Finite Group Databasebeta

← #16786917

Subgroups — #16786917

Order 3328.

440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 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#13130079C212yesmaximal, complemented
2#13129779C212yesmaximal, complemented
3#13129765C212yesmaximal
4#13117550C211yesmaximal, characteristic
5C2×#88201C411yescharacteristic
6C2×Q8×Dic13C2211yescharacteristic
7C2×#88147C411yescharacteristic
8#11719119—22no—
9C13×#391C411yescharacteristic
10#11718637C2211yescharacteristic
11#11718600C2211yescharacteristic
12#11718593C411yescharacteristic
13#11718330—22no—
14#11718329—22no—
15#11718285—22no—
16#11718285C2214yes—
17C22×Q8×C13C2×C411yescharacteristic
18C22×D52C2×C411yescharacteristic
19C22×C4×D13—21no—
20C2×C13×C4⋊C4C2×C411yescharacteristic
21C2×C42×C13C2×C411yescharacteristic
22#88201—22no—
23#88201C814yescomplemented
24Q8×Dic13—42no—
25C2×#6103—22no—
26C2×#6103—21no—
27C2×#6103—22no—
28C2×C4⋊Dic13D412yes—
29C2×C4⋊Dic13C2311yescharacteristic
30C2×#6101—22no—
31C2×C4×Dic13C2×C411yescharacteristic
32C2×C4×Dic13C2×C411yescharacteristic
33C2×C4×Dic13D412yes—
34#88147—42no—
35C22⋊#1057—24no—
36C4⋊#1057—42no—
37C4×#1057—42no—
38C13×#95—22no—
39#88071—21no—
40#88071—21no—
41#11480—131nomaximal, Sylow
42C23×D13—21no—
43C2×Q8×C13—41no—
44C2×Q8×C13—41no—
45C2×Q8×C13C2×C812yes—
46C2×Q8×C13M1612yes—
47C22×C4×C13—21no—
48C22×C4×C13—21no—
49C22×C4×C13C22⋊C412yes—
50C22×C4×C13C22×C411yescharacteristic
51C22×Dic13—21no—
52C22×Dic13C22⋊C411yescharacteristic
53C22×Dic13C22⋊C411yescharacteristic
54C22×Dic13C2×D412yes—
55C2×D52—21no—
56C2×D52—41no—
57C2×D52—41no—
58C2×D52C2×C812yes—
59C2×C4×D13—42no—
60C2×C4×D13—82no—
61C2×#1057—44no—
62C13×C4⋊C4—41no—
63C42×C13—41no—
64#6103—28no—
65#6103—42no—
66#6103—82no—
67C4⋊Dic13—42no—
68C4⋊Dic13—41no—
69#6101—82no—
70C4×Dic13—21no—
71C4×Dic13—22no—
72C4×Dic13—22no—
73C4×Dic13—22no—
74C4×Dic13C2×C812yes—
75#2326—131no—
76#1461—132no—
77#1325—132no—
78#1318—132no—
79C23×C13C2×C22⋊C411yescharacteristic
80C22×D13—22no—
81C22×D13—42no—
82C22×D13—42no—
83C22×D13—41no—
84C22×D13—41no—
85Q8×C13—42no—
86Q8×C13—42no—
87C2×C4×C13—21no—
88C2×C4×C13—22no—
89C2×C4×C13—21no—
90C2×C4×C13—22no—
91C2×C4×C13—42no—
92C2×C4×C13—41no—
93C2×C4×C13—41no—
94C2×C4×C13—41no—
95C2×C4×C13—41no—
96C2×C4×C13C22⋊C814yes—
97C2×C4×C13C22×C811yescharacteristic
98C2×C4×C13C2×M1611yescharacteristic
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—21no—
102C2×Dic13—41no—
103C2×Dic13—41no—
104C2×Dic13—41no—
105C2×Dic13—42no—
106C2×Dic13—41no—
107C2×Dic13#9912yes—
108D52—42no—
109D52—42no—
110C4×D13—84no—
111#1057—84no—
112C2×#124—131no—
113C2×C4×Q8—131no—
114C2×C4×D4—131no—
115#431—262no—
116#391#26111yescharacteristic
117#390—131no—
118#388—262no—
119#385—262no—
120#384—131no—
121#376—131no—
122#324—134no—
123#324—262no—
124C22×C13—21no—
125C22×C13—21no—
126C22×C13C2×C22⋊C811yescharacteristic
127C22×C13C2×#9911yescharacteristic
128C22×C13#41311yescharacteristic
129C2×D13—44no—
130C2×D13—42no—
131C2×D13—82no—
132C2×D13—81no—
133C2×D13—81no—
134C4×C13—21no—
135C4×C13—21no—
136C4×C13—24no—
137C4×C13—81no—
138C4×C13—81no—
139Dic13—22no—
140Dic13—42no—
141Dic13—42no—
142Dic13—41no—
143C22×Q8C2×#26111yescharacteristic
144C22×D4—131no—
145C23×C4—261no—
146#124—134no—
147#124—262no—
148C4×Q8—522no—
149C4×D4—522no—
150C2×C4⋊C4—132no—
151C2×C4⋊C4—131no—
152C2×C4⋊C4—262no—
153C2×C4⋊C4C2×#26111yescharacteristic
154C2×C22⋊C4—262no—
155C2×C22⋊C4—261no—
156C2×C22⋊C4—262no—
157C2×C42—131no—
158C2×C42—132no—
159C2×C42—131no—
160C2×C42C2×#26111yescharacteristic
161C4⋊C8—522no—
162C22⋊C8—264no—
163C4×C8—522no—
164#95—22no—
165#95—261no—
166#95—261no—
167C2×C13—21no—
168C2×C13—21no—
169C2×C13#145911yescharacteristic
170C2×C13#212811yescharacteristic
171C2×C13#213311yescharacteristic
172D13—82no—
173C24—261no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8D13⋊C812yes—
177C2×Q8#611812yes—
178C2×D4—132no—
179C2×D4—261no—
180C2×D4—521no—
181C2×D4—521no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—131no—
185C22×C4—131no—
186C22×C4—132no—
187C22×C4—261no—
188C22×C4—522no—
189C22×C4—1042no—
190C22×C4C22⋊#26112yes—
191C22×C4C22×#26111yescharacteristic
192C2×C8—524no—
193C4⋊C4—41no—
194C4⋊C4—521no—
195C4⋊C4—522no—
196C4⋊C4—1042no—
197C22⋊C4—268no—
198C22⋊C4—522no—
199C22⋊C4—1042no—
200C42—41no—
201C42—132no—
202C42—262no—
203C42—262no—
204C42—262no—
205C42—261no—
206C13#1148011yescharacteristic, complemented, Sylow
207C23—262no—
208C23—522no—
209C23—522no—
210C23—521no—
211C23—521no—
212C23C2×C22⋊#26111yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—522no—
216D4—522no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—132no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—261no—
230C2×C4—521no—
231C2×C4—521no—
232C2×C4—521no—
233C2×C4—522no—
234C2×C4—521no—
235C2×C4—1044no—
236C2×C4#8811114yes—
237C2×C4C2×D13⋊C811yescharacteristic
238C2×C4C2×#611811yescharacteristic
239C8—1044no—
240C22—21no—
241C22—21no—
242C22—524no—
243C22—522no—
244C22—1042no—
245C22—1041no—
246C22—1041no—
247C22C2×#8811111yescharacteristic
248C22D13⋊#9911yescharacteristic
249C22#1171914811yescharacteristic
250C4—24no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—81no—
255C4—262no—
256C4—522no—
257C4—522no—
258C4—521no—
259C2—21no—
260C2—21no—
261C2—1042no—
262C2#1313007411yescharacteristic
263C2#1313111611yescharacteristic
264C2#1313113411yescharacteristic
subgroupG/H
centerC22D13⋊#99
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC13×#391C4
socleC22×C13C2×#99
prime radicalC22×D52C2×C4
cosocleC2×C4⋊Dic13C23
hypercenter#391#261
hypocenterC13#11480
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #13130079C2semidirect
2. #13129779C2semidirect
23. #88201C8semidirect
206. C13#11480semidirect

Maximal subgroups

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

subgroupindex
#131300792
#131297792
#131297652
#131175502
#1148013