Finite Group Databasebeta

← #28299439

Subgroups — #28299439

Order 13568.

528 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290429C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal
4#20277254C211yesmaximal, characteristic
5C2×C4×#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133205C411yescharacteristic
9C2×#13133182C411yescharacteristic
10C2×#13133179C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810524C2211yescharacteristic
13#16810512—22no—
14#16810470—22no—
15C42×Dic53C2211yescharacteristic
16C53×#382C411yescharacteristic
17C22×#88324—24no—
18C4×#88324—28no—
19C22×Dic106C2×C411yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133205—22no—
23#13133205C814yescomplemented
24#13133182—42no—
25#13133179—42no—
26C2×#11719919—21no—
27C2×#11719921—22no—
28C2×#11719921—21no—
29C2×#11719921—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35#13133124—22no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#95×C53—22no—
40C23×D53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53C22⋊C412yes—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53—21no—
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C2×D412yes—
49C2×D212—21no—
50C2×D212—41no—
51C2×D212—41no—
52C2×D212C2×C812yes—
53C2×C4×D53—42no—
54C2×C4×D53—44no—
55C2×Dic106—21no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106C2×C812yes—
59C2×#88324—28no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C42×C53C2×C812yes—
65C42×C53M1612yes—
66#11719921—28no—
67#11719921—42no—
68#11719921—44no—
69#11719919—42no—
70C4×Dic53—22no—
71C4×Dic53—24no—
72C4×Dic53—24no—
73C4×Dic53—41no—
74C4×Dic53—42no—
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—22no—
77C22×D53—41no—
78C22×D53—41no—
79C22×D53—42no—
80C22×D53—42no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53C22⋊C814yes—
90C2×C4×C53C22×C811yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53#10012yes—
100C2×Dic53#10414yes—
101C2×Dic53#13514yes—
102D212—42no—
103D212—42no—
104C4×D53—84no—
105Dic106—42no—
106Dic106—42no—
107#88324—48no—
108#7567—531nomaximal, Sylow
109C22×C53C2×C22⋊C811yescharacteristic
110C22×C53#41011yescharacteristic
111C22×C53C2×#10011yescharacteristic
112C22×C53#41612yes—
113C22×C53C2×#10412yes—
114C2×D53—42no—
115C2×D53—44no—
116C2×D53—81no—
117C2×D53—82no—
118C2×D53—81no—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—24no—
122C4×C53—81no—
123C4×C53—81no—
124Dic53—24no—
125Dic53—41no—
126Dic53—42no—
127Dic53#32914yes—
128C4×#124—531no—
129#1744—532no—
130#1272—532no—
131#1270—532no—
132C2×C53#146011yescharacteristic
133C2×C53#147612yes—
134C2×C53#150012yes—
135C2×C53C2×#32911yescharacteristic
136C2×C53#154311yescharacteristic
137D53—82no—
138C2×#124—531no—
139C2×C4×Q8—531no—
140C2×C4×D4—531no—
141C2×C4×C8—1062no—
142#391—531no—
143#390—531no—
144#388—1062no—
145#382#615711yescharacteristic
146C4×C22⋊C4—1062no—
147C43—531no—
148#336—534no—
149#336—1062no—
150C53#756711yescharacteristic, complemented, Sylow
151C22×Q8—531no—
152C22×D4—531no—
153C23×C4—1061no—
154C22×C8—1064no—
155#124—534no—
156#124—1062no—
157C4×Q8—2122no—
158C4×D4—2122no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4C2×#615711yescharacteristic
161C2×C4⋊C4C2×#615711yescharacteristic
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C22⋊C4—1062no—
165C2×C42—532no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C4×C8—1068no—
172#95—22no—
173#95—1062no—
174C24—1061no—
175C2×Q8—532no—
176C2×Q8—1061no—
177C2×Q8—2121no—
178C2×Q8—2121no—
179C2×D4—532no—
180C2×D4—1061no—
181C2×D4—2121no—
182C2×D4—2121no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—531no—
186C22×C4—532no—
187C22×C4—531no—
188C22×C4—1061no—
189C22×C4—2122no—
190C22×C4—2124no—
191C22×C4C22⋊#615712yes—
192C22×C4C22×#615711yescharacteristic
193C2×C8—1068no—
194C2×C8—2124no—
195C2×C8—2124no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—2122no—
199C22⋊C4—1068no—
200C22⋊C4—2124no—
201C22⋊C4—2122no—
202C42—1064no—
203C42—1062no—
204C42—1064no—
205C42—2121no—
206C42—2122no—
207C42D53⋊C812yes—
208C42#1171993612yes—
209C23—1062no—
210C23—2121no—
211C23—2121no—
212C23—2122no—
213C23—2122no—
214C23C2×C22⋊#615711yescharacteristic
215Q8—2122no—
216Q8—2122no—
217D4—2122no—
218D4—2122no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—532no—
228C2×C4—534no—
229C2×C4—534no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1061no—
234C2×C4—1062no—
235C2×C4—4244no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4#1313327414yes—
239C2×C4C2×D53⋊C811yescharacteristic
240C2×C4C2×#1171993611yescharacteristic
241C8—2128no—
242C22—4242no—
243C22—4241no—
244C22—4241no—
245C22—2122no—
246C22—2124no—
247C22C2×#1313327411yescharacteristic
248C22#1681160411yescharacteristic
249C22D53⋊#10011yescharacteristic
250C22#1681161312yes—
251C22D53⋊#10412yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—81no—
256C4—81no—
257C4—534no—
258C4—1064no—
259C4—2122no—
260C4—2121no—
261C2—4242no—
262C2#2028995611yescharacteristic
263C2#2029000812yes—
264C2#2029008912yes—
265C2D53⋊#32911yescharacteristic
266C2#2029021911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7567
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #20290429C2semidirect
2. #20289536C2semidirect
23. #13133205C8semidirect
150. C53#7567semidirect

Maximal subgroups

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

subgroupindex
#202904292
#202895362
#202895322
#202772542
#756753