Finite Group Databasebeta

← #15757918

Subgroups — #15757918

Order 3328.

800 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 147 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×#11718640C211yesmaximal, characteristic
2#13114770C212yesmaximal
3C2×#11718120C211yesmaximal, characteristic
4#13112714C212yesmaximal
5C13×#2316C211yesmaximal, characteristic
6C23×Dic26C2211yescharacteristic
7C2×C13×#117—21no—
8C22×C13×C4⋊C4C2211yescharacteristic
9C22×#6101—21no—
10C13×#389—24no—
11C13×#380—21no—
12C13×#380C2212yes—
13C2×#88071—21no—
14C2×#88071C2211yescharacteristic
15C2×C22⋊#1055C2212yes—
16#11718640—22no—
17#11718640C414yescomplemented
18C2×#88050—24no—
19#11718120—22no—
20C23×C4×C13D411yescharacteristic
21C23×C4×C13D411yescharacteristic
22C23×C4×C13C2311yescharacteristic
23C23×Dic13—21no—
24C23×Dic13—21no—
25C22×Dic26—22no—
26C22×Dic26—22no—
27C22×Dic26—21no—
28C22×Dic26—44no—
29C22×Dic26C2×C412yes—
30C13×#117—42no—
31C13×#117—42no—
32C2×C13×C4⋊C4—22no—
33C2×C13×C4⋊C4—22no—
34C2×C13×C4⋊C4—22no—
35C2×C13×C4⋊C4—22no—
36C2×C13×C4⋊C4C2×C412yes—
37C2×C13×C4⋊C4D412yes—
38C2×C13×C4⋊C4D414yes—
39C2×C13×C22⋊C4—22no—
40C2×C42×C13—22no—
41C2×#6101—24no—
42C2×#6101—44no—
43C22×#1055—22no—
44C13×#95—22no—
45C13×#95—22no—
46#88071—24no—
47#88071—21no—
48#88071C2×C412yes—
49C22⋊#1055—22no—
50C22⋊#1055—22no—
51#88050—48no—
52#9015—131nomaximal, Sylow
53C24×C13C2×D411yescharacteristic
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—24no—
61C22×C4×C13—42no—
62C22×C4×C13—42no—
63C22×C4×C13C22⋊C412yes—
64C22×C4×C13C22⋊C412yes—
65C22×C4×C13SD1612yes—
66C22×C4×C13Q1612yes—
67C22×C4×C13C22×C411yescharacteristic
68C22×C4×C13C2×D411yescharacteristic
69C22×C4×C13C2×D412yes—
70C22×C4×C13#4112yes—
71C22×Dic13—24no—
72C22×Dic13—24no—
73C22×Dic13—22no—
74C22×Dic13—22no—
75C22×Dic13—42no—
76C22×Dic13—42no—
77C22×Dic13—41no—
78C22×Dic13—41no—
79C22×Dic13—41no—
80C22×Dic13—41no—
81C2×Dic26—28no—
82C2×Dic26—24no—
83C2×Dic26—44no—
84C2×Dic26—48no—
85C2×Dic26—44no—
86C2×Dic26—44no—
87C2×Dic26—42no—
88C13×C4⋊C4—24no—
89C13×C4⋊C4—24no—
90C13×C4⋊C4—44no—
91C13×C4⋊C4—44no—
92C13×C4⋊C4—82no—
93C13×C22⋊C4—42no—
94C13×C22⋊C4—42no—
95C42×C13—44no—
96#6101—48no—
97C2×#1055—44no—
98C2×#1055—42no—
99C2×#395—131no—
100#2316D1311yescharacteristic
101#1787—132no—
102C2×#324—131no—
103#1294—132no—
104C23×C13—22no—
105C23×C13—22no—
106C23×C13C2×C22⋊C411yescharacteristic
107C23×C13#12011yescharacteristic
108C23×C13#12311yescharacteristic
109C23×C13C2×SD1611yescharacteristic
110C23×C13C2×Q1611yescharacteristic
111C23×C13AGL(1,ℤ/8)11yescharacteristic
112C23×C13#13711yescharacteristic
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—24no—
118C2×C4×C13—22no—
119C2×C4×C13—22no—
120C2×C4×C13—22no—
121C2×C4×C13—41no—
122C2×C4×C13—41no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—44no—
127C2×C4×C13—44no—
128C2×C4×C13—44no—
129C2×C4×C13—81no—
130C2×C4×C13#10314yescomplemented
131C2×C4×C13C4×D412yes—
132C2×C4×C13#12112yes—
133C2×Dic13—24no—
134C2×Dic13—24no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—44no—
138C2×Dic13—44no—
139C2×Dic13—44no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—44no—
143Dic26—28no—
144Dic26—48no—
145Dic26—44no—
146#1055—82no—
147C23×Q8—131no—
148C2×#117—21no—
149C22×C4⋊C4—261no—
150C22×C4⋊C4C2×D1311yescharacteristic
151C2×#103—264no—
152C2×C22⋊C8—132no—
153#395—134no—
154#395—262no—
155#389—24no—
156#380—21no—
157#380C2×D1312yes—
158C2×#95—131no—
159C2×#95—261no—
160#324—262no—
161C22×C13—22no—
162C22×C13—22no—
163C22×C13—22no—
164C22×C13—22no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—21no—
168C22×C13—21no—
169C22×C13#38611yescharacteristic
170C22×C13C2×#10311yescharacteristic
171C22×C13#41711yescharacteristic
172C22×C13#45011yescharacteristic
173C22×C13#45111yescharacteristic
174C22×C13#45612yes—
175C22×C13B(4,𝔽2)12yes—
176C22×C13#48311yescharacteristic
177C22×C13#48411yescharacteristic
178C4×C13—24no—
179C4×C13—22no—
180C4×C13—22no—
181C4×C13—42no—
182C4×C13—84no—
183Dic13—44no—
184Dic13—44no—
185C22×Q8—132no—
186C22×Q8—261no—
187C22×Q8—262no—
188C22×Q8—262no—
189C22×Q8—524no—
190C23×C4—261no—
191C23×C4—261no—
192C23×C4C22⋊D1311yescharacteristic
193C23×C4C22⋊D1311yescharacteristic
194C23×C4C22×D1311yescharacteristic
195C22×C8—262no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—264no—
203C2×C4⋊C4—524no—
204C2×C4⋊C4C4×D1312yes—
205C2×C4⋊C4D5212yes—
206C2×C4⋊C4C22⋊D1314yes—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209#103—528no—
210C22⋊C8—262no—
211C22⋊C8—262no—
212#95—22no—
213#95—22no—
214#95—132no—
215#95—264no—
216#95—261no—
217C2×C13—21no—
218C2×C13—21no—
219C2×C13—21no—
220C2×C13—21no—
221C2×C13#160412yes—
222C2×C13#160512yes—
223C2×C13#187211yescharacteristic
224C2×C13#188611yescharacteristic
225C2×C13#188911yescharacteristic
226C24C2×C22⋊D1311yescharacteristic
227C2×Q8—268no—
228C2×Q8—264no—
229C2×Q8—524no—
230C2×Q8—524no—
231C2×Q8—524no—
232C2×Q8—528no—
233C2×Q8—522no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—264no—
244C22×C4—264no—
245C22×C4—262no—
246C22×C4—262no—
247C22×C4—522no—
248C22×C4—522no—
249C22×C4—521no—
250C22×C4—521no—
251C22×C4—521no—
252C22×C4—521no—
253C22×C4#610312yes—
254C22×C4#610312yes—
255C22×C4#610512yes—
256C22×C4#610712yes—
257C22×C4C2×C4×D1311yescharacteristic
258C22×C4C2×D5211yescharacteristic
259C22×C4#612712yes—
260C22×C4C2×C22⋊D1312yes—
261C2×C8—524no—
262C2×C8—522no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—528no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—44no—
272C13#901511yescharacteristic, complemented, Sylow
273C23—22no—
274C23—22no—
275C23C2×#610311yescharacteristic
276C23#8818311yescharacteristic
277C23#8818611yescharacteristic
278C23C2×#610511yescharacteristic
279C23#8819611yescharacteristic
280C23C2×#610711yescharacteristic
281C23#8820711yescharacteristic
282Q8—268no—
283Q8—528no—
284Q8—524no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4—264no—
303C2×C4—264no—
304C2×C4—522no—
305C2×C4—524no—
306C2×C4—524no—
307C2×C4—522no—
308C2×C4—524no—
309C2×C4—522no—
310C2×C4—522no—
311C2×C4—524no—
312C2×C4#8805014yescomplemented
313C2×C4C4×C22⋊D1312yes—
314C2×C4#8818412yes—
315C8—1042no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#1171862711yescharacteristic
325C22C2×#8805011yescharacteristic
326C22#1171863111yescharacteristic
327C22#1171863211yescharacteristic
328C22#1171887411yescharacteristic
329C22#1171887911yescharacteristic
330C22#1171888911yescharacteristic
331C22#1171890512yes—
332C22#1171890712yes—
333C4—24no—
334C4—22no—
335C4—22no—
336C4—42no—
337C4—84no—
338C4—524no—
339C4—524no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1311380512yes—
345C2#1311381012yes—
346C2#1311514311yescharacteristic
347C2#1311522911yescharacteristic
348C2#1311524411yescharacteristic
subgroupG/H
centerC23#88207
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2316C2
socleC23×C13#120
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#2316D13
hypocenterC13#9015
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
17. #11718640C4semidirect
130. C2×C4×C13#103semidirect
272. C13#9015semidirect
312. C2×C4#88050semidirect

Maximal subgroups

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

subgroupindex
C2×#117186402
#131147702
C2×#117181202
#131127142
C13×#23162
#901513