Finite Group Databasebeta

← #27280878

Subgroups — #27280878

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#20284508C211yesmaximal, characteristic
2#20283755C211yesmaximal, characteristic
3#20274726C211yesmaximal, characteristic
4#20273279C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#1481C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810920—22no—
10#16810910C412yescomplemented
11#16810900C412yescomplemented
12#16810702—22no—
13#16810700—22no—
14#16810676C2211yescharacteristic
15#16810666—21no—
16#16810647C2211yescharacteristic
17#16810627—21no—
18C2×#13133139—21no—
19#16810585—22no—
20#16810489C2211yescharacteristic
21#16810386—21no—
22#16810360—22no—
23C53×#533C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×Q8⋊C8—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719917—21no—
30C2×#11719912—24no—
31C4×D212—21no—
32C4×D212—21no—
33C4×D212C2×C412yes—
34C4×D212D412yes—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719921—21no—
38#13133150—22no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322C2×C412yes—
43#13133139—42no—
44#13133137—22no—
45#13133137—22no—
46C22⋊#88322D412yes—
47#13133129—24no—
48C4×#88322—21no—
49#13133124—21no—
50#13133124—21no—
51C22×C8×C53—21no—
52C4⋊Q8×C53C2×C411yescharacteristic
53C4⋊Q8×C53C2×C411yescharacteristic
54#125×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59#117×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65C4×C8×C53D412yes—
66C23×D53—21no—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—41no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212C22⋊C412yes—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—82no—
83C2×C8×C53—22no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—41no—
95C42×C53C22×C411yescharacteristic
96C42×C53C2×D411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53#4111yescharacteristic
99#11719921—22no—
100#11719921—22no—
101C4⋊Dic53—21no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C4⋊Dic53C22⋊C412yes—
105#11719917—41no—
106#11719917—41no—
107C2×#88322—22no—
108C2×#88322—22no—
109#11719912—84no—
110C23×C53#12111yescharacteristic
111C22×D53—22no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—81no—
115C22×D53—81no—
116Q8×C53—42no—
117Q8×C53—81no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53C2×C22⋊C411yescharacteristic
123C2×C4×C53C4×D411yescharacteristic
124C2×C4×C53C4×D411yescharacteristic
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53#12111yescharacteristic
127C2×C4×C53#12311yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×C4×C53#13711yescharacteristic
132C2×Dic53—22no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137D212—22no—
138D212—22no—
139D212—41no—
140D212—41no—
141C4×D53—44no—
142C4×D53—82no—
143C8×C53—42no—
144C8×C53—81no—
145#88322—42no—
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—81no—
163C4×C53—81no—
164C4×C53#41711yescharacteristic
165C4×C53#41911yescharacteristic
166C4×C53#46811yescharacteristic
167C4×C53#47011yescharacteristic
168Dic53—42no—
169Dic53—161no—
170#3077—531no—
171#2975—531no—
172#1804—531no—
173#1518—531no—
174#1481D5311yescharacteristic, complemented
175#1301—531no—
176#1286—531no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53#188111yescharacteristic
180C2×C53#189411yescharacteristic
181C2×C53#190411yescharacteristic
182D53—82no—
183#533C2×D5311yescharacteristic
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#423—531no—
194C2×C4⋊C8C2×D5311yescharacteristic
195#418—1061no—
196C2×#102—1061no—
197#407—1062no—
198#382—531no—
199#340—1061no—
200#328—1062no—
201Q8⋊C8—22no—
202C53#930711yescharacteristic, complemented, Sylow
203C22×D4—531no—
204C23×C4—1061no—
205C2×M16—1064no—
206C2×M16—1061no—
207C22×C8—21no—
208C4⋊Q8C4×D5311yescharacteristic
209C4⋊Q8C4×D5311yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×D4—532no—
216C4×D4—532no—
217C4×D4—1061no—
218C4×D4—1061no—
219#117—21no—
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—532no—
230C4⋊C8—1061no—
231C4⋊C8—1061no—
232#102—1062no—
233#102—1062no—
234#102—2122no—
235C22⋊C8—21no—
236C22⋊C8—532no—
237C22⋊C8—1064no—
238C4×C8—1061no—
239C4×C8C22⋊D5312yes—
240#95—1061no—
241#95—1061no—
242C24—1061no—
243C2×Q8—21no—
244C2×Q8—21no—
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×C4#1171994511yescharacteristic
257C22×C4D4×D5311yescharacteristic
258C22×C4D4×D5311yescharacteristic
259M16—4244no—
260M16—2121no—
261M16—2121no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—1062no—
267C2×C8—1062no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—532no—
275C4⋊C4—1061no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C22⋊C4—41no—
279C22⋊C4—1062no—
280C22⋊C4—1062no—
281C42—41no—
282C42C2×C4×D5311yescharacteristic
283C42D4×D5311yescharacteristic
284C42#1171994711yescharacteristic
285C42C2×C22⋊D5311yescharacteristic
286C23—1062no—
287C23—4241no—
288C23—4241no—
289C23—2121no—
290C23—2121no—
291C23#1313319011yescharacteristic
292Q8—42no—
293Q8—81no—
294D4—1062no—
295D4—1062no—
296D4—2121no—
297D4—2121no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—1062no—
303C2×C4—4242no—
304C2×C4—4241no—
305C2×C4—4241no—
306C2×C4—4241no—
307C2×C4—4241no—
308C2×C4—2124no—
309C2×C4C22⋊C4×D5311yescharacteristic
310C2×C4#1313317711yescharacteristic
311C2×C4C4×C22⋊D5311yescharacteristic
312C2×C4#1313318511yescharacteristic
313C2×C4#1313318811yescharacteristic
314C2×C4#1313319911yescharacteristic
315C2×C4#1313324411yescharacteristic
316C2×C4#1313324611yescharacteristic
317C2×C4#1313324911yescharacteristic
318C2×C4#1313324911yescharacteristic
319C8—42no—
320C8—81no—
321C8—2122no—
322C8—2122no—
323C22—21no—
324C22—21no—
325C22—4242no—
326C22—4242no—
327C22—2122no—
328C22—2122no—
329C22—2122no—
330C22#1681049911yescharacteristic
331C22#1681087211yescharacteristic
332C22#1681088011yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—81no—
338C4—81no—
339C4—8481no—
340C4—2122no—
341C4#1681062311yescharacteristic
342C4#1681063011yescharacteristic
343C4#1681086611yescharacteristic
344C4#1681087811yescharacteristic
345C2—21no—
346C2—21no—
347C2—4242no—
348C2#2027508011yescharacteristic
349C2#2027514511yescharacteristic
350C2#2027519911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1481C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1481D53
hypocenterC53#9307
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
4. #20273279C2semidirect
5. #20272612C2semidirect
6. #20272574C2semidirect
7. C53×#1481C2semidirect
10. #16810910C4semidirect
11. #16810900C4semidirect
174. #1481D53semidirect
202. C53#9307semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202837552
#202747262
#202732792
#202726122
#202725742
C53×#14812
#930753