Finite Group Databasebeta

← #27275452

Subgroups — #27275452

Order 13568.

1174 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 179 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#20274625C212yesmaximal
2C2×#16810349C214yesmaximal, complemented
3C22×C22⋊Q8×C53C211yesmaximal, characteristic
4C2×C22⋊#88322—24no—
5C2×C22⋊#88322C2214yes—
6#16810349—28no—
7C23×Q8×C53C411yescharacteristic
8C24×C4×C53C2211yescharacteristic
9C2×C22⋊Q8×C53—24no—
10C2×C22⋊Q8×C53—28no—
11C22×C4⋊C4×C53C411yescharacteristic
12C22×C4⋊C4×C53C2212yes—
13C22×C22⋊C4×C53C412yes—
14C22×#88322—24no—
15C22⋊#88322—28no—
16C22⋊#88322—48no—
17C25×C53C2×C411yescharacteristic
18C22×Q8×C53—24no—
19C22×Q8×C53—22no—
20C22×Q8×C53—44no—
21C23×C4×C53—24no—
22C23×C4×C53—44no—
23C23×C4×C53C2×C412yes—
24C23×C4×C53C2×C411yescharacteristic
25C23×C4×C53C2×C412yes—
26C23×C4×C53D414yes—
27C23×C4×C53C2311yescharacteristic
28C22⋊Q8×C53—232no—
29C2×C4⋊C4×C53—24no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—24no—
32C2×C4⋊C4×C53—24no—
33C2×C4⋊C4×C53D418yes—
34C2×C22⋊C4×C53—24no—
35C2×C22⋊C4×C53—24no—
36C2×C22⋊C4×C53—24no—
37C24×C53—24no—
38C24×C53—22no—
39C24×C53—24no—
40C24×C53—42no—
41C24×C53C22⋊C412yes—
42C24×C53C22×C411yescharacteristic
43C2×Q8×C53—28no—
44C2×Q8×C53—416no—
45C2×Q8×C53—44no—
46C2×Q8×C53—44no—
47C22×C4×C53—22no—
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—28no—
54C22×C4×C53—28no—
55C22×C4×C53—28no—
56C22×C4×C53—44no—
57C22×C4×C53—42no—
58C22×C4×C53—416no—
59C22×C4×C53—44no—
60C22×C4×C53C22⋊C414yes—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C414yes—
63C22×C4×C53C2×D412yes—
64C22×C4×C53C2×D414yes—
65C4⋊C4×C53—28no—
66C4⋊C4×C53—216no—
67C22⋊C4×C53—216no—
68C2×#88322—48no—
69C2×#88322—44no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—28no—
73C23×C53—22no—
74C23×C53—24no—
75C23×C53—24no—
76C23×C53—22no—
77C23×C53—416no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53#10114yes—
81C23×C53C2×C22⋊C412yes—
82C23×C53C2×C22⋊C411yescharacteristic
83C23×C53#13114yes—
84Q8×C53—416no—
85C2×C4×C53—28no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—24no—
89C2×C4×C53—24no—
90C2×C4×C53—24no—
91C2×C4×C53—28no—
92C2×C4×C53—44no—
93C2×C4×C53—416no—
94C2×C4×C53—44no—
95C2×C4×C53—44no—
96C2×C4×C53—44no—
97C2×C4×C53—44no—
98C2×C4×C53—44no—
99C2×C4×C53#9914yes—
100C2×C4×C53#12014yes—
101#88322—84no—
102#8308—531nomaximal, Sylow
103C22×C53—28no—
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—28no—
107C22×C53—24no—
108C22×C53—416no—
109C22×C53—44no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53#37911yescharacteristic
113C22×C53#40814yes—
114C22×C53#40812yes—
115C22×C53C2×#9912yes—
116C22×C53C2×#10112yes—
117C4×C53—24no—
118C4×C53—24no—
119C4×C53—44no—
120C4×C53—44no—
121C22×C22⋊Q8D5311yescharacteristic
122#1779—532no—
123C2×#324—534no—
124C2×C53—24no—
125C2×C53—44no—
126C2×C53#147214yes—
127C2×C53#178011yescharacteristic
128C2×C53#178111yescharacteristic
129C2×C53#178411yescharacteristic
130C23×Q8Dic5311yescharacteristic
131C24×C4C2×D5311yescharacteristic
132C2×C22⋊Q8—24no—
133C2×C22⋊Q8—28no—
134C22×C4⋊C4Dic5311yescharacteristic
135C22×C4⋊C4C2×D5312yes—
136C22×C22⋊C4Dic5312yes—
137C2×C22⋊C8—534no—
138C2×C22⋊C8—1064no—
139#324—1068no—
140C53#830811yescharacteristic, complemented, Sylow
141C25C2×Dic5311yescharacteristic
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—44no—
145C23×C4—24no—
146C23×C4—44no—
147C23×C4C2×Dic5312yes—
148C23×C4C2×Dic5311yescharacteristic
149C23×C4C2×Dic5312yes—
150C23×C4C22⋊D5314yes—
151C23×C4C22×D5311yescharacteristic
152C22×C8—1064no—
153C22⋊Q8—232no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊D5318yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C22⋊C8—1068no—
163C22⋊C8—2128no—
164C24—24no—
165C24—22no—
166C24—24no—
167C24—42no—
168C24C22⋊Dic5312yes—
169C24C22×Dic5311yescharacteristic
170C2×Q8—28no—
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—28no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4C22⋊Dic5314yes—
188C22×C4C22⋊Dic5312yes—
189C22×C4C22⋊Dic5314yes—
190C22×C4C2×C22⋊D5312yes—
191C22×C4C2×C22⋊D5314yes—
192C2×C8—2128no—
193C2×C8—2124no—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C22⋊C4—216no—
197C23—22no—
198C23—22no—
199C23—28no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—416no—
205C23—44no—
206C23—44no—
207C23#1313313614yes—
208C23C2×C22⋊Dic5312yes—
209C23C2×C22⋊Dic5311yescharacteristic
210C23#1313322414yes—
211Q8—416no—
212C2×C4—28no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—28no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—416no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4#1313313214yes—
227C2×C4#1313318614yes—
228C8—4244no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—28no—
233C22—24no—
234C22—416no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22#1681047911yescharacteristic
239C22#1681059014yes—
240C22#1681059012yes—
241C22C2×#1313313212yes—
242C22C2×#1313313612yes—
243C4—24no—
244C4—24no—
245C4—44no—
246C4—44no—
247C2—24no—
248C2—44no—
249C2#2027307914yes—
250C2#2027462711yescharacteristic
251C2#2027463011yescharacteristic
252C2#2027463911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×C22⋊Q8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenterC22×C22⋊Q8D53
hypocenterC53#8308
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16810349C2semidirect
140. C53#8308semidirect

Maximal subgroups

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

subgroupindex
#202746252
C2×#168103492
C22×C22⋊Q8×C532
#830853