Finite Group Databasebeta

← #28298246

Subgroups — #28298246

Order 13568.

368 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 78 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#20289553C212yesmaximal
2#20272983C211yesmaximal, characteristic
3C2×C22⋊#88324—22no—
4C2×#13133124—22no—
5C2×#13133124C411yescharacteristic
6C2×#13133124C2211yescharacteristic
7C22×C4⋊C4×C53C411yescharacteristic
8C22×#88324—42no—
9C22⋊#88324—24no—
10C22⋊#88324—42no—
11C23×Dic53—21no—
12C23×Dic53D412yes—
13#13133124—41no—
14#13133124—44no—
15#13133124—41no—
16C23×C4×C53—21no—
17C23×C4×C53C2×C411yescharacteristic
18C2×C4⋊C4×C53—21no—
19C2×C4⋊C4×C53—41no—
20C2×C4⋊C4×C53—41no—
21C2×C4⋊C4×C53C812yescomplemented
22C24×C53C22⋊C411yescharacteristic
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53—22no—
26C22×C4×C53—42no—
27C22×C4×C53—42no—
28C22×C4×C53—42no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53C2×C811yescharacteristic
32C22×C4×C53M1611yescharacteristic
33C22×Dic53—22no—
34C22×Dic53—22no—
35C22×Dic53—22no—
36C22×Dic53—42no—
37C22×Dic53—42no—
38C22×Dic53—42no—
39C22×Dic53—41no—
40C22×Dic53—41no—
41C22×Dic53—42no—
42C22×Dic53SD1612yes—
43C22×Dic53Q1612yes—
44C2×#88324—44no—
45C2×#88324—82no—
46C2×#88324—82no—
47C4⋊C4×C53—42no—
48C4⋊C4×C53—42no—
49C23×C53—22no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53C22⋊C811yescharacteristic
53C23×C53#9911yescharacteristic
54C23×C53#10011yescharacteristic
55C23×C53#10312yes—
56C23×C53#10412yes—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—42no—
60C2×C4×C53—42no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—44no—
64C2×C4×C53—42no—
65C2×C4×C53—82no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—42no—
71C2×Dic53—42no—
72C2×Dic53—42no—
73C2×Dic53—42no—
74C2×Dic53—42no—
75C2×Dic53—44no—
76C2×Dic53—82no—
77C2×Dic53—81no—
78C2×Dic53—82no—
79C2×Dic53—81no—
80#88324—84no—
81#7132—531nomaximal, Sylow
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53#32311yescharacteristic
91C22×C53Q8⋊C812yes—
92C22×C53#32812yes—
93C22×C53#32911yescharacteristic
94C22×C53#33311yescharacteristic
95C22×C53#35212yes—
96C22×C53#35311yescharacteristic
97C22×C53#35511yescharacteristic
98C4×C53—41no—
99C4×C53—41no—
100C4×C53—82no—
101Dic53—42no—
102Dic53—42no—
103Dic53—82no—
104#1445—531no—
105#1280—532no—
106C2×C53—21no—
107C2×C53—22no—
108C2×C53—21no—
109C2×C53#127211yescharacteristic
110C2×C53#131711yescharacteristic
111C2×C53#131811yescharacteristic
112C2×C53#135212yes—
113C2×C53#135312yes—
114C22×C4⋊C4#615711yescharacteristic
115C2×C22⋊C8—1062no—
116C2×#95—531no—
117C2×#95—531no—
118C2×#95—1062no—
119C53#713211yescharacteristic, complemented, Sylow
120C23×C4—21no—
121C23×C4—532no—
122C23×C4—1061no—
123C23×C4C2×#615711yescharacteristic
124C22×C8—2122no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4#8832412yescomplemented
129C22⋊C8—1064no—
130C22⋊C8—2122no—
131#95—2121no—
132#95—2121no—
133#95—2124no—
134C24C22⋊#615711yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—532no—
144C22×C4—532no—
145C22×C4—1062no—
146C22×C4—1062no—
147C22×C4—1062no—
148C22×C4—2122no—
149C22×C4—2121no—
150C22×C4—2121no—
151C22×C4—2122no—
152C22×C4—2122no—
153C22×C4—2122no—
154C22×C4C2×#8832411yescharacteristic
155C22×C4#1171994011yescharacteristic
156C2×C8—4242no—
157C2×C8—4242no—
158C2×C8—2124no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23C22⋊#8832411yescharacteristic
165C23#1313327511yescharacteristic
166C23#1313327711yescharacteristic
167C23#1313328012yes—
168C23#1313328212yes—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—1062no—
181C2×C4—1062no—
182C2×C4—4241no—
183C2×C4—4242no—
184C2×C4—4241no—
185C2×C4—4242no—
186C2×C4—2122no—
187C2×C4—2124no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4—2122no—
192C8—4244no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22#1681146611yescharacteristic
202C22Q8⋊#8832412yes—
203C22#1681147712yes—
204C22#1681147911yescharacteristic
205C22#1681148311yescharacteristic
206C22#1681152512yes—
207C22#1681152711yescharacteristic
208C22#1681153111yescharacteristic
209C4—41no—
210C4—41no—
211C4—82no—
212C4—4242no—
213C4—2122no—
214C4—2122no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2#2028953511yescharacteristic
219C2#2028964111yescharacteristic
220C2#2028964311yescharacteristic
221C2#2028971212yes—
222C2#2028971412yes—
subgroupG/H
centerC23#13133275
commutatorC22×C4×C53C2×C8
FrattiniC23×C4C2×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×#13133124C22
hypercenterC22×C4⋊C4#6157
hypocenterC53#7132
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
21. C2×C4⋊C4×C53C8semidirect
119. C53#7132semidirect
128. C2×C4⋊C4#88324semidirect

Maximal subgroups

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

subgroupindex
#202895532
#202729832
#713253