Finite Group Databasebeta

← #28298468

Subgroups — #28298468

Order 13568.

364 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C4×#13133277C211yesmaximal, characteristic
2#20289950C211yesmaximal, characteristic
3#20275007C211yesmaximal, characteristic
4C2×#13133277C412yes—
5D53⋊(C22⋊C8)—21no—
6C4×#11719940—21no—
7#16811503—21no—
8#16811469—21no—
9#16811466C412yes—
10C2×C4×C22⋊D53C2211yescharacteristic
11C2×C22⋊#88322C411yescharacteristic
12C2×#13133129—21no—
13#16810377—21no—
14C23×C8×C53C411yescharacteristic
15C2×#11719940—22no—
16C2×D53⋊C8—41no—
17#13133277—22no—
18#13133277C814yescomplemented
19#13133274—41no—
20C22⋊#88324—22no—
21#13133272—41no—
22C4×#88324—41no—
23C22×C22⋊D53C2×C411yescharacteristic
24C22×C4×D53D411yescharacteristic
25C22×#88322—21no—
26C4×C22⋊D53—41no—
27C4×C22⋊D53—41no—
28C2×#11719919D411yescharacteristic
29C2×C22⋊Dic53C2×C411yescharacteristic
30C2×#11719921Q811yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C22⋊#88322—41no—
33#13133129—42no—
34C23×C4×C53C2×C411yescharacteristic
35C22×C8×C53—21no—
36C22×C8×C53—41no—
37C22×C8×C53—42no—
38C24×C53C4211yescharacteristic
39C23×D53C4⋊C411yescharacteristic
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C4⋊C411yescharacteristic
46C2×C22⋊D53—41no—
47C2×C22⋊D53—41no—
48C2×C22⋊D53C2×C812yes—
49C2×C22⋊D53M1612yes—
50C22×Dic53C22⋊C411yescharacteristic
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C4⋊C411yescharacteristic
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—41no—
56C2×C4×D53—41no—
57#11719940—24no—
58#11719940—42no—
59C2×#88324—42no—
60C2×#88324—42no—
61D53⋊C8—81no—
62D53⋊C8—81no—
63C2×C8×C53—22no—
64C2×C8×C53—41no—
65C2×C8×C53—42no—
66C2×C8×C53—41no—
67C2×C8×C53—42no—
68C22⋊Dic53—41no—
69#11719921—41no—
70#11719919—41no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C2×#88322—41no—
74C2×#88322—41no—
75C2×#88322—41no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—41no—
79C23×C53—41no—
80C23×C53#9511yescharacteristic
81C23×C53C4×C811yescharacteristic
82C23×C53#9711yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53C4⋊C812yes—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53#9911yescharacteristic
96C2×C4×C53#9911yescharacteristic
97C22⋊D53—42no—
98C22⋊D53—42no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53C22⋊C812yes—
106C4×D53—41no—
107C4×D53—41no—
108C4×D53—81no—
109#88324—44no—
110#88324—82no—
111C8×C53—42no—
112#88322—81no—
113#7203—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53#33611yescharacteristic
124C22×C53#34211yescharacteristic
125C22×C53#34411yescharacteristic
126C2×D53—22no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—81no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53AΣL(1,𝔽16)11yescharacteristic
134C4×C53#35211yescharacteristic
135Dic53—22no—
136Dic53—41no—
137Dic53—41no—
138#1869—531no—
139C4×#100—531no—
140#1459—531no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—41no—
144C2×C53—41no—
145C2×C53#128211yescharacteristic
146C2×C53#139011yescharacteristic
147C2×C53#139211yescharacteristic
148D53—42no—
149C23×C8#615711yescharacteristic
150C2×C4×D4—531no—
151C2×#100—532no—
152C2×C22⋊C8—531no—
153C2×C22⋊C8—1061no—
154C2×C22⋊C8—1061no—
155C4×M16—1061no—
156#343—1061no—
157#336—1061no—
158#324—1061no—
159#323—532no—
160C53#720311yescharacteristic, complemented, Sylow
161C22×D4—531no—
162C23×C4—531no—
163C23×C4C2×#615711yescharacteristic
164C2×M16—1062no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—41no—
168C22×C8—1061no—
169C22×C8—2121no—
170C4×D4—2121no—
171C4×D4—2121no—
172C2×C4⋊C4—531no—
173C2×C22⋊C4—531no—
174C2×C22⋊C4—531no—
175C2×C42—531no—
176#100—534no—
177#100—1062no—
178C22⋊C8—1062no—
179C22⋊C8—2122no—
180C22⋊C8—2121no—
181C22⋊C8—2121no—
182#97—2121no—
183C4×C8—2121no—
184C24—531no—
185C24C4×#615711yescharacteristic
186C2×D4—532no—
187C2×D4—532no—
188C2×D4—2121no—
189C2×D4—2121no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1061no—
198C22×C4—1061no—
199C22×C4—2121no—
200C22×C4—2121no—
201C22×C4C4⋊#615711yescharacteristic
202C22×C4C22⋊#615711yescharacteristic
203M16—1064no—
204M16—2122no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—4241no—
211C2×C8—4241no—
212C2×C8—2121no—
213C2×C8—2121no—
214C2×C8—2122no—
215C2×C8—2121no—
216C2×C8—2122no—
217C4⋊C4—2121no—
218C22⋊C4—2121no—
219C22⋊C4—2121no—
220C42—1061no—
221C42—1061no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—532no—
227C23—1061no—
228C23—1061no—
229C23—2121no—
230C23—2121no—
231C23(C2×#6157)⋊C411yescharacteristic
232C23C8×#615711yescharacteristic
233C23#1313327111yescharacteristic
234D4—2122no—
235D4—2122no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—532no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—4241no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4#1313327511yescharacteristic
254C2×C4#1313327611yescharacteristic
255C8—42no—
256C8—4241no—
257C8—4242no—
258C8—2124no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—41no—
268C22—1062no—
269C22—4241no—
270C22—2122no—
271C22—2121no—
272C22—2121no—
273C22(C2×#6157)⋊C811yescharacteristic
274C22#1681150111yescharacteristic
275C22#1681150511yescharacteristic
276C4—21no—
277C4—41no—
278C4—1062no—
279C4—2121no—
280C4—2121no—
281C4#1681152311yescharacteristic
282C4#1681152511yescharacteristic
283C2—21no—
284C2—21no—
285C2—41no—
286C2—41no—
287C2—2122no—
288C2#2028955711yescharacteristic
289C2#2028979911yescharacteristic
290C2#2028980611yescharacteristic
subgroupG/H
centerC2×C4#13133275
commutatorC23×C53C4×C8
FrattiniC23×C4C2×#6157
FittingC23×C8×C53C4
socleC22×C53#342
prime radicalC22×C22⋊D53C2×C4
cosocleC2×C4×C22⋊D53C22
hypercenterC23×C8#6157
hypocenterC53#7203
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
18. #13133277C8semidirect
160. C53#7203semidirect

Maximal subgroups

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

subgroupindex
C4×#131332772
#202899502
#202750072
#720353