Finite Group Databasebeta

← #28299558

Subgroups — #28299558

Order 13568.

492 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290631C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289532C212yesmaximal
4C4×#13133206C211yesmaximal, characteristic
5C2×#13133284—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133206C411yescharacteristic
9C2×C4×D212C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810524C2211yescharacteristic
12#16810513—22no—
13#16810488C411yescharacteristic
14C4×#11719921—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C22×#88324—24no—
18#13133284—44no—
19C22×D212C2×C411yescharacteristic
20C22×C4×D53—21no—
21#13133206—22no—
22#13133206C814yescomplemented
23C4×D212—42no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×#11719921—21no—
26C2×#11719921—22no—
27C2×#11719921—22no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53D412yes—
31C2×C4×Dic53C2311yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C4×Q8×C53—42no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C42×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C23×D53—21no—
41C2×Q8×C53—21no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×Q8×C53C2×C812yes—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—21no—
51C22×Dic53C22⋊C412yes—
52C22×Dic53C2×D412yes—
53C2×D212—21no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212C2×C812yes—
57C2×C4×D53—44no—
58C2×C4×D53—42no—
59C2×#88324—44no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—41no—
64C42×C53—22no—
65C42×C53—42no—
66#11719921—28no—
67#11719921—42no—
68#11719921—42no—
69#11719921—42no—
70C4⋊Dic53—41no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—42no—
74C4×Dic53—41no—
75C4×Dic53C2×C812yes—
76C4×Dic53M1612yes—
77C23×C53C2×C22⋊C411yescharacteristic
78C22×D53—22no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53—42no—
82C22×D53—42no—
83Q8×C53—42no—
84Q8×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53#10012yes—
96C2×C4×C53C22×C811yescharacteristic
97C2×C4×C53C2×M1611yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—41no—
103C2×Dic53—42no—
104C2×Dic53C22⋊C814yes—
105C2×Dic53AGL(1,ℤ/8)12yes—
106C2×Dic53#13712yes—
107D212—42no—
108D212—42no—
109C4×D53—84no—
110#88324—84no—
111#7601—531nomaximal, Sylow
112C22×C53C2×C22⋊C811yescharacteristic
113C22×C53#41011yescharacteristic
114C22×C53C2×#10011yescharacteristic
115C22×C53#41712yes—
116C22×C53#42112yes—
117C2×D53—42no—
118C2×D53—44no—
119C2×D53—81no—
120C2×D53—82no—
121C2×D53—81no—
122C4×C53—21no—
123C4×C53—21no—
124C4×C53—22no—
125C4×C53—41no—
126C4×C53—42no—
127C4×C53—81no—
128Dic53—24no—
129Dic53—22no—
130Dic53—22no—
131Dic53—81no—
132C4×#124—531no—
133#1831—532no—
134#1272—532no—
135#1270—532no—
136C2×C53#146011yescharacteristic
137C2×C53#147712yes—
138C2×C53#151212yes—
139C2×C53#153311yescharacteristic
140C2×C53#156011yescharacteristic
141D53—82no—
142C2×#124—531no—
143C2×C4×Q8#615711yescharacteristic
144C2×C4×D4—531no—
145C2×C4⋊C8—1062no—
146#391—531no—
147#390—531no—
148#388—1062no—
149#382—531no—
150C4×C22⋊C4—1062no—
151C43—531no—
152#336—534no—
153#336—1062no—
154C53#760111yescharacteristic, complemented, Sylow
155C22×Q8C2×#615711yescharacteristic
156C22×D4—531no—
157C23×C4—1061no—
158C22×C8—1064no—
159#124—534no—
160#124—1062no—
161C4×Q8—42no—
162C4×D4—2122no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4C2×#615711yescharacteristic
166C2×C22⋊C4—1062no—
167C2×C22⋊C4—1062no—
168C2×C22⋊C4—1061no—
169C2×C42—21no—
170C2×C42—531no—
171C2×C42—532no—
172C2×C42—532no—
173C2×C42—531no—
174C2×C42C2×#615711yescharacteristic
175C4⋊C8—2124no—
176#95—1062no—
177#95—1062no—
178C24—1061no—
179C2×Q8—21no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8D53⋊C812yes—
183C2×D4—532no—
184C2×D4—1061no—
185C2×D4—2121no—
186C2×D4—2121no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—532no—
190C22×C4—532no—
191C22×C4—1061no—
192C22×C4—2124no—
193C22×C4—2122no—
194C22×C4C22⋊#615711yescharacteristic
195C22×C4C22⋊#615711yescharacteristic
196C22×C4C22×#615711yescharacteristic
197C2×C8—2124no—
198C2×C8—2124no—
199C2×C8—2124no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—2121no—
203C22⋊C4—1068no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C22⋊C4—2122no—
207C42—22no—
208C42—42no—
209C42—532no—
210C42—532no—
211C42—1062no—
212C42—1062no—
213C42—2121no—
214C42—2122no—
215C23—1062no—
216C23—2121no—
217C23—2121no—
218C23—2122no—
219C23—2122no—
220C23C2×C22⋊#615711yescharacteristic
221Q8—42no—
222Q8—42no—
223D4—2122no—
224D4—2122no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—534no—
236C2×C4—532no—
237C2×C4—532no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—4244no—
243C2×C4—2121no—
244C2×C4—2122no—
245C2×C4#1313327712yes—
246C2×C4C2×D53⋊C811yescharacteristic
247C2×C4D53⋊M1611yescharacteristic
248C8—4244no—
249C22—4242no—
250C22—4241no—
251C22—4241no—
252C22—2122no—
253C22—2124no—
254C22D53⋊(C22⋊C8)11yescharacteristic
255C22#1681160211yescharacteristic
256C22C2×#1313327711yescharacteristic
257C22#1681161412yes—
258C22#1681162212yes—
259C4—22no—
260C4—21no—
261C4—21no—
262C4—42no—
263C4—41no—
264C4—81no—
265C4—1062no—
266C4—1062no—
267C4—1064no—
268C4—4241no—
269C2—4242no—
270C2#2028995411yescharacteristic
271C2#2029001312yes—
272C2#2029012812yes—
273C2#2029019611yescharacteristic
274C2#2029025211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7601
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #20290631C2semidirect
2. #20289535C2semidirect
22. #13133206C8semidirect
154. C53#7601semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202895352
#202895322
C4×#131332062
#760153