Finite Group Databasebeta

← #16789429

Subgroups — #16789429

Order 3328.

456 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 98 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#13131087C212yesmaximal
2#13130591C212yesmaximal, complemented
3#13129927C211yesmaximal, characteristic, complemented
4#13129924C211yesmaximal, characteristic, complemented
5#13123539C211yesmaximal, characteristic
6C22×C13×M16C411yescharacteristic, complemented
7#11719496C2211yescharacteristic
8C2×#88155C2211yescharacteristic
9#11719132C2212yes—
10#11719116C2212yes—
11#11719089—22no—
12#11718868C411yescharacteristic, complemented
13#11718789—22no—
14#11718788—22no—
15#11718771C2211yescharacteristic
16#11718763C411yescharacteristic, complemented
17#11718762C411yescharacteristic, complemented
18#11718306C414yescomplemented
19#11718288C414yescomplemented
20#11718266—22no—
21C23×C4×C13C2×C411yescharacteristic
22C2×#6127C2311yescharacteristic
23#88237—22no—
24D13⋊M16—22no—
25C2×C13×M16—21no—
26C2×C13×M16—21no—
27C2×C13×M16—42no—
28C22×C8×C13C2×C411yescharacteristic
29C22×C8×C13C2×C411yescharacteristic
30#88207C2×C412yes—
31#88184—21no—
32#88183C2×C412yes—
33C4×C22⋊D13—22no—
34#88180C2×C411yescharacteristic
35#88178—21no—
36C2×#6099C2×C411yescharacteristic
37C2×#6099C2×C411yescharacteristic
38#88155—42no—
39C2×#6094D412yes—
40C2×C8×D13D412yes—
41#88121C2×C414yes—
42#88119C2×C414yes—
43#88113—22no—
44#88107—22no—
45#88100—42no—
46C8×#261—42no—
47C22⋊#1055—22no—
48#88060—22no—
49#88055—21no—
50#88054—22no—
51C8×Dic13—21no—
52#12624—131nomaximal, Sylow
53C24×C13C4211yescharacteristic
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—41no—
57C22×C4×C13—41no—
58C22×C4×C13C4211yescharacteristic
59C22×C4×C13C22×C411yescharacteristic
60C2×C22⋊D13—22no—
61C2×C22⋊D13C22×C412yes—
62#6127—41no—
63#6127—41no—
64C2×D52#4111yescharacteristic
65C2×C4×D13C2×D412yes—
66C2×Dic26#4111yescharacteristic
67C22⋊#261—24no—
68#6122—24no—
69C4⋊#261—42no—
70C4×#261—42no—
71#6118—42no—
72D13⋊C8—42no—
73C13×M16—42no—
74C13×M16—42no—
75C2×C8×C13—22no—
76C2×C8×C13—21no—
77C2×C8×C13—41no—
78C2×C8×C13C22⋊C411yescharacteristic
79C2×C8×C13C22⋊C411yescharacteristic
80C22⋊Dic13—22no—
81C22⋊Dic13C4212yes—
82#6103—22no—
83C4⋊Dic13—21no—
84#6101—22no—
85C4×Dic13—21no—
86#6099—41no—
87#6099—41no—
88C2×#1055—21no—
89C2×#1055C22⋊C412yes—
90#6094—42no—
91C8×D13—22no—
92C8×D13—22no—
93#2930—131no—
94#2120—132no—
95#1757—132no—
96#1392—131no—
97#1391—131no—
98C23×C13—21no—
99C23×C13—41no—
100C23×C13—81no—
101C23×C13C2×C4211yescharacteristic
102C22×D13C4×D412yes—
103C2×#261—44no—
104C2×C4×C13—21no—
105C2×C4×C13—21no—
106C2×C4×C13—21no—
107C2×C4×C13—42no—
108C2×C4×C13—41no—
109C2×C4×C13—41no—
110C2×C4×C13—41no—
111C2×C4×C13C2×C22⋊C411yescharacteristic
112C2×C4×C13#11711yescharacteristic
113C22⋊D13—42no—
114C22⋊D13—82no—
115C2×Dic13—22no—
116C2×Dic13C4×D412yes—
117D52—41no—
118C4×D13—22no—
119C4×D13—22no—
120Dic26—41no—
121#1057—44no—
122C8×C13—21no—
123C8×C13—41no—
124C8×C13#9912yes—
125#1055—22no—
126#1055—41no—
127C2×#131—131no—
128C22×M16#26111yescharacteristic, complemented
129#525—131no—
130#436—262no—
131#435—262no—
132#424—131no—
133#413—132no—
134#410—132no—
135#408—131no—
136#407—131no—
137#405—131no—
138#405—262no—
139#352—134no—
140AΣL(1,𝔽16)—134no—
141#344—262no—
142C22×C13—21no—
143C22×C13—41no—
144C22×C13—41no—
145C22×C13—41no—
146C22×C13—41no—
147C22×C13—82no—
148C22×C13C4×C22⋊C411yescharacteristic
149C2×D13—22no—
150C2×D13—42no—
151#261—84no—
152C4×C13—21no—
153C4×C13—41no—
154C4×C13C2×#9911yescharacteristic
155C4×C13#41011yescharacteristic
156Dic13—22no—
157Dic13—42no—
158C2×#41—131no—
159C23×C4C2×#26111yescharacteristic
160#131—522no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—42no—
164C2×M16—131no—
165C2×M16—132no—
166C2×M16—131no—
167C2×M16—262no—
168C22×C8—132no—
169C22×C8C2×#26111yescharacteristic
170C22×C8C2×#26111yescharacteristic
171#123—132no—
172C22⋊Q8—261no—
173#121—261no—
174#120—132no—
175C4×D4—262no—
176#117—131no—
177#117—262no—
178C4⋊C8—262no—
179#101—262no—
180#100—134no—
181#99—134no—
182#99—262no—
183C22⋊C8—262no—
184C22⋊C8—262no—
185#97—261no—
186#97—522no—
187C4×C8—261no—
188C4×C8—522no—
189C2×C13—21no—
190C2×C13—41no—
191C2×C13—81no—
192C2×C13C4×#9911yescharacteristic
193D13—42no—
194C24C4×#26111yescharacteristic
195#41—521no—
196#41—521no—
197C2×Q8—131no—
198C2×D4—132no—
199C2×D4—131no—
200C2×D4—262no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—132no—
206C22×C4C4×#26111yescharacteristic
207C22×C4C22×#26111yescharacteristic
208M16—42no—
209M16—42no—
210M16—264no—
211M16—522no—
212M16—521no—
213M16—522no—
214M16—521no—
215C2×C8—22no—
216C2×C8—21no—
217C2×C8—41no—
218C2×C8—132no—
219C2×C8—262no—
220C2×C8—262no—
221C2×C8—261no—
222C2×C8—522no—
223C2×C8C22⋊#26111yescharacteristic
224C2×C8C22⋊#26111yescharacteristic
225C4⋊C4—262no—
226C4⋊C4—261no—
227C4⋊C4—522no—
228C22⋊C4—132no—
229C22⋊C4—262no—
230C22⋊C4—262no—
231C22⋊C4—264no—
232C42—261no—
233C42—522no—
234C13#1262411yescharacteristic, complemented, Sylow
235C23—21no—
236C23—41no—
237C23—81no—
238C23—132no—
239C23C2×C4×#26111yescharacteristic
240Q8—521no—
241D4—521no—
242D4—522no—
243D4—1042no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—132no—
252C2×C4—262no—
253C2×C4—262no—
254C2×C4—262no—
255C2×C4—524no—
256C2×C4#8823711yescharacteristic
257C2×C4C2×C22⋊#26111yescharacteristic
258C8—21no—
259C8—41no—
260C8—262no—
261C8—524no—
262C8—521no—
263C8#8811912yes—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—82no—
270C22—262no—
271C22—522no—
272C22C4×C22⋊#26111yescharacteristic
273C4—21no—
274C4—41no—
275C4—262no—
276C4—522no—
277C4—1044no—
278C4#1171911611yescharacteristic
279C4C2×#8811911yescharacteristic
280C2—21no—
281C2—41no—
282C2—81no—
283C2—522no—
284C2C4×#8811911yescharacteristic
subgroupG/H
centerC4C2×#88119
commutatorC23×C13C2×C42
FrattiniC22×C4C22×#261
FittingC22×C13×M16C4
socleC2×C13C4×#99
prime radical#11719496C22
cosocleC2×#6127C23
hypercenterC22×M16#261
hypocenterC13#12624
LCSG ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. #13130591C2semidirect
3. #13129927C2semidirect
4. #13129924C2semidirect
6. C22×C13×M16C4semidirect
12. #11718868C4semidirect
16. #11718763C4semidirect
17. #11718762C4semidirect
18. #11718306C4semidirect
19. #11718288C4semidirect
128. C22×M16#261semidirect
234. C13#12624semidirect

Maximal subgroups

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

subgroupindex
#131310872
#131305912
#131299272
#131299242
#131235392
#1262413