Finite Group Databasebeta

← #28307926

Subgroups — #28307926

Order 13568.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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#20289946C212yesmaximal
2#20289645C214yesmaximal, complemented
3#20277301C211yesmaximal, characteristic
4#16811639—22no—
5#16811468C2214yes—
6#16811466—22no—
7C2×#13133176C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810530—22no—
10#16810530C412yes—
11#16810530C2212yes—
12#16810486—22no—
13C23×Q8×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17#13133176—42no—
18#13133176—42no—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53C2×C411yescharacteristic
21C2×C4⋊Dic53C2311yescharacteristic
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53D412yes—
24C2×C4×Dic53C2×C412yes—
25C2×C4×Dic53D412yes—
26#13133124—24no—
27#13133124—24no—
28C22×Q8×C53—21no—
29C22×Q8×C53—22no—
30C22×Q8×C53—42no—
31C22×Q8×C53—42no—
32C22×Q8×C53C814yescomplemented
33C22×Q8×C53C2×C412yes—
34C23×C4×C53—21no—
35C23×C4×C53C2×C411yescharacteristic
36C24×C53C22⋊C411yescharacteristic
37C2×Q8×C53—24no—
38C2×Q8×C53—22no—
39C2×Q8×C53—48no—
40C2×Q8×C53—48no—
41C2×Q8×C53—42no—
42C2×Q8×C53—42no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C2×Q8×C53C2×C814yes—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—42no—
51C22×C4×C53—42no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C2×C812yes—
54C22×C4×C53M1612yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C2×D412yes—
59C2×#88324—44no—
60C22⋊Dic53—42no—
61C22⋊Dic53—42no—
62C4⋊Dic53—41no—
63C4⋊Dic53—41no—
64C4⋊Dic53—42no—
65C4×Dic53—22no—
66C4×Dic53—22no—
67C4×Dic53—42no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53C22⋊C812yes—
73C23×C53C2×C22⋊C411yescharacteristic
74Q8×C53—44no—
75Q8×C53—44no—
76Q8×C53—48no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—24no—
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—48no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53C22⋊C812yes—
86C2×C4×C53#9912yes—
87C2×C4×C53#10012yes—
88C2×C4×C53C22×C811yescharacteristic
89C2×C4×C53C2×M1611yescharacteristic
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97#88324—84no—
98#11434—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—22no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53C2×C22⋊C811yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10011yescharacteristic
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—44no—
112C4×C53#32314yes—
113Dic53—42no—
114Dic53—42no—
115Dic53—82no—
116#2310—531no—
117#1458—532no—
118#1319—534no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—42no—
122C2×C53C2×#32311yescharacteristic
123C2×C53#212911yescharacteristic
124C2×C53#213511yescharacteristic
125C23×Q8#615711yescharacteristic
126C2×#117—531no—
127C2×#117—531no—
128#431—1062no—
129#391—532no—
130#391—532no—
131#391—1062no—
132#381—1062no—
133#324—534no—
134#323—1062no—
135C53#1143411yescharacteristic, complemented, Sylow
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8#8832414yescomplemented
141C22×Q8C2×#615712yes—
142C23×C4—21no—
143C23×C4C2×#615711yescharacteristic
144#117—2122no—
145#117—2122no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—1061no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—1061no—
151C2×C42—532no—
152C2×C42—532no—
153C4⋊C8—2122no—
154C22⋊C8—1064no—
155C4×C8—2122no—
156#95—1064no—
157#95—1064no—
158C24C22⋊#615711yescharacteristic
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C2×#8832414yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—532no—
175C22×C4—532no—
176C22×C4—1062no—
177C22×C4C2×#8832412yes—
178C22×C4#1171994012yes—
179C22×C4C22⋊#615711yescharacteristic
180C22×C4C22×#615711yescharacteristic
181C2×C8—2124no—
182C4⋊C4—2121no—
183C4⋊C4—2121no—
184C4⋊C4—2122no—
185C22⋊C4—2122no—
186C22⋊C4—2122no—
187C42—1062no—
188C42—1062no—
189C42—2122no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊#8832412yes—
195C23C2×C22⋊#615711yescharacteristic
196Q8—44no—
197Q8—44no—
198Q8—48no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4C22⋊#8832412yes—
215C2×C4#1313327512yes—
216C2×C4#1313327712yes—
217C2×C4C22×#8832411yescharacteristic
218C2×C4C2×#1171994011yescharacteristic
219C8—4244no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×C22⋊#8832411yescharacteristic
228C22C2×#1313327511yescharacteristic
229C22C2×#1313327711yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—4242no—
234C4—2122no—
235C4—2122no—
236C4#1681146614yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1681146611yescharacteristic
241C2#2029099711yescharacteristic
242C2#2029101711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC23×Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#11434
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #20289645C2semidirect
32. C22×Q8×C53C8semidirect
135. C53#11434semidirect
140. C22×Q8#88324semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896452
#202773012
#1143453