Finite Group Databasebeta

← #27293208

Subgroups — #27293208

Order 13568.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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#20273029C212yesmaximal
2#20272649C214yesmaximal, complemented
3C53×#2310C211yesmaximal, characteristic
4#16810676—22no—
5#16810349C2214yes—
6#16810347—22no—
7C23×Q8×C53C411yescharacteristic
8C2×#117×C53C411yescharacteristic
9C2×#117×C53C2211yescharacteristic
10C53×#391—22no—
11C53×#391C412yes—
12C53×#391C2212yes—
13C53×#381—22no—
14C4⋊#88322—42no—
15C22⋊#88322—24no—
16C4×#88322—42no—
17C22×Q8×C53—21no—
18C22×Q8×C53—22no—
19C22×Q8×C53—42no—
20C22×Q8×C53—42no—
21C22×Q8×C53C814yescomplemented
22C22×Q8×C53C2×C412yes—
23C23×C4×C53—21no—
24C23×C4×C53C2×C411yescharacteristic
25#117×C53—42no—
26#117×C53—42no—
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53C2×C411yescharacteristic
29C2×C4⋊C4×C53C2311yescharacteristic
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53D412yes—
32C2×C42×C53C2×C412yes—
33C2×C42×C53D412yes—
34#95×C53—24no—
35#95×C53—24no—
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—22no—
51C22×C4×C53—42no—
52C22×C4×C53—42no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C2×C812yes—
56C22×C4×C53M1612yes—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D412yes—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—42no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C42×C53—22no—
65C42×C53—22no—
66C42×C53—42no—
67C2×#88322—44no—
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—22no—
83C2×C4×C53—22no—
84C2×C4×C53—48no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53C22⋊C812yes—
93C2×C4×C53#9912yes—
94C2×C4×C53#10012yes—
95C2×C4×C53C22×C811yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97#88322—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—42no—
112C4×C53—42no—
113C4×C53—44no—
114C4×C53—82no—
115C4×C53#32314yes—
116#2310D5311yescharacteristic
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×Q8Dic5311yescharacteristic
126C2×#117Dic5311yescharacteristic
127C2×#117C2×D5311yescharacteristic
128#431—1062no—
129#391—22no—
130#391Dic5312yes—
131#391C2×D5312yes—
132#381—22no—
133#324—534no—
134#323—1062no—
135C53#1143411yescharacteristic, complemented, Sylow
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8#8832214yescomplemented
141C22×Q8C2×Dic5312yes—
142C23×C4—21no—
143C23×C4C2×Dic5311yescharacteristic
144#117—42no—
145#117—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C2×Dic5311yescharacteristic
148C2×C4⋊C4C22×D5311yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4C22⋊D5312yes—
151C2×C42C2×Dic5312yes—
152C2×C42C22⋊D5312yes—
153C4⋊C8—2122no—
154C22⋊C8—1064no—
155C4×C8—2122no—
156#95—24no—
157#95—24no—
158C24C22⋊Dic5311yescharacteristic
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×#8832214yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4C2×#8832212yes—
176C22×C4#1171991712yes—
177C22×C4C22⋊Dic5312yes—
178C22×C4C22⋊Dic5311yescharacteristic
179C22×C4C22×Dic5311yescharacteristic
180C22×C4C2×C22⋊D5312yes—
181C2×C8—2124no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C42—22no—
188C42—22no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊#8832212yes—
195C23C2×C22⋊Dic5311yescharacteristic
196Q8—44no—
197Q8—44no—
198Q8—48no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—48no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4C22⋊#8832212yes—
215C2×C4#1313313212yes—
216C2×C4#1313313412yes—
217C2×C4C22×#8832211yescharacteristic
218C2×C4C2×#1171991711yescharacteristic
219C8—4244no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×C22⋊#8832211yescharacteristic
228C22C2×#1313313211yescharacteristic
229C22C2×#1313313411yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—42no—
235C4—82no—
236C4#1681034714yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1681034711yescharacteristic
241C2#2027639011yescharacteristic
242C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2310C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2310D53
hypocenterC53#11434
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #20272649C2semidirect
21. C22×Q8×C53C8semidirect
135. C53#11434semidirect
140. C22×Q8#88322semidirect

Maximal subgroups

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

subgroupindex
#202730292
#202726492
C53×#23102
#1143453