Finite Group Databasebeta

← #27287174

Subgroups — #27287174

Order 13568.

392 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 89 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#20282990C211yesmaximal, characteristic
2#20275688C212yesmaximal
3#20275534C211yesmaximal, characteristic
4#20272597C212yesmaximal
5C53×#1295C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133140—22no—
8C2×#13133128C2211yescharacteristic
9#16810557—22no—
10#16810556—22no—
11#16810555—22no—
12#16810489C2212yes—
13#16810386—22no—
14#16810386—22no—
15C2×#97×C53C2212yes—
16C53×#382C2211yescharacteristic
17C22×Dic106—21no—
18C22×#88322—21no—
19#13133217—21no—
20#13133217—22no—
21#13133217Q812yes—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27C2×#11719919—22no—
28#13133140—44no—
29#13133128—22no—
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33C22×C8×C53—22no—
34C2×C4⋊C4×C53D412yes—
35C2×C42×C53C2311yescharacteristic
36#97×C53—22no—
37#97×C53—22no—
38#95×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53C2×D412yes—
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53—41no—
45C2×Dic106—22no—
46C2×Dic106—42no—
47C2×Dic106—42no—
48C2×Dic106—41no—
49C2×C8×C53—24no—
50C2×C8×C53—42no—
51C2×C8×C53—42no—
52C4⋊C4×C53—42no—
53C42×C53C2×Q811yescharacteristic
54C42×C53#4112yes—
55C42×C53#4111yescharacteristic
56C4⋊Dic53—24no—
57C4⋊Dic53—42no—
58C4⋊Dic53—42no—
59C4⋊Dic53—42no—
60C4⋊Dic53—82no—
61#11719919—82no—
62#11719919—82no—
63C2×#88322—22no—
64C2×#88322—41no—
65C2×#88322—41no—
66C23×C53#12011yescharacteristic
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53C22⋊Q812yes—
75C2×C4×C53C22⋊Q811yescharacteristic
76C2×C4×C53#12412yes—
77C2×C4×C53#12411yescharacteristic
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87Dic106—44no—
88Dic106—82no—
89C8×C53—44no—
90#88322—42no—
91#10361—531nomaximal, Sylow
92C22×C53#39311yescharacteristic
93C22×C53#45311yescharacteristic
94C22×C53#45411yescharacteristic
95C22×C53#45512yes—
96C22×C53#45612yes—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—82no—
102Dic53—82no—
103Dic53—82no—
104Dic53—82no—
105#2845—531no—
106#1993—532no—
107#1960—531no—
108#1295—532no—
109#1295D5311yescharacteristic
110C2×C53#166112yes—
111C2×C53#166211yescharacteristic
112C2×C53#166311yescharacteristic
113C2×C53#202711yescharacteristic
114C2×C53#202812yes—
115C2×C4⋊Q8—531no—
116C2×#103—1062no—
117C2×#97—531no—
118C2×#97C2×D5312yes—
119#398—1062no—
120#398—1062no—
121#398—1062no—
122#382—532no—
123#382C2×D5311yescharacteristic
124#340—1062no—
125#340—1062no—
126C53#1036111yescharacteristic, complemented, Sylow
127C22×Q8—1061no—
128C22×C8—22no—
129C22×C8—1061no—
130C4⋊Q8—532no—
131C4⋊Q8—1062no—
132C4⋊Q8—1061no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—1062no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4—1062no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4D21212yes—
140C2×C42C22×D5311yescharacteristic
141#103—2124no—
142#97—22no—
143#97—22no—
144#97—1062no—
145#95—22no—
146#95—1062no—
147#95—1062no—
148#95—1062no—
149C2×Q8—1062no—
150C2×Q8—2122no—
151C2×Q8—2122no—
152C2×Q8—2121no—
153C22×C4—22no—
154C22×C4—1062no—
155C22×C4—1062no—
156C22×C4—2121no—
157C22×C4C2×D21211yescharacteristic
158C22×C4D4×D5312yes—
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—1062no—
163C2×C8—2121no—
164C2×C8—2121no—
165C4⋊C4—42no—
166C4⋊C4—1064no—
167C4⋊C4—4242no—
168C4⋊C4—4242no—
169C4⋊C4—4242no—
170C4⋊C4—2122no—
171C4⋊C4—2122no—
172C4⋊C4—2122no—
173C42#1171994512yes—
174C42#1171994711yescharacteristic
175C42Q8×D5311yescharacteristic
176C23#1313318711yescharacteristic
177Q8—4242no—
178Q8—2124no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—4241no—
187C2×C4—4241no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4#1313319512yes—
196C2×C4#1313319711yescharacteristic
197C2×C4#1313320312yes—
198C2×C4#1313320511yescharacteristic
199C8—44no—
200C8—2122no—
201C22#1681053611yescharacteristic
202C22#1681079311yescharacteristic
203C22#1681079811yescharacteristic
204C22#1681080212yes—
205C22#1681080712yes—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—4242no—
212C4—4242no—
213C4—4242no—
214C2#2027403312yes—
215C2#2027403911yescharacteristic
216C2#2027404411yescharacteristic
217C2#2027587211yescharacteristic
218C2#2027587512yes—
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10361
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
126. C53#10361semidirect

Maximal subgroups

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

subgroupindex
#202829902
#202756882
#202755342
#202725972
C53×#12952
#1036153