Finite Group Databasebeta

← #16786733

Subgroups — #16786733

Order 3328.

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#13130067C212yesmaximal
2#13129762C214yesmaximal
3#13117422C211yesmaximal, characteristic
4C23×Q8×C13C411yescharacteristic
5C2×#88180C411yescharacteristic
6C2×#88180C2211yescharacteristic
7#11719119—22no—
8#11718891—22no—
9#11718891—22no—
10#11718872C412yes—
11#11718872C2212yes—
12#11718305C2214yes—
13#11718285—22no—
14C22×Q8×C13—21no—
15C22×Q8×C13—21no—
16C22×Q8×C13—22no—
17C22×Q8×C13—42no—
18C22×Q8×C13—42no—
19C22×Q8×C13C814yescomplemented
20C23×C4×C13C2×C412yes—
21C23×C4×C13C2×C411yescharacteristic
22C2×C22⋊Dic13C2×C412yes—
23C2×C22⋊Dic13C2×C411yescharacteristic
24C2×C22⋊Dic13C2311yescharacteristic
25#88180—42no—
26#88180—42no—
27C2×C4⋊Dic13—21no—
28C2×C4⋊Dic13D412yes—
29C2×C4×Dic13—21no—
30C2×C4×Dic13D412yes—
31C22⋊#1057—24no—
32C4⋊#1057—42no—
33C4×#1057—42no—
34#88071—24no—
35#88071—24no—
36#11428—131nomaximal, Sylow
37C24×C13C22×C411yescharacteristic
38C2×Q8×C13—24no—
39C2×Q8×C13—24no—
40C2×Q8×C13—48no—
41C2×Q8×C13—42no—
42C2×Q8×C13—42no—
43C2×Q8×C13—48no—
44C2×Q8×C13—42no—
45C2×Q8×C13—42no—
46C22×C4×C13—22no—
47C22×C4×C13—22no—
48C22×C4×C13—22no—
49C22×C4×C13—22no—
50C22×C4×C13—42no—
51C22×C4×C13—42no—
52C22×C4×C13C22⋊C411yescharacteristic
53C22×C4×C13C22⋊C411yescharacteristic
54C22×C4×C13C2×C812yes—
55C22×C4×C13C2×C814yes—
56C22×C4×C13M1612yes—
57C22×Dic13—22no—
58C22×Dic13C22⋊C412yes—
59C22×Dic13C2×D412yes—
60C2×#1057—44no—
61C22⋊Dic13—41no—
62C22⋊Dic13—41no—
63C22⋊Dic13—42no—
64C4⋊Dic13—42no—
65C4⋊Dic13—42no—
66C4×Dic13—22no—
67C4×Dic13—22no—
68C4×Dic13—42no—
69#2310—131no—
70#1458—132no—
71#1317—134no—
72C23×C13—21no—
73C23×C13—21no—
74C23×C13—42no—
75C23×C13C2×C22⋊C411yescharacteristic
76C23×C13C22×C811yescharacteristic
77C23×C13C2×M1611yescharacteristic
78Q8×C13—48no—
79Q8×C13—48no—
80C2×C4×C13—21no—
81C2×C4×C13—21no—
82C2×C4×C13—24no—
83C2×C4×C13—24no—
84C2×C4×C13—24no—
85C2×C4×C13—44no—
86C2×C4×C13—42no—
87C2×C4×C13—42no—
88C2×C4×C13—44no—
89C2×C4×C13C22⋊C812yes—
90C2×C4×C13C22⋊C812yes—
91C2×C4×C13#9912yes—
92C2×C4×C13#10012yes—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—42no—
96C2×Dic13—42no—
97C2×Dic13—44no—
98C2×Dic13—42no—
99#1057—84no—
100C23×Q8#26111yescharacteristic
101C2×#117—131no—
102C2×#117—131no—
103#431—262no—
104#391—262no—
105#391—262no—
106#381—132no—
107#381—132no—
108#324—262no—
109#323—134no—
110C22×C13—22no—
111C22×C13—21no—
112C22×C13—21no—
113C22×C13—41no—
114C22×C13—41no—
115C22×C13—42no—
116C22×C13—42no—
117C22×C13C2×C22⋊C811yescharacteristic
118C22×C13C2×#9911yescharacteristic
119C22×C13C2×#10011yescharacteristic
120C4×C13—22no—
121C4×C13—44no—
122C4×C13#32314yes—
123Dic13—42no—
124Dic13—42no—
125Dic13—82no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8#105714yescomplemented
132C23×C4C2×#26112yes—
133C23×C4C2×#26111yescharacteristic
134#117—522no—
135#117—522no—
136C2×C4⋊C4—132no—
137C2×C4⋊C4—261no—
138C2×C22⋊C4—131no—
139C2×C22⋊C4—131no—
140C2×C22⋊C4—132no—
141C2×C42—132no—
142C2×C42—261no—
143C4⋊C8—522no—
144C22⋊C8—264no—
145C4×C8—522no—
146#95—264no—
147#95—264no—
148C2×C13—21no—
149C2×C13—21no—
150C2×C13—41no—
151C2×C13—41no—
152C2×C13C2×#32311yescharacteristic
153C2×C13#212312yes—
154C24C22×#26111yescharacteristic
155C2×Q8—24no—
156C2×Q8—24no—
157C2×Q8—48no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—48no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—132no—
170C22×C4—132no—
171C22×C4—262no—
172C22×C4C2×#105712yes—
173C22×C4C2×#105714yes—
174C22×C4#612212yes—
175C22×C4C22⋊#26111yescharacteristic
176C22×C4C22⋊#26111yescharacteristic
177C2×C8—524no—
178C4⋊C4—522no—
179C4⋊C4—522no—
180C22⋊C4—521no—
181C22⋊C4—521no—
182C22⋊C4—522no—
183C42—262no—
184C42—262no—
185C42—522no—
186C13#1142811yescharacteristic, complemented, Sylow
187C23—21no—
188C23—21no—
189C23—42no—
190C23C22×#105711yescharacteristic
191C23C2×#612211yescharacteristic
192C23C2×C22⋊#26111yescharacteristic
193Q8—48no—
194Q8—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—262no—
205C2×C4—262no—
206C2×C4—522no—
207C2×C4—522no—
208C2×C4—524no—
209C2×C4—522no—
210C2×C4#8811912yes—
211C2×C4C22⋊#105712yes—
212C2×C4C22⋊#105712yes—
213C2×C4#8812112yes—
214C8—1044no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—42no—
222C22C2×#8811911yescharacteristic
223C22C2×C22⋊#105711yescharacteristic
224C22C2×#8812111yescharacteristic
225C4—22no—
226C4—44no—
227C4—522no—
228C4—522no—
229C4—1042no—
230C4#1171830514yes—
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2C2×#1171830511yescharacteristic
236C2#1313109812yes—
subgroupG/H
centerC22C2×#88119
commutatorC23×C13C22×C8
FrattiniC24C22×#261
FittingC23×Q8×C13C4
socleC22×C13C2×#99
prime radicalC24×C13C22×C4
cosocleC2×C22⋊Dic13C23
hypercenterC23×Q8#261
hypocenterC13#11428
LCSG ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
19. C22×Q8×C13C8semidirect
131. C22×Q8#1057semidirect
186. C13#11428semidirect

Maximal subgroups

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

subgroupindex
#131300672
#131297622
#131174222
#1142813