Finite Group Databasebeta

← #27271386

Subgroups — #27271386

Order 13568.

470 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 67 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#20274633C211yesmaximal, characteristic
2#20274633C211yesmaximal, characteristic
3C53×#1785C211yesmaximal, characteristic
4C2×#13133132—21no—
5C2×#13133132—21no—
6C2×#13133132—21no—
7C2×#13133132C412yescomplemented
8#16810486—21no—
9#16810486C412yescomplemented
10C22×#41×C53C2211yescharacteristic
11C2×#101×C53C412yescomplemented
12C2×#100×C53—21no—
13C53×#407—21no—
14C2×C22⋊Dic53—22no—
15C2×C22⋊Dic53—22no—
16#13133132—42no—
17#13133132—42no—
18#13133132—42no—
19#13133132—42no—
20#13133124—42no—
21#13133124—42no—
22C2×#41×C53—42no—
23C2×#41×C53—42no—
24C2×#41×C53—42no—
25C22×Q8×C53C2×C411yescharacteristic
26C22×D4×C53C2×C411yescharacteristic
27C22×D4×C53D411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C23×C4×C53Q811yescharacteristic
32C2×M16×C53—22no—
33#101×C53—42no—
34#100×C53—42no—
35C22⋊C8×C53—42no—
36C24×C53C22⋊C411yescharacteristic
37C24×C53C22⋊C411yescharacteristic
38C24×C53C4⋊C411yescharacteristic
39#41×C53—416no—
40C2×Q8×C53—22no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C4211yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C4⋊C411yescharacteristic
64C22×C4×C53C4⋊C411yescharacteristic
65C22×Dic53—42no—
66C22×Dic53—42no—
67M16×C53—42no—
68M16×C53—42no—
69C2×C8×C53—42no—
70C22⋊Dic53—42no—
71C22⋊Dic53—42no—
72C22⋊Dic53—42no—
73C22⋊Dic53—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—81no—
82C23×C53—81no—
83C23×C53—81no—
84C23×C53#9511yescharacteristic
85C23×C53#9912yes—
86C23×C53#9912yes—
87Q8×C53—44no—
88D4×C53—44no—
89D4×C53—44no—
90D4×C53—44no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—44no—
103C2×C4×C53—44no—
104C2×C4×C53—44no—
105C2×C4×C53#9912yes—
106C2×Dic53—82no—
107C2×Dic53—82no—
108C2×Dic53—84no—
109C2×Dic53—84no—
110C8×C53—82no—
111#7313—531nomaximal, Sylow
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—81no—
120C22×C53—81no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53#34211yescharacteristic
126C22×C53#34211yescharacteristic
127C22×C53#34211yescharacteristic
128C4×C53—22no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132Dic53—162no—
133Dic53—162no—
134#1785D5311yescharacteristic
135#1782—531no—
136#1782—531no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53—42no—
141C2×C53—42no—
142C2×C53—81no—
143C2×C53#130411yescharacteristic
144C22×#41C2×D5311yescharacteristic
145C2×#101Dic5312yescomplemented
146C2×#100—21no—
147C2×#99—532no—
148C2×#99—1061no—
149C2×#99—1061no—
150C2×#99—1061no—
151#407—21no—
152#381—532no—
153#381—1061no—
154C53#731311yescharacteristic, complemented, Sylow
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C22×Q8C2×Dic5311yescharacteristic
159C22×D4C4×D5311yescharacteristic
160C22×D4D21211yescharacteristic
161C22×D4C22⋊D5311yescharacteristic
162C23×C4Dic10611yescharacteristic
163C23×C4C4×D5311yescharacteristic
164C23×C4C22⋊D5311yescharacteristic
165C2×M16—22no—
166C2×C22⋊C4—1062no—
167C2×C22⋊C4—1062no—
168#101—42no—
169#100—42no—
170#99—2122no—
171#99—2122no—
172#99—2122no—
173#99—2122no—
174C22⋊C8—42no—
175#95—2122no—
176#95—2122no—
177C24#1171991911yescharacteristic
178C24#1171992111yescharacteristic
179C24#1171992111yescharacteristic
180#41—416no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—2122no—
203C22×C4—2122no—
204C22×C4C4×Dic5311yescharacteristic
205C22×C4#1171991911yescharacteristic
206C22×C4C4⋊Dic5311yescharacteristic
207C22×C4C22⋊Dic5311yescharacteristic
208M16—42no—
209M16—42no—
210C2×C8—42no—
211C22⋊C4—2122no—
212C22⋊C4—2122no—
213C22⋊C4—2122no—
214C22⋊C4—2122no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23#1313312411yescharacteristic
226C23#1313313112yes—
227C23#1313313112yes—
228Q8—44no—
229D4—44no—
230D4—44no—
231D4—44no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—4242no—
247C2×C4—4242no—
248C2×C4—4244no—
249C2×C4—4244no—
250C2×C4#1313313212yes—
251C8—82no—
252C22—22no—
253C22—22no—
254C22—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#1681038911yescharacteristic
266C22#1681038911yescharacteristic
267C22#1681039011yescharacteristic
268C4—22no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—8482no—
273C4—8482no—
274C2—21no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2—42no—
279C2—81no—
280C2#2027261911yescharacteristic
subgroupG/H
centerC2#20272619
commutatorC22×C4×C53C42
FrattiniC22×#41C2×D53
FittingC53×#1785C2
socleC2×C53#1304
prime radical, cosocleC22×#41×C53C22
hypercenter#1785D53
hypocenterC53#7313
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
7. C2×#13133132C4semidirect
9. #16810486C4semidirect
11. C2×#101×C53C4semidirect
145. C2×#101Dic53semidirect
154. C53#7313semidirect

Maximal subgroups

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

subgroupindex
#202746332
C53×#17852
#731353