Finite Group Databasebeta

← #28299547

Subgroups — #28299547

Order 13568.

514 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290569C212yesmaximal
2#20289534C214yesmaximal, complemented
3#20277197C211yesmaximal, characteristic
4C2×C4⋊#88324—22no—
5C2×C22⋊#88324—22no—
6#16811489C2214yes—
7C2×#13133176C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810519C412yes—
10#16810519C2212yes—
11#16810495—22no—
12C4×#11719919—22no—
13C2×#117×C53C411yescharacteristic
14C22×#88324—24no—
15C4⋊#88324—44no—
16C22⋊#88324—44no—
17C23×Dic53D412yes—
18#13133176—22no—
19#13133176C814yescomplemented
20C2×C4⋊Dic53C2×C412yes—
21C2×#11719919—22no—
22C2×#11719919—22no—
23C2×C22⋊Dic53—21no—
24C2×C4×Dic53—21no—
25C2×C4×Dic53—42no—
26C2×C4×Dic53C2×C411yescharacteristic
27C2×C4×Dic53D412yes—
28C2×C4×Dic53C2311yescharacteristic
29#13133124—22no—
30#13133124—22no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53C2×C412yes—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53—21no—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C2×C812yes—
42C22×C4×C53M1612yes—
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53—22no—
47C22×Dic53—42no—
48C22×Dic53—42no—
49C22×Dic53C22⋊C412yes—
50C22×Dic53C2×D412yes—
51C2×#88324—44no—
52C2×#88324—44no—
53C2×#88324—44no—
54C4⋊C4×C53—42no—
55C22⋊C4×C53—42no—
56C42×C53—42no—
57C22⋊Dic53—24no—
58C4⋊Dic53—22no—
59C4⋊Dic53C2×C814yes—
60#11719919—44no—
61#11719919—42no—
62#11719919—42no—
63C4×Dic53—24no—
64C4×Dic53—24no—
65C4×Dic53—41no—
66C4×Dic53—41no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53C22⋊C812yes—
72C23×C53C2×C22⋊C411yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—24no—
76C2×C4×C53—24no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53C22⋊C812yes—
81C2×C4×C53#9912yes—
82C2×C4×C53#10112yes—
83C2×C4×C53C22×C811yescharacteristic
84C2×C4×C53C2×M1611yescharacteristic
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—44no—
92C2×Dic53—44no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53AGL(1,ℤ/8)12yes—
96C2×Dic53#13712yes—
97#88324—84no—
98#7598—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53—24no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—42no—
104C22×C53C2×C22⋊C811yescharacteristic
105C22×C53C2×#9911yescharacteristic
106C22×C53C2×#10111yescharacteristic
107C22×C53#41712yes—
108C22×C53#42112yes—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—82no—
112C4×C53#32414yes—
113Dic53—22no—
114Dic53—22no—
115Dic53—44no—
116Dic53—42no—
117#2296—531no—
118#1797—532no—
119#1271—534no—
120C2×C53—42no—
121C2×C53C2×#32411yescharacteristic
122C2×C53#147712yes—
123C2×C53#151312yes—
124C2×C53#153711yescharacteristic
125C2×C53#156111yescharacteristic
126C2×#117—531no—
127C2×#117#615711yescharacteristic
128C22×C42—531no—
129C2×C4⋊C8—1062no—
130C2×C22⋊C8—1062no—
131#389—532no—
132#389—532no—
133#385—1062no—
134C4×C4⋊C4—1062no—
135#336—534no—
136C53#759811yescharacteristic, complemented, Sylow
137C23×C4—532no—
138C23×C4C2×#615711yescharacteristic
139C22×C8—1064no—
140#117—42no—
141#117—534no—
142#117—1062no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—1062no—
145C2×C4⋊C4—1062no—
146C2×C4⋊C4C2×#615712yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—1061no—
149C2×C42—21no—
150C2×C42—532no—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42—1061no—
154C2×C42—2122no—
155C4⋊C8—2124no—
156C22⋊C8—2124no—
157#95—1062no—
158#95—1062no—
159C24C22⋊#615711yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1062no—
166C22×C4—1062no—
167C22×C4—1062no—
168C22×C4—2122no—
169C22×C4—2122no—
170C22×C4D53⋊C812yes—
171C22×C4#1171993612yes—
172C22×C4C22⋊#615711yescharacteristic
173C22×C4C22×#615711yescharacteristic
174C2×C8—2124no—
175C2×C8—2124no—
176C2×C8—2124no—
177C4⋊C4—42no—
178C4⋊C4—534no—
179C4⋊C4—1062no—
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C4⋊C4—2124no—
183C22⋊C4—42no—
184C22⋊C4—1064no—
185C42—42no—
186C42—1064no—
187C42—1064no—
188C42—2121no—
189C42—2121no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23#1313327412yes—
195C23C2×C22⋊#615711yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—532no—
204C2×C4—532no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1064no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—2124no—
212C2×C4—2124no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4#1313327412yes—
216C2×C4#1313327512yes—
217C2×C4#1313327812yes—
218C2×C4C2×D53⋊C811yescharacteristic
219C2×C4C2×#1171993611yescharacteristic
220C8—4244no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×#1313327411yescharacteristic
227C22C2×#1313327511yescharacteristic
228C22C2×#1313327811yescharacteristic
229C22#1681161412yes—
230C22#1681162012yes—
231C4—21no—
232C4—21no—
233C4—82no—
234C4—1062no—
235C4—1062no—
236C4—2122no—
237C4—2124no—
238C4#1681146914yes—
239C2—42no—
240C2C2×#1681146911yescharacteristic
241C2#2029001212yes—
242C2#2029013012yes—
243C2#2029020611yescharacteristic
244C2#2029025411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#7598
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289534C2semidirect
19. #13133176C8semidirect
136. C53#7598semidirect

Maximal subgroups

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

subgroupindex
#202905692
#202895342
#202771972
#759853