Finite Group Databasebeta

← #28298514

Subgroups — #28298514

Order 13568.

290 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 52 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#20290612C211yesmaximal, characteristic
2#20289968C211yesmaximal, characteristic
3#20274482C211yesmaximal, characteristic
4#16811643—21no—
5#16811566—21no—
6#16811501—21no—
7#16811501C412yes—
8#16811469C412yescomplemented
9#16811467—21no—
10C2×#13133174C2211yescharacteristic
11C2×#13133129—21no—
12C2×#13133128C411yescharacteristic, complemented
13#16810377—21no—
14C2×C4×C8×C53C411yescharacteristic, complemented
15C2×C22⋊#6157—22no—
16#13133284—41no—
17C22⋊#88324—22no—
18#13133272—41no—
19(C2×#6157)⋊C4—41no—
20(C2×#6157)⋊C4—41no—
21C22×C4×D53D411yescharacteristic
22C22×#88322—21no—
23#13133174—41no—
24#13133174—41no—
25C2×#11719919C2×C411yescharacteristic
26C2×#11719919Q811yescharacteristic
27C2×#11719921C2×C411yescharacteristic
28C2×#11719921D411yescharacteristic
29C2×C4×Dic53D411yescharacteristic
30#13133129—24no—
31#13133128—41no—
32C22×C8×C53—21no—
33C2×C42×C53C2×C411yescharacteristic
34C4×C8×C53—41no—
35C23×D53C22⋊C411yescharacteristic
36C22×#6157—42no—
37C22×C4×C53C4211yescharacteristic
38C22×C4×C53C22⋊C411yescharacteristic
39C22×C4×C53C4⋊C411yescharacteristic
40C22×Dic53C22⋊C411yescharacteristic
41C22×Dic53C4⋊C411yescharacteristic
42C22×Dic53C4⋊C411yescharacteristic
43C2×C4×D53—22no—
44C2×C4×D53—21no—
45C2×C4×D53—21no—
46C2×C4×D53—41no—
47C22⋊#6157—42no—
48C22⋊#6157—42no—
49C2×#88324—42no—
50C2×C8×C53—22no—
51C2×C8×C53—41no—
52C2×C8×C53—41no—
53C42×C53—21no—
54C42×C53—21no—
55#11719921—41no—
56#11719921—41no—
57#11719919—41no—
58#11719919—41no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C2×#88322—22no—
62C2×#88322—41no—
63C2×#88322—41no—
64C23×C53#9511yescharacteristic
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—41no—
68C22×D53—41no—
69C22×D53#10412yes—
70C2×#6157—82no—
71C2×#6157—82no—
72C2×#6157—82no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53#9911yescharacteristic
80C2×C4×C53#9911yescharacteristic
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—41no—
87C2×Dic53—41no—
88C4×D53—42no—
89C4×D53—42no—
90#88324—82no—
91C8×C53—42no—
92#88322—42no—
93#7214—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—21no—
96C22×C53H(3,ℤ/4)11yescharacteristic
97C22×C53#33911yescharacteristic
98C22×C53#34211yescharacteristic
99C2×D53—22no—
100C2×D53—41no—
101C2×D53—41no—
102C2×D53—42no—
103C2×D53—81no—
104#6157—162no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109Dic53—41no—
110Dic53—41no—
111Dic53—41no—
112#1820—531no—
113#1750—531no—
114#1464—531no—
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#128411yescharacteristic
118C2×C53#139111yescharacteristic
119C2×C53#139511yescharacteristic
120D53—42no—
121C2×#117—531no—
122#432—1061no—
123C2×C22⋊C8—1061no—
124C2×#97—531no—
125C2×C4×C8#615711yescharacteristic, complemented
126#381—1061no—
127#342—532no—
128#342—1061no—
129#336—1061no—
130#324—532no—
131#323—1061no—
132C53#721411yescharacteristic, complemented, Sylow
133C23×C4—531no—
134C22×C8—21no—
135C22×C8—1061no—
136#117—2121no—
137#117—2121no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C22⋊C4—531no—
141C2×C22⋊C4—531no—
142C2×C22⋊C4—1062no—
143C2×C42—531no—
144C2×C42C2×#615711yescharacteristic
145C4⋊C8—2121no—
146C22⋊C8—1064no—
147C22⋊C8—1062no—
148#97—2121no—
149#97—2121no—
150C4×C8—41no—
151#95—2121no—
152#95—2121no—
153C24—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—1062no—
158C22×C4—1061no—
159C22×C4—1061no—
160C22×C4—2121no—
161C22×C4—2122no—
162C22×C4C4×#615711yescharacteristic
163C22×C4C4⋊#615711yescharacteristic
164C22×C4C22⋊#615711yescharacteristic
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—1062no—
169C2×C8—2122no—
170C2×C8—2121no—
171C2×C8—2121no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C22⋊C4—2121no—
175C22⋊C4—2121no—
176C22⋊C4—2122no—
177C22⋊C4—2122no—
178C42—21no—
179C42—21no—
180C42—1061no—
181C42—1061no—
182C23—532no—
183C23—1061no—
184C23—1061no—
185C23—2121no—
186C23—2121no—
187C23(C2×#6157)⋊C411yescharacteristic
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—4242no—
200C2×C4—4242no—
201C2×C4—4242no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4#1313327511yescharacteristic
207C2×C4#1313327611yescharacteristic
208C8—42no—
209C8—4242no—
210C8—2122no—
211C22—21no—
212C22—21no—
213C22—1062no—
214C22—4241no—
215C22—2122no—
216C22—2121no—
217C22—2121no—
218C22#1681149011yescharacteristic
219C22#1681149411yescharacteristic
220C22#1681150111yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—8482no—
226C4—2121no—
227C4—2121no—
228C4—2121no—
229C2—21no—
230C2—21no—
231C2—2122no—
232C2#2028956311yescharacteristic
233C2#2028980311yescharacteristic
234C2#2028981511yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC2×C42C2×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#342
prime radicalC23×D53C22⋊C4
cosocleC2×#13133174C22
hypercenterC2×C4×C8#6157
hypocenterC53#7214
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
8. #16811469C4semidirect
12. C2×#13133128C4semidirect
14. C2×C4×C8×C53C4semidirect
125. C2×C4×C8#6157semidirect
132. C53#7214semidirect

Maximal subgroups

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

subgroupindex
#202906122
#202899682
#202744822
#721453