Finite Group Databasebeta

← #27300893

Subgroups — #27300893

Order 13568.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277827C211yesmaximal, characteristic
2#20275104C211yesmaximal, characteristic
3#20274998C212yesmaximal
4#20272832C212yesmaximal
5C2×C53×#354C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810632—21no—
8C2×#13133134C2211yescharacteristic
9#16810530—21no—
10#16810516—22no—
11#16810503C2212yes—
12#16810498—22no—
13C22⋊C4×Dic53—22no—
14#16810469—21no—
15#16810389—22no—
16#16810383—21no—
17C2×#124×C53C2211yescharacteristic
18C2×#99×C53C2212yes—
19C53×#354C414yescomplemented
20C23×Dic53D412yes—
21C2×#11719917—21no—
22D4×Dic53—42no—
23#13133176—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—42no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53D412yes—
29C2×C4×Dic53—22no—
30C2×C4×Dic53—21no—
31#13133141—42no—
32#13133134—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124—21no—
36#13133124—22no—
37#13133124—42no—
38C22×Q8×C53D411yescharacteristic
39C22×D4×C53C2311yescharacteristic
40#124×C53—22no—
41#124×C53C2×C412yes—
42#124×C53Q812yes—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C42×C53D411yescharacteristic
46#99×C53—22no—
47#99×C53C2×C414yes—
48C24×C53C2×D412yes—
49C2×Q8×C53—21no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C4⋊C412yescomplemented
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53C22×C411yescharacteristic
56C2×D4×C53C2×Q811yescharacteristic
57C2×D4×C53#4112yes—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—44no—
67C22×Dic53—42no—
68C22⋊C4×C53—24no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—42no—
72C42×C53—21no—
73C42×C53C4⋊C412yescomplemented
74C22⋊Dic53—44no—
75C22⋊Dic53—42no—
76C22⋊Dic53—41no—
77C22⋊Dic53—42no—
78C4⋊Dic53—22no—
79C4⋊Dic53—81no—
80C4⋊Dic53—84no—
81C4×Dic53—44no—
82C4×Dic53—41no—
83C4×Dic53—81no—
84#11719917—41no—
85#11719917—41no—
86C2×#88322—41no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53C4×D412yes—
92C23×C53#12011yescharacteristic
93C23×C53C22⋊Q812yes—
94Q8×C53—42no—
95Q8×C53—81no—
96D4×C53—41no—
97D4×C53—41no—
98D4×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—44no—
107C2×C4×C53—82no—
108C2×C4×C53—82no—
109C2×C4×C53C2×C4⋊C411yescharacteristic
110C2×C4×C53#12311yescharacteristic
111C2×Dic53—24no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—44no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—82no—
120C2×Dic53—82no—
121C2×Dic53—82no—
122#88322—81no—
123#12965—531nomaximal, Sylow
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—22no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—82no—
132C22×C53#38511yescharacteristic
133C22×C53B(4,𝔽2)11yescharacteristic
134C22×C53#45811yescharacteristic
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53—84no—
140Dic53—44no—
141Dic53—82no—
142Dic53—82no—
143#2372—531no—
144C2×#354D5311yescharacteristic
145#1885—531no—
146#1867—532no—
147#1390—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—42no—
151C2×C53—42no—
152C2×C53#186711yescharacteristic
153C2×C53#219712yes—
154C2×#124C2×D5311yescharacteristic
155#518—531no—
156#419—1061no—
157C2×#100—531no—
158C2×#99C2×D5312yes—
159#391—1061no—
160#388—1062no—
161#386—532no—
162#385—1062no—
163C4×C22⋊C4—1062no—
164#376—1061no—
165#354Dic5314yescomplemented
166#342—1062no—
167#339—1061no—
168C53#1296511yescharacteristic, complemented, Sylow
169C22×Q8D21211yescharacteristic
170C22×D4C22×D5311yescharacteristic
171C23×C4—532no—
172C2×M16—1061no—
173#124—22no—
174#124Dic10612yes—
175#124C2×Dic5312yes—
176C4×D4—2122no—
177#117—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—2122no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—1062no—
184C2×C22⋊C4—1062no—
185C2×C42—1061no—
186C2×C42—1062no—
187C2×C42D21211yescharacteristic
188#103—2122no—
189#100—1062no—
190#99—22no—
191#99C2×Dic5314yes—
192#95—1061no—
193#95—1061no—
194#95—1062no—
195#95—1062no—
196#95—2122no—
197C24D4×D5312yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C4⋊Dic5312yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×Dic10611yescharacteristic
205C2×D4#1171994712yes—
206C2×D4C22×Dic5311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—2124no—
215C22×C4—2122no—
216C22×C4C2×D21211yescharacteristic
217M16—2121no—
218M16—2121no—
219C2×C8—2121no—
220C4⋊C4—1062no—
221C4⋊C4—4241no—
222C4⋊C4—4244no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—2124no—
228C22⋊C4—2122no—
229C22⋊C4—2121no—
230C22⋊C4—2122no—
231C42—21no—
232C42—4241no—
233C42—2121no—
234C42—2124no—
235C42C4⋊Dic5312yescomplemented
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23D4×Dic5312yes—
241C23#1313318711yescharacteristic
242C23#1313319412yes—
243Q8—42no—
244Q8—81no—
245D4—41no—
246D4—42no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—1064no—
259C2×C4—4242no—
260C2×C4—4242no—
261C2×C4—4242no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2124no—
269C2×C4C2×C4⋊Dic5311yescharacteristic
270C2×C4#1313320111yescharacteristic
271C8—4241no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22#1681049811yescharacteristic
281C22#1681081111yescharacteristic
282C22#1681081411yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—84no—
288C4—4242no—
289C4—4242no—
290C4—2124no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—42no—
295C2#2027499811yescharacteristic
296C2#2027670812yes—
subgroupG/H
centerC22#16810811
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#354C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#354D53
hypocenterC53#12965
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
19. C53×#354C4semidirect
52. C2×Q8×C53C4⋊C4semidirect
73. C42×C53C4⋊C4semidirect
165. #354Dic53semidirect
168. C53#12965semidirect
201. C2×Q8C4⋊Dic53semidirect
235. C42C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778272
#202751042
#202749982
#202728322
C2×C53×#3542
#1296553