Finite Group Databasebeta

← #27273277

Subgroups — #27273277

Order 13568.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#20276258C211yesmaximal, characteristic, complemented
2#20276118C212yesmaximal, complemented
3#20272541C211yesmaximal, characteristic, complemented
4#20272536C212yesmaximal, complemented
5C53×#2733C211yesmaximal, characteristic
6C2×#13133148—21no—
7C2×#13133141—21no—
8C2×#13133138—22no—
9#16810493C2211yescharacteristic
10#16810385—22no—
11#16810378C2211yescharacteristic
12#16810378C2212yes—
13C2×#124×C53—21no—
14C53×#394—21no—
15C53×#392—22no—
16C53×#391C2211yescharacteristic
17C53×#390C2212yes—
18C53×#388—22no—
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—21no—
23#13133148—42no—
24#13133141—42no—
25#13133138—44no—
26C22×Q8×C53D411yescharacteristic
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53D412yes—
30#124×C53—42no—
31#124×C53—42no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D411yescharacteristic
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53—21no—
39C2×C42×C53C2311yescharacteristic
40#95×C53—21no—
41#95×C53—22no—
42C24×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53SD1612yes—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—42no—
52C2×D4×C53—81no—
53C2×D4×C53—81no—
54C2×D4×C53—81no—
55C2×D4×C53—82no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D412yes—
61C22×Dic53—41no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—44no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—82no—
70C42×C53—41no—
71C42×C53—42no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74C4⋊Dic53—42no—
75C4⋊Dic53—42no—
76C2×#88322—22no—
77C2×#88322—41no—
78C2×#88322—41no—
79C2×#88322—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53—42no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53#12011yescharacteristic
90Q8×C53—42no—
91Q8×C53—81no—
92D4×C53—44no—
93D4×C53—82no—
94D4×C53—84no—
95D4×C53—84no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53C2×SD1611yescharacteristic
109C2×C4×C53#13512yes—
110C2×C4×C53AGL(1,ℤ/8)12yes—
111C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
112C2×Dic53—82no—
113C2×Dic53—81no—
114#88322—42no—
115#88322—82no—
116#7810—531nomaximal, Sylow
117C22×C53—82no—
118C22×C53—82no—
119C22×C53—82no—
120C22×C53—82no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53#44912yes—
125C22×C53#45011yescharacteristic
126C22×C53#45311yescharacteristic
127C22×C53#45512yes—
128C22×C53#45811yescharacteristic
129C4×C53—22no—
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53—82no—
135C4×C53—81no—
136Dic53—161no—
137#2733D5311yescharacteristic
138#2096—531no—
139#2071—532no—
140#1272—531no—
141#1270—532no—
142C2×C53—162no—
143C2×C53#161212yes—
144C2×C53#161311yescharacteristic
145C2×C53#162411yescharacteristic
146C2×C53#164212yes—
147C2×C53#165911yescharacteristic
148C2×#124—21no—
149C2×#106—1061no—
150C2×#103—1061no—
151C2×#102—1062no—
152#394—21no—
153#392—22no—
154#391C2×D5311yescharacteristic
155#390C2×D5312yes—
156#388—22no—
157#384—531no—
158#340—1062no—
159#336—532no—
160#336—531no—
161C53#781011yescharacteristic, complemented, Sylow
162C22×Q8C22⋊D5311yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22⋊D5312yes—
166C22×C8—1061no—
167C22×C8—1062no—
168#124—42no—
169#124—42no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4C22⋊D5312yes—
173C2×C4⋊C4C22⋊D5311yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42—21no—
179C2×C42C22×D5311yescharacteristic
180#106—2122no—
181#103—2122no—
182#102—2124no—
183#95—21no—
184#95—22no—
185C24—22no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#1171992312yes—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—2121no—
203C22×C4C2×C22⋊D5311yescharacteristic
204C22×C4C2×C22⋊D5312yes—
205C2×C8—1062no—
206C2×C8—2122no—
207C2×C8—2122no—
208C2×C8—2122no—
209C2×C8—2121no—
210C2×C8—2121no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C22⋊C4—44no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—82no—
223C42—41no—
224C42—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23#1313318611yescharacteristic
233Q8—42no—
234Q8—81no—
235D4—44no—
236D4—82no—
237D4—84no—
238D4—84no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—4242no—
252C2×C4—4241no—
253C2×C4C2×#1171992311yescharacteristic
254C2×C4#1313323712yes—
255C2×C4#1313324212yes—
256C2×C4#1313324311yescharacteristic
257C8—4242no—
258C8—2122no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22#1681076812yes—
267C22#1681077411yescharacteristic
268C22#1681079111yescharacteristic
269C22#1681080112yes—
270C22#1681081311yescharacteristic
271C4—22no—
272C4—22no—
273C4—21no—
274C4—21no—
275C4—42no—
276C4—81no—
277C4—82no—
278C4—8481no—
279C2—162no—
280C2#2027372912yes—
281C2#2027373511yescharacteristic
282C2#2027379811yescharacteristic
283C2#2027391512yes—
284C2#2027402111yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2733C2
socleC23×C53#120
hypercenter#2733D53
hypocenterC53#7810
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #20276258C2semidirect
2. #20276118C2semidirect
3. #20272541C2semidirect
4. #20272536C2semidirect
161. C53#7810semidirect

Maximal subgroups

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

subgroupindex
#202762582
#202761182
#202725412
#202725362
C53×#27332
#781053