Finite Group Databasebeta

← #27288593

Subgroups — #27288593

Order 13568.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#20282987C211yesmaximal, characteristic
2#20274884C211yesmaximal, characteristic
3#20274849C212yesmaximal
4#20272597C212yesmaximal
5C53×#1295C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133148—21no—
8C2×#13133128C2211yescharacteristic
9#16810565—22no—
10#16810562—21no—
11#16810544—22no—
12#16810543—21no—
13#16810493C2211yescharacteristic
14#16810489C2212yes—
15#16810386—22no—
16#16810386—22no—
17C2×#97×C53C2212yes—
18C53×#382C2211yescharacteristic
19C22×Dic106—21no—
20C22×#88322—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53D412yes—
25C2×C4⋊Dic53D412yes—
26C2×#11719919—22no—
27#13133150—42no—
28#13133148—42no—
29#13133128—21no—
30#13133128Q812yescomplemented
31#13133124—21no—
32#13133124—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×C8×C53—22no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C42×C53C2311yescharacteristic
40#97×C53—22no—
41#97×C53Q814yescomplemented
42#95×C53—22no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53C2×D411yescharacteristic
46C22×C4×C53C2×D412yes—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—41no—
50C2×Dic106—81no—
51C2×Dic106—82no—
52C2×C8×C53—24no—
53C2×C8×C53—42no—
54C2×C8×C53—42no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—41no—
57C42×C53C2×Q812yes—
58C42×C53C2×Q811yescharacteristic
59C42×C53#4111yescharacteristic
60C4⋊Dic53—42no—
61C4⋊Dic53—42no—
62C4⋊Dic53—41no—
63C4⋊Dic53—41no—
64C4⋊Dic53—44no—
65C4⋊Dic53—81no—
66#11719919—82no—
67C2×#88322—22no—
68C2×#88322—22no—
69C2×#88322—22no—
70C23×C53#12011yescharacteristic
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53C22⋊Q812yes—
80C2×C4×C53C22⋊Q811yescharacteristic
81C2×C4×C53C4⋊Q812yes—
82C2×C4×C53C4⋊Q811yescharacteristic
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—81no—
90C2×Dic53—82no—
91Dic106—84no—
92C8×C53—44no—
93#88322—24no—
94#10584—531nomaximal, Sylow
95C22×C53#39511yescharacteristic
96C22×C53#45311yescharacteristic
97C22×C53#45412yes—
98C22×C53#45512yes—
99C22×C53#45611yescharacteristic
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—22no—
104C4×C53—81no—
105C4×C53—81no—
106Dic53—82no—
107Dic53—82no—
108Dic53—161no—
109#2844—531no—
110#1847—531no—
111#1836—532no—
112#1295—532no—
113#1295D5311yescharacteristic
114C2×C53#165711yescharacteristic
115C2×C53#165911yescharacteristic
116C2×C53#166412yes—
117C2×C53#206011yescharacteristic
118C2×C53#206112yes—
119C2×#107—1061no—
120C2×#106—1061no—
121C2×#97—531no—
122C2×#97C2×D5312yes—
123#400—1062no—
124#400—1061no—
125#395—1062no—
126#395—1061no—
127#384—531no—
128#382—532no—
129#382C2×D5311yescharacteristic
130#340—1062no—
131#340—1062no—
132C53#1058411yescharacteristic, complemented, Sylow
133C22×Q8—1061no—
134C22×C8—22no—
135C22×C8—1061no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4—1061no—
140C2×C4⋊C4—1061no—
141C2×C4⋊C4—1062no—
142C2×C4⋊C4D21211yescharacteristic
143C2×C4⋊C4D21211yescharacteristic
144C2×C42C22×D5311yescharacteristic
145#107—2122no—
146#106—2122no—
147#97—22no—
148#97—532no—
149#97—1061no—
150#97Dic10614yescomplemented
151#95—22no—
152#95—1062no—
153#95—1061no—
154#95—1061no—
155#95—1062no—
156#95—1062no—
157C2×Q8—4241no—
158C2×Q8—4242no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—1062no—
162C22×C4—1062no—
163C22×C4—2121no—
164C22×C4C2×D21211yescharacteristic
165C22×C4D4×D5312yes—
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—1062no—
170C2×C8—1062no—
171C2×C8—1062no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—4242no—
175C4⋊C4—4241no—
176C4⋊C4—2121no—
177C4⋊C4—2121no—
178C4⋊C4—2124no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C42C2×Dic10612yes—
182C42#1171994711yescharacteristic
183C42Q8×D5311yescharacteristic
184C23#1313318711yescharacteristic
185Q8—4244no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—4241no—
195C2×C4—4242no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4#1313319412yes—
203C2×C4#1313319711yescharacteristic
204C2×C4#1313321612yes—
205C2×C4#1313321711yescharacteristic
206C8—44no—
207C8—1064no—
208C22#1681054411yescharacteristic
209C22#1681079311yescharacteristic
210C22#1681079712yes—
211C22#1681080212yes—
212C22#1681080811yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—8481no—
220C4—4242no—
221C4—4242no—
222C2#2027401411yescharacteristic
223C2#2027402211yescharacteristic
224C2#2027405112yes—
225C2#2027606111yescharacteristic
226C2#2027606412yes—
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socle, prime radicalC23×C53#120
hypercenter#1295D53
hypocenterC53#10584
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
30. #13133128Q8semidirect
41. #97×C53Q8semidirect
132. C53#10584semidirect
150. #97Dic106semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202748842
#202748492
#202725972
C53×#12952
#1058453