Finite Group Databasebeta

← #28298885

Subgroups — #28298885

Order 13568.

356 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 38 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#20289641C212yesmaximal, complemented
2#20274633C211yesmaximal, characteristic
3#16811466—22no—
4C2×#13133132—22no—
5#16810486C411yescharacteristic
6#16810486C2211yescharacteristic
7C22×#41×C53C411yescharacteristic
8C22⋊#88324—42no—
9C2×C22⋊Dic53—21no—
10C2×C22⋊Dic53D412yes—
11#13133132—44no—
12#13133124—41no—
13#13133124—41no—
14C2×#41×C53—41no—
15C2×#41×C53—41no—
16C2×#41×C53—44no—
17C22×Q8×C53C811yescharacteristic, complemented
18C22×D4×C53—21no—
19C22×D4×C53C811yescharacteristic, complemented
20C23×C4×C53—21no—
21C23×C4×C53C2×C411yescharacteristic
22C24×C53—21no—
23C24×C53C22⋊C411yescharacteristic
24#41×C53—44no—
25#41×C53—44no—
26#41×C53—44no—
27#41×C53—44no—
28C2×Q8×C53—42no—
29C2×Q8×C53—41no—
30C2×D4×C53—24no—
31C2×D4×C53—41no—
32C2×D4×C53—42no—
33C2×D4×C53—41no—
34C2×D4×C53—41no—
35C2×D4×C53—42no—
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53—21no—
39C22×C4×C53—42no—
40C22×C4×C53—42no—
41C22×C4×C53—42no—
42C22×C4×C53—42no—
43C22×C4×C53C2×C811yescharacteristic
44C22×C4×C53M1611yescharacteristic
45C22×Dic53—22no—
46C22×Dic53—41no—
47C2×#88324—82no—
48C22⋊Dic53—42no—
49C22⋊Dic53—42no—
50C23×C53—22no—
51C23×C53—22no—
52C23×C53—41no—
53C23×C53—41no—
54C23×C53—81no—
55C23×C53—82no—
56C23×C53C22⋊C811yescharacteristic
57C23×C53#9911yescharacteristic
58C23×C53#10011yescharacteristic
59Q8×C53—42no—
60Q8×C53—42no—
61D4×C53—44no—
62D4×C53—42no—
63D4×C53—42no—
64D4×C53—44no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—42no—
70C2×C4×C53—48no—
71C2×C4×C53—41no—
72C2×C4×C53—42no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×Dic53—42no—
76C2×Dic53—82no—
77C2×Dic53—81no—
78C2×Dic53—82no—
79#88324—162no—
80#7339—531nomaximal, Sylow
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—42no—
87C22×C53—82no—
88C22×C53—82no—
89C22×C53—82no—
90C22×C53—82no—
91C22×C53#32311yescharacteristic
92C22×C53AΣL(1,𝔽16)12yes—
93C4×C53—41no—
94C4×C53—42no—
95C4×C53—41no—
96Dic53—82no—
97Dic53—161no—
98#1782—531no—
99#1317—532no—
100C2×C53—22no—
101C2×C53—21no—
102C2×C53—41no—
103C2×C53—41no—
104C2×C53—82no—
105C2×C53#131611yescharacteristic
106C22×#41#615711yescharacteristic
107C2×#99—1062no—
108#381—531no—
109#381—531no—
110#323—1062no—
111C53#733911yescharacteristic, complemented, Sylow
112C2×#41—41no—
113C2×#41—41no—
114C2×#41—44no—
115C22×Q8#8832411yescharacteristic, complemented
116C22×D4—21no—
117C22×D4#8832411yescharacteristic, complemented
118C23×C4—21no—
119C23×C4C2×#615711yescharacteristic
120C2×C22⋊C4—532no—
121C2×C22⋊C4—1061no—
122#99—2124no—
123C22⋊C8—2122no—
124#95—2121no—
125#95—2121no—
126C24—21no—
127C24C22⋊#615711yescharacteristic
128#41—44no—
129#41—44no—
130#41—44no—
131#41—44no—
132C2×Q8—42no—
133C2×Q8—41no—
134C2×D4—24no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—1062no—
148C22×C4—2121no—
149C22×C4C2×#8832411yescharacteristic
150C22×C4#1171994011yescharacteristic
151C2×C8—4242no—
152C22⋊C4—2122no—
153C22⋊C4—2122no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—82no—
160C23C22⋊#8832411yescharacteristic
161C23#1313327511yescharacteristic
162C23#1313327711yescharacteristic
163Q8—42no—
164Q8—42no—
165D4—44no—
166D4—42no—
167D4—42no—
168D4—44no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—48no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—4241no—
180C2×C4—4242no—
181C2×C4—4242no—
182C2×C4—2122no—
183C8—8482no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—82no—
194C22#1681146611yescharacteristic
195C22#1681152312yes—
196C4—41no—
197C4—42no—
198C4—41no—
199C4—8481no—
200C4—4242no—
201C2—22no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—82no—
206C2#2028963911yescharacteristic
subgroupG/H
centerC2#20289639
commutatorC22×C4×C53C2×C8
FrattiniC23×C4C2×#6157
Fitting, prime radicalC22×#41×C53C4
socleC2×C53#1316
cosocle#16810486C22
hypercenterC22×#41#6157
hypocenterC53#7339
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #20289641C2semidirect
17. C22×Q8×C53C8semidirect
19. C22×D4×C53C8semidirect
111. C53#7339semidirect
115. C22×Q8#88324semidirect
117. C22×D4#88324semidirect

Maximal subgroups

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

subgroupindex
#202896412
#202746332
#733953