Finite Group Databasebeta

← #28299684

Subgroups — #28299684

Order 13568.

636 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290432C212yesmaximal, complemented
2#20289531C214yesmaximal, complemented
3#127×Dic53C211yesmaximal, characteristic
4C2×#13133272—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×D4×Dic53—22no—
8C2×D4×Dic53C412yes—
9#16810526—22no—
10#16810526C2212yes—
11#16810493C411yescharacteristic
12C42×Dic53C2211yescharacteristic
13C2×#127×C53C411yescharacteristic
14C22×#88324—24no—
15#13133272—28no—
16C23×Dic53—22no—
17D4×Dic53—48no—
18D4×Dic53—44no—
19C2×C4⋊Dic53—22no—
20C2×C4⋊Dic53C2×C412yes—
21C2×C22⋊Dic53—22no—
22C2×C22⋊Dic53—24no—
23C2×C4×Dic53C2×C411yescharacteristic
24C2×C4×Dic53D412yes—
25C2×C4×Dic53D412yes—
26C2×C4×Dic53C2311yescharacteristic
27C22×D4×C53—22no—
28C22×D4×C53C2×C412yes—
29#127×C53—24no—
30C2×C42×C53C2×C411yescharacteristic
31C24×C53—22no—
32C2×D4×C53—28no—
33C2×D4×C53—22no—
34C2×D4×C53—44no—
35C2×D4×C53—42no—
36C2×D4×C53—42no—
37C2×D4×C53—44no—
38C2×D4×C53M1614yes—
39C22×C4×C53C22⋊C412yes—
40C22×C4×C53C22×C411yescharacteristic
41C22×Dic53—22no—
42C22×Dic53—48no—
43C22×Dic53—44no—
44C22×Dic53C22⋊C411yescharacteristic
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C2×D412yes—
47C2×#88324—28no—
48C2×#88324—44no—
49C2×#88324—44no—
50C42×C53—22no—
51C22⋊Dic53—44no—
52C22⋊Dic53—48no—
53C4⋊Dic53—44no—
54C4⋊Dic53—42no—
55C4×Dic53—24no—
56C4×Dic53—22no—
57C4×Dic53—24no—
58C4×Dic53—41no—
59C23×C53—24no—
60C23×C53—42no—
61C23×C53—42no—
62C23×C53—44no—
63C23×C53—44no—
64C23×C53C2×C22⋊C411yescharacteristic
65D4×C53—44no—
66D4×C53—44no—
67D4×C53—416no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53#10414yes—
73C2×C4×C53C2×M1612yes—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—22no—
79C2×Dic53—42no—
80C2×Dic53—44no—
81C2×Dic53—88no—
82C2×Dic53#10012yes—
83C2×Dic53AGL(1,ℤ/8)14yes—
84#88324—48no—
85#7637—531nomaximal, Sylow
86C22×C53—44no—
87C22×C53—48no—
88C22×C53—82no—
89C22×C53—84no—
90C22×C53—82no—
91C22×C53#40711yescharacteristic
92C22×C53#41011yescharacteristic
93C22×C53C2×#10011yescharacteristic
94C22×C53#41812yes—
95C22×C53C2×#10412yes—
96C4×C53—22no—
97C4×C53—24no—
98Dic53—22no—
99Dic53—24no—
100Dic53—41no—
101Dic53—82no—
102C4×#127—531no—
103#1745—532no—
104#1270—534no—
105C2×C53—84no—
106C2×C53#146611yescharacteristic
107C2×C53#149012yes—
108C2×C53#150512yes—
109C2×C53#152511yescharacteristic
110C2×C53#155511yescharacteristic
111C2×#127#615711yescharacteristic
112C2×C4×D4—532no—
113C2×C4×D4—1062no—
114C2×#97—1062no—
115#390—532no—
116#390—1062no—
117#384—531no—
118C43—531no—
119#336—534no—
120#336—1062no—
121C53#763711yescharacteristic, complemented, Sylow
122C22×D4—22no—
123C22×D4C2×#615712yes—
124C23×C4—1062no—
125C22×C8—1064no—
126#127—24no—
127C4×D4—2128no—
128C4×D4—2124no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—1062no—
131C2×C22⋊C4—1062no—
132C2×C22⋊C4—1064no—
133C2×C42—532no—
134C2×C42—532no—
135C2×C42—531no—
136C2×C42—531no—
137C2×C42C2×#615711yescharacteristic
138#97—1068no—
139C24—22no—
140C2×D4—28no—
141C2×D4—22no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4#1171994014yes—
147C22×C4—531no—
148C22×C4—532no—
149C22×C4—531no—
150C22×C4—1062no—
151C22×C4—2124no—
152C22×C4—2128no—
153C22×C4C22⋊#615712yes—
154C22×C4C22×#615711yescharacteristic
155C2×C8—1068no—
156C2×C8—2124no—
157C2×C8—2124no—
158C4⋊C4—2122no—
159C4⋊C4—2124no—
160C22⋊C4—2128no—
161C22⋊C4—2124no—
162C42—22no—
163C42—1062no—
164C42—1064no—
165C42—1064no—
166C42—2121no—
167C23—24no—
168C23—42no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23C2×C22⋊#615711yescharacteristic
173D4—44no—
174D4—416no—
175D4—44no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—532no—
181C2×C4—534no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1061no—
186C2×C4—1062no—
187C2×C4—4248no—
188C2×C4—2122no—
189C2×C4—2124no—
190C2×C4#1313328114yes—
191C2×C4C2×#1171994012yes—
192C8—2128no—
193C22—48no—
194C22—44no—
195C22—84no—
196C22—82no—
197C22—82no—
198C22#1681159311yescharacteristic
199C22#1681160311yescharacteristic
200C22D53⋊#10011yescharacteristic
201C22#1681161512yes—
202C22C2×#1313328112yes—
203C4—24no—
204C4—22no—
205C4—1064no—
206C4—1062no—
207C4—4242no—
208C4—2121no—
209C2—84no—
210C2#2028997511yescharacteristic
211C2#2029005412yes—
212C2#2029010412yes—
213C2#2029017011yescharacteristic
214C2#2029024311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#127#6157
hypocenterC53#7637
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290432C2semidirect
2. #20289531C2semidirect
121. C53#7637semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202895312
#127×Dic532
#763753