Finite Group Databasebeta

← #28308118

Subgroups — #28308118

Order 13568.

418 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 96 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#20291077C212yesmaximal, complemented
2#20289636C212yesmaximal, complemented
3#20289635C212yesmaximal, complemented
4C2×C8×C22⋊D53C211yesmaximal, characteristic
5#16811544—22no—
6C22⋊#11719910C2214yes—
7#16811512—22no—
8C22×C8×D53C411yescharacteristic
9C2×C4×C22⋊D53C411yescharacteristic
10C8×C22⋊D53—22no—
11C8×C22⋊D53—22no—
12C2×#13133146C411yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14C2×#13133129C2211yescharacteristic
15C2×C8×Dic53C2211yescharacteristic
16C23×C8×C53C411yescharacteristic
17C2×#11719910—24no—
18C22×C22⋊D53C811yescharacteristic
19C22×C4×D53C2×C411yescharacteristic
20C22×#88322C2×C411yescharacteristic
21C22×#88322C2311yescharacteristic
22C2×C8×D53—21no—
23C2×C8×D53—21no—
24C2×C8×D53—42no—
25C4×C22⋊D53—22no—
26C4×C22⋊D53—22no—
27C2×#11719919C811yescharacteristic
28C2×C22⋊Dic53C811yescharacteristic
29C2×#11719921C811yescharacteristic
30C2×C4×Dic53C2×C411yescharacteristic
31#13133146—22no—
32C22⋊#88322—22no—
33#13133129—22no—
34C8×Dic53D412yes—
35C8×Dic53D412yes—
36C23×C4×C53C2×C411yescharacteristic
37C22×C8×C53—21no—
38C22×C8×C53—21no—
39C22×C8×C53—42no—
40C22×C8×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42C24×C53C2×C811yescharacteristic
43C23×D53C2×C811yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53C2×C811yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C22×Dic53C2×C811yescharacteristic
54C22×Dic53C2×C811yescharacteristic
55C22×Dic53C2×C811yescharacteristic
56C2×C4×D53—21no—
57C2×C4×D53—21no—
58C2×C4×D53—41no—
59C2×C4×D53—41no—
60C2×C8×C53—22no—
61C2×C8×C53—21no—
62C2×C8×C53—42no—
63C2×C8×C53—42no—
64C2×C8×C53—41no—
65C2×C8×C53C22⋊C411yescharacteristic
66C2×C8×C53C22⋊C411yescharacteristic
67C22⋊Dic53—22no—
68#11719921—22no—
69#11719919—22no—
70C4×Dic53C22⋊C412yes—
71C4×Dic53M1612yes—
72C2×#88322—21no—
73C2×#88322—41no—
74C2×#88322—41no—
75C2×#88322C2×D412yes—
76C8×D53—42no—
77C8×D53—42no—
78#11719910—44no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—41no—
82C23×C53—41no—
83C23×C53M3212yes—
84C23×C53C22×C811yescharacteristic
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—41no—
88C22×D53—41no—
89C22×D53M3212yes—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C22⋊D53—28no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53C22⋊C812yes—
106C2×Dic53C22⋊C812yes—
107C4×D53—42no—
108C4×D53—41no—
109C4×D53—41no—
110C8×C53—21no—
111C8×C53—41no—
112C8×C53#10012yes—
113#88322—22no—
114#88322—41no—
115#11488—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53C2×C22⋊C811yescharacteristic
124C22×C53C2×M3211yescharacteristic
125C22×C53C2×M3211yescharacteristic
126C2×D53—22no—
127C2×D53—42no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—42no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53#41011yescharacteristic
134C4×C53C2×#10011yescharacteristic
135Dic53—22no—
136Dic53—22no—
137Dic53—41no—
138C2×C8×D4—531no—
139#2163—532no—
140#1314—532no—
141#1314—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—42no—
145C2×C53#146011yescharacteristic
146C2×C53#211211yescharacteristic
147C2×C53#211511yescharacteristic
148D53—42no—
149C23×C8—531no—
150C23×C8#615711yescharacteristic
151C2×C4×D4—531no—
152C8×D4—1062no—
153C8×D4—1062no—
154C2×C4⋊C8—531no—
155C2×C22⋊C8—531no—
156C2×C22⋊C8—531no—
157C2×C4×C8—531no—
158C4⋊C16—1062no—
159C22⋊C16—534no—
160#346—1062no—
161C53#1148811yescharacteristic, complemented, Sylow
162C22×D4—531no—
163C23×C4—531no—
164C23×C4C2×#615711yescharacteristic
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—531no—
169C22×C8—531no—
170C22×C8—1061no—
171C22×C8—1061no—
172C22×C8—2122no—
173C22×C8C2×#615711yescharacteristic
174C22×C8C2×#615711yescharacteristic
175C4×D4—1062no—
176C4×D4—1062no—
177C2×C4⋊C4—531no—
178C2×C22⋊C4—531no—
179C2×C22⋊C4—531no—
180C2×C42—531no—
181C2×C16—1064no—
182C4⋊C8—1062no—
183C22⋊C8—1062no—
184C22⋊C8—1062no—
185C4×C8—532no—
186C4×C8—532no—
187C24—531no—
188C24D53⋊C811yescharacteristic
189C2×D4—1062no—
190C2×D4—1062no—
191C2×D4—1062no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—2121no—
202C22×C4—2121no—
203C22×C4D53⋊C811yescharacteristic
204C22×C4C22×#615711yescharacteristic
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—532no—
211C2×C8—1061no—
212C2×C8—2122no—
213C2×C8—2121no—
214C2×C8—2122no—
215C2×C8—2121no—
216C2×C8C22⋊#615711yescharacteristic
217C2×C8C22⋊#615711yescharacteristic
218C4⋊C4—1062no—
219C22⋊C4—1062no—
220C22⋊C4—1062no—
221C42—532no—
222C42—532no—
223C16—2124no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—532no—
229C23—1061no—
230C23—1061no—
231C23—2121no—
232C23—2121no—
233C23#1313329212yes—
234C23C2×D53⋊C811yescharacteristic
235D4—1068no—
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—532no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—2122no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4C2×C22⋊#615711yescharacteristic
254C2×C4D53⋊M1611yescharacteristic
255C8—21no—
256C8—41no—
257C8—1062no—
258C8—2121no—
259C8#1313327712yes—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—1062no—
268C22—2122no—
269C22—2122no—
270C22—2121no—
271C22—2121no—
272C22D53⋊(C22⋊C8)11yescharacteristic
273C22D53⋊M3211yescharacteristic
274C22C2×#1313329211yescharacteristic
275C4—21no—
276C4—41no—
277C4—1062no—
278C4—1062no—
279C4—2121no—
280C4#1681160211yescharacteristic
281C4C2×#1313327711yescharacteristic
282C2—21no—
283C2—21no—
284C2—42no—
285C2—2122no—
286C2#2028995411yescharacteristic
287C2#2029094311yescharacteristic
288C2#2029095211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC23×C8×C53C4
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×#88322C23
hypercenterC23×C8#6157
hypocenterC53#11488
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
1. #20291077C2semidirect
2. #20289636C2semidirect
3. #20289635C2semidirect
161. C53#11488semidirect

Maximal subgroups

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

subgroupindex
#202910772
#202896362
#202896352
C2×C8×C22⋊D532
#1148853