Finite Group Databasebeta

← #27280876

Subgroups — #27280876

Order 13568.

464 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284514C211yesmaximal, characteristic
2#20283755C211yesmaximal, characteristic
3#20274726C211yesmaximal, characteristic
4#20273280C211yesmaximal, characteristic, complemented
5#20273117C211yesmaximal, characteristic, complemented
6#20272575C211yesmaximal, characteristic, complemented
7C53×#1301C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810924—22no—
11#16810914C412yescomplemented
12#16810904C412yescomplemented
13#16810702—22no—
14#16810700—22no—
15#16810676C2211yescharacteristic
16#16810665—21no—
17#16810629—21no—
18C2×#13133139—21no—
19#16810585—22no—
20#16810489C2211yescharacteristic
21#16810386—21no—
22#16810361—22no—
23#16810353—22no—
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719912—24no—
30#13133217C2×C411yescharacteristic
31#13133217C2×C411yescharacteristic
32#13133209—21no—
33#13133196—21no—
34#13133196—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212C2×C412yes—
40C4×D212D412yes—
41#13133176—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719921—21no—
45#13133151—22no—
46#13133149—22no—
47C4⋊#88322—21no—
48#13133139—22no—
49#13133139—22no—
50#13133139—42no—
51C22⋊#88322—21no—
52#13133129—24no—
53C4×#88322D412yes—
54#13133124—21no—
55#13133124—21no—
56C2×M16×C53—21no—
57C22×C8×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53C2×C412yes—
64C22⋊C8×C53D412yes—
65C4×C8×C53—21no—
66C23×D53—21no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—41no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212C22⋊C412yes—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—82no—
81C2×Dic106—21no—
82C2×Dic106—21no—
83M16×C53—41no—
84M16×C53—41no—
85C2×C8×C53—22no—
86C2×C8×C53—22no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C42×C53C22×C411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53#4111yescharacteristic
94C22⋊Dic53—41no—
95#11719921—22no—
96#11719921—22no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53C22⋊C412yes—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—41no—
108C2×#88322—22no—
109#11719912—84no—
110C23×C53#12111yescharacteristic
111C22×D53—22no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—81no—
115C22×D53—81no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53C4×D411yescharacteristic
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53#12011yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C53#13711yescharacteristic
126C2×C4×C53#13711yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C2×Dic53—22no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—81no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135D212—22no—
136D212—22no—
137D212—41no—
138D212—41no—
139C4×D53—44no—
140C4×D53—82no—
141Dic106—42no—
142Dic106—81no—
143C8×C53—42no—
144C8×C53—42no—
145C8×C53—81no—
146#88322—42no—
147#9307—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53#38611yescharacteristic
151C22×C53#46811yescharacteristic
152C22×C53#47011yescharacteristic
153C2×D53—42no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—82no—
157C2×D53—82no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53#41711yescharacteristic
163C4×C53#41911yescharacteristic
164C4×C53#46811yescharacteristic
165C4×C53#47011yescharacteristic
166Dic53—42no—
167Dic53—81no—
168Dic53—81no—
169Dic53—161no—
170#3077—531no—
171#2975—531no—
172#1804—531no—
173#1518—531no—
174#1481—531no—
175#1301D5311yescharacteristic, complemented
176#1286—531no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53#188111yescharacteristic
180C2×C53#189411yescharacteristic
181C2×C53#190411yescharacteristic
182D53—82no—
183#533—531no—
184C2×C4×D4—531no—
185#478—1062no—
186#476—532no—
187#474—532no—
188#436—1062no—
189#436—1062no—
190#431—531no—
191#431C2×D5311yescharacteristic
192#428—1061no—
193#423C2×D5311yescharacteristic
194C2×C4⋊C8C2×D5311yescharacteristic
195#418—1061no—
196C2×#102—1061no—
197#407—1062no—
198#382—531no—
199#340—1061no—
200#328—1062no—
201Q8⋊C8—1062no—
202C53#930711yescharacteristic, complemented, Sylow
203C22×D4—531no—
204C23×C4—1061no—
205C2×M16—21no—
206C2×M16—1064no—
207C22×C8—21no—
208C4⋊Q8—531no—
209C4⋊Q8—531no—
210#125—1061no—
211C22⋊Q8—1061no—
212C22⋊Q8—1061no—
213C4×Q8—1061no—
214C4×Q8—1061no—
215C4×D4—532no—
216C4×D4—532no—
217C4×D4—1061no—
218C4×D4—1061no—
219#117—1061no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C42C22×D5311yescharacteristic
224#107—1062no—
225#106—1062no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—1061no—
231C4⋊C8C4×D5312yes—
232#102—1062no—
233#102—1062no—
234#102—2122no—
235C22⋊C8—1064no—
236C22⋊C8—1061no—
237C22⋊C8D21212yes—
238C4×C8—21no—
239C4×C8—532no—
240#95—1061no—
241#95—1061no—
242C24—1061no—
243C2×Q8—1061no—
244C2×Q8—1061no—
245C2×D4—532no—
246C2×D4—1061no—
247C2×D4—1061no—
248C2×D4—1061no—
249C2×D4—1061no—
250C2×D4—1061no—
251C22×C4—1061no—
252C22×C4—1062no—
253C22×C4—1062no—
254C22×C4—4242no—
255C22×C4—2121no—
256C22×C4C2×D21211yescharacteristic
257C22×C4D4×D5311yescharacteristic
258C22×C4#1171994911yescharacteristic
259M16—41no—
260M16—41no—
261M16—4244no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—1062no—
268C4⋊C4—532no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C22⋊C4—1062no—
279C22⋊C4—1062no—
280C22⋊C4—2121no—
281C42—2121no—
282C42C2×C4×D5311yescharacteristic
283C42#1171994511yescharacteristic
284C42D4×D5311yescharacteristic
285C42D4×D5311yescharacteristic
286C23—1062no—
287C23—4241no—
288C23—4241no—
289C23—2121no—
290C23—2121no—
291C23#1313319211yescharacteristic
292Q8—4241no—
293Q8—2122no—
294D4—1062no—
295D4—1062no—
296D4—2121no—
297D4—2121no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—1062no—
301C2×C4—4242no—
302C2×C4—4241no—
303C2×C4—4241no—
304C2×C4—4241no—
305C2×C4—4241no—
306C2×C4—2124no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4C22⋊C4×D5311yescharacteristic
310C2×C4#1313317911yescharacteristic
311C2×C4C4×D21211yescharacteristic
312C2×C4#1313318711yescharacteristic
313C2×C4#1313319011yescharacteristic
314C2×C4#1313320211yescharacteristic
315C2×C4#1313324111yescharacteristic
316C2×C4#1313324611yescharacteristic
317C2×C4#1313324611yescharacteristic
318C2×C4#1313324911yescharacteristic
319C8—42no—
320C8—42no—
321C8—81no—
322C8—2122no—
323C22—21no—
324C22—21no—
325C22—4242no—
326C22—4242no—
327C22—2122no—
328C22—2122no—
329C22—2122no—
330C22#1681050111yescharacteristic
331C22#1681087011yescharacteristic
332C22#1681088211yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—8481no—
338C4—4241no—
339C4—4241no—
340C4—2122no—
341C4#1681062111yescharacteristic
342C4#1681063011yescharacteristic
343C4#1681086811yescharacteristic
344C4#1681088011yescharacteristic
345C2—21no—
346C2—21no—
347C2—4242no—
348C2#2027507811yescharacteristic
349C2#2027514711yescharacteristic
350C2#2027520111yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1301C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1301D53
hypocenterC53#9307
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
4. #20273280C2semidirect
5. #20273117C2semidirect
6. #20272575C2semidirect
7. C53×#1301C2semidirect
11. #16810914C4semidirect
12. #16810904C4semidirect
175. #1301D53semidirect
202. C53#9307semidirect

Maximal subgroups

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

subgroupindex
#202845142
#202837552
#202747262
#202732802
#202731172
#202725752
C53×#13012
#930753