Finite Group Databasebeta

← #27302387

Subgroups — #27302387

Order 13568.

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

normal subgroups only

This group has 540 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#20279023C211yesmaximal, characteristic
2C2×#16810414C214yesmaximal
3C4×#13133132C211yesmaximal, characteristic
4C53×#1800C211yesmaximal, characteristic
5C2×#13133198C2212yes—
6C22×C4⋊Dic53C411yescharacteristic, complemented
7C2×#13133132C2212yes—
8#16810553—21no—
9#16810548—21no—
10#16810521—21no—
11#16810484C412yescomplemented
12#16810483—21no—
13#16810483C411yescharacteristic, complemented
14C4×C22⋊Dic53—21no—
15#16810414—28no—
16#16810390—21no—
17C2×C4×D4×C53C2211yescharacteristic
18C53×#423—21no—
19C2×#100×C53C2212yes—
20C53×#343—21no—
21C23×Dic53C2×C412yes—
22#13133198—24no—
23#13133198—24no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53—41no—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53C2×C412yes—
33C2×C4×Dic53—41no—
34#13133132—22no—
35#13133132D414yes—
36#13133124—22no—
37#13133124—21no—
38#13133124—22no—
39#13133124—41no—
40C22×D4×C53C2311yescharacteristic
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C2×M16×C53—22no—
44C4×D4×C53—41no—
45C4×D4×C53—41no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C4⋊C8×C53—41no—
51C4⋊C8×C53—41no—
52#100×C53—24no—
53C24×C53C22×C411yescharacteristic
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C2×D412yes—
58C2×D4×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—24no—
73C22×Dic53—42no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76M16×C53—42no—
77M16×C53—42no—
78C2×C8×C53—42no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—24no—
85C22⋊Dic53—24no—
86C22⋊Dic53—42no—
87C22⋊Dic53—44no—
88C4⋊Dic53—42no—
89C4⋊Dic53—42no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—42no—
94C4×Dic53—82no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C4×D412yes—
105C23×C53#12112yes—
106D4×C53—42no—
107D4×C53—42no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—81no—
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#12411yescharacteristic
124C2×C4×C53#12711yescharacteristic
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—44no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53—84no—
135C8×C53—82no—
136#13220—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53#39011yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C22×C53#41011yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53#35212yescomplemented
159Dic53—82no—
160Dic53—84no—
161Dic53—82no—
162#2486—531no—
163C2×#352—534no—
164#1800D5311yescharacteristic
165C4×#99—531no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—42no—
169C2×C53—42no—
170C2×C53#196411yescharacteristic
171C2×C53C2×#35211yescharacteristic
172C2×C53#212311yescharacteristic
173C2×#123—532no—
174C2×C4×D4C2×D5311yescharacteristic
175C22×C4⋊C4—531no—
176#423—21no—
177C2×#100C2×D5312yes—
178C2×#99—532no—
179#398—1061no—
180#397—1061no—
181#389—1061no—
182#381—532no—
183#380—531no—
184#380—1061no—
185C4×C22⋊C4—1061no—
186#352—1068no—
187#343—21no—
188#342—1061no—
189C53#1322011yescharacteristic, complemented, Sylow
190C22×D4C22×D5311yescharacteristic
191C23×C4—532no—
192C23×C4C4×D5311yescharacteristic
193C23×C4D21211yescharacteristic
194C2×M16—22no—
195#123—1064no—
196#123—1064no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—2121no—
204C2×C4⋊C4—2121no—
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C22⋊C4—532no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C22⋊C4C4×D5311yescharacteristic
210C2×C22⋊C4D21211yescharacteristic
211C2×C42—2121no—
212C2×C42C22⋊D5311yescharacteristic
213C4⋊C8—41no—
214C4⋊C8—41no—
215#100—24no—
216#99—534no—
217#99—1062no—
218#95—1061no—
219#95—1062no—
220#95—1062no—
221#95—2121no—
222C24C2×C4×D5311yescharacteristic
223C24C2×D21211yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D5312yes—
227C2×D4#1171994912yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1064no—
235C22×C4—1062no—
236C22×C4—1062no—
237C22×C4—2122no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4#1171992111yescharacteristic
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C4⋊C4—41no—
249C4⋊C4—2122no—
250C4⋊C4—2122no—
251C4⋊C4—2121no—
252C4⋊C4—2121no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—2122no—
260C22⋊C4—2124no—
261C42—21no—
262C42—21no—
263C42—4242no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×#1171992111yescharacteristic
273C23#1313317912yes—
274C23#1313319212yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—4242no—
291C2×C4—4242no—
292C2×C4—4244no—
293C2×C4—2124no—
294C2×C4—2122no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4#1313313111yescharacteristic
301C2×C4#1313313111yescharacteristic
302C2×C4#1313320611yescharacteristic
303C2×C4#1313321311yescharacteristic
304C8—82no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1681052411yescharacteristic
318C22C2×#1313313111yescharacteristic
319C22#1681059611yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—4242no—
327C4—4244no—
328C4—4242no—
329C4#1681041412yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2027555011yescharacteristic
335C2C2×#1681041411yescharacteristic
336C2#2027637011yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1800D53
hypocenterC53#13220
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic53C4semidirect
11. #16810484C4semidirect
13. #16810483C4semidirect
158. C4×C53#352semidirect
189. C53#13220semidirect
329. C4#16810414semidirect

Maximal subgroups

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

subgroupindex
#202790232
C2×#168104142
C4×#131331322
C53×#18002
#1322053