Finite Group Databasebeta

← #15779728

Subgroups — #15779728

Order 3328.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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×#11718133C211yesmaximal, characteristic
2#13115225C211yesmaximal, characteristic
3#13115116C212yesmaximal
4#13112954C212yesmaximal
5C13×#2379C211yesmaximal, characteristic
6C2×C13×#124C2211yescharacteristic
7#11719503C2211yescharacteristic
8C13×#391—21no—
9C13×#388—22no—
10C13×#386C2212yes—
11C13×#385—22no—
12C13×#382—21no—
13#11718886—21no—
14C2×#88074C2212yes—
15C2×#88073C2211yescharacteristic
16#11718871—22no—
17#11718133C414yescomplemented
18#11718119—22no—
19#11718115—21no—
20C22×Q8×C13D411yescharacteristic
21C22×D4×C13C2311yescharacteristic
22C23×C4×C13D412yes—
23C13×#124—22no—
24C13×#124C2×C412yes—
25C13×#124Q812yes—
26C2×C13×C4⋊C4—21no—
27C2×C13×C4⋊C4—21no—
28C2×C13×C4⋊C4—21no—
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4—22no—
31C2×C13×C22⋊C4D412yes—
32C2×C42×C13—21no—
33C2×C42×C13D411yescharacteristic
34C2×C22⋊Dic13—22no—
35C2×C22⋊Dic13—22no—
36D4×Dic13—41no—
37D4×Dic13—41no—
38#88180—21no—
39C2×#6101—42no—
40C2×#6099—21no—
41C13×#95—22no—
42C13×#95—22no—
43C13×#95—22no—
44#88075—41no—
45#88075—41no—
46#88074—22no—
47#88074C2×C414yes—
48#88073—22no—
49#88071—42no—
50#12967—131nomaximal, Sylow
51C24×C13C2×D412yes—
52C2×Q8×C13—21no—
53C2×Q8×C13—41no—
54C2×Q8×C13—41no—
55C2×Q8×C13C4⋊C412yescomplemented
56C2×D4×C13—41no—
57C2×D4×C13—41no—
58C2×D4×C13C22×C411yescharacteristic
59C2×D4×C13C2×Q811yescharacteristic
60C2×D4×C13#4112yes—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—22no—
64C22×C4×C13—22no—
65C22×C4×C13—42no—
66C22×C4×C13—82no—
67C22×C4×C13C2×D411yescharacteristic
68C22×Dic13—22no—
69C22×Dic13—22no—
70C22×Dic13—42no—
71C13×C4⋊C4—81no—
72C13×C4⋊C4—81no—
73C13×C4⋊C4—81no—
74C13×C22⋊C4—24no—
75C13×C22⋊C4—24no—
76C13×C22⋊C4—42no—
77C13×C22⋊C4—82no—
78C42×C13—21no—
79C42×C13—81no—
80C42×C13C4⋊C412yescomplemented
81C22⋊Dic13—24no—
82C22⋊Dic13—42no—
83C22⋊Dic13—42no—
84C22⋊Dic13—41no—
85C4⋊Dic13—21no—
86C4⋊Dic13—21no—
87#6101—84no—
88C4×Dic13—41no—
89#6099—41no—
90#6099—41no—
91C2×#1055—41no—
92#2379D1311yescharacteristic
93C2×#354—131no—
94#1885—131no—
95#1867—132no—
96#1390—132no—
97C23×C13—22no—
98C23×C13—22no—
99C23×C13—42no—
100C23×C13—42no—
101C23×C13C4×D412yes—
102C23×C13#12011yescharacteristic
103C23×C13C22⋊Q812yes—
104Q8×C13—42no—
105Q8×C13—81no—
106D4×C13—41no—
107D4×C13—41no—
108D4×C13—42no—
109C2×C4×C13—21no—
110C2×C4×C13—21no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C2×C4×C13—41no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—44no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13—82no—
125C2×C4×C13C2×C4⋊C411yescharacteristic
126C2×C4×C13#12311yescharacteristic
127C2×Dic13—44no—
128C2×Dic13—41no—
129C2×Dic13—41no—
130C2×Dic13—82no—
131C2×Dic13—82no—
132C2×Dic13—82no—
133C2×Dic13—82no—
134#1055—81no—
135C2×#124C2×D1311yescharacteristic
136#518—131no—
137#419—261no—
138C2×#100—131no—
139C2×#99—132no—
140#391—21no—
141#388—22no—
142#386C2×D1312yes—
143#385—22no—
144#385—262no—
145#382—21no—
146#354—134no—
147#342—262no—
148#339—261no—
149C22×C13—21no—
150C22×C13—21no—
151C22×C13—22no—
152C22×C13—22no—
153C22×C13—42no—
154C22×C13—42no—
155C22×C13—42no—
156C22×C13—42no—
157C22×C13—82no—
158C22×C13#38511yescharacteristic
159C22×C13B(4,𝔽2)11yescharacteristic
160C22×C13#45811yescharacteristic
161C4×C13—21no—
162C4×C13—21no—
163C4×C13—42no—
164C4×C13—42no—
165C4×C13—82no—
166C4×C13—82no—
167Dic13—84no—
168Dic13—81no—
169Dic13—81no—
170C22×Q8C22⋊D1311yescharacteristic
171C22×D4C22×D1311yescharacteristic
172C23×C4C22⋊D1312yes—
173C2×M16—261no—
174#124—22no—
175#124Dic2612yes—
176#124C4×D1312yes—
177C4×D4—521no—
178C4×D4—521no—
179#117—261no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—522no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—262no—
187C2×C22⋊C4—262no—
188C2×C22⋊C4C22⋊D1312yes—
189C2×C42—21no—
190C2×C42C22⋊D1311yescharacteristic
191#103—521no—
192#103—521no—
193#100—262no—
194#99—134no—
195#99—262no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—522no—
200C2×C13—21no—
201C2×C13—21no—
202C2×C13—42no—
203C2×C13—42no—
204C2×C13#186711yescharacteristic
205C2×C13#219911yescharacteristic
206C2×C13#220111yescharacteristic
207C24C2×C22⋊D1312yes—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8#610112yescomplemented
212C2×D4—41no—
213C2×D4—41no—
214C2×D4C2×Dic2611yescharacteristic
215C2×D4C2×C4×D1311yescharacteristic
216C2×D4#612712yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—262no—
224C22×C4—262no—
225C22×C4—522no—
226C22×C4C2×C22⋊D1311yescharacteristic
227M16—521no—
228M16—521no—
229C2×C8—521no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—261no—
234C4⋊C4—261no—
235C4⋊C4—1044no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—42no—
239C22⋊C4—82no—
240C22⋊C4—264no—
241C22⋊C4—522no—
242C22⋊C4—522no—
243C22⋊C4—521no—
244C42—21no—
245C42—81no—
246C42—521no—
247C42#610112yescomplemented
248C13#1296711yescharacteristic, complemented, Sylow
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23#8817812yes—
254C23C4×C22⋊D1312yes—
255C23#8820711yescharacteristic
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—524no—
278C2×C4—521no—
279C2×C4—521no—
280C2×C4—1042no—
281C2×C4—1042no—
282C2×C4—1042no—
283C2×C4—1042no—
284C2×C4C2×#610111yescharacteristic
285C2×C4#8818311yescharacteristic
286C8—1041no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1171887111yescharacteristic
297C22#1171890411yescharacteristic
298C22#1171890511yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—82no—
305C4—1044no—
306C4—1041no—
307C4—1041no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#1311511711yescharacteristic
313C2#1311683711yescharacteristic
314C2#1311684911yescharacteristic
subgroupG/H
centerC22#11718905
commutatorC2×D4×C13C22×C4
FrattiniC22×D4C22×D13
FittingC13×#2379C2
socleC22×C13B(4,𝔽2)
prime radical, cosocleC22×D4×C13C23
hypercenter#2379D13
hypocenterC13#12967
LCSG ▷ C2×D4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2379

Complements

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

NH ≅ G/Ntype
17. #11718133C4semidirect
55. C2×Q8×C13C4⋊C4semidirect
80. C42×C13C4⋊C4semidirect
211. C2×Q8#6101semidirect
247. C42#6101semidirect
248. C13#12967semidirect

Maximal subgroups

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

subgroupindex
C2×#117181332
#131152252
#131151162
#131129542
C13×#23792
#1296713