Finite Group Databasebeta

← #27288591

Subgroups — #27288591

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#20274884C211yesmaximal, characteristic
2#20274848C212yesmaximal
3#20272596C211yesmaximal, characteristic
4#20272596C212yesmaximal
5C53×#2844C211yesmaximal, characteristic
6C2×#13133150—21no—
7C2×#13133148—21no—
8C2×#13133128C2212yes—
9C2×#13133128C2211yescharacteristic
10#16810493C2211yescharacteristic
11#16810385—22no—
12#16810385—22no—
13C53×#400—22no—
14C53×#400—21no—
15C53×#395—22no—
16C53×#395—21no—
17C53×#382C2212yes—
18C53×#382C2211yescharacteristic
19C22×#88322—21no—
20C22×#88322—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—21no—
23#13133150—42no—
24#13133148—42no—
25#13133128—21no—
26#13133128—22no—
27#13133128Q812yescomplemented
28#13133128Q814yescomplemented
29C22×Q8×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D411yescharacteristic
35C2×C4⋊C4×C53D411yescharacteristic
36C2×C42×C53C2311yescharacteristic
37#95×C53—22no—
38#95×C53—21no—
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53—22no—
43C2×Q8×C53—81no—
44C2×Q8×C53—82no—
45C22×C4×C53—21no—
46C22×C4×C53—22no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—41no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—81no—
57C4⋊C4×C53—82no—
58C42×C53C2×Q811yescharacteristic
59C42×C53C2×Q812yes—
60C42×C53#4111yescharacteristic
61C4⋊Dic53—41no—
62C4⋊Dic53—41no—
63C4⋊Dic53—44no—
64C2×#88322—22no—
65C2×#88322—22no—
66C2×#88322—22no—
67C2×#88322—24no—
68C2×#88322—42no—
69C2×#88322—42no—
70C23×C53#12011yescharacteristic
71Q8×C53—84no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—42no—
78C2×C4×C53—41no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53C22⋊Q811yescharacteristic
87C2×C4×C53C22⋊Q812yes—
88C2×C4×C53C4⋊Q811yescharacteristic
89C2×C4×C53C4⋊Q812yes—
90C2×Dic53—81no—
91C2×Dic53—82no—
92#88322—24no—
93#88322—44no—
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—
106C4×C53—82no—
107C4×C53—82no—
108Dic53—161no—
109#2844D5311yescharacteristic
110#1847—531no—
111#1836—532no—
112#1295—532no—
113#1295—531no—
114C2×C53#165711yescharacteristic
115C2×C53#165911yescharacteristic
116C2×C53#166412yes—
117C2×C53#206011yescharacteristic
118C2×C53#206112yes—
119C2×#107—1061no—
120C2×#106—1061no—
121C2×#97—532no—
122C2×#97—531no—
123#400—22no—
124#400—21no—
125#395—22no—
126#395—21no—
127#384—531no—
128#382C2×D5312yes—
129#382C2×D5311yescharacteristic
130#340—1062no—
131#340—1062no—
132C53#1058411yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C22×C8—1061no—
135C22×C8—1062no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4—1061no—
140C2×C4⋊C4C22⋊D5312yes—
141C2×C4⋊C4C22⋊D5312yes—
142C2×C4⋊C4C22⋊D5311yescharacteristic
143C2×C4⋊C4C22⋊D5311yescharacteristic
144C2×C42C22×D5311yescharacteristic
145#107—2122no—
146#106—2122no—
147#97—532no—
148#97—534no—
149#97—1061no—
150#97—1062no—
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—2121no—
164C22×C4C2×C22⋊D5311yescharacteristic
165C22×C4C2×C22⋊D5312yes—
166C2×C8—1062no—
167C2×C8—1062no—
168C2×C8—1062no—
169C2×C8—1064no—
170C2×C8—2122no—
171C2×C8—2122no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—2121no—
179C4⋊C4—2121no—
180C4⋊C4—2124no—
181C42Q8×D5311yescharacteristic
182C42Q8×D5312yes—
183C42#1171994911yescharacteristic
184C23#1313318611yescharacteristic
185Q8—84no—
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—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—4241no—
201C2×C4—4242no—
202C2×C4#1313319311yescharacteristic
203C2×C4#1313319312yes—
204C2×C4#1313321511yescharacteristic
205C2×C4#1313321512yes—
206C8—1064no—
207C8—2124no—
208C22#1681054211yescharacteristic
209C22#1681079111yescharacteristic
210C22#1681079612yes—
211C22#1681080112yes—
212C22#1681080611yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—8481no—
222C2#2027401311yescharacteristic
223C2#2027402111yescharacteristic
224C2#2027404912yes—
225C2#2027605711yescharacteristic
226C2#2027606212yes—
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2844C2
socle, prime radicalC23×C53#120
hypercenter#2844D53
hypocenterC53#10584
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
27. #13133128Q8semidirect
28. #13133128Q8semidirect
132. C53#10584semidirect

Maximal subgroups

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

subgroupindex
#202748842
#202748482
#202725962
C53×#28442
#1058453