Finite Group Databasebeta

← #27280873

Subgroups — #27280873

Order 13568.

464 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 111 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#20284512C211yesmaximal, characteristic
2#20283756C211yesmaximal, characteristic
3#20273278C211yesmaximal, characteristic, complemented
4#20273117C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#1804C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810923—22no—
10#16810913C412yescomplemented
11#16810903C412yescomplemented
12#16810701—22no—
13#16810701—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810663—21no—
17#16810647C2211yescharacteristic
18#16810628—21no—
19#16810587—22no—
20#16810359—22no—
21#16810353—22no—
22C2×C4×D4×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#102×C53—21no—
25C53×#382C2211yescharacteristic
26C53×#340—21no—
27C2×#11719917—24no—
28C2×#11719917—21no—
29#13133217C2×C411yescharacteristic
30#13133217C2×C411yescharacteristic
31#13133209—21no—
32#13133196—21no—
33#13133196—21no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36#13133176—21no—
37#13133150—22no—
38#13133148—22no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322C2×C412yes—
43#13133138—22no—
44#13133138—22no—
45C22⋊#88322—24no—
46C22⋊#88322—21no—
47C22⋊#88322D412yes—
48C4×#88322—21no—
49C4×#88322D412yes—
50C22×D4×C53D411yescharacteristic
51C23×C4×C53—21no—
52C22×C8×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53C2×C412yes—
56C4×D4×C53D412yes—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63#102×C53—42no—
64#95×C53—21no—
65#95×C53—21no—
66C24×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53C22⋊C412yes—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53—82no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C2×Dic106—21no—
82C2×Dic106—21no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53C22⋊C412yes—
90C22⋊C4×C53—22no—
91C22⋊C4×C53—22no—
92C42×C53C22×C411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53#4111yescharacteristic
96C22⋊Dic53—41no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101#11719919—41no—
102#11719919—41no—
103C4×Dic53—41no—
104#11719917—41no—
105#11719917—41no—
106#11719917—84no—
107C2×#88322—22no—
108C2×#88322—22no—
109C2×#88322—22no—
110C23×C53—22no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—81no—
114C23×C53—81no—
115C23×C53#12111yescharacteristic
116D4×C53—22no—
117D4×C53—22no—
118D4×C53—41no—
119D4×C53—41no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—44no—
124C2×C4×C53—82no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×Dic53—41no—
140C2×Dic53—41no—
141Dic106—42no—
142Dic106—81no—
143C8×C53—81no—
144#88322—42no—
145#88322—42no—
146#88322—42no—
147#9307—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—82no—
154C22×C53—82no—
155C22×C53#38611yescharacteristic
156C22×C53#46811yescharacteristic
157C22×C53#47011yescharacteristic
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—42no—
163C4×C53—161no—
164C4×C53#41711yescharacteristic
165C4×C53#41911yescharacteristic
166C4×C53#46811yescharacteristic
167C4×C53#47011yescharacteristic
168Dic53—81no—
169Dic53—81no—
170#3077—531no—
171#2975—531no—
172#1804D5311yescharacteristic
173#1518—531no—
174#1481—531no—
175#1301—531no—
176#1286—531no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—82no—
180C2×C53#188111yescharacteristic
181C2×C53#189411yescharacteristic
182C2×C53#190411yescharacteristic
183#533—531no—
184C2×C4×D4C2×D5311yescharacteristic
185#478—1062no—
186#476—532no—
187#474—532no—
188#436—1062no—
189#436—1062no—
190#431—531no—
191#431—531no—
192#428—1061no—
193#423—531no—
194C2×C4⋊C8C2×D5311yescharacteristic
195#418—1061no—
196C2×#102—21no—
197#407—1062no—
198#382C2×D5311yescharacteristic
199#340—21no—
200#328—1062no—
201Q8⋊C8—1062no—
202C53#930711yescharacteristic, complemented, Sylow
203C22×D4C22⋊D5311yescharacteristic
204C23×C4—21no—
205C2×M16—1064no—
206C2×M16—1061no—
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—21no—
216C4×D4—21no—
217C4×D4C4×D5312yes—
218C4×D4D21212yes—
219#117—1061no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D5311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C42C22×D5311yescharacteristic
224#107—1062no—
225#106—1062no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—532no—
229C4⋊C8—1061no—
230C4⋊C8—1061no—
231C4⋊C8—1061no—
232#102—42no—
233#102—1062no—
234#102—1062no—
235C22⋊C8—532no—
236C22⋊C8—1064no—
237C22⋊C8—1061no—
238C4×C8—532no—
239C4×C8—1061no—
240#95—21no—
241#95—21no—
242C24—21no—
243C2×Q8—1061no—
244C2×Q8—1061no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4#1171992112yes—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4D4×D5311yescharacteristic
257C22×C4#1171994711yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259M16—4244no—
260M16—2121no—
261M16—2121no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—1062no—
266C2×C8—1062no—
267C2×C8—1062no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4#1171992112yes—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—2121no—
281C42—2121no—
282C42C2×C4×D5311yescharacteristic
283C42C2×D21211yescharacteristic
284C42D4×D5311yescharacteristic
285C42#1171994711yescharacteristic
286C23—22no—
287C23—41no—
288C23—41no—
289C23—81no—
290C23—81no—
291C23#1313318911yescharacteristic
292Q8—4241no—
293Q8—2122no—
294D4—22no—
295D4—22no—
296D4—41no—
297D4—41no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4C2×#1171992111yescharacteristic
310C2×C4#1313317711yescharacteristic
311C2×C4#1313317711yescharacteristic
312C2×C4#1313318711yescharacteristic
313C2×C4#1313318911yescharacteristic
314C2×C4#1313320111yescharacteristic
315C2×C4#1313324311yescharacteristic
316C2×C4#1313324711yescharacteristic
317C2×C4#1313324711yescharacteristic
318C2×C4#1313324811yescharacteristic
319C8—81no—
320C8—2122no—
321C8—2122no—
322C8—2122no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22#1681050011yescharacteristic
331C22#1681086711yescharacteristic
332C22#1681087911yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—161no—
339C4—4241no—
340C4—4241no—
341C4#1681062211yescharacteristic
342C4#1681063111yescharacteristic
343C4#1681086711yescharacteristic
344C4#1681087911yescharacteristic
345C2—21no—
346C2—21no—
347C2—82no—
348C2#2027507511yescharacteristic
349C2#2027514611yescharacteristic
350C2#2027520011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1804C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#1804D53
hypocenterC53#9307
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
3. #20273278C2semidirect
4. #20273117C2semidirect
5. #20272612C2semidirect
6. #20272573C2semidirect
10. #16810913C4semidirect
11. #16810903C4semidirect
202. C53#9307semidirect

Maximal subgroups

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

subgroupindex
#202845122
#202837562
#202732782
#202731172
#202726122
#202725732
C53×#18042
#930753