Finite Group Databasebeta

← #28299569

Subgroups — #28299569

Order 13568.

440 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 106 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#20290643C212yesmaximal
2#20289536C212yesmaximal
3#20289535C212yesmaximal
4C4×#13133215C211yesmaximal, characteristic
5#16811489—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133215C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810528C2211yescharacteristic
12#16810518—22no—
13#16810492C411yescharacteristic
14C4×#11719919—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C22×#88324—24no—
18C22×Dic106C2×C411yescharacteristic
19#13133215—22no—
20#13133215C814yescomplemented
21C4×Dic106—42no—
22C2×C4⋊Dic53C2×C411yescharacteristic
23C2×#11719919—21no—
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53C2×C411yescharacteristic
28C2×C4×Dic53D412yes—
29C2×C4×Dic53D412yes—
30C2×C4×Dic53C2311yescharacteristic
31#13133124—22no—
32#13133124—22no—
33C22×Q8×C53C2×C411yescharacteristic
34C4×Q8×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C42×C53—21no—
38C2×C42×C53C2×C411yescharacteristic
39C2×Q8×C53—21no—
40C2×Q8×C53—41no—
41C2×Q8×C53—41no—
42C2×Q8×C53C2×C812yes—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—21no—
49C22×Dic53—21no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C2×D412yes—
52C2×Dic106—21no—
53C2×Dic106—41no—
54C2×Dic106—41no—
55C2×Dic106C2×C812yes—
56C2×#88324—44no—
57C2×#88324—44no—
58C2×#88324—44no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—41no—
61C42×C53—22no—
62C42×C53—42no—
63C4⋊Dic53—41no—
64#11719919—28no—
65#11719919—44no—
66#11719919—42no—
67C4×Dic53—24no—
68C4×Dic53—42no—
69C4×Dic53—41no—
70C4×Dic53—42no—
71C4×Dic53C2×C812yes—
72C4×Dic53M1612yes—
73C23×C53C2×C22⋊C411yescharacteristic
74Q8×C53—42no—
75Q8×C53—42no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53#10012yes—
87C2×C4×C53C22×C811yescharacteristic
88C2×C4×C53C2×M1611yescharacteristic
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—41no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—42no—
98C2×Dic53C22⋊C814yes—
99C2×Dic53#13714yes—
100Dic106—42no—
101Dic106—42no—
102#88324—84no—
103#7605—531nomaximal, Sylow
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53#41011yescharacteristic
106C22×C53C2×#10011yescharacteristic
107C22×C53#41912yes—
108C22×C53#42112yes—
109C4×C53—22no—
110C4×C53—21no—
111C4×C53—21no—
112C4×C53—41no—
113C4×C53—42no—
114C4×C53—81no—
115Dic53—24no—
116Dic53—24no—
117Dic53—42no—
118Dic53—81no—
119C4×C4⋊Q8—531no—
120#1839—532no—
121#1272—532no—
122#1272—532no—
123C2×C53#146011yescharacteristic
124C2×C53#147912yes—
125C2×C53#151512yes—
126C2×C53#153611yescharacteristic
127C2×C53#155811yescharacteristic
128C2×C4⋊Q8—531no—
129C2×C4×Q8—531no—
130C2×C4×Q8#615711yescharacteristic
131#391—531no—
132#391—531no—
133#389—1062no—
134#384—531no—
135C4×C4⋊C4—1062no—
136C43—531no—
137#336—534no—
138#336—1062no—
139#336—1062no—
140C53#760511yescharacteristic, complemented, Sylow
141C22×Q8—531no—
142C22×Q8C2×#615711yescharacteristic
143C22×C8—1064no—
144C4⋊Q8—534no—
145C4⋊Q8—1062no—
146C4×Q8—42no—
147C4×Q8—2122no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4—1062no—
152C2×C4⋊C4—1061no—
153C2×C4⋊C4C2×#615711yescharacteristic
154C2×C42—21no—
155C2×C42—531no—
156C2×C42—532no—
157C2×C42—532no—
158C2×C42—531no—
159C2×C42—1061no—
160C2×C42C2×#615711yescharacteristic
161#95—1062no—
162#95—1062no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—532no—
167C2×Q8—1061no—
168C2×Q8—2121no—
169C2×Q8—2121no—
170C2×Q8D53⋊C812yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—1061no—
176C22×C4—1061no—
177C22×C4C22⋊#615711yescharacteristic
178C22×C4C22⋊#615711yescharacteristic
179C22×C4C22×#615711yescharacteristic
180C2×C8—2124no—
181C2×C8—2124no—
182C2×C8—2124no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—1068no—
186C4⋊C4—2121no—
187C4⋊C4—2122no—
188C4⋊C4—2124no—
189C42—22no—
190C42—42no—
191C42—532no—
192C42—532no—
193C42—1064no—
194C42—2121no—
195C42—2122no—
196C42—2122no—
197C23C2×C22⋊#615711yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—2122no—
201Q8—2122no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—534no—
213C2×C4—534no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—2121no—
220C2×C4—2122no—
221C2×C4—2121no—
222C2×C4—2121no—
223C2×C4#1313327712yes—
224C2×C4C2×D53⋊C811yescharacteristic
225C2×C4D53⋊M1611yescharacteristic
226C8—4244no—
227C22D53⋊(C22⋊C8)11yescharacteristic
228C22#1681160211yescharacteristic
229C22C2×#1313327711yescharacteristic
230C22#1681161612yes—
231C22#1681162012yes—
232C4—21no—
233C4—21no—
234C4—22no—
235C4—41no—
236C4—42no—
237C4—81no—
238C4—1064no—
239C4—1064no—
240C4—4241no—
241C4—2122no—
242C2#2028995411yescharacteristic
243C2#2029001812yes—
244C2#2029013712yes—
245C2#2029020311yescharacteristic
246C2#2029024811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7605
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
20. #13133215C8semidirect
140. C53#7605semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202895362
#202895352
C4×#131332152
#760553