Finite Group Databasebeta

← #16789427

Subgroups — #16789427

Order 3328.

470 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 96 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#13130592C212yesmaximal, complemented
3#13129928C211yesmaximal, characteristic, complemented
4#13129925C211yesmaximal, characteristic, complemented
5#13123526C211yesmaximal, characteristic
6#11719496C2211yescharacteristic
7C23×#1055C411yescharacteristic, complemented
8C2×#88161C2211yescharacteristic
9#11719132C2212yes—
10#11719116C2212yes—
11#11719089—22no—
12C13×#407C411yescharacteristic, complemented
13#11718790C2211yescharacteristic
14#11718783C411yescharacteristic, complemented
15#11718780C411yescharacteristic, complemented
16#11718391—22no—
17#11718386—22no—
18#11718306C414yescomplemented
19#11718288C414yescomplemented
20#11718266—22no—
21C23×C4×C13C2×C411yescharacteristic
22C2×#6127C2311yescharacteristic
23#88237—22no—
24D13⋊M16—22no—
25C2×C13×M16C2×C411yescharacteristic
26C2×C13×M16C2×C411yescharacteristic
27#88207C2×C412yes—
28#88184—21no—
29#88183C2×C412yes—
30C4×C22⋊D13—22no—
31#88180C2×C411yescharacteristic
32#88178—21no—
33C22×#1055—21no—
34C22×#1055—21no—
35C22×#1055—41no—
36C22×#1055—41no—
37C22×#1055C2×C411yescharacteristic
38C22×#1055C2×C411yescharacteristic
39#88161—41no—
40#88161—41no—
41C2×#6094D412yes—
42C2×C8×D13D412yes—
43#88121C2×C414yes—
44#88119C2×C414yes—
45#88113—22no—
46#88107—22no—
47#88100—42no—
48C8×#261—42no—
49C13×C22⋊C8—22no—
50#88060—22no—
51#88055—21no—
52#88054—22no—
53C8×Dic13—21no—
54#12623—131nomaximal, Sylow
55C24×C13C4211yescharacteristic
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—41no—
59C22×C4×C13—41no—
60C22×C4×C13C4211yescharacteristic
61C22×C4×C13C22×C411yescharacteristic
62C2×C22⋊D13—22no—
63C2×C22⋊D13C22×C412yes—
64#6127—41no—
65#6127—41no—
66C2×D52#4111yescharacteristic
67C2×C4×D13C2×D412yes—
68C2×Dic26#4111yescharacteristic
69C22⋊#261—24no—
70#6122—24no—
71C4⋊#261—42no—
72C4×#261—42no—
73#6118—42no—
74D13⋊C8—42no—
75C13×M16—41no—
76C13×M16—41no—
77C2×C8×C13—21no—
78C2×C8×C13C22⋊C412yes—
79C22⋊Dic13—22no—
80C22⋊Dic13C4212yes—
81#6103—22no—
82C4⋊Dic13—21no—
83#6101—22no—
84C4×Dic13—21no—
85C2×#1055—21no—
86C2×#1055—21no—
87C2×#1055—21no—
88C2×#1055—42no—
89C2×#1055—41no—
90C2×#1055—41no—
91C2×#1055—41no—
92C2×#1055C22⋊C411yescharacteristic
93C2×#1055C22⋊C411yescharacteristic
94#6094—42no—
95C8×D13—22no—
96C8×D13—22no—
97#2929—131no—
98#2120—132no—
99#1757—132no—
100#1392—131no—
101#1391—131no—
102C23×C13—21no—
103C23×C13—41no—
104C23×C13—42no—
105C23×C13C2×C4211yescharacteristic
106C22×D13C4×D412yes—
107C2×#261—44no—
108C2×C4×C13—21no—
109C2×C4×C13—21no—
110C2×C4×C13—21no—
111C2×C4×C13—42no—
112C2×C4×C13—41no—
113C2×C4×C13—41no—
114C2×C4×C13—41no—
115C2×C4×C13C2×C22⋊C411yescharacteristic
116C2×C4×C13#11711yescharacteristic
117C22⋊D13—42no—
118C22⋊D13—44no—
119C2×Dic13—22no—
120C2×Dic13C4×D412yes—
121D52—41no—
122C4×D13—22no—
123C4×D13—22no—
124Dic26—41no—
125#1057—44no—
126C8×C13—22no—
127C8×C13—41no—
128#1055—21no—
129#1055—41no—
130#1055#9911yescharacteristic
131#1055#9911yescharacteristic
132C2×#131—131no—
133C23×C8—131no—
134#525—131no—
135#435—262no—
136C8×D4—262no—
137#424—131no—
138#413—132no—
139#410—132no—
140#408—131no—
141#407#26111yescharacteristic, complemented
142#405—131no—
143#405—262no—
144#352—134no—
145AΣL(1,𝔽16)—134no—
146#344—262no—
147C22×C13—22no—
148C22×C13—21no—
149C22×C13—41no—
150C22×C13—41no—
151C22×C13—42no—
152C22×C13—42no—
153C22×C13—41no—
154C22×C13C4×C22⋊C411yescharacteristic
155C2×D13—22no—
156C2×D13—42no—
157#261—84no—
158C4×C13—21no—
159C4×C13—41no—
160C4×C13C2×#9911yescharacteristic
161C4×C13#41011yescharacteristic
162Dic13—22no—
163Dic13—42no—
164C2×#41—131no—
165C23×C4C2×#26111yescharacteristic
166#131—521no—
167#131—521no—
168C2×M16—132no—
169C2×M16—262no—
170C2×M16C2×#26111yescharacteristic
171C2×M16C2×#26111yescharacteristic
172C22×C8—131no—
173C22×C8—132no—
174C22×C8—131no—
175C22×C8—261no—
176C22×C8—261no—
177C22×C8—521no—
178C22×C8—521no—
179#123—132no—
180C22⋊Q8—261no—
181#121—261no—
182#120—132no—
183C4×D4—262no—
184#117—131no—
185#117—262no—
186C4⋊C8—262no—
187#101—262no—
188#100—134no—
189#99—134no—
190#99—262no—
191C22⋊C8—22no—
192C22⋊C8—262no—
193#97—261no—
194#97—522no—
195C4×C8—261no—
196C4×C8—522no—
197C2×C13—21no—
198C2×C13—42no—
199C2×C13—41no—
200C2×C13C4×#9911yescharacteristic
201D13—42no—
202C24C4×#26111yescharacteristic
203#41—521no—
204#41—521no—
205C2×Q8—131no—
206C2×D4—132no—
207C2×D4—131no—
208C2×D4—262no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—132no—
214C22×C4C4×#26111yescharacteristic
215C22×C4C22×#26111yescharacteristic
216M16—41no—
217M16—41no—
218M16—264no—
219M16—522no—
220M16—522no—
221C2×C8—21no—
222C2×C8—131no—
223C2×C8—131no—
224C2×C8—262no—
225C2×C8—262no—
226C2×C8—261no—
227C2×C8—261no—
228C2×C8—261no—
229C2×C8—521no—
230C2×C8—521no—
231C2×C8—522no—
232C2×C8—522no—
233C2×C8—521no—
234C2×C8C22⋊#26112yes—
235C4⋊C4—262no—
236C4⋊C4—261no—
237C4⋊C4—522no—
238C22⋊C4—132no—
239C22⋊C4—262no—
240C22⋊C4—262no—
241C22⋊C4—264no—
242C42—261no—
243C42—522no—
244C13#1262311yescharacteristic, complemented, Sylow
245C23—21no—
246C23—41no—
247C23—42no—
248C23—132no—
249C23C2×C4×#26111yescharacteristic
250Q8—521no—
251D4—521no—
252D4—522no—
253D4—524no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—132no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—524no—
266C2×C4#8823711yescharacteristic
267C2×C4C2×C22⋊#26111yescharacteristic
268C8—22no—
269C8—41no—
270C8—131no—
271C8—131no—
272C8—261no—
273C8—524no—
274C8—521no—
275C22—22no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—262no—
283C22—522no—
284C22C4×C22⋊#26111yescharacteristic
285C4—21no—
286C4—41no—
287C4—262no—
288C4—522no—
289C4—1044no—
290C4D13⋊#9911yescharacteristic
291C4#1171914211yescharacteristic
292C2—21no—
293C2—42no—
294C2—41no—
295C2—522no—
296C2#1313058311yescharacteristic
subgroupG/H
centerC4D13⋊#99
commutatorC23×C13C2×C42
FrattiniC22×C4C22×#261
FittingC13×#407C4
socleC2×C13C4×#99
prime radical#11719496C22
cosocleC2×#6127C23
hypercenter#407#261
hypocenterC13#12623
LCSG ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #13130592C2semidirect
3. #13129928C2semidirect
4. #13129925C2semidirect
7. C23×#1055C4semidirect
12. C13×#407C4semidirect
14. #11718783C4semidirect
15. #11718780C4semidirect
18. #11718306C4semidirect
19. #11718288C4semidirect
141. #407#261semidirect
244. C13#12623semidirect

Maximal subgroups

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

subgroupindex
#131310872
#131305922
#131299282
#131299252
#131235262
#1262313