Finite Group Databasebeta

← #27299601

Subgroups — #27299601

Order 13568.

1018 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16810422C214yesmaximal, complemented
2#20274639C212yesmaximal
3C22×C4⋊Q8×C53C211yesmaximal, characteristic
4C2×#13133136—22no—
5C2×#13133136C2214yes—
6#16810587—22no—
7#16810422—28no—
8C23×Q8×C53C411yescharacteristic
9C23×Q8×C53C2211yescharacteristic
10C2×C4⋊Q8×C53—22no—
11C2×C4⋊Q8×C53—28no—
12C2×C4⋊Q8×C53C412yes—
13C2×C4⋊Q8×C53C2212yes—
14C22×C4⋊C4×C53—22no—
15C22×C42×C53C411yescharacteristic
16C2×#11719917—24no—
17#13133136—28no—
18#13133136—44no—
19C22⋊#88322—44no—
20C22×Q8×C53—24no—
21C22×Q8×C53—24no—
22C22×Q8×C53—21no—
23C22×Q8×C53—42no—
24C22×Q8×C53—42no—
25C22×Q8×C53—42no—
26C22×Q8×C53—42no—
27C22×Q8×C53C2×C412yes—
28C22×Q8×C53C2×C411yescharacteristic
29C22×Q8×C53D412yes—
30C22×Q8×C53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C23×C4×C53D412yes—
35C4⋊Q8×C53—216no—
36C4⋊Q8×C53—24no—
37C4⋊Q8×C53—24no—
38C4⋊Q8×C53—24no—
39C4⋊Q8×C53D418yes—
40C2×C4⋊C4×C53—216no—
41C2×C4⋊C4×C53—24no—
42C2×C4⋊C4×C53—24no—
43C2×C42×C53—24no—
44C2×C42×C53—21no—
45C2×C42×C53C2×C412yes—
46C24×C53C22⋊C411yescharacteristic
47C2×Q8×C53—24no—
48C2×Q8×C53—24no—
49C2×Q8×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—24no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—42no—
55C2×Q8×C53—41no—
56C2×Q8×C53—41no—
57C2×Q8×C53—416no—
58C2×Q8×C53—416no—
59C2×Q8×C53—42no—
60C2×Q8×C53—42no—
61C2×Q8×C53—42no—
62C2×Q8×C53—42no—
63C2×Q8×C53C22⋊C414yes—
64C2×Q8×C53C2×D414yes—
65C22×C4×C53—24no—
66C22×C4×C53—28no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—24no—
72C22×C4×C53—24no—
73C22×C4×C53—22no—
74C22×C4×C53—24no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53C22⋊C412yes—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D412yes—
82C4⋊C4×C53—216no—
83C4⋊C4×C53—216no—
84C42×C53—24no—
85C42×C53—22no—
86C42×C53C22⋊C414yes—
87#11719917—44no—
88#11719917—44no—
89C2×#88322—44no—
90C23×C53—24no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53#9912yes—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—44no—
96Q8×C53—48no—
97Q8×C53—48no—
98Q8×C53—44no—
99Q8×C53—44no—
100Q8×C53—44no—
101C2×C4×C53—216no—
102C2×C4×C53—28no—
103C2×C4×C53—28no—
104C2×C4×C53—24no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—48no—
116C2×C4×C53—48no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53#9912yes—
120C2×C4×C53C2×C22⋊C412yes—
121C2×C4×C53#12014yes—
122#88322—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—28no—
134C4×C53—24no—
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—44no—
139C22×C4⋊Q8D5311yescharacteristic
140C2×#356—534no—
141#1784—532no—
142C2×C53—21no—
143C2×C53—24no—
144C2×C53—21no—
145C2×C53#178111yescharacteristic
146C2×C53C2×#35612yes—
147C23×Q8Dic5311yescharacteristic
148C23×Q8C2×D5311yescharacteristic
149C2×C4⋊Q8—22no—
150C2×C4⋊Q8—28no—
151C2×C4⋊Q8Dic5312yes—
152C2×C4⋊Q8C2×D5312yes—
153C22×C4⋊C4—22no—
154C22×C42Dic5311yescharacteristic
155C2×#101—534no—
156C2×#101—1062no—
157#407—1062no—
158#356—1068no—
159C53#1270011yescharacteristic, complemented, Sylow
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8C2×Dic5312yes—
168C22×Q8C2×Dic5311yescharacteristic
169C22×Q8C22⋊D5312yes—
170C22×Q8C22×D5311yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×Dic5311yescharacteristic
174C23×C4C22⋊D5312yes—
175C2×M16—1064no—
176C4⋊Q8—216no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8C22⋊D5318yes—
181C2×C4⋊C4—216no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42C2×Dic5312yes—
187#101—1068no—
188#101—2124no—
189C22⋊C8—2124no—
190C24C22⋊Dic5311yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊Dic5314yes—
208C2×Q8C2×C22⋊D5314yes—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22⋊Dic5312yes—
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22×Dic5311yescharacteristic
225C22×C4C2×C22⋊D5312yes—
226M16—2124no—
227M16—2124no—
228C2×C8—2124no—
229C4⋊C4—216no—
230C4⋊C4—216no—
231C42—24no—
232C42—22no—
233C42C22⋊Dic5314yes—
234C23—21no—
235C23—24no—
236C23—21no—
237C23#1313313212yes—
238C23C2×C22⋊Dic5311yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—48no—
243Q8—44no—
244Q8—44no—
245C2×C4—216no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4#1313313212yes—
264C2×C4C2×C22⋊Dic5312yes—
265C2×C4#1313318614yes—
266C8—4244no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#1681042214yescomplemented
273C22#1681047911yescharacteristic
274C22C2×#1313313211yescharacteristic
275C22C2×#1313313211yescharacteristic
276C4—28no—
277C4—24no—
278C4—22no—
279C4—22no—
280C4—42no—
281C4—44no—
282C2—21no—
283C2—21no—
284C2—24no—
285C2#2027463011yescharacteristic
286C2C2×#1681042212yes—
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC22×C4⋊Q8×C53C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC22×C4⋊Q8D53
hypocenterC53#12700
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810422C2semidirect
129. C22×C53#356semidirect
159. C53#12700semidirect
272. C22#16810422semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202746392
C22×C4⋊Q8×C532
#1270053