Finite Group Databasebeta

← #15779743

Subgroups — #15779743

Order 3328.

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

normal subgroups only

This group has 472 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×#11718134C211yesmaximal, characteristic
2#13115240C211yesmaximal, characteristic
3#13115050C212yesmaximal, complemented
4#13112963C212yesmaximal, complemented
5C13×#2379C211yesmaximal, characteristic
6C2×C13×#124C2211yescharacteristic
7#11719517C2211yescharacteristic
8C13×#391C2212yes—
9C13×#388—22no—
10C13×#386—21no—
11C13×#385—22no—
12C13×#382—21no—
13C2×#88076C2211yescharacteristic
14C2×#88076C2212yes—
15#11718876—21no—
16#11718625—22no—
17#11718134C414yescomplemented
18#11718115—21no—
19#11718115—22no—
20C22×Q8×C13C2311yescharacteristic
21C22×D4×C13D411yescharacteristic
22C23×C4×C13—21no—
23C13×#124—22no—
24C13×#124C2×C412yes—
25C13×#124Q812yes—
26C2×C13×C4⋊C4—21no—
27C2×C13×C4⋊C4—21no—
28C2×C13×C4⋊C4D412yes—
29C2×C13×C22⋊C4—22no—
30C2×C13×C22⋊C4—22no—
31C2×C13×C22⋊C4—21no—
32C2×C42×C13D411yescharacteristic
33C2×C42×C13D412yes—
34Q8×Dic13—41no—
35Q8×Dic13—41no—
36#88180—21no—
37#88180—22no—
38C2×#6099—22no—
39C2×#6099—21no—
40C13×#95—22no—
41C13×#95—22no—
42C13×#95—22no—
43#88076—22no—
44#88076—22no—
45#88076C2×C414yescomplemented
46#88072—41no—
47#88072—41no—
48#88048—42no—
49#88047—42no—
50#12970—131nomaximal, Sylow
51C24×C13—21no—
52C2×Q8×C13—41no—
53C2×Q8×C13—41no—
54C2×Q8×C13C22×C411yescharacteristic
55C2×Q8×C13C2×Q811yescharacteristic
56C2×Q8×C13#4112yes—
57C2×D4×C13—21no—
58C2×D4×C13—41no—
59C2×D4×C13—41no—
60C2×D4×C13C4⋊C412yes—
61C22×C4×C13—21no—
62C22×C4×C13—22no—
63C22×C4×C13—22no—
64C22×C4×C13—82no—
65C22×C4×C13—81no—
66C22×C4×C13C2×D411yescharacteristic
67C22×C4×C13C2×D412yes—
68C13×C4⋊C4—42no—
69C13×C4⋊C4—81no—
70C13×C4⋊C4—81no—
71C13×C22⋊C4—24no—
72C13×C22⋊C4—24no—
73C13×C22⋊C4—82no—
74C13×C22⋊C4—81no—
75C42×C13—21no—
76C42×C13—42no—
77C42×C13C4⋊C412yes—
78C22⋊Dic13—41no—
79C4⋊Dic13—21no—
80C4⋊Dic13—21no—
81C4⋊Dic13—22no—
82C4⋊Dic13—22no—
83C4×Dic13—42no—
84C4×Dic13—41no—
85#6099—24no—
86#6099—42no—
87#6099—41no—
88#6099—41no—
89C2×#1055—42no—
90C2×#1055—41no—
91#2379D1311yescharacteristic
92C2×#355—131no—
93#1888—131no—
94#1857—132no—
95#1394—132no—
96C23×C13—22no—
97C23×C13—42no—
98C23×C13—42no—
99C23×C13—41no—
100C23×C13—41no—
101C23×C13#12011yescharacteristic
102Q8×C13—41no—
103Q8×C13—41no—
104Q8×C13—42no—
105D4×C13—42no—
106D4×C13—81no—
107C2×C4×C13—22no—
108C2×C4×C13—21no—
109C2×C4×C13—22no—
110C2×C4×C13—21no—
111C2×C4×C13—22no—
112C2×C4×C13—41no—
113C2×C4×C13—41no—
114C2×C4×C13—42no—
115C2×C4×C13—42no—
116C2×C4×C13—42no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—84no—
121C2×C4×C13C2×C4⋊C411yescharacteristic
122C2×C4×C13C4×D412yes—
123C2×C4×C13C22⋊Q812yes—
124C2×C4×C13#12311yescharacteristic
125C2×Dic13—41no—
126C2×Dic13—41no—
127#1055—44no—
128#1055—81no—
129C2×#124C2×D1311yescharacteristic
130#519—131no—
131#428—262no—
132#418—261no—
133C2×#101—132no—
134C2×#101—131no—
135#391C2×D1312yes—
136#388—22no—
137#386—21no—
138#385—22no—
139#382—21no—
140#355—134no—
141#339—262no—
142#339—261no—
143C22×C13—21no—
144C22×C13—21no—
145C22×C13—44no—
146C22×C13—42no—
147C22×C13—82no—
148C22×C13—81no—
149C22×C13—81no—
150C22×C13#38511yescharacteristic
151C22×C13B(4,𝔽2)11yescharacteristic
152C22×C13#45811yescharacteristic
153C4×C13—22no—
154C4×C13—21no—
155C4×C13—21no—
156C4×C13—22no—
157C4×C13—42no—
158C4×C13—82no—
159C4×C13—82no—
160Dic13—81no—
161Dic13—81no—
162C22×Q8C22×D1311yescharacteristic
163C22×D4C22⋊D1311yescharacteristic
164C23×C4—21no—
165C2×M16—262no—
166C2×M16—261no—
167#124—22no—
168#124Dic2612yes—
169#124C4×D1312yes—
170C4×Q8—521no—
171C4×Q8—521no—
172#117—261no—
173#117—262no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C22⋊D1312yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C42C22⋊D1311yescharacteristic
181C2×C42C22⋊D1312yes—
182#107—522no—
183#106—522no—
184#102—521no—
185#102—521no—
186#101—134no—
187#101—262no—
188#101—262no—
189#95—22no—
190#95—22no—
191#95—22no—
192C2×C13—21no—
193C2×C13—21no—
194C2×C13—82no—
195C2×C13#186711yescharacteristic
196C2×C13#220211yescharacteristic
197C2×C13#220311yescharacteristic
198C24—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C2×Dic2611yescharacteristic
202C2×Q8C2×C4×D1311yescharacteristic
203C2×Q8#612712yes—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#610112yes—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—82no—
212C22×C4—81no—
213C22×C4C2×C22⋊D1311yescharacteristic
214C22×C4C2×C22⋊D1312yes—
215M16—264no—
216M16—522no—
217M16—521no—
218M16—521no—
219C2×C8—522no—
220C2×C8—521no—
221C4⋊C4—42no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—261no—
225C4⋊C4—261no—
226C4⋊C4—262no—
227C4⋊C4—262no—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—82no—
231C22⋊C4—81no—
232C22⋊C4—521no—
233C42—21no—
234C42—42no—
235C42—522no—
236C42—521no—
237C42#610112yes—
238C13#1297011yescharacteristic, complemented, Sylow
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#8820711yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4—521no—
265C2×C4—521no—
266C2×C4C2×#610111yescharacteristic
267C2×C4#8817812yes—
268C2×C4C4×C22⋊D1312yes—
269C2×C4#8818311yescharacteristic
270C8—524no—
271C8—1041no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1171887111yescharacteristic
280C22#1171890411yescharacteristic
281C22#1171890511yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—82no—
289C4—1041no—
290C4—1041no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#1311511711yescharacteristic
295C2#1311685311yescharacteristic
296C2#1311685711yescharacteristic
subgroupG/H
centerC22#11718905
commutatorC2×Q8×C13C22×C4
FrattiniC22×Q8C22×D13
FittingC13×#2379C2
socleC22×C13B(4,𝔽2)
prime radicalC22×D4×C13D4
cosocleC22×Q8×C13C23
hypercenter#2379D13
hypocenterC13#12970
LCSG ▷ C2×Q8×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2379

Complements

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

NH ≅ G/Ntype
3. #13115050C2semidirect
4. #13112963C2semidirect
17. #11718134C4semidirect
45. #88076C2×C4semidirect
238. C13#12970semidirect

Maximal subgroups

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

subgroupindex
C2×#117181342
#131152402
#131150502
#131129632
C13×#23792
#1297013