Finite Group Databasebeta

← #28299679

Subgroups — #28299679

Order 13568.

520 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 98 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#20290500C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289532C212yesmaximal
4#20277254C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×#13133272—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—
18#13133284—28no—
19#13133272—44no—
20C22×Dic106C2×C411yescharacteristic
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23#13133205—22no—
24#13133205—22no—
25#13133182—42no—
26#13133179—42no—
27C2×#11719919—21no—
28C2×#11719921—22no—
29C2×#11719921—21no—
30C2×#11719921—22no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53D412yes—
35C2×C4×Dic53C2311yescharacteristic
36#13133124—22no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40#95×C53—22no—
41C23×D53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53C22⋊C412yes—
45C22×C4×C53C22×C411yescharacteristic
46C22×Dic53—21no—
47C22×Dic53C22⋊C411yescharacteristic
48C22×Dic53C22⋊C411yescharacteristic
49C22×Dic53C2×D412yes—
50C2×D212—21no—
51C2×D212—41no—
52C2×D212—41no—
53C2×D212M1612yes—
54C2×C4×D53—42no—
55C2×C4×D53—44no—
56C2×Dic106—21no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106M1612yes—
60C2×#88324—28no—
61C2×#88324—44no—
62C2×#88324—44no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C42×C53—21no—
66C42×C53—21no—
67#11719921—28no—
68#11719921—42no—
69#11719921—42no—
70#11719921—42no—
71#11719919—42no—
72C4×Dic53—22no—
73C4×Dic53—24no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—41no—
77C4×Dic53—42no—
78C23×C53C2×C22⋊C411yescharacteristic
79C22×D53—22no—
80C22×D53—41no—
81C22×D53—41no—
82C22×D53—42no—
83C22×D53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53#10012yes—
103C2×Dic53#10414yes—
104C2×Dic53AGL(1,ℤ/8)12yes—
105C2×Dic53#13712yes—
106D212—42no—
107D212—42no—
108C4×D53—84no—
109Dic106—42no—
110Dic106—42no—
111#88324—48no—
112#7635—531nomaximal, Sylow
113C22×C53#40711yescharacteristic
114C22×C53#41011yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C22×C53#41712yes—
117C22×C53C2×#10412yes—
118C2×D53—42no—
119C2×D53—44no—
120C2×D53—81no—
121C2×D53—82no—
122C2×D53—81no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—81no—
127C4×C53—81no—
128Dic53—22no—
129Dic53—22no—
130Dic53—41no—
131Dic53—42no—
132Dic53#32914yes—
133C4×#124—531no—
134#1767—532no—
135#1272—532no—
136#1270—532no—
137C2×C53#146611yescharacteristic
138C2×C53#148912yes—
139C2×C53#150912yes—
140C2×C53C2×#32911yescharacteristic
141C2×C53#156011yescharacteristic
142D53—82no—
143C2×#124—531no—
144C2×C4×Q8—531no—
145C2×C4×D4—531no—
146C2×C4⋊C8—1062no—
147C2×#97—1062no—
148#391—531no—
149#390—531no—
150#388—1062no—
151#382#615711yescharacteristic
152C4×C22⋊C4—1062no—
153C43—531no—
154#336—534no—
155C53#763511yescharacteristic, complemented, Sylow
156C22×Q8—531no—
157C22×D4—531no—
158C23×C4—1061no—
159C22×C8—1064no—
160#124—1062no—
161#124—1062no—
162C4×Q8—2122no—
163C4×D4—2122no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4C2×#615711yescharacteristic
166C2×C4⋊C4C2×#615711yescharacteristic
167C2×C22⋊C4—1062no—
168C2×C22⋊C4—1061no—
169C2×C22⋊C4—1062no—
170C2×C42—532no—
171C2×C42—532no—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42—1061no—
175C2×C42C2×#615711yescharacteristic
176C4⋊C8—1068no—
177#97—2124no—
178#95—22no—
179#95—1062no—
180C24—1061no—
181C2×Q8—532no—
182C2×Q8—1061no—
183C2×Q8—2121no—
184C2×Q8—2121no—
185C2×D4—532no—
186C2×D4—1061no—
187C2×D4—2121no—
188C2×D4—2121no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—531no—
192C22×C4—532no—
193C22×C4—531no—
194C22×C4—1061no—
195C22×C4—2122no—
196C22×C4—2124no—
197C22×C4C22⋊#615712yes—
198C22×C4C22×#615711yescharacteristic
199C2×C8—1068no—
200C2×C8—2124no—
201C2×C8—2124no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—2122no—
205C22⋊C4—1068no—
206C22⋊C4—2122no—
207C22⋊C4—2122no—
208C22⋊C4—2122no—
209C42—21no—
210C42—21no—
211C42—1062no—
212C42—1064no—
213C42—1062no—
214C42—1062no—
215C42—2121no—
216C42—2122no—
217C23—1062no—
218C23—2121no—
219C23—2121no—
220C23—2122no—
221C23—2122no—
222C23C2×C22⋊#615711yescharacteristic
223Q8—2122no—
224Q8—2122no—
225D4—2122no—
226D4—2122no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—532no—
237C2×C4—534no—
238C2×C4—532no—
239C2×C4—532no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1061no—
244C2×C4—1062no—
245C2×C4—4244no—
246C2×C4—2121no—
247C2×C4—2121no—
248C2×C4D53⋊M1611yescharacteristic
249C2×C4D53⋊M1611yescharacteristic
250C8—2128no—
251C22—4242no—
252C22—4241no—
253C22—4241no—
254C22—2124no—
255C22—2122no—
256C22#1681159511yescharacteristic
257C22#1681160411yescharacteristic
258C22D53⋊#10011yescharacteristic
259C22#1681161412yes—
260C22D53⋊#10412yes—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—81no—
265C4—81no—
266C4—534no—
267C4—1062no—
268C4—1062no—
269C4—2122no—
270C4—2121no—
271C2—4242no—
272C2#2028997611yescharacteristic
273C2#2029005212yes—
274C2#2029011812yes—
275C2D53⋊#32911yescharacteristic
276C2#2029025311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7635
LCSG ▷ C22×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. #20290500C2semidirect
2. #20289536C2semidirect
155. C53#7635semidirect

Maximal subgroups

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

subgroupindex
#202905002
#202895362
#202895322
#202772542
#763553