Finite Group Databasebeta

← #27272204

Subgroups — #27272204

Order 13568.

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

normal subgroups only

This group has 492 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#20277262C211yesmaximal, characteristic
2#20274830C212yesmaximal, complemented
3#20272540C212yesmaximal, complemented
4#20272536C211yesmaximal, characteristic
5C53×#1270C211yesmaximal, characteristic
6C2×#13133204C411yescharacteristic
7C2×#13133182C411yescharacteristic
8C2×D4×Dic53C411yescharacteristic
9C2×#13133146—22no—
10#16810529C2211yescharacteristic
11#16810511—22no—
12#16810488C411yescharacteristic
13C22⋊C4×Dic53—22no—
14C42×Dic53C2211yescharacteristic
15#16810378C2212yes—
16#16810377—22no—
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C23×Dic53—21no—
21C22×#88322—22no—
22#13133204—22no—
23#13133204C814yescomplemented
24#13133182—42no—
25D4×Dic53—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53D412yes—
35#13133146—44no—
36#13133124—22no—
37#13133124—22no—
38C22×D4×C53C2×C411yescharacteristic
39C22×C8×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43C24×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53C2×C812yes—
48C22×C4×C53—21no—
49C22×C4×C53C22×C411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53—44no—
54C22×Dic53—42no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C412yes—
57C22×Dic53C22⋊C411yescharacteristic
58C2×Dic106—21no—
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106C2×C812yes—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C2×C8×C53—42no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68C42×C53—41no—
69C22⋊Dic53—28no—
70C22⋊Dic53—42no—
71C4⋊Dic53—41no—
72#11719919—42no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53C2×C812yes—
79C4×Dic53M1612yes—
80C2×#88322—42no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89D4×C53—42no—
90D4×C53—42no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53C22×C811yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)12yes—
100C2×C4×C53#13712yes—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—84no—
112C2×Dic53C22⋊C814yes—
113C2×Dic53#10012yes—
114Dic106—42no—
115Dic106—42no—
116C8×C53—82no—
117#88322—82no—
118#7601—531nomaximal, Sylow
119C22×C53—42no—
120C22×C53—44no—
121C22×C53—81no—
122C22×C53—82no—
123C22×C53—81no—
124C22×C53C2×C22⋊C811yescharacteristic
125C22×C53#41011yescharacteristic
126C22×C53C2×#10011yescharacteristic
127C22×C53#41712yes—
128C22×C53#42112yes—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—81no—
134Dic53—24no—
135Dic53—22no—
136Dic53—41no—
137Dic53—42no—
138Dic53—81no—
139C4×#124—531no—
140#1831—532no—
141#1272—532no—
142#1270—531no—
143#1270D5311yescharacteristic
144C2×C53—82no—
145C2×C53#146011yescharacteristic
146C2×C53#147712yes—
147C2×C53#151212yes—
148C2×C53#153311yescharacteristic
149C2×C53#156011yescharacteristic
150C2×#124—531no—
151C2×C4×Q8—531no—
152C2×C4×D4—531no—
153C2×C4⋊C8—1062no—
154#391—531no—
155#390C2×D5311yescharacteristic
156#388—1062no—
157#382—531no—
158C4×C22⋊C4—1062no—
159C43—531no—
160#336—532no—
161#336—1062no—
162#336C2×D5312yes—
163C53#760111yescharacteristic, complemented, Sylow
164C22×Q8—531no—
165C22×D4C4×D5311yescharacteristic
166C23×C4—1061no—
167C22×C8—22no—
168C22×C8—1062no—
169#124—534no—
170#124—1062no—
171C4×Q8—2122no—
172C4×D4—2122no—
173C2×C4⋊C4—531no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1061no—
179C2×C42—531no—
180C2×C42—531no—
181C2×C42—532no—
182C2×C42—532no—
183C2×C42—1061no—
184C2×C42C22×D5311yescharacteristic
185C4⋊C8—2124no—
186#95—1062no—
187#95—1062no—
188C24—21no—
189C2×Q8—532no—
190C2×Q8—1061no—
191C2×Q8—2121no—
192C2×Q8—2121no—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C8×D5312yes—
197C22×C4—21no—
198C22×C4—532no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—2122no—
204C22×C4—2124no—
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4D4×D5312yes—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—2122no—
211C2×C8—2122no—
212C2×C8—2122no—
213C4⋊C4—41no—
214C4⋊C4—2121no—
215C4⋊C4—2122no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—1068no—
219C22⋊C4—2122no—
220C42—41no—
221C42—532no—
222C42—532no—
223C42—1062no—
224C42—1062no—
225C42—1062no—
226C42—2122no—
227C42—2122no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23C22⋊C4×D5311yescharacteristic
234Q8—2122no—
235Q8—2122no—
236D4—42no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—534no—
245C2×C4—532no—
246C2×C4—1062no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1062no—
252C2×C4—4244no—
253C2×C4—2121no—
254C2×C4—2122no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4C2×C8×D5311yescharacteristic
258C2×C4M16×D5311yescharacteristic
259C2×C4#1313324112yes—
260C2×C4#1313324612yes—
261C8—82no—
262C8—4242no—
263C22—42no—
264C22—44no—
265C22—81no—
266C22—82no—
267C22—81no—
268C22C22⋊C8×D5311yescharacteristic
269C22#1681059511yescharacteristic
270C22#100×D5311yescharacteristic
271C22#1681062112yes—
272C22#1681063812yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—81no—
278C4—1062no—
279C4—1064no—
280C4—4241no—
281C4—2121no—
282C4—2122no—
283C2—82no—
284C2#2027303411yescharacteristic
285C2#2027309912yes—
286C2#2027325012yes—
287C2#2027333711yescharacteristic
288C2#2027344711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1270D53
hypocenterC53#7601
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20274830C2semidirect
3. #20272540C2semidirect
23. #13133204C8semidirect
163. C53#7601semidirect

Maximal subgroups

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

subgroupindex
#202772622
#202748302
#202725402
#202725362
C53×#12702
#760153