Finite Group Databasebeta

← #28308197

Subgroups — #28308197

Order 13568.

446 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289966C212yesmaximal
2#20289657C212yesmaximal, complemented
3#20289655C212yesmaximal, complemented
4#20277122C211yesmaximal, characteristic
5#16811643—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×#13133176C411yescharacteristic
9C2×#13133176C2211yescharacteristic
10#16810493C411yescharacteristic
11#16810493C2211yescharacteristic
12#16810489C411yescharacteristic
13#16810489C2211yescharacteristic
14#16810483—22no—
15C4×C4⋊Dic53—22no—
16C22×C42×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20#13133176—41no—
21#13133176—41no—
22#13133176—41no—
23#13133176—41no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2311yescharacteristic
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53D412yes—
32C2×C4×Dic53—21no—
33C2×C4×Dic53D412yes—
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53—21no—
37C23×C4×C53C2×C411yescharacteristic
38C2×C42×C53—22no—
39C2×C42×C53—22no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C24×C53C22⋊C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—24no—
51C22×C4×C53—21no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×#88324—44no—
61C42×C53—24no—
62C42×C53—21no—
63C42×C53—21no—
64C42×C53M1612yes—
65C42×C53M1612yes—
66C22⋊Dic53—42no—
67C22⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—44no—
71C4⋊Dic53—44no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—41no—
77C4×Dic53—41no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53#10112yes—
83C23×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—48no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53#9911yescharacteristic
100C2×C4×C53#9911yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53C2×M1611yescharacteristic
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111#88324—84no—
112#11513—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—22no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53#40711yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10111yescharacteristic
123C4×C53—24no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—42no—
129Dic53—42no—
130Dic53—42no—
131Dic53—81no—
132Dic53—81no—
133#2285—531no—
134#1464—532no—
135#1325—532no—
136#1324—532no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—42no—
140C2×C53#146311yescharacteristic
141C2×C53#212611yescharacteristic
142C2×C53#213211yescharacteristic
143C2×#117—531no—
144C2×#117—531no—
145C22×C42#615711yescharacteristic
146#432—1062no—
147#384—531no—
148#384—531no—
149#382—531no—
150#382—531no—
151#380—1062no—
152C4×C4⋊C4—1062no—
153#324—534no—
154#323—1062no—
155C53#1151311yescharacteristic, complemented, Sylow
156C23×C4—21no—
157C23×C4C2×#615711yescharacteristic
158#117—2121no—
159#117—2121no—
160#117—2121no—
161#117—2121no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4—1062no—
168C2×C22⋊C4—532no—
169C2×C22⋊C4—1061no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—532no—
174C2×C42—1061no—
175C2×C42C2×#615711yescharacteristic
176C2×C42C2×#615711yescharacteristic
177C4⋊C8—2122no—
178C22⋊C8—1064no—
179#97—2122no—
180#95—1062no—
181#95—1062no—
182C24C22⋊#615711yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—532no—
194C22×C4—532no—
195C22×C4—1061no—
196C22×C4—1061no—
197C22×C4C22⋊#615711yescharacteristic
198C22×C4C22×#615711yescharacteristic
199C2×C8—2124no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2124no—
205C4⋊C4—2124no—
206C22⋊C4—2122no—
207C22⋊C4—2122no—
208C42—24no—
209C42—21no—
210C42—21no—
211C42—1062no—
212C42—1062no—
213C42—2121no—
214C42—2121no—
215C42#1171994012yes—
216C42#1171994012yes—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23#1313327812yes—
222C23C2×C22⋊#615711yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—48no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4#1313327611yescharacteristic
247C2×C4#1313327611yescharacteristic
248C2×C4C2×#1171994011yescharacteristic
249C2×C4C2×#1171994011yescharacteristic
250C8—4244no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22#1681159311yescharacteristic
259C22C2×#1313327611yescharacteristic
260C22C2×#1313327811yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—4241no—
268C4—4241no—
269C4—2122no—
270C4—2122no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2#2028996311yescharacteristic
275C2#2029098811yescharacteristic
276C2#2029100811yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11513
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289657C2semidirect
3. #20289655C2semidirect
155. C53#11513semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896572
#202896552
#202771222
#1151353