Finite Group Databasebeta

← #15779753

Subgroups — #15779753

Order 3328.

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

normal subgroups only

This group has 492 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×#2372C211yesmaximal, characteristic
6C2×C13×#124C2211yescharacteristic
7#11719517C2211yescharacteristic
8C13×#391C2212yes—
9C13×#388—22no—
10C13×#386—21no—
11C4×C13×C22⋊C4—22no—
12C13×#376—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⋊C4D412yes—
27C2×C13×C22⋊C4—22no—
28C2×C13×C22⋊C4—22no—
29C2×C13×C22⋊C4—21no—
30C2×C42×C13—22no—
31C2×C42×C13D411yescharacteristic
32C2×C42×C13D412yes—
33Q8×Dic13—42no—
34#88180—21no—
35#88180—22no—
36C2×#6099—22no—
37C2×#6099—21no—
38C13×#95—21no—
39C13×#95—21no—
40C13×#95—22no—
41C13×#95—22no—
42#88076—22no—
43#88076—22no—
44#88076C2×C414yescomplemented
45#88072—42no—
46#88048—42no—
47#88047—42no—
48#12972—131nomaximal, Sylow
49C24×C13—21no—
50C2×Q8×C13—41no—
51C2×Q8×C13—41no—
52C2×Q8×C13C22×C411yescharacteristic
53C2×Q8×C13C2×Q811yescharacteristic
54C2×Q8×C13#4112yes—
55C2×D4×C13—21no—
56C2×D4×C13—41no—
57C2×D4×C13—41no—
58C2×D4×C13C4⋊C412yes—
59C22×C4×C13—22no—
60C22×C4×C13—21no—
61C22×C4×C13—22no—
62C22×C4×C13—44no—
63C22×C4×C13—81no—
64C22×C4×C13C2×D411yescharacteristic
65C22×C4×C13C2×D412yes—
66C13×C4⋊C4—42no—
67C13×C22⋊C4—24no—
68C13×C22⋊C4—24no—
69C13×C22⋊C4—44no—
70C13×C22⋊C4—81no—
71C42×C13—21no—
72C42×C13—42no—
73C42×C13—44no—
74C42×C13C4⋊C412yes—
75C22⋊Dic13—41no—
76C4⋊Dic13—22no—
77C4⋊Dic13—22no—
78C4⋊Dic13—22no—
79C4×Dic13—42no—
80C4×Dic13—41no—
81#6099—24no—
82#6099—42no—
83#6099—41no—
84#6099—41no—
85C2×#1055—42no—
86C2×#1055—41no—
87#2372D1311yescharacteristic
88C2×#355—131no—
89#1888—131no—
90#1857—132no—
91#1394—132no—
92C23×C13—22no—
93C23×C13—42no—
94C23×C13—42no—
95C23×C13—41no—
96C23×C13—41no—
97C23×C13#12011yescharacteristic
98Q8×C13—41no—
99Q8×C13—41no—
100Q8×C13—42no—
101D4×C13—42no—
102D4×C13—81no—
103C2×C4×C13—22no—
104C2×C4×C13—21no—
105C2×C4×C13—22no—
106C2×C4×C13—21no—
107C2×C4×C13—24no—
108C2×C4×C13—22no—
109C2×C4×C13—42no—
110C2×C4×C13—42no—
111C2×C4×C13—42no—
112C2×C4×C13—42no—
113C2×C4×C13—42no—
114C2×C4×C13—42no—
115C2×C4×C13—84no—
116C2×C4×C13C2×C4⋊C411yescharacteristic
117C2×C4×C13C4×D412yes—
118C2×C4×C13C22⋊Q812yes—
119C2×C4×C13#12311yescharacteristic
120C2×Dic13—42no—
121#1055—44no—
122#1055—81no—
123C2×#124C2×D1311yescharacteristic
124#519—131no—
125#428—262no—
126#418—261no—
127C2×#101—132no—
128C2×#101—131no—
129#391C2×D1312yes—
130#388—22no—
131#386—21no—
132C4×C22⋊C4—22no—
133#376—21no—
134#355—134no—
135#339—262no—
136#339—261no—
137C22×C13—22no—
138C22×C13—44no—
139C22×C13—42no—
140C22×C13—81no—
141C22×C13—81no—
142C22×C13—82no—
143C22×C13#38511yescharacteristic
144C22×C13B(4,𝔽2)11yescharacteristic
145C22×C13#45811yescharacteristic
146C4×C13—22no—
147C4×C13—21no—
148C4×C13—21no—
149C4×C13—22no—
150C4×C13—42no—
151C4×C13—44no—
152C4×C13—82no—
153Dic13—82no—
154C22×Q8C22×D1311yescharacteristic
155C22×D4C22⋊D1311yescharacteristic
156C23×C4—21no—
157C2×M16—262no—
158C2×M16—261no—
159#124—22no—
160#124Dic2612yes—
161#124C4×D1312yes—
162C4×Q8—522no—
163#117—261no—
164#117—262no—
165C2×C4⋊C4C22⋊D1312yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C42—22no—
170C2×C42C22⋊D1311yescharacteristic
171C2×C42C22⋊D1312yes—
172#107—522no—
173#106—522no—
174#102—522no—
175#101—134no—
176#101—262no—
177#101—262no—
178#95—21no—
179#95—21no—
180#95—22no—
181#95—22no—
182C2×C13—21no—
183C2×C13—21no—
184C2×C13—82no—
185C2×C13#186711yescharacteristic
186C2×C13#220412yes—
187C24—21no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8C2×Dic2611yescharacteristic
191C2×Q8C2×C4×D1311yescharacteristic
192C2×Q8#612712yes—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4#610112yes—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—44no—
201C22×C4—81no—
202C22×C4C2×C22⋊D1311yescharacteristic
203C22×C4C2×C22⋊D1312yes—
204M16—264no—
205M16—522no—
206M16—521no—
207M16—521no—
208C2×C8—522no—
209C2×C8—521no—
210C4⋊C4—42no—
211C4⋊C4—262no—
212C4⋊C4—262no—
213C4⋊C4—262no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—44no—
217C22⋊C4—81no—
218C22⋊C4—521no—
219C42—21no—
220C42—42no—
221C42—44no—
222C42—522no—
223C42—521no—
224C42#610112yes—
225C13#1297211yescharacteristic, complemented, Sylow
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#8820711yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—522no—
251C2×C4C2×#610111yescharacteristic
252C2×C4#8817812yes—
253C2×C4C4×C22⋊D1312yes—
254C2×C4#8818311yescharacteristic
255C8—524no—
256C8—1041no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1171887111yescharacteristic
264C22#1171890411yescharacteristic
265C22#1171890511yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—44no—
272C4—82no—
273C4—1042no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#1311511711yescharacteristic
278C2#1311686212yes—
subgroupG/H
centerC22#11718905
commutatorC2×Q8×C13C22×C4
FrattiniC22×Q8C22×D13
FittingC13×#2372C2
socleC22×C13B(4,𝔽2)
prime radicalC22×D4×C13D4
cosocleC22×Q8×C13C23
hypercenter#2372D13
hypocenterC13#12972
LCSG ▷ C2×Q8×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2372

Complements

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

NH ≅ G/Ntype
3. #13115050C2semidirect
4. #13112963C2semidirect
17. #11718134C4semidirect
44. #88076C2×C4semidirect
225. C13#12972semidirect

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×#23722
#1297213