Finite Group Databasebeta

← #15778109

Subgroups — #15778109

Order 3328.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#11718134C212yesmaximal
2C4×#88076C212yesmaximal, complemented
3#13112963C212yesmaximal, complemented
4C13×#2326C211yesmaximal, characteristic
5C2×C13×#124C2211yescharacteristic
6C2×C4×Q8×C13C2211yescharacteristic
7C2×C4×D4×C13C411yescharacteristic
8C13×#391C2211yescharacteristic
9C13×#390C411yescharacteristic
10C13×#388—22no—
11C13×#385—22no—
12C13×#384C411yescharacteristic
13C13×#376C411yescharacteristic
14C2×#88076C2214yes—
15C4×#6099—22no—
16#11718134C418yescomplemented
17#11718122—22no—
18C22×Q8×C13C2311yescharacteristic
19C22×D4×C13C2×C411yescharacteristic
20C23×C4×C13—21no—
21C13×#124—22no—
22C13×#124C2×C414yes—
23C4×Q8×C13—42no—
24C4×D4×C13—42no—
25C2×C13×C4⋊C4—22no—
26C2×C13×C4⋊C4C2×C411yescharacteristic
27C2×C13×C4⋊C4C2×C411yescharacteristic
28C2×C13×C4⋊C4D412yes—
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4—21no—
31C2×C13×C22⋊C4—22no—
32C2×C42×C13C2×C411yescharacteristic
33C2×C42×C13C2×C411yescharacteristic
34C2×C42×C13C2×C411yescharacteristic
35C2×C42×C13D412yes—
36C2×#6099—24no—
37C13×#95—21no—
38C13×#95—21no—
39C13×#95—22no—
40#88076—24no—
41#88076C2×C418yescomplemented
42#88043—42no—
43C4×#1055—42no—
44#12632—131nomaximal, Sylow
45C24×C13—21no—
46C2×Q8×C13—41no—
47C2×Q8×C13—41no—
48C2×Q8×C13C22×C412yes—
49C2×Q8×C13#4112yes—
50C2×D4×C13—21no—
51C2×D4×C13—41no—
52C2×D4×C13—41no—
53C2×D4×C13C4212yes—
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13—42no—
58C22×C4×C13—82no—
59C22×C4×C13C22⋊C411yescharacteristic
60C22×C4×C13C22⋊C412yes—
61C22×C4×C13C22⋊C411yescharacteristic
62C22×C4×C13C22×C411yescharacteristic
63C22×C4×C13C2×D412yes—
64C13×C4⋊C4—41no—
65C13×C4⋊C4—42no—
66C13×C4⋊C4—41no—
67C13×C4⋊C4—82no—
68C13×C22⋊C4—28no—
69C13×C22⋊C4—42no—
70C13×C22⋊C4—82no—
71C42×C13—22no—
72C42×C13—22no—
73C42×C13—22no—
74C42×C13—21no—
75C42×C13—41no—
76C42×C13C4212yes—
77#6099—28no—
78#6099—44no—
79C2×#1055—44no—
80#2326D1311yescharacteristic
81C2×#355—132no—
82C4×#101—132no—
83#1394—132no—
84C23×C13—22no—
85C23×C13—42no—
86C23×C13—42no—
87C23×C13—41no—
88C23×C13—41no—
89C23×C13C2×C22⋊C411yescharacteristic
90Q8×C13—42no—
91Q8×C13—42no—
92D4×C13—42no—
93D4×C13—42no—
94C2×C4×C13—22no—
95C2×C4×C13—21no—
96C2×C4×C13—22no—
97C2×C4×C13—21no—
98C2×C4×C13—22no—
99C2×C4×C13—21no—
100C2×C4×C13—22no—
101C2×C4×C13—41no—
102C2×C4×C13—41no—
103C2×C4×C13—41no—
104C2×C4×C13—42no—
105C2×C4×C13—41no—
106C2×C4×C13—41no—
107C2×C4×C13—42no—
108C2×C4×C13—41no—
109C2×C4×C13—41no—
110C2×C4×C13—41no—
111C2×C4×C13—84no—
112C2×C4×C13#9912yes—
113C2×C4×C13C2×C4211yescharacteristic
114C2×C4×C13#11711yescharacteristic
115C2×C4×C13C4×D414yes—
116#1055—48no—
117C2×#124C2×D1311yescharacteristic
118C2×C4×Q8C2×D1311yescharacteristic
119C2×C4×D4Dic1311yescharacteristic
120C2×#101—134no—
121C4×M16—262no—
122#391C2×D1311yescharacteristic
123#390Dic1311yescharacteristic
124#388—22no—
125#385—22no—
126#384Dic1311yescharacteristic
127#376Dic1311yescharacteristic
128#355—138no—
129#343—262no—
130C22×C13—21no—
131C22×C13—21no—
132C22×C13—44no—
133C22×C13—42no—
134C22×C13—82no—
135C22×C13—81no—
136C22×C13—81no—
137C22×C13C4×C22⋊C411yescharacteristic
138C22×C13C2×#9911yescharacteristic
139C22×C13#41011yescharacteristic
140C4×C13—22no—
141C4×C13—24no—
142C4×C13—21no—
143C4×C13—21no—
144C4×C13—42no—
145C4×C13—42no—
146C4×C13—41no—
147C4×C13—81no—
148C4×C13—81no—
149C22×Q8C22×D1311yescharacteristic
150C22×D4C2×Dic1311yescharacteristic
151C23×C4—21no—
152C2×M16—264no—
153#124—22no—
154#124C4×D1314yes—
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic1311yescharacteristic
159C2×C4⋊C4C2×Dic1311yescharacteristic
160C2×C4⋊C4C22⋊D1312yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C2×Dic1311yescharacteristic
165C2×C42C2×Dic1311yescharacteristic
166C2×C42C2×Dic1311yescharacteristic
167C2×C42C22⋊D1312yes—
168#101—138no—
169#101—264no—
170#97—522no—
171C4×C8—522no—
172#95—21no—
173#95—21no—
174#95—22no—
175C2×C13—21no—
176C2×C13—21no—
177C2×C13—82no—
178C2×C13C4×#9911yescharacteristic
179C2×C13#213311yescharacteristic
180C2×C13#213411yescharacteristic
181C24—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8C2×C4×D1312yes—
185C2×Q8#612712yes—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4C4×Dic1312yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4C22⋊Dic1311yescharacteristic
196C22×C4C22⋊Dic1312yes—
197C22×C4C22⋊Dic1311yescharacteristic
198C22×C4C22×Dic1311yescharacteristic
199C22×C4C2×C22⋊D1312yes—
200M16—268no—
201M16—524no—
202C2×C8—524no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—82no—
207C22⋊C4—28no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—41no—
215C42C4×Dic1312yes—
216C13#1263211yescharacteristic, complemented, Sylow
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊Dic1311yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#8807412yes—
246C2×C4C2×C4×Dic1311yescharacteristic
247C2×C4#8818011yescharacteristic
248C2×C4C4×C22⋊D1314yes—
249C8—528no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊Dic1311yescharacteristic
258C22C2×#8807411yescharacteristic
259C22#1171890111yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—22no—
264C4—42no—
265C4—42no—
266C4—41no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#8807411yescharacteristic
273C2#1311652511yescharacteristic
274C2#1311652811yescharacteristic
subgroupG/H
centerC22C2×#88074
commutatorC2×C4×C13C2×C42
FrattiniC22×Q8C22×D13
FittingC13×#2326C2
socleC22×C13C2×#99
prime radicalC22×D4×C13C2×C4
cosocleC22×Q8×C13C23
hypercenter#2326D13
hypocenterC13#12632
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C4×#88076C2semidirect
3. #13112963C2semidirect
16. #11718134C4semidirect
41. #88076C2×C4semidirect
216. C13#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#117181342
C4×#880762
#131129632
C13×#23262
#1263213