Finite Group Databasebeta

← #15778401

Subgroups — #15778401

Order 3328.

976 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 139 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×#11718131C214yesmaximal, complemented
2#13114764C211yesmaximal, characteristic
3#13114754C211yesmaximal, characteristic
4C2×C13×#536C211yesmaximal, characteristic
5C22×C13×#41C2211yescharacteristic
6C23×Q8×C13C411yescharacteristic, complemented
7C13×#536—28no—
8C2×C13×C4⋊Q8—21no—
9C2×C13×#124—21no—
10C2×C13×#123C412yescomplemented
11C2×C13×#123C2212yes—
12C2×C13×C22⋊Q8—21no—
13C2×C13×C22⋊Q8—21no—
14C2×C13×#117C411yescharacteristic, complemented
15C2×#88076—21no—
16C2×#88074—21no—
17C2×#88074C2212yes—
18C2×#88073C2212yes—
19#11718872—21no—
20#11718868—21no—
21#11718131—28no—
22C2×C13×#41—24no—
23C2×C13×#41—42no—
24C2×C13×#41—42no—
25C22×Q8×C13—21no—
26C22×Q8×C13—21no—
27C22×Q8×C13—24no—
28C22×Q8×C13—42no—
29C22×Q8×C13—42no—
30C22×Q8×C13D411yescharacteristic
31C22×D4×C13C2×C411yescharacteristic
32C22×D4×C13D411yescharacteristic
33C22×D4×C13C2311yescharacteristic
34C23×C4×C13C2×C411yescharacteristic
35C23×C4×C13C2×C412yes—
36C23×C4×C13D411yescharacteristic
37C23×C4×C13D411yescharacteristic
38C13×C4⋊Q8—28no—
39C13×#124—28no—
40C13×#123—24no—
41C13×#123—24no—
42C13×#123—24no—
43C13×#123D418yes—
44C13×C22⋊Q8—28no—
45C13×C22⋊Q8—28no—
46C13×#117—24no—
47C2×C13×C4⋊C4—21no—
48C2×C13×C4⋊C4—22no—
49C2×C13×C4⋊C4—22no—
50C2×C13×C22⋊C4—22no—
51C2×C13×C22⋊C4—22no—
52C2×C13×C22⋊C4C2×C412yes—
53C2×C42×C13—21no—
54C2×C22⋊Dic13—22no—
55C2×#6099—22no—
56#88076—42no—
57#88074—24no—
58#88074—42no—
59#88073—24no—
60#88071—42no—
61C22⋊#1055—42no—
62#12692—131nomaximal, Sylow
63C24×C13C22⋊C411yescharacteristic
64C24×C13C22×C411yescharacteristic
65C24×C13C2×D411yescharacteristic
66C13×#41—416no—
67C2×Q8×C13—24no—
68C2×Q8×C13—24no—
69C2×Q8×C13—22no—
70C2×Q8×C13—22no—
71C2×Q8×C13—41no—
72C2×Q8×C13—416no—
73C2×Q8×C13—42no—
74C2×Q8×C13—42no—
75C2×Q8×C13—42no—
76C2×Q8×C13—42no—
77C2×D4×C13—22no—
78C2×D4×C13—22no—
79C2×D4×C13—22no—
80C2×D4×C13—41no—
81C2×D4×C13—41no—
82C2×D4×C13—41no—
83C2×D4×C13—41no—
84C2×D4×C13—41no—
85C2×D4×C13C2×D414yes—
86C22×C4×C13—22no—
87C22×C4×C13—22no—
88C22×C4×C13—22no—
89C22×C4×C13—22no—
90C22×C4×C13—22no—
91C22×C4×C13—22no—
92C22×C4×C13—22no—
93C22×C4×C13—22no—
94C22×C4×C13—22no—
95C22×C4×C13—22no—
96C22×C4×C13—41no—
97C22×C4×C13—41no—
98C22×C4×C13—42no—
99C22×C4×C13—42no—
100C22×C4×C13C22⋊C411yescharacteristic
101C22×C4×C13C22⋊C411yescharacteristic
102C22×C4×C13C22⋊C414yes—
103C22×C4×C13C22⋊C411yescharacteristic
104C22×C4×C13C2×D411yescharacteristic
105C22×Dic13—42no—
106C13×C4⋊C4—24no—
107C13×C4⋊C4—28no—
108C13×C4⋊C4—28no—
109C13×C22⋊C4—28no—
110C13×C22⋊C4—22no—
111C13×C22⋊C4—28no—
112C13×C22⋊C4C22⋊C414yes—
113C42×C13—24no—
114C22⋊Dic13—42no—
115C22⋊Dic13—42no—
116#6099—42no—
117#6099—42no—
118C2×#1055—42no—
119C2×#536D1311yescharacteristic
120C2×#352—134no—
121#1785—131no—
122#1782—131no—
123C23×C13—21no—
124C23×C13—21no—
125C23×C13—21no—
126C23×C13—21no—
127C23×C13—21no—
128C23×C13—21no—
129C23×C13—21no—
130C23×C13—42no—
131C23×C13—42no—
132C23×C13—42no—
133C23×C13C2×C22⋊C411yescharacteristic
134C23×C13C2×C22⋊C412yes—
135C23×C13#12012yes—
136Q8×C13—44no—
137Q8×C13—48no—
138Q8×C13—48no—
139D4×C13—44no—
140D4×C13—44no—
141D4×C13—44no—
142C2×C4×C13—28no—
143C2×C4×C13—21no—
144C2×C4×C13—21no—
145C2×C4×C13—21no—
146C2×C4×C13—21no—
147C2×C4×C13—21no—
148C2×C4×C13—21no—
149C2×C4×C13—24no—
150C2×C4×C13—24no—
151C2×C4×C13—21no—
152C2×C4×C13—21no—
153C2×C4×C13—21no—
154C2×C4×C13—44no—
155C2×C4×C13—44no—
156C2×C4×C13—44no—
157C2×C4×C13—44no—
158C2×C4×C13—42no—
159C2×C4×C13—42no—
160C2×C4×C13—44no—
161C2×C4×C13#9912yes—
162C2×C4×C13#9912yes—
163C2×C4×C13#12012yes—
164C2×Dic13—82no—
165C2×Dic13—84no—
166#1055—82no—
167C22×#41C2×D1311yescharacteristic
168C23×Q8Dic1311yescharacteristic, complemented
169#536—28no—
170C2×C4⋊Q8—21no—
171C2×#124—21no—
172C2×#123Dic1312yescomplemented
173C2×#123C2×D1312yes—
174C2×C22⋊Q8—21no—
175C2×C22⋊Q8—21no—
176C2×#117Dic1311yescharacteristic, complemented
177C2×#101—261no—
178C2×#100—132no—
179C2×#99—132no—
180C2×#99—261no—
181#407—261no—
182#381—261no—
183#352—268no—
184C22×C13—22no—
185C22×C13—22no—
186C22×C13—22no—
187C22×C13—42no—
188C22×C13—42no—
189C22×C13—42no—
190C22×C13—42no—
191C22×C13—42no—
192C22×C13—42no—
193C22×C13—42no—
194C22×C13—41no—
195C22×C13—81no—
196C22×C13—81no—
197C22×C13#37911yescharacteristic
198C22×C13C2×#9911yescharacteristic
199C22×C13C2×#9911yescharacteristic
200C4×C13—22no—
201C4×C13—22no—
202C4×C13—22no—
203C4×C13—44no—
204C4×C13—41no—
205C4×C13—44no—
206Dic13—162no—
207C2×#41—24no—
208C2×#41—42no—
209C2×#41—42no—
210C22×Q8—21no—
211C22×Q8—24no—
212C22×Q8—21no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8C22⋊D1311yescharacteristic
216C22×D4C2×Dic1311yescharacteristic
217C22×D4C22⋊D1311yescharacteristic
218C22×D4C22×D1311yescharacteristic
219C23×C4C2×Dic1311yescharacteristic
220C23×C4C2×Dic1312yes—
221C23×C4C22⋊D1311yescharacteristic
222C23×C4C22⋊D1311yescharacteristic
223C2×M16—262no—
224C4⋊Q8—28no—
225#124—28no—
226#123—24no—
227#123—24no—
228#123—24no—
229#123C22⋊D1318yes—
230C22⋊Q8—28no—
231C22⋊Q8—28no—
232#117—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—262no—
239C2×C22⋊C4C2×Dic1312yes—
240C2×C42—21no—
241#101—522no—
242#100—264no—
243#99—264no—
244#99—522no—
245C22⋊C8—522no—
246#95—522no—
247C2×C13—21no—
248C2×C13—21no—
249C2×C13—42no—
250C2×C13—42no—
251C2×C13—81no—
252C2×C13#178111yescharacteristic
253C2×C13#212312yes—
254C24C22⋊Dic1311yescharacteristic
255C24C22×Dic1311yescharacteristic
256C24C2×C22⋊D1311yescharacteristic
257#41—416no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—416no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4C2×C22⋊D1314yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—522no—
292C22×C4C22⋊Dic1311yescharacteristic
293C22×C4C22⋊Dic1311yescharacteristic
294C22×C4C22⋊Dic1314yes—
295C22×C4C22⋊Dic1311yescharacteristic
296C22×C4C2×C22⋊D1311yescharacteristic
297M16—522no—
298M16—522no—
299C2×C8—522no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—22no—
305C22⋊C4—28no—
306C22⋊C4—522no—
307C22⋊C4—522no—
308C22⋊C4C22⋊Dic1314yes—
309C42—24no—
310C13#1269211yescharacteristic, complemented, Sylow
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23C2×C22⋊Dic1311yescharacteristic
322C23C2×C22⋊Dic1312yes—
323C23#8820712yes—
324Q8—44no—
325Q8—48no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—1042no—
350C2×C4—1044no—
351C2×C4#8807412yes—
352C2×C4#8807412yes—
353C2×C4#8820712yes—
354C8—1042no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22C2×#8807411yescharacteristic
369C22C2×#8807411yescharacteristic
370C22#1171890811yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—2082no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#1311475111yescharacteristic
384C2#1311649012yes—
subgroupG/H
centerC22C2×#88074
commutatorC24×C13C22×C4
FrattiniC22×D4C22×D13
FittingC2×C13×#536C2
socleC22×C13C2×#99
prime radicalC22×C13×#41C22
cosocleC22×D4×C13C23
hypercenterC2×#536D13
hypocenterC13#12692
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×#11718131C2semidirect
6. C23×Q8×C13C4semidirect
10. C2×C13×#123C4semidirect
14. C2×C13×#117C4semidirect
168. C23×Q8Dic13semidirect
172. C2×#123Dic13semidirect
176. C2×#117Dic13semidirect
310. C13#12692semidirect

Maximal subgroups

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

subgroupindex
C2×#117181312
#131147642
#131147542
C2×C13×#5362
#1269213