Finite Group Databasebeta

← #27272218

Subgroups — #27272218

Order 13568.

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

normal subgroups only

This group has 440 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#20277284C211yesmaximal, characteristic
2#20274857C212yesmaximal
3#20272541C211yesmaximal, characteristic
4#20272540C212yesmaximal
5C53×#1272C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133182C411yescharacteristic
9#16810529C2211yescharacteristic
10#16810520—22no—
11#16810492C411yescharacteristic
12#16810474—22no—
13C42×Dic53C2211yescharacteristic
14#16810378C2212yes—
15#16810377—22no—
16#16810377—22no—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×#88322—22no—
21#13133215—22no—
22#13133215C814yescomplemented
23Q8×Dic53—42no—
24#13133182—42no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×#11719919—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C2×C4×Dic53D412yes—
36#13133124—22no—
37C22×Q8×C53C2×C411yescharacteristic
38C22×C8×C53—22no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41#95×C53—22no—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53C2×C812yes—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C22⋊C412yes—
54C22×Dic53C22⋊C411yescharacteristic
55C2×Dic106—21no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106C2×C812yes—
59C2×C8×C53—42no—
60C2×C8×C53—42no—
61C2×C8×C53—42no—
62C4⋊C4×C53—41no—
63C42×C53—41no—
64C4⋊Dic53—42no—
65C4⋊Dic53—41no—
66#11719919—28no—
67#11719919—44no—
68#11719919—42no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—42no—
73C4×Dic53—42no—
74C4×Dic53—42no—
75C4×Dic53C2×C812yes—
76C4×Dic53M1612yes—
77C2×#88322—42no—
78C2×#88322—42no—
79C2×#88322—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53C22×C811yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53#13712yes—
95C2×C4×C53#13712yes—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53C22⋊C814yes—
107C2×Dic53#10012yes—
108Dic106—42no—
109Dic106—42no—
110C8×C53—82no—
111#88322—82no—
112#7605—531nomaximal, Sylow
113C22×C53C2×C22⋊C811yescharacteristic
114C22×C53#41011yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C22×C53#41912yes—
117C22×C53#42112yes—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—81no—
124Dic53—22no—
125Dic53—24no—
126Dic53—41no—
127Dic53—42no—
128Dic53—81no—
129C4×C4⋊Q8—531no—
130#1839—532no—
131#1272—531no—
132#1272—532no—
133#1272D5311yescharacteristic
134C2×C53#146011yescharacteristic
135C2×C53#147912yes—
136C2×C53#151512yes—
137C2×C53#153611yescharacteristic
138C2×C53#155811yescharacteristic
139C2×C4⋊Q8—531no—
140C2×C4×Q8—531no—
141C2×C4×Q8—531no—
142#391—531no—
143#391C2×D5311yescharacteristic
144#389—1062no—
145#384—531no—
146C4×C4⋊C4—1062no—
147C43—531no—
148#336—532no—
149#336—1062no—
150#336—1062no—
151#336C2×D5312yes—
152C53#760511yescharacteristic, complemented, Sylow
153C22×Q8—531no—
154C22×Q8C4×D5311yescharacteristic
155C22×C8—22no—
156C22×C8—1062no—
157C4⋊Q8—534no—
158C4⋊Q8—1062no—
159C4×Q8—2122no—
160C4×Q8—2122no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4C4×D5311yescharacteristic
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—532no—
170C2×C42—532no—
171C2×C42—1061no—
172C2×C42—1061no—
173C2×C42C22×D5311yescharacteristic
174#95—22no—
175#95—1062no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—532no—
180C2×Q8—1061no—
181C2×Q8—2121no—
182C2×Q8—2121no—
183C2×Q8C8×D5312yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—531no—
187C22×C4—532no—
188C22×C4—531no—
189C22×C4—1061no—
190C22×C4—1061no—
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4D4×D5312yes—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—2122no—
197C2×C8—2122no—
198C2×C8—2122no—
199C4⋊C4—41no—
200C4⋊C4—1068no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C4⋊C4—2122no—
204C4⋊C4—2124no—
205C42—41no—
206C42—532no—
207C42—532no—
208C42—1062no—
209C42—1062no—
210C42—1062no—
211C42—2122no—
212C42—2122no—
213C42—2122no—
214C23C22⋊C4×D5311yescharacteristic
215Q8—42no—
216Q8—42no—
217Q8—2122no—
218Q8—2122no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—534no—
229C2×C4—532no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1061no—
235C2×C4—1062no—
236C2×C4—2121no—
237C2×C4—2122no—
238C2×C4—2121no—
239C2×C4—2121no—
240C2×C4C2×C8×D5311yescharacteristic
241C2×C4M16×D5311yescharacteristic
242C2×C4#1313324612yes—
243C2×C4#1313324912yes—
244C8—82no—
245C8—4242no—
246C22C22⋊C8×D5311yescharacteristic
247C22#1681059511yescharacteristic
248C22#100×D5311yescharacteristic
249C22#1681063012yes—
250C22#1681063812yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—42no—
256C4—81no—
257C4—1062no—
258C4—1064no—
259C4—4241no—
260C4—2122no—
261C4—2121no—
262C2#2027303411yescharacteristic
263C2#2027310812yes—
264C2#2027326412yes—
265C2#2027334811yescharacteristic
266C2#2027343911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7605
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
22. #13133215C8semidirect
152. C53#7605semidirect

Maximal subgroups

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

subgroupindex
#202772842
#202748572
#202725412
#202725402
C53×#12722
#760553