Finite Group Databasebeta

← #28307908

Subgroups — #28307908

Order 13568.

526 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289946C212yesmaximal
2#20289642C214yesmaximal
3#20277299C211yesmaximal, characteristic
4#16811639—22no—
5#16811468—22no—
6#16811466C2214yes—
7C23×Dic106C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810529—22no—
10#16810528—22no—
11#16810484C412yes—
12#16810484C2212yes—
13C2×#117×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17C22×Dic106—21no—
18C22×Dic106—21no—
19C22×Dic106—22no—
20C22×Dic106—42no—
21C22×Dic106—42no—
22C22×Dic106C814yescomplemented
23C23×Dic53C2×C412yes—
24#13133176—42no—
25C2×C4⋊Dic53D412yes—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53C2311yescharacteristic
28C2×C4×Dic53D412yes—
29#13133124—24no—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53C2×C412yes—
35C2×C42×C53—21no—
36C24×C53C22×C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C22⋊C411yescharacteristic
42C22×C4×C53C2×C812yes—
43C22×C4×C53M1612yes—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53—42no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C2×C814yes—
50C22×Dic53C2×D412yes—
51C2×Dic106—24no—
52C2×Dic106—24no—
53C2×Dic106—42no—
54C2×Dic106—42no—
55C2×Dic106—48no—
56C2×Dic106—42no—
57C2×Dic106—42no—
58C2×Dic106—48no—
59C2×#88324—44no—
60C4⋊C4×C53—42no—
61C22⋊C4×C53—42no—
62C42×C53—42no—
63C22⋊Dic53—41no—
64C22⋊Dic53—41no—
65C4⋊Dic53—42no—
66C4×Dic53—22no—
67C4×Dic53—22no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—42no—
71C23×C53C2×C22⋊C411yescharacteristic
72C23×C53C22×C811yescharacteristic
73C23×C53C2×M1611yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—24no—
77C2×C4×C53—24no—
78C2×C4×C53—44no—
79C2×C4×C53—42no—
80C2×C4×C53C22⋊C812yes—
81C2×C4×C53C22⋊C812yes—
82C2×C4×C53#9912yes—
83C2×C4×C53#10012yes—
84C2×Dic53—24no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—44no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—44no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93Dic106—48no—
94Dic106—48no—
95#88324—84no—
96#11428—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53C2×#9911yescharacteristic
106C22×C53C2×#10011yescharacteristic
107C4×C53—22no—
108C4×C53—82no—
109C4×C53#32314yes—
110Dic53—42no—
111Dic53—42no—
112Dic53—44no—
113#2310—531no—
114#1458—532no—
115#1317—534no—
116C2×C53—21no—
117C2×C53—21no—
118C2×C53—41no—
119C2×C53—41no—
120C2×C53C2×#32311yescharacteristic
121C2×C53#212312yes—
122C23×Q8—531no—
123C2×#117—531no—
124C2×#117#615711yescharacteristic
125#431—1062no—
126#391—1062no—
127#391—1062no—
128#381—532no—
129#381—532no—
130#324—1062no—
131#323—534no—
132C53#1142811yescharacteristic, complemented, Sylow
133C22×Q8—534no—
134C22×Q8—1062no—
135C22×Q8—1061no—
136C22×Q8—1061no—
137C22×Q8—2122no—
138C22×Q8—2122no—
139C23×C4—532no—
140C23×C4C2×#615711yescharacteristic
141#117—42no—
142#117—2122no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—532no—
145C2×C22⋊C4—531no—
146C2×C22⋊C4—531no—
147C2×C22⋊C4C2×#615712yes—
148C2×C42—21no—
149C2×C42—532no—
150C4⋊C8—2122no—
151C22⋊C8—1064no—
152C4×C8—2122no—
153#95—1064no—
154#95—1064no—
155C24C22×#615711yescharacteristic
156C2×Q8—1064no—
157C2×Q8—1064no—
158C2×Q8—2128no—
159C2×Q8—2122no—
160C2×Q8—2122no—
161C2×Q8—2128no—
162C2×Q8—2122no—
163C2×Q8—2122no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—532no—
168C22×C4—532no—
169C22×C4—534no—
170C22×C4—1062no—
171C22×C4—1062no—
172C22×C4—2122no—
173C22×C4—2122no—
174C22×C4D53⋊C812yes—
175C22×C4#1171993612yes—
176C22×C4C22⋊#615711yescharacteristic
177C22×C4C22⋊#615711yescharacteristic
178C2×C8—2124no—
179C4⋊C4—42no—
180C4⋊C4—2122no—
181C22⋊C4—42no—
182C22⋊C4—2121no—
183C22⋊C4—2121no—
184C42—42no—
185C42—1062no—
186C42—1062no—
187C23—21no—
188C23—21no—
189C23—42no—
190C23C2×C22⋊#615711yescharacteristic
191C23C2×D53⋊C811yescharacteristic
192C23C2×#1171993611yescharacteristic
193Q8—2128no—
194Q8—2128no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—1064no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—2124no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C2×C4—2122no—
209C2×C4—2124no—
210C2×C4#1313327412yes—
211C2×C4#1313327412yes—
212C2×C4#1313327612yes—
213C2×C4#1313327712yes—
214C8—4244no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—42no—
222C22C2×#1313327411yescharacteristic
223C22C2×#1313327611yescharacteristic
224C22C2×#1313327711yescharacteristic
225C4—22no—
226C4—82no—
227C4—2124no—
228C4—2122no—
229C4—2122no—
230C4#1681146714yes—
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2C2×#1681146711yescharacteristic
236C2#2029097912yes—
subgroupG/H
centerC22C2×#13133276
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11428
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
22. C22×Dic106C8semidirect
132. C53#11428semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896422
#202772992
#1142853