Finite Group Databasebeta

← #28299692

Subgroups — #28299692

Order 13568.

862 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 128 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#20290571C212yesmaximal, complemented
2#20289531C214yesmaximal, complemented
3C2×#16810526C211yesmaximal, characteristic
4C2×#13133284—22no—
5C2×C22⋊#88324—22no—
6#16811489C2214yescomplemented
7C22×C4⋊Dic53C411yescharacteristic
8C22×C22⋊Dic53—22no—
9C22×C4×Dic53C2211yescharacteristic
10#16810526—22no—
11#16810526C412yes—
12#16810526C2212yes—
13C23×D4×C53C411yescharacteristic
14C22×#88324—24no—
15#13133284—44no—
16C22⋊#88324—44no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—42no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53—24no—
24C2×C22⋊Dic53—44no—
25C2×C22⋊Dic53—44no—
26C2×C4×Dic53—42no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53C2311yescharacteristic
30C25×C53—21no—
31C22×D4×C53—24no—
32C22×D4×C53—21no—
33C22×D4×C53—42no—
34C22×D4×C53—42no—
35C22×D4×C53C2×C412yes—
36C23×C4×C53C2×C411yescharacteristic
37C24×C53—24no—
38C24×C53—22no—
39C24×C53—42no—
40C24×C53—44no—
41C24×C53—42no—
42C24×C53—44no—
43C24×C53C22⋊C411yescharacteristic
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—416no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53M1614yes—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—42no—
59C22×Dic53—42no—
60C22×Dic53—42no—
61C22×Dic53—44no—
62C22×Dic53C22⋊C412yes—
63C22×Dic53D814yes—
64C22×Dic53SD1614yes—
65C22×Dic53C2×D412yes—
66C2×#88324—44no—
67C2×#88324—44no—
68C2×#88324—44no—
69C22⋊Dic53—48no—
70C22⋊Dic53—44no—
71C22⋊Dic53—44no—
72C4⋊Dic53—42no—
73C4⋊Dic53—42no—
74C4×Dic53—24no—
75C4×Dic53—41no—
76C4×Dic53—41no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—24no—
81C23×C53—48no—
82C23×C53—416no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—44no—
86C23×C53—44no—
87C23×C53—88no—
88C23×C53—84no—
89C23×C53—84no—
90C23×C53#10012yes—
91C23×C53#10214yes—
92C23×C53C2×C22⋊C411yescharacteristic
93D4×C53—44no—
94D4×C53—48no—
95D4×C53—44no—
96C2×C4×C53—28no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53#9912yes—
102C2×C4×C53C2×M1612yes—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—44no—
108C2×Dic53—44no—
109C2×Dic53—42no—
110C2×Dic53—44no—
111C2×Dic53—84no—
112C2×Dic53#10214yes—
113C2×Dic53C2×D812yes—
114C2×Dic53C2×SD1612yes—
115#88324—84no—
116#7640—531nomaximal, Sylow
117C22×C53—22no—
118C22×C53—24no—
119C22×C53—24no—
120C22×C53—22no—
121C22×C53—44no—
122C22×C53—48no—
123C22×C53—816no—
124C22×C53—82no—
125C22×C53—84no—
126C22×C53—82no—
127C22×C53#33114yes—
128C22×C53#40711yescharacteristic
129C22×C53C2×#9911yescharacteristic
130C22×C53C2×#10011yescharacteristic
131C22×C53C2×#10212yes—
132C22×C53#42112yes—
133C4×C53—22no—
134C4×C53—22no—
135Dic53—22no—
136Dic53—24no—
137Dic53—22no—
138Dic53—82no—
139C2×#390—531no—
140#1797—532no—
141#1270—534no—
142C2×C53—24no—
143C2×C53—84no—
144C2×C53#146211yescharacteristic
145C2×C53#148612yes—
146C2×C53#151912yes—
147C2×C53#153311yescharacteristic
148C2×C53C2×#33111yescharacteristic
149C23×D4#615711yescharacteristic
150C22×C4⋊C4—531no—
151C22×C22⋊C4—1062no—
152C22×C42—531no—
153C2×C4⋊C8—1062no—
154C2×C22⋊C8—1062no—
155#390—532no—
156#390—532no—
157#390—1062no—
158#336—534no—
159C53#764011yescharacteristic, complemented, Sylow
160C25—21no—
161C22×D4—24no—
162C22×D4—21no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4C2×#615712yes—
166C23×C4—532no—
167C23×C4—1061no—
168C23×C4C2×#615711yescharacteristic
169C22×C8—1064no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1061no—
172C2×C4⋊C4—2122no—
173C2×C22⋊C4—1064no—
174C2×C22⋊C4—1064no—
175C2×C22⋊C4—2124no—
176C2×C22⋊C4—2124no—
177C2×C42—532no—
178C2×C42—531no—
179C2×C42—531no—
180C2×C42—2122no—
181C4⋊C8—2124no—
182C22⋊C8—2124no—
183C24—24no—
184C24—22no—
185C24—42no—
186C24—44no—
187C24—42no—
188C24—44no—
189C24C22⋊#615711yescharacteristic
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—416no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4#1171994014yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—534no—
204C22×C4—534no—
205C22×C4—1062no—
206C22×C4—2122no—
207C22×C4—2122no—
208C22×C4—2122no—
209C22×C4—2124no—
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212C2×C8—2124no—
213C2×C8—2124no—
214C2×C8—2124no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C22⋊C4—2128no—
218C22⋊C4—2124no—
219C22⋊C4—2124no—
220C42—1064no—
221C42—2121no—
222C42—2121no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—24no—
227C23—48no—
228C23—416no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—44no—
233C23—88no—
234C23—84no—
235C23—84no—
236C23#1313327712yes—
237C23#1313327914yes—
238C23C2×C22⋊#615711yescharacteristic
239D4—44no—
240D4—48no—
241D4—44no—
242C2×C4—28no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—534no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—4244no—
255C2×C4—2124no—
256C2×C4—2124no—
257C2×C4—2122no—
258C2×C4—2124no—
259C2×C4#1313327512yes—
260C2×C4C2×#1171994012yes—
261C8—4244no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—44no—
267C22—48no—
268C22—816no—
269C22—84no—
270C22—82no—
271C22—82no—
272C22#1681148114yes—
273C22#1681159311yescharacteristic
274C22C2×#1313327511yescharacteristic
275C22C2×#1313327711yescharacteristic
276C22C2×#1313327912yes—
277C22#1681162112yes—
278C4—22no—
279C4—22no—
280C4—1062no—
281C4—1064no—
282C4—1062no—
283C4—4242no—
284C2—24no—
285C2—84no—
286C2#2028996011yescharacteristic
287C2#2029004012yes—
288C2#2029015112yes—
289C2#2029019411yescharacteristic
290C2C2×#1681148111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC23×D4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC23×D4#6157
hypocenterC53#7640
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20290571C2semidirect
2. #20289531C2semidirect
6. #16811489C22semidirect
159. C53#7640semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895312
C2×#168105262
#764053