121×C53—22no—
50#121×C53—22no—
51#121×C53D412yes—
52#121×C53D412yes—
53#120×C53—22no—
54C4×Q8×C53—22no—
55C4×Q8×C53—22no—
56C4×D4×C53—22no—
57C4×D4×C53—22no—
58C4×D4×C53—24no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61#117×C53—24no—
62#117×C53—22no—
63#117×C53—22no—
64#117×C53—22no—
65#117×C53C2×C412yes—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53C2×C411yescharacteristic
68C2×C4⋊C4×C53C2×C411yescharacteristic
69C2×C22⋊C4×C53D412yes—
70C2×C42×C53D412yes—
71C24×C53C2×D411yescharacteristic
72#41×C53—44no—
73C2×Q8×C53—22no—
74C2×Q8×C53C22⋊C412yes—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53C22⋊C412yes—
80C22×C4×C53—24no—
81C22×C4×C53—22no—
82C22×C4×C53—42no—
83C22×C4×C53C22⋊C412yes—
84C22×C4×C53C22⋊C412yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D412yes—
89C22×Dic53—42no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
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×C53C22⋊C412yes—
99C4⋊C4×C53C22⋊C412yes—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C42×C53—24no—
109C42×C53—22no—
110C42×C53—22no—
111C42×C53—22no—
112C22⋊Dic53—44no—
113C2×#88322—42no—
114C2×#88322—42no—
115C2×#88322—41no—
116C2×#88322—41no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12012yes—
123Q8×C53—42no—
124Q8×C53—82no—
125D4×C53—42no—
126D4×C53—44no—
127D4×C53—82no—
128D4×C53—82no—
129D4×C53—82no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—44no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—84no—
143C2×C4×C53C2×C22⋊C412yes—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149#88322—81no—
150#88322—81no—
151#8463—531nomaximal, Sylow
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—82no—
161C22×C53#37911yescharacteristic
162C22×C53#41012yes—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—42no—
167C4×C53—42no—
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—42no—
171Dic53—162no—
172#3493D5311yescharacteristic
173#1820—531no—
174#1818—531no—
175#1516—532no—
176#1516—532no—
177C2×C53—22no—
178C2×C53—42no—
179C2×C53—82no—
180C2×C53#178211yescharacteristic
181C2×C53#179411yescharacteristic
182C2×C53#179411yescharacteristic
183#556—22no—
184#555—22no—
185#553—22no—
186#552—22no—
187#544Dic5311yescharacteristic, complemented
188#538—22no—
189#537Dic5311yescharacteristic, complemented
190#529—22no—
191#525C2×D5312yes—
192#520Dic5312yescomplemented
193C2×#117C2×D5311yescharacteristic
194C2×C22⋊C8—531no—
195C2×C22⋊C8—531no—
196#381—532no—
197#342—1062no—
198#336—1062no—
199#328—1062no—
200#328—1062no—
201#327—1062no—
202#327—1062no—
203C53#846311yescharacteristic, complemented, Sylow
204C2×#41C2×Dic5312yes—
205C23×C4C22×D5311yescharacteristic
206C22×C8—1061no—
207C22×C8—1061no—
208C4⋊Q8—22no—
209#126—24no—
210#126—24no—
211#125—22no—
212#125—22no—
213#124—22no—
214#124—24no—
215#123—24no—
216#123—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8C22⋊D5312yes—
220C22⋊Q8C22⋊D5312yes—
221#121—24no—
222#121—22no—
223#121—22no—
224#121C22⋊D5312yes—
225#121C22⋊D5312yes—
226#120—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—24no—
234#117—24no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117C2×Dic5312yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×Dic5311yescharacteristic
241C2×C4⋊C4C2×Dic5311yescharacteristic
242C2×C22⋊C4—1062no—
243C2×C22⋊C4C22⋊D5312yes—
244C2×C42C22⋊D5312yes—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247#95—1062no—
248#95—1062no—
249C24C2×C22⋊D5311yescharacteristic
250#41—44no—
251C2×Q8—22no—
252C2×Q8C22⋊Dic5312yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊Dic5312yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—2122no—
262C22×C4C22⋊Dic5312yes—
263C22×C4C22⋊Dic5312yes—
264C22×C4C22×Dic5311yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C2×C22⋊D5312yes—
267C22×C4C2×C22⋊D5312yes—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2121no—
271C2×C8—2121no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C22⋊Dic5312yes—
281C4⋊C4C22⋊Dic5312yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—2124no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic5311yescharacteristic
300C23#1313318612yes—
301Q8—42no—
302Q8—82no—
303D4—42no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—4242no—
322C2×C4—4242no—
323C2×C4—4242no—
324C2×C4C2×C22⋊Dic5312yes—
325C2×C4#1313318611yescharacteristic
326C2×C4#1313318611yescharacteristic
327C8—4241no—
328C8—4241no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22#1681047911yescharacteristic
339C22#1681059712yes—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—8482no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2#2027463311yescharacteristic
353C2#2027467911yescharacteristic
354C2#2027467911yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3493C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#3493D53
hypocenterC53#8463
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
1. #20274792C2semidirect
2. #20274786C2semidirect
3. #20273271C2semidirect
4. #20273271C2semidirect
19. C53×#544C4semidirect
21. C53×#537C4semidirect
24. C53×#520C4semidirect
187. #544Dic53semidirect
189. #537Dic53semidirect
192. #520Dic53semidirect
203. C53#8463semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202747862
#202732712
C53×#34932
#846353