Finite Group Databasebeta

← #28310915

Subgroups — #28310915

Order 13568.

1018 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×#16811533C214yesmaximal, complemented
2#20290549C212yesmaximal
3C22×#13133215C211yesmaximal, characteristic
4C2×#13133278—22no—
5C2×#13133278C2214yes—
6#16811593—22no—
7#16811533—28no—
8C23×Dic106C2211yescharacteristic
9C2×#13133215—22no—
10C2×#13133215—28no—
11C2×#13133215C412yes—
12C2×#13133215C2212yes—
13C22×#11719919—22no—
14C22×C4×Dic53C411yescharacteristic
15C23×Q8×C53C411yescharacteristic
16C2×#11719940—24no—
17#13133278—28no—
18#13133278—44no—
19C22⋊#88324—44no—
20C22×Dic106—24no—
21C22×Dic106—42no—
22C22×Dic106—42no—
23C22×Dic106C2×C411yescharacteristic
24C22×Dic106D412yes—
25C22×Dic106C2311yescharacteristic
26C23×Dic53—21no—
27C23×Dic53D412yes—
28#13133215—216no—
29#13133215—24no—
30#13133215—24no—
31#13133215—24no—
32#13133215D418yes—
33C2×#11719919—216no—
34C2×#11719919—24no—
35C2×#11719919—24no—
36C2×C4×Dic53—24no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C412yes—
39C22×Q8×C53—24no—
40C22×Q8×C53—21no—
41C22×Q8×C53—42no—
42C22×Q8×C53—42no—
43C22×Q8×C53C2×C412yes—
44C23×C4×C53—21no—
45C23×C4×C53C2×C411yescharacteristic
46C24×C53C22⋊C411yescharacteristic
47C2×Q8×C53—24no—
48C2×Q8×C53—22no—
49C2×Q8×C53—416no—
50C2×Q8×C53—42no—
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53C2×Q8×C53—42no—
54C2×Q8×C53C22⋊C414yes—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—24no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—24no—
65C22×Dic53—28no—
66C22×Dic53—22no—
67C22×Dic53—24no—
68C22×Dic53—24no—
69C22×Dic53—42no—
70C22×Dic53C22⋊C412yes—
71C22×Dic53C2×D412yes—
72C2×Dic106—24no—
73C2×Dic106—22no—
74C2×Dic106—24no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—42no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106—416no—
81C2×Dic106C2×D414yes—
82#11719940—44no—
83#11719940—44no—
84C2×#88324—44no—
85#11719919—216no—
86#11719919—216no—
87C4×Dic53—24no—
88C4×Dic53—22no—
89C4×Dic53C22⋊C414yes—
90C23×C53—24no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53#9912yes—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—48no—
96Q8×C53—44no—
97Q8×C53—44no—
98C2×C4×C53—28no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—24no—
104C2×C4×C53—48no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53#9912yes—
108C2×C4×C53C2×C22⋊C412yes—
109C2×Dic53—216no—
110C2×Dic53—28no—
111C2×Dic53—24no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—48no—
118C2×Dic53#12014yes—
119Dic106—44no—
120Dic106—48no—
121Dic106—44no—
122#88324—84no—
123#12700—531nomaximal, Sylow
124C22×C53—24no—
125C22×C53—24no—
126C22×C53—22no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53#35614yescomplemented
130C22×C53#37911yescharacteristic
131C22×C53C2×#9911yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—44no—
136Dic53—28no—
137Dic53—24no—
138Dic53—42no—
139C22×C4⋊Q8—531no—
140C2×#356—534no—
141#1784—532no—
142C2×C53—21no—
143C2×C53—24no—
144C2×C53—21no—
145C2×C53#178111yescharacteristic
146C2×C53C2×#35612yes—
147C23×Q8—531no—
148C23×Q8#615711yescharacteristic
149C2×C4⋊Q8—532no—
150C2×C4⋊Q8—532no—
151C2×C4⋊Q8—1068no—
152C2×C4⋊Q8—1062no—
153C22×C4⋊C4—1062no—
154C22×C42—531no—
155C2×#101—534no—
156C2×#101—1062no—
157#407—1062no—
158#356—1068no—
159C53#1270011yescharacteristic, complemented, Sylow
160C22×Q8—24no—
161C22×Q8—21no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—532no—
165C22×Q8—531no—
166C22×Q8—531no—
167C22×Q8—1064no—
168C22×Q8—2122no—
169C22×Q8—2122no—
170C22×Q8C2×#615712yes—
171C23×C4—21no—
172C23×C4—532no—
173C23×C4—1061no—
174C23×C4C2×#615711yescharacteristic
175C2×M16—1064no—
176C4⋊Q8—538no—
177C4⋊Q8—1064no—
178C4⋊Q8—1064no—
179C4⋊Q8—10616no—
180C4⋊Q8—1064no—
181C2×C4⋊C4—1064no—
182C2×C4⋊C4—1064no—
183C2×C4⋊C4—10616no—
184C2×C42—532no—
185C2×C42—1064no—
186C2×C42—1061no—
187#101—1068no—
188#101—2124no—
189C22⋊C8—2124no—
190C24C22⋊#615711yescharacteristic
191C2×Q8—24no—
192C2×Q8—22no—
193C2×Q8—416no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—534no—
199C2×Q8—1064no—
200C2×Q8—1064no—
201C2×Q8—1062no—
202C2×Q8—2121no—
203C2×Q8—2121no—
204C2×Q8—2122no—
205C2×Q8—2121no—
206C2×Q8—2121no—
207C2×Q8—21216no—
208C2×Q8C22⋊#615714yes—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—532no—
217C22×C4—532no—
218C22×C4—1064no—
219C22×C4—1064no—
220C22×C4—1068no—
221C22×C4—1062no—
222C22×C4—1064no—
223C22×C4—2122no—
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226M16—2124no—
227M16—2124no—
228C2×C8—2124no—
229C4⋊C4—10616no—
230C4⋊C4—10616no—
231C42—534no—
232C42—1064no—
233C42—1062no—
234C23—24no—
235C23—21no—
236C23—21no—
237C23#1313327512yes—
238C23C2×C22⋊#615711yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—2124no—
243Q8—2128no—
244Q8—2124no—
245C2×C4—28no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—534no—
255C2×C4—10616no—
256C2×C4—1068no—
257C2×C4—1064no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—2128no—
264C2×C4#1313327512yes—
265C2×C4C2×C22⋊#615712yes—
266C8—4244no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#1681153314yescomplemented
273C22#1681156411yescharacteristic
274C22C2×#1313327511yescharacteristic
275C22C2×#1313327511yescharacteristic
276C4—22no—
277C4—22no—
278C4—44no—
279C4—1068no—
280C4—1064no—
281C4—2122no—
282C2—21no—
283C2—24no—
284C2—21no—
285C2#2029054111yescharacteristic
286C2C2×#1681153312yes—
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC23×Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Dic106C23
hypercenterC23×Q8#6157
hypocenterC53#12700
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16811533C2semidirect
129. C22×C53#356semidirect
159. C53#12700semidirect
272. C22#16811533semidirect

Maximal subgroups

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

subgroupindex
C2×#168115332
#202905492
C22×#131332152
#1270053