Finite Group Databasebeta

← #27300361

Subgroups — #27300361

Order 13568.

642 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 107 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#20274643C212yesmaximal
2#20272842C214yesmaximal, complemented
3C53×#2297C211yesmaximal, characteristic
4C2×#13133136C2214yes—
5C2×#13133134—22no—
6#16810587—22no—
7C22×#41×C53C2211yescharacteristic
8C2×#117×C53C411yescharacteristic
9C22×C42×C53C411yescharacteristic
10C53×#391C2212yes—
11C53×#390C412yes—
12C53×#386—22no—
13C4×C22⋊C4×C53—22no—
14C2×#11719917—24no—
15#13133136—28no—
16#13133134—44no—
17C22⋊#88322—44no—
18C2×#41×C53—24no—
19C2×#41×C53—42no—
20C2×#41×C53—42no—
21C22×Q8×C53C2311yescharacteristic
22C22×D4×C53C2×C411yescharacteristic
23C22×D4×C53D412yes—
24C23×C4×C53—21no—
25C23×C4×C53C2×C411yescharacteristic
26C23×C4×C53D412yes—
27#117×C53—42no—
28C2×C4⋊C4×C53C2×C412yes—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53—24no—
31C2×C42×C53—21no—
32C2×C42×C53—21no—
33C2×C42×C53—42no—
34C2×C42×C53C2×C412yes—
35#95×C53—24no—
36C24×C53C22⋊C412yes—
37C24×C53C22⋊C411yescharacteristic
38#41×C53—416no—
39C2×Q8×C53—41no—
40C2×Q8×C53—41no—
41C2×Q8×C53#4114yes—
42C2×D4×C53—24no—
43C2×D4×C53—22no—
44C2×D4×C53—24no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53—42no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—24no—
53C22×C4×C53—24no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—44no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D412yes—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—48no—
63C22⋊C4×C53—42no—
64C42×C53—42no—
65C42×C53—42no—
66C42×C53—42no—
67#11719917—44no—
68#11719917—44no—
69C2×#88322—44no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—42no—
76C23×C53—44no—
77C23×C53#9912yes—
78C23×C53C2×C22⋊C411yescharacteristic
79Q8×C53—44no—
80D4×C53—48no—
81D4×C53—44no—
82C2×C4×C53—28no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—24no—
90C2×C4×C53—48no—
91C2×C4×C53—42no—
92C2×C4×C53—48no—
93C2×C4×C53—42no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53#9912yes—
97C2×C4×C53#11712yes—
98C2×C4×C53#12314yes—
99#88322—84no—
100#12861—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—42no—
104C22×C53—42no—
105C22×C53—44no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—44no—
109C22×C53—82no—
110C22×C53#38111yescharacteristic
111C22×C53C2×#9911yescharacteristic
112C22×C53C2×#9911yescharacteristic
113C4×C53—24no—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—44no—
117C4×C53—82no—
118#2297D5311yescharacteristic
119#1785—532no—
120#1394—534no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—42no—
124C2×C53—82no—
125C2×C53#181911yescharacteristic
126C2×C53#213212yes—
127C22×#41C2×D5311yescharacteristic
128C2×#117Dic5311yescharacteristic
129C22×C42Dic5311yescharacteristic
130C2×#101—534no—
131C2×#100—1062no—
132#407—1062no—
133#391C2×D5312yes—
134#390Dic5312yes—
135#386—22no—
136C4×C22⋊C4—22no—
137C53#1286111yescharacteristic, complemented, Sylow
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8C22×D5311yescharacteristic
142C22×D4C2×Dic5311yescharacteristic
143C22×D4C22⋊D5312yes—
144C23×C4—21no—
145C23×C4C2×Dic5311yescharacteristic
146C23×C4C22⋊D5312yes—
147C2×M16—1064no—
148#117—42no—
149C2×C4⋊C4C2×Dic5312yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2×Dic5312yes—
156#101—1068no—
157#100—2124no—
158C22⋊C8—2124no—
159#95—24no—
160C24C22⋊Dic5312yes—
161C24C22⋊Dic5311yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1171994514yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊Dic5311yescharacteristic
183C22×C4C22×Dic5311yescharacteristic
184C22×C4C2×C22⋊D5312yes—
185M16—2124no—
186M16—2124no—
187C2×C8—2124no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23#1313313212yes—
202C23C2×C22⋊Dic5311yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4#1313313212yes—
221C2×C4#1313317612yes—
222C2×C4#1313320014yes—
223C8—4244no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22#1681048611yescharacteristic
234C22C2×#1313313211yescharacteristic
235C22C2×#1313313211yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2#2027478911yescharacteristic
246C2#2027640012yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2297C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×Q8×C53C23
hypercenter#2297D53
hypocenterC53#12861
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #20272842C2semidirect
137. C53#12861semidirect

Maximal subgroups

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

subgroupindex
#202746432
#202728422
C53×#22972
#1286153