Finite Group Databasebeta

← #27277514

Subgroups — #27277514

Order 13568.

630 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 155 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#384×D53C211yesmaximal, characteristic
2#20274692C212yesmaximal
3#20272599C212yesmaximal, complemented
4#20272598C211yesmaximal, characteristic, complemented
5C53×#1296C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D53—22no—
7C2×C4⋊C4×D53C412yes—
8C2×C42×D53C2211yescharacteristic
9C2×C4⋊#88322C2212yes—
10C2×#13133129—22no—
11#16810493C2211yescharacteristic
12#16810492—22no—
13#16810492C412yes—
14#16810377—22no—
15C2×C4⋊C8×C53C2212yes—
16C53×#384C2211yescharacteristic
17C22×C4×D53—22no—
18C22×C4×D53C2×C411yescharacteristic
19C22×C4×D53D412yes—
20C22×#88322—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53C2×C412yes—
23C2×#11719919—24no—
24C2×#11719919—22no—
25C4⋊C4×D53—44no—
26C4⋊C4×D53—44no—
27C4⋊C4×D53—44no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C42×D53—22no—
31C42×D53—21no—
32C42×D53—21no—
33C4⋊#88322—22no—
34C4⋊#88322—22no—
35#13133129—44no—
36C22×C8×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C42×C53C2311yescharacteristic
40C4⋊C8×C53—22no—
41C4⋊C8×C53—22no—
42C23×D53C22⋊C411yescharacteristic
43C22×C4×C53—22no—
44C22×C4×C53C22×C411yescharacteristic
45C22×C4×C53C2×D412yes—
46C22×Dic53—22no—
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C22⋊C412yes—
49C2×C4×D53—22no—
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—22no—
53C2×C4×D53—22no—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—48no—
58C2×C4×D53—44no—
59C2×C4×D53SD1614yes—
60C2×C4×D53Q1614yes—
61C2×C8×C53—42no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C42×C53#4112yes—
68C42×C53#4111yescharacteristic
69C42×C53#4111yescharacteristic
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72C4⋊Dic53—42no—
73#11719919—44no—
74#11719919—48no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C2×#88322—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—22no—
86C22×D53#10012yes—
87C22×D53#10314yes—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—44no—
92C2×C4×C53#11711yescharacteristic
93C2×C4×C53#11711yescharacteristic
94C2×C4×C53#12312yes—
95C2×C4×C53#12312yes—
96C2×C4×C53C2×SD1612yes—
97C2×C4×C53C2×Q1612yes—
98C2×C4×C53AGL(1,ℤ/8)12yes—
99C2×C4×C53AGL(1,ℤ/8)12yes—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—42no—
105C2×Dic53—44no—
106C2×Dic53#10112yes—
107C2×Dic53#10314yes—
108C4×D53—24no—
109C4×D53—24no—
110C4×D53—24no—
111C4×D53—24no—
112C4×D53—24no—
113C4×D53—24no—
114C4×D53—88no—
115C8×C53—82no—
116#88322—82no—
117#8725—531nomaximal, Sylow
118C22×C53#38111yescharacteristic
119C22×C53C2×#10011yescharacteristic
120C22×C53C2×#10111yescharacteristic
121C22×C53C2×#10312yes—
122C22×C53#41812yes—
123C2×D53—24no—
124C2×D53—22no—
125C2×D53—22no—
126C2×D53—24no—
127C2×D53#33314yes—
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—82no—
131C4×C53#48112yes—
132C4×C53#48112yes—
133C4×C53#48412yes—
134C4×C53#48412yes—
135Dic53—22no—
136Dic53—24no—
137Dic53—82no—
138C2×#384—531no—
139#1797—532no—
140#1296—532no—
141#1296—531no—
142#1296D5311yescharacteristic, complemented
143C2×C53C2×#33311yescharacteristic
144C2×C53#154912yes—
145C2×C53#155511yescharacteristic
146C2×C53#182111yescharacteristic
147C2×C53#182412yes—
148D53—24no—
149C22×C4⋊C4—532no—
150C22×C4⋊C4—1062no—
151C22×C42—531no—
152C2×C4⋊C8—532no—
153C2×C4⋊C8C2×D5312yes—
154C2×C22⋊C8—1062no—
155#384—532no—
156#384—531no—
157#384—1062no—
158#384C2×D5311yescharacteristic
159#336—1062no—
160C53#872511yescharacteristic, complemented, Sylow
161C23×C4—532no—
162C23×C4—531no—
163C23×C4—1062no—
164C22×C8—22no—
165C22×C8—1062no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—1062no—
169C2×C4⋊C4—1064no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4—2124no—
172C2×C4⋊C4—2124no—
173C2×C4⋊C4—2124no—
174C2×C4⋊C4C4×D5312yes—
175C2×C42—531no—
176C2×C42—532no—
177C2×C42—1062no—
178C2×C42—1061no—
179C2×C42—1061no—
180C2×C42C22×D5311yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—1062no—
184C4⋊C8—1062no—
185C22⋊C8—2124no—
186C24—531no—
187C22×C4—22no—
188C22×C4—531no—
189C22×C4—532no—
190C22×C4—534no—
191C22×C4—534no—
192C22×C4—1062no—
193C22×C4—1062no—
194C22×C4—1061no—
195C22×C4—1061no—
196C22×C4—1062no—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—2124no—
202C22×C4—2128no—
203C22×C4C2×C4×D5311yescharacteristic
204C22×C4D4×D5312yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—2122no—
209C2×C8—2122no—
210C2×C8—2122no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—2122no—
215C4⋊C4—2124no—
216C4⋊C4—2128no—
217C4⋊C4—2122no—
218C4⋊C4—2122no—
219C42—1061no—
220C42—1061no—
221C42—1062no—
222C42—1062no—
223C42#1171994512yes—
224C42#1171994711yescharacteristic
225C42#1171994911yescharacteristic
226C23—532no—
227C23—534no—
228C23—1061no—
229C23—1062no—
230C23—1061no—
231C23C22⋊C4×D5311yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—532no—
237C2×C4—534no—
238C2×C4—1064no—
239C2×C4—1064no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—1061no—
243C2×C4—1062no—
244C2×C4—1064no—
245C2×C4—1064no—
246C2×C4—1064no—
247C2×C4—1064no—
248C2×C4—4248no—
249C2×C4—2122no—
250C2×C4—2124no—
251C2×C4#1313317411yescharacteristic
252C2×C4#1313317511yescharacteristic
253C2×C4#1313319912yes—
254C2×C4#1313320212yes—
255C2×C4SD16×D5312yes—
256C2×C4Q16×D5312yes—
257C2×C4#1313324112yes—
258C2×C4#1313324412yes—
259C8—82no—
260C8—4242no—
261C22—534no—
262C22—1062no—
263C22—1062no—
264C22—1064no—
265C22—1064no—
266C22#1681048511yescharacteristic
267C22#100×D5311yescharacteristic
268C22#101×D5311yescharacteristic
269C22#103×D5312yes—
270C22#1681062612yes—
271C4—21no—
272C4—21no—
273C4—82no—
274C4—1062no—
275C4—1064no—
276C4—4242no—
277C4#1681094012yes—
278C4#1681094312yes—
279C4#1681095712yes—
280C4#1681096012yes—
281C2—1064no—
282C2#333×D5311yescharacteristic
283C2#2027339812yes—
284C2#2027342711yescharacteristic
285C2#2027479411yescharacteristic
286C2#2027480412yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#8725
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20272599C2semidirect
4. #20272598C2semidirect
5. C53×#1296C2semidirect
142. #1296D53semidirect
160. C53#8725semidirect

Maximal subgroups

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

subgroupindex
#384×D532
#202746922
#202725992
#202725982
C53×#12962
#872553