Finite Group Databasebeta

← #28299709

Subgroups — #28299709

Order 13568.

630 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290606C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289531C212yesmaximal, complemented
4#20277206C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133176C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10#16810530C411yescharacteristic
11#16810530C2211yescharacteristic
12#16810526C411yescharacteristic
13#16810526C2211yescharacteristic
14#16810503—22no—
15C4×C22⋊Dic53—22no—
16C22×#41×C53C411yescharacteristic
17C22×#88324—24no—
18C22⋊#88324—44no—
19C23×Dic53D412yes—
20#13133176—42no—
21C2×C4⋊Dic53C2×C411yescharacteristic
22C2×C4⋊Dic53C2×C411yescharacteristic
23C2×C22⋊Dic53—21no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53—42no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31#13133124—22no—
32#13133124—22no—
33C2×#41×C53—22no—
34C2×#41×C53—22no—
35C2×#41×C53—42no—
36C2×#41×C53—42no—
37C22×Q8×C53C2×C411yescharacteristic
38C22×D4×C53—21no—
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53—21no—
41C23×C4×C53C2×C411yescharacteristic
42C24×C53—21no—
43C24×C53C22⋊C411yescharacteristic
44#41×C53—48no—
45#41×C53—48no—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53M1612yes—
50C2×D4×C53—24no—
51C2×D4×C53—21no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53M1612yes—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—24no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×Dic53—24no—
68C22×Dic53—24no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—42no—
72C22×Dic53C22⋊C412yes—
73C22×Dic53C2×D412yes—
74C2×#88324—44no—
75C2×#88324—44no—
76C2×#88324—44no—
77C22⋊Dic53—44no—
78C22⋊Dic53—42no—
79C22⋊Dic53—44no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4×Dic53—41no—
83C4×Dic53—42no—
84C4×Dic53—41no—
85C4×Dic53—42no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53#9912yes—
96C23×C53C2×C22⋊C411yescharacteristic
97Q8×C53—42no—
98Q8×C53—42no—
99D4×C53—42no—
100D4×C53—42no—
101D4×C53—48no—
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—48no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53#10012yes—
113C2×C4×C53C2×M1611yescharacteristic
114C2×C4×C53C2×M1611yescharacteristic
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—48no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53#13514yes—
126#88324—84no—
127#7645—531nomaximal, Sylow
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—44no—
135C22×C53—81no—
136C22×C53—81no—
137C22×C53—82no—
138C22×C53#40711yescharacteristic
139C22×C53C2×#9911yescharacteristic
140C22×C53C2×#10011yescharacteristic
141C22×C53#41612yes—
142C22×C53#42112yes—
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147Dic53—24no—
148Dic53—42no—
149Dic53—81no—
150Dic53—81no—
151#2297—531no—
152#1817—532no—
153#1272—532no—
154#1270—532no—
155C2×C53—42no—
156C2×C53—82no—
157C2×C53#146211yescharacteristic
158C2×C53#148812yes—
159C2×C53#151612yes—
160C2×C53#153611yescharacteristic
161C2×C53#154311yescharacteristic
162C22×#41#615711yescharacteristic
163C2×#117—531no—
164C22×C42—531no—
165C2×C22⋊C8—1062no—
166#391—531no—
167#391—531no—
168#390—531no—
169#390—531no—
170#386—1062no—
171C4×C22⋊C4—1062no—
172#336—534no—
173#336—1062no—
174C53#764511yescharacteristic, complemented, Sylow
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8C2×#615711yescharacteristic
180C22×D4—21no—
181C22×D4C2×#615711yescharacteristic
182C23×C4—21no—
183C23×C4—532no—
184C23×C4C2×#615711yescharacteristic
185C22×C8—1064no—
186#117—2122no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—531no—
189C2×C22⋊C4—1062no—
190C2×C22⋊C4—1062no—
191C2×C22⋊C4—1061no—
192C2×C42—532no—
193C2×C42—531no—
194C2×C42—531no—
195C2×C42—1061no—
196C2×C42—2122no—
197C22⋊C8—2124no—
198#95—1062no—
199#95—1062no—
200C24—21no—
201C24C22⋊#615711yescharacteristic
202#41—48no—
203#41—48no—
204C2×Q8—21no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8#1171994012yes—
208C2×D4—24no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4#1171994012yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—532no—
224C22×C4—532no—
225C22×C4—1064no—
226C22×C4—1064no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—2122no—
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232C2×C8—2124no—
233C2×C8—2124no—
234C2×C8—2124no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C22⋊C4—2124no—
238C22⋊C4—2124no—
239C22⋊C4—2122no—
240C42—2122no—
241C42—2121no—
242C42—2121no—
243C42—2122no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23#1313327512yes—
254C23C2×C22⋊#615711yescharacteristic
255Q8—42no—
256Q8—42no—
257D4—42no—
258D4—42no—
259D4—48no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—48no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—534no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—2128no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313327712yes—
282C2×C4C2×#1171994011yescharacteristic
283C2×C4C2×#1171994011yescharacteristic
284C8—4244no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22#1681159311yescharacteristic
296C22C2×#1313327511yescharacteristic
297C22C2×#1313327711yescharacteristic
298C22#1681161312yes—
299C22#1681162012yes—
300C4—22no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—1064no—
305C4—4241no—
306C4—4241no—
307C4—2122no—
308C2—42no—
309C2—82no—
310C2#2028996011yescharacteristic
311C2#2029004612yes—
312C2#2029014112yes—
313C2#2029020311yescharacteristic
314C2#2029021811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC22×#41×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×#41#6157
hypocenterC53#7645
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #20290606C2semidirect
2. #20289535C2semidirect
3. #20289531C2semidirect
174. C53#7645semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202895352
#202895312
#202772062
#764553