Finite Group Databasebeta

← #28299731

Subgroups — #28299731

Order 13568.

624 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 122 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#20290632C212yesmaximal, complemented
2#20289532C212yesmaximal, complemented
3#20289531C212yesmaximal, complemented
4C4×#13133213C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133213C411yescharacteristic
9C2×C4×D212C411yescharacteristic
10C2×C4×C22⋊D53—22no—
11#16810526C2211yescharacteristic
12#16810524C2211yescharacteristic
13#16810523—22no—
14#16810492C411yescharacteristic
15C42×Dic53C2211yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17C22×#88324—24no—
18C4⋊#88324—44no—
19C22×D212C2×C411yescharacteristic
20C22×C22⋊D53—22no—
21C22×C4×D53—21no—
22#13133213—22no—
23#13133213—22no—
24C4×D212—42no—
25C4×C22⋊D53—44no—
26C4×C22⋊D53—44no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30C2×#11719921—21no—
31C2×#11719921—22no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C2×C4×Dic53D412yes—
35C2×C4×Dic53C2311yescharacteristic
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53—21no—
38C4×D4×C53—42no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C24×C53—21no—
43C23×D53—21no—
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53M1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—42no—
50C22×C4×C53—44no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—28no—
55C2×C22⋊D53—44no—
56C2×C22⋊D53—44no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×D212—21no—
61C2×D212—41no—
62C2×D212—41no—
63C2×D212M1612yes—
64C2×C4×D53—42no—
65C2×C4×D53—44no—
66C2×#88324—44no—
67C2×#88324—44no—
68C2×#88324—44no—
69C4⋊C4×C53—41no—
70C22⋊C4×C53—42no—
71C42×C53—22no—
72C22⋊Dic53—42no—
73C22⋊Dic53—42no—
74#11719921—42no—
75#11719921—42no—
76#11719921—42no—
77C4⋊Dic53—41no—
78#11719919—44no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—42no—
82C4×Dic53—41no—
83C4×Dic53D814yes—
84C4×Dic53SD1614yes—
85C23×C53—22no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—22no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—42no—
95C22×D53—42no—
96D4×C53—42no—
97D4×C53—42no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—41no—
103C2×C4×C53—42no—
104C2×C4×C53—84no—
105C2×C4×C53#10012yes—
106C2×C4×C53#10214yes—
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C22⋊D53—416no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53#10214yes—
117C2×Dic53C2×D812yes—
118C2×Dic53C2×SD1612yes—
119D212—42no—
120D212—42no—
121C4×D53—84no—
122#88324—84no—
123#7651—531nomaximal, Sylow
124C22×C53—42no—
125C22×C53—44no—
126C22×C53—81no—
127C22×C53—81no—
128C22×C53—82no—
129C22×C53#40711yescharacteristic
130C22×C53#41011yescharacteristic
131C22×C53C2×#10011yescharacteristic
132C22×C53C2×#10212yes—
133C22×C53#42112yes—
134C2×D53—42no—
135C2×D53—44no—
136C2×D53—81no—
137C2×D53—81no—
138C2×D53—82no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—81no—
143C4×C53#33114yes—
144Dic53—24no—
145Dic53—22no—
146Dic53—22no—
147Dic53—81no—
148C4×#127—531no—
149#1831—532no—
150#1270—532no—
151#1270—532no—
152C2×C53—82no—
153C2×C53#146611yescharacteristic
154C2×C53#148612yes—
155C2×C53#151712yes—
156C2×C53#153611yescharacteristic
157C2×C53C2×#33111yescharacteristic
158D53—82no—
159C2×#127—531no—
160C2×C4×D4—531no—
161C2×C4×D4—1062no—
162C2×C4×D4#615711yescharacteristic
163C2×C4⋊C8—1062no—
164#390—531no—
165#390—531no—
166#390—1062no—
167#384—531no—
168C43—531no—
169#336—534no—
170#336—1062no—
171C53#765111yescharacteristic, complemented, Sylow
172C22×D4—531no—
173C22×D4—1062no—
174C22×D4C2×#615711yescharacteristic
175C23×C4—21no—
176C23×C4—1061no—
177C22×C8—1064no—
178#127—1062no—
179#127—1062no—
180C4×D4—42no—
181C4×D4—2122no—
182C4×D4—2124no—
183C4×D4—2124no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1062no—
186C2×C4⋊C4C2×#615711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1061no—
191C2×C42—532no—
192C2×C42—531no—
193C2×C42—532no—
194C2×C42—531no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—2124no—
197C24—21no—
198C24—1061no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—532no—
203C2×D4—1068no—
204C2×D4—1061no—
205C2×D4—2121no—
206C2×D4—2121no—
207C2×D4—2124no—
208C2×D4—2124no—
209C2×D4#1171993612yes—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—532no—
214C22×C4—532no—
215C22×C4—1061no—
216C22×C4—2122no—
217C22×C4—2124no—
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22×#615711yescharacteristic
221C2×C8—2124no—
222C2×C8—2124no—
223C2×C8—2124no—
224C4⋊C4—41no—
225C4⋊C4—2121no—
226C4⋊C4—2124no—
227C22⋊C4—42no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C22⋊C4—2122no—
232C22⋊C4—2122no—
233C42—22no—
234C42—534no—
235C42—534no—
236C42—1061no—
237C42—1061no—
238C42—2121no—
239C42—2122no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—1062no—
246C23—2121no—
247C23—2121no—
248C23—2122no—
249C23—2122no—
250C23C2×C22⋊#615711yescharacteristic
251D4—42no—
252D4—42no—
253D4—2122no—
254D4—2122no—
255D4—21216no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—534no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—4244no—
271C2×C4—2121no—
272C2×C4—2122no—
273C2×C4#1313327712yes—
274C2×C4#1313327914yes—
275C2×C4D53⋊M1611yescharacteristic
276C2×C4C2×#1171993611yescharacteristic
277C8—4244no—
278C22—42no—
279C22—44no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—4242no—
284C22—4241no—
285C22—4241no—
286C22—2122no—
287C22—2124no—
288C22#1681159411yescharacteristic
289C22#1681160211yescharacteristic
290C22C2×#1313327711yescharacteristic
291C22C2×#1313327912yes—
292C22#1681162012yes—
293C4—21no—
294C4—21no—
295C4—41no—
296C4—81no—
297C4—1062no—
298C4—1062no—
299C4—1064no—
300C4—4241no—
301C4#1681148114yes—
302C2—82no—
303C2—4242no—
304C2#2028997411yescharacteristic
305C2#2029004112yes—
306C2#2029014512yes—
307C2#2029020311yescharacteristic
308C2C2×#1681148111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7651
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290632C2semidirect
2. #20289532C2semidirect
3. #20289531C2semidirect
171. C53#7651semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202895322
#202895312
C4×#131332132
#765153