Finite Group Databasebeta

← #27285080

Subgroups — #27285080

Order 13568.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284719C212yesmaximal
2#20275513C211yesmaximal, characteristic
3#20273118C212yesmaximal, complemented
4#20272546C211yesmaximal, characteristic, complemented
5C53×#3162C211yesmaximal, characteristic, complemented
6C2×C4×D212C2211yescharacteristic
7#16810951—22no—
8#16810910—22no—
9#16810825—22no—
10#16810773—22no—
11C4×#11719924—24no—
12#16810676C2212yes—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133129—22no—
15Q8⋊#88322—24no—
16C53×#533C2212yes—
17C53×#501—22no—
18C53×#501—22no—
19C53×#427—22no—
20C2×#97×C53C2211yescharacteristic
21C22×D212D411yescharacteristic
22C22×C4×D53—21no—
23C2×#11719924—24no—
24C22×#88322—21no—
25C4×D212—22no—
26C4×D212—22no—
27C2×C4⋊Dic53D411yescharacteristic
28C2×#11719921—21no—
29#13133148—24no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133141—24no—
33#13133137—24no—
34C22⋊#88322—22no—
35#13133129—44no—
36C4×#88322D414yes—
37C22×C8×C53D412yes—
38C4⋊Q8×C53D412yes—
39C4⋊Q8×C53D412yes—
40#125×C53—22no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45#117×C53—22no—
46C2×C42×C53C2311yescharacteristic
47#107×C53—22no—
48#107×C53—22no—
49#106×C53—24no—
50#97×C53—22no—
51C23×D53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—22no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4111yescharacteristic
57C22×Dic53—21no—
58C2×D212—22no—
59C2×D212—42no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53—42no—
63C2×C4×D53—42no—
64C2×C8×C53—22no—
65C2×C8×C53—22no—
66C2×C8×C53SD1614yescomplemented
67C4⋊C4×C53—22no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C42×C53—42no—
75C42×C53C2×D412yes—
76C42×C53C2×D412yes—
77#11719924—48no—
78#11719921—42no—
79C4⋊Dic53—22no—
80C2×#88322—24no—
81C2×#88322—41no—
82C2×#88322—42no—
83C23×C53#12111yescharacteristic
84C22×D53—22no—
85C22×D53—41no—
86C22×D53—42no—
87C22×D53—41no—
88C22×D53—42no—
89Q8×C53—44no—
90Q8×C53—44no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53#12012yes—
96C2×C4×C53#12011yescharacteristic
97C2×C4×C53#12112yes—
98C2×C4×C53#12711yescharacteristic
99C2×C4×C53C2×SD1612yes—
100C2×C4×C53#13712yes—
101C2×Dic53—41no—
102C2×Dic53—42no—
103D212—24no—
104D212—42no—
105C4×D53—82no—
106C4×D53—81no—
107C4×D53—81no—
108C8×C53—24no—
109#88322—44no—
110#88322—81no—
111#10036—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#39211yescharacteristic
115C22×C53#46112yes—
116C2×D53—44no—
117C2×D53—42no—
118C2×D53—82no—
119C2×D53—81no—
120C2×D53—81no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—82no—
124C4×C53—82no—
125C4×C53#44812yes—
126C4×C53#49211yescharacteristic
127C4×C53#49711yescharacteristic
128Dic53—81no—
129#3162D5311yescharacteristic, complemented
130#3101—532no—
131#1955—531no—
132#1481—532no—
133#1275—531no—
134C2×C53—22no—
135C2×C53#201011yescharacteristic
136C2×C53#201311yescharacteristic
137C2×C53#201711yescharacteristic
138D53—82no—
139#533C2×D5312yes—
140C2×C4×D4—531no—
141#501—22no—
142#501—22no—
143#483—1062no—
144#476—1062no—
145#461—1062no—
146#450—1062no—
147C4×SD16—1064no—
148#431—532no—
149#427—22no—
150C2×C4⋊C8—531no—
151C2×C22⋊C8—1062no—
152C2×#97C2×D5311yescharacteristic
153Q8⋊C8—1064no—
154C53#1003611yescharacteristic, complemented, Sylow
155C22×D4—531no—
156C23×C4—1061no—
157C2×SD16—1064no—
158C22×C8—1061no—
159C22×C8C22⋊D5312yes—
160C4⋊Q8C22⋊D5312yes—
161C4⋊Q8C22⋊D5312yes—
162#125—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×D4—1062no—
168C4×D4—1062no—
169#117—22no—
170C2×C4⋊C4—531no—
171C2×C22⋊C4—1061no—
172C2×C42C22×D5311yescharacteristic
173#107—22no—
174#107—22no—
175#106—24no—
176#106—1064no—
177C4⋊C8—1062no—
178C4⋊C8—1062no—
179#103—1064no—
180#102—1064no—
181C22⋊C8—1062no—
182C22⋊C8—2124no—
183#97—22no—
184C4×C8—534no—
185C24—1061no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×D4—1062no—
189C2×D4—2122no—
190C22×C4—1061no—
191C22×C4—2121no—
192C22×C4—2121no—
193C22×C4—2122no—
194C22×C4—2122no—
195C22×C4D4×D5311yescharacteristic
196C22×C4#1171994711yescharacteristic
197C22×C4C2×C22⋊D5311yescharacteristic
198SD16—2128no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—1064no—
202C2×C8—2121no—
203C2×C8—2122no—
204C2×C8#1171992414yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—1062no—
212C22⋊C4—42no—
213C22⋊C4—2122no—
214C42—42no—
215C42D4×D5312yes—
216C42C2×C22⋊D5312yes—
217C23—1062no—
218C23—2121no—
219C23—2122no—
220C23—2121no—
221C23—2122no—
222C23#1313318811yescharacteristic
223Q8—44no—
224Q8—44no—
225D4—1064no—
226D4—2122no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—4241no—
232C2×C4—4242no—
233C2×C4—4241no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4#1313318512yes—
237C2×C4#1313318611yescharacteristic
238C2×C4#1313318912yes—
239C2×C4#1313321311yescharacteristic
240C2×C4C2×#1171992412yes—
241C2×C4#1313324812yes—
242C8—24no—
243C8—4241no—
244C8—2124no—
245C22—21no—
246C22—21no—
247C22—4242no—
248C22—4241no—
249C22—4241no—
250C22—2124no—
251C22—2122no—
252C22#1681053111yescharacteristic
253C22#1681082512yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—4241no—
259C4#1681076212yes—
260C4#1681099711yescharacteristic
261C4#1681101511yescharacteristic
262C2—22no—
263C2—4242no—
264C2#2027577911yescharacteristic
265C2#2027579811yescharacteristic
266C2#2027581811yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3162C2
socleC22×C53#392
prime radicalC22×D212D4
hypercenter#3162D53
hypocenterC53#10036
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #20273118C2semidirect
4. #20272546C2semidirect
5. C53×#3162C2semidirect
66. C2×C8×C53SD16semidirect
129. #3162D53semidirect
154. C53#10036semidirect
204. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202755132
#202731182
#202725462
C53×#31622
#1003653