Finite Group Databasebeta

← #16786753

Subgroups — #16786753

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#13129766C214yesmaximal, complemented
3#13117422C211yesmaximal, characteristic
4C23×Q8×C13C411yescharacteristic
5C2×#88180C411yescharacteristic
6C2×#88180C2211yescharacteristic
7#11719119—22no—
8#11718891—22no—
9#11718891C412yes—
10#11718891C2212yes—
11#11718872—22no—
12#11718305—22no—
13#11718285C2214yes—
14C22×Q8×C13—21no—
15C22×Q8×C13—22no—
16C22×Q8×C13—42no—
17C22×Q8×C13—42no—
18C22×Q8×C13C814yescomplemented
19C22×Q8×C13C2×C412yes—
20C23×C4×C13—21no—
21C23×C4×C13C2×C411yescharacteristic
22C2×C22⋊Dic13—21no—
23C2×C22⋊Dic13D412yes—
24#88180—42no—
25#88180—42no—
26C2×C4⋊Dic13—21no—
27C2×C4⋊Dic13C2×C411yescharacteristic
28C2×C4⋊Dic13C2311yescharacteristic
29C2×C4×Dic13C2×C412yes—
30C2×C4×Dic13D412yes—
31C22⋊#1057—24no—
32C4⋊#1057—42no—
33C4×#1057—42no—
34#88071—24no—
35#88071—24no—
36#11434—131nomaximal, Sylow
37C24×C13C22⋊C411yescharacteristic
38C2×Q8×C13—24no—
39C2×Q8×C13—22no—
40C2×Q8×C13—48no—
41C2×Q8×C13—48no—
42C2×Q8×C13—42no—
43C2×Q8×C13—42no—
44C2×Q8×C13—42no—
45C2×Q8×C13—42no—
46C2×Q8×C13C2×C814yes—
47C22×C4×C13—22no—
48C22×C4×C13—22no—
49C22×C4×C13—24no—
50C22×C4×C13—22no—
51C22×C4×C13—42no—
52C22×C4×C13—42no—
53C22×C4×C13C22⋊C411yescharacteristic
54C22×C4×C13C2×C812yes—
55C22×C4×C13M1612yes—
56C22×C4×C13C22×C411yescharacteristic
57C22×Dic13—22no—
58C22×Dic13C22⋊C412yes—
59C22×Dic13C2×D412yes—
60C2×#1057—44no—
61C22⋊Dic13—42no—
62C22⋊Dic13—42no—
63C4⋊Dic13—41no—
64C4⋊Dic13—41no—
65C4⋊Dic13—42no—
66C4×Dic13—22no—
67C4×Dic13—22no—
68C4×Dic13—42no—
69#2310—131no—
70#1458—132no—
71#1319—134no—
72C23×C13—21no—
73C23×C13—21no—
74C23×C13—41no—
75C23×C13—41no—
76C23×C13C22⋊C812yes—
77C23×C13C2×C22⋊C411yescharacteristic
78Q8×C13—44no—
79Q8×C13—44no—
80Q8×C13—48no—
81C2×C4×C13—21no—
82C2×C4×C13—21no—
83C2×C4×C13—24no—
84C2×C4×C13—24no—
85C2×C4×C13—24no—
86C2×C4×C13—48no—
87C2×C4×C13—42no—
88C2×C4×C13—42no—
89C2×C4×C13C22⋊C812yes—
90C2×C4×C13#9912yes—
91C2×C4×C13#10012yes—
92C2×C4×C13C22×C811yescharacteristic
93C2×C4×C13C2×M1611yescharacteristic
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—42no—
97C2×Dic13—42no—
98C2×Dic13—42no—
99C2×Dic13—42no—
100C2×Dic13—42no—
101#1057—84no—
102C23×Q8#26111yescharacteristic
103C2×#117—131no—
104C2×#117—131no—
105#431—262no—
106#391—132no—
107#391—132no—
108#391—262no—
109#381—262no—
110#324—134no—
111#323—262no—
112C22×C13—21no—
113C22×C13—21no—
114C22×C13—22no—
115C22×C13—42no—
116C22×C13—42no—
117C22×C13—41no—
118C22×C13—41no—
119C22×C13C2×C22⋊C811yescharacteristic
120C22×C13C2×#9911yescharacteristic
121C22×C13C2×#10011yescharacteristic
122C4×C13—21no—
123C4×C13—21no—
124C4×C13—44no—
125C4×C13#32314yes—
126Dic13—42no—
127Dic13—42no—
128Dic13—82no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8#105714yescomplemented
134C22×Q8C2×#26112yes—
135C23×C4—21no—
136C23×C4C2×#26111yescharacteristic
137#117—522no—
138#117—522no—
139C2×C4⋊C4—131no—
140C2×C4⋊C4—131no—
141C2×C4⋊C4—261no—
142C2×C22⋊C4—132no—
143C2×C22⋊C4—261no—
144C2×C42—132no—
145C2×C42—132no—
146C4⋊C8—522no—
147C22⋊C8—264no—
148C4×C8—522no—
149#95—264no—
150#95—264no—
151C2×C13—21no—
152C2×C13—21no—
153C2×C13—42no—
154C2×C13C2×#32311yescharacteristic
155C2×C13#212911yescharacteristic
156C2×C13#213511yescharacteristic
157C24C22⋊#26111yescharacteristic
158C2×Q8—24no—
159C2×Q8—22no—
160C2×Q8—48no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8C2×#105714yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—132no—
174C22×C4—132no—
175C22×C4—262no—
176C22×C4C2×#105712yes—
177C22×C4#612212yes—
178C22×C4C22⋊#26111yescharacteristic
179C22×C4C22×#26111yescharacteristic
180C2×C8—524no—
181C4⋊C4—521no—
182C4⋊C4—521no—
183C4⋊C4—522no—
184C22⋊C4—522no—
185C22⋊C4—522no—
186C42—262no—
187C42—262no—
188C42—522no—
189C13#1143411yescharacteristic, complemented, Sylow
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊#105712yes—
195C23C2×C22⋊#26111yescharacteristic
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—262no—
208C2×C4—262no—
209C2×C4—522no—
210C2×C4—522no—
211C2×C4—522no—
212C2×C4—522no—
213C2×C4—522no—
214C2×C4#8811912yes—
215C2×C4C22⋊#105712yes—
216C2×C4#8812112yes—
217C2×C4C22×#105711yescharacteristic
218C2×C4C2×#612211yescharacteristic
219C8—1044no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×#8811911yescharacteristic
228C22C2×C22⋊#105711yescharacteristic
229C22C2×#8812111yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—522no—
234C4—522no—
235C4—1042no—
236C4#1171830514yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1171830511yescharacteristic
241C2#1313111811yescharacteristic
242C2#1313113811yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC23×Q8×C13C4
socleC22×C13C2×#99
prime radicalC24×C13C22⋊C4
cosocleC2×C4⋊Dic13C23
hypercenterC23×Q8#261
hypocenterC13#11434
LCSG ▷ C2×C4×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
2. #13129766C2semidirect
18. C22×Q8×C13C8semidirect
133. C22×Q8#1057semidirect
189. C13#11434semidirect

Maximal subgroups

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

subgroupindex
#131300672
#131297662
#131174222
#1143413