Finite Group Databasebeta

← C22×#16810412 #27536806

Subgroups — C22×#16810412 #27536806

Order 13568.

1486 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 291 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
1C22×#13133198C211yesmaximal, characteristic
2C22×#13133134C211yesmaximal, characteristic
3C2×#16810412C2112yesmaximal, direct product factor
4C22×#99×C53C211yesmaximal, characteristic
5C24×Dic53C411yescharacteristic, complemented
6C22×#11719917—21no—
7C2×#13133198—23no—
8C2×#13133198—23no—
9C2×#13133198C416yescomplemented
10C2×#13133198C2216yes—
11C22×#11719919—21no—
12C22×C22⋊Dic53—21no—
13C22×C22⋊Dic53C411yescharacteristic, complemented
14C2×#13133134—23no—
15C2×#13133134C2216yes—
16#16810412C22116yesdirect product factor
17C23×D4×C53C2211yescharacteristic
18C22×C22⋊C4×C53—21no—
19C2×#99×C53—23no—
20C2×#99×C53C2216yes—
21C23×Dic53—21no—
22C23×Dic53—21no—
23C23×Dic53—23no—
24C23×Dic53—43no—
25C23×Dic53—41no—
26C23×Dic53C2×C416yescomplemented
27C2×#11719917—26no—
28C2×#11719917—43no—
29C22×#88322—41no—
30#13133198—212no—
31#13133198—26no—
32#13133198—26no—
33#13133198C2×C4112yescomplemented
34#13133198C2314yes—
35C2×#11719919—26no—
36C2×#11719919—26no—
37C2×C22⋊Dic53—26no—
38C2×C22⋊Dic53—23no—
39C2×C22⋊Dic53—26no—
40C2×C22⋊Dic53C2×C416yescomplemented
41#13133134—26no—
42#13133134C2314yes—
43C25×C53C2×C411yescharacteristic
44C25×C53D411yescharacteristic
45C22×D4×C53—41no—
46C22×D4×C53—43no—
47C22×D4×C53C2×C413yes—
48C22×D4×C53D413yes—
49C22×D4×C53D413yes—
50C22×D4×C53C2313yes—
51C23×C4×C53—41no—
52C23×C4×C53D411yescharacteristic
53C2×C22⋊C4×C53—26no—
54C2×C22⋊C4×C53—43no—
55#99×C53—26no—
56#99×C53C2314yes—
57C24×C53—21no—
58C24×C53—23no—
59C24×C53—21no—
60C24×C53—23no—
61C24×C53—44no—
62C24×C53—44no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C413yes—
65C24×C53C22⋊C413yes—
66C24×C53C22×C413yes—
67C24×C53C2×D413yes—
68C2×D4×C53—43no—
69C2×D4×C53—43no—
70C2×D4×C53—43no—
71C2×D4×C53—43no—
72C2×D4×C53—43no—
73C2×D4×C53—43no—
74C2×D4×C53—43no—
75C2×D4×C53—43no—
76C2×D4×C53C22⋊C416yes—
77C2×D4×C53C22×C413yes—
78C2×D4×C53C2×D413yes—
79C2×D4×C53C2×D413yes—
80C2×D4×C53C2411yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—23no—
83C22×C4×C53—46no—
84C22×C4×C53—84no—
85C22×C4×C53C22⋊C413yes—
86C22×C4×C53C2×D413yes—
87C22×Dic53—26no—
88C22×Dic53—26no—
89C22×Dic53—26no—
90C22×Dic53—44no—
91C22×Dic53—46no—
92C22×Dic53—46no—
93C22×Dic53—43no—
94C22×Dic53—41no—
95C22×Dic53—46no—
96C22×Dic53—46no—
97C22×Dic53C22×C414yescomplemented
98C22⋊C4×C53—24no—
99C22⋊C4×C53—46no—
100C22⋊Dic53—212no—
101C22⋊Dic53—26no—
102C22⋊Dic53—24no—
103C22⋊Dic53C22×C414yescomplemented
104#11719919—212no—
105#11719919—24no—
106#11719917—24no—
107#11719917—46no—
108C2×#88322—46no—
109C23×C53—24no—
110C23×C53—23no—
111C23×C53—23no—
112C23×C53—23no—
113C23×C53—23no—
114C23×C53—23no—
115C23×C53—23no—
116C23×C53—23no—
117C23×C53—23no—
118C23×C53—46no—
119C23×C53—46no—
120C23×C53—412no—
121C23×C53—46no—
122C23×C53—46no—
123C23×C53—44no—
124C23×C53—44no—
125C23×C53—86no—
126C23×C53#9913yes—
127C23×C53#9911yescharacteristic
128C23×C53C2×C22⋊C413yes—
129C23×C53C2×C22⋊C413yes—
130C23×C53C2×C22⋊C413yes—
131C23×C53C23×C411yescharacteristic
132C23×C53C22×D411yescharacteristic
133D4×C53—46no—
134D4×C53—43no—
135D4×C53—43no—
136D4×C53—43no—
137D4×C53—41no—
138C2×C4×C53—23no—
139C2×C4×C53—23no—
140C2×C4×C53—23no—
141C2×C4×C53—23no—
142C2×C4×C53—44no—
143C2×C4×C53—812no—
144C2×C4×C53C2×C22⋊C413yes—
145C2×C4×C53C22×D411yescharacteristic
146C2×Dic53—24no—
147C2×Dic53—24no—
148C2×Dic53—412no—
149C2×Dic53—412no—
150C2×Dic53—46no—
151C2×Dic53—46no—
152C2×Dic53—44no—
153#88322—44no—
154C22×#352—531nomaximal, Sylow
155C22×C53—26no—
156C22×C53—26no—
157C22×C53—23no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—23no—
161C22×C53—43no—
162C22×C53—43no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—46no—
166C22×C53—46no—
167C22×C53—46no—
168C22×C53—46no—
169C22×C53—46no—
170C22×C53—46no—
171C22×C53—83no—
172C22×C53—83no—
173C22×C53#35214yescomplemented
174C22×C53C2×#9913yes—
175C22×C53C2×#9913yes—
176C22×C53C22×C22⋊C411yescharacteristic
177C4×C53—21no—
178C4×C53—23no—
179C4×C53—84no—
180Dic53—44no—
181Dic53—44no—
182C22×#123—531no—
183C22×#100—531no—
184C22×#99D5311yescharacteristic
185C2×#352—5312no—
186C2×C53—21no—
187C2×C53—23no—
188C2×C53—41no—
189C2×C53—43no—
190C2×C53—41no—
191C2×C53—43no—
192C2×C53C2×#35216yescomplemented
193C2×C53C22×#9911yescharacteristic
194C23×D4C2×D5311yescharacteristic
195C24×C4—531no—
196C22×M16—1061no—
197C2×#123—536no—
198C2×#123—536no—
199C2×#123—1063no—
200C2×#123—1063no—
201C22×C4⋊C4—1061no—
202C22×C22⋊C4—21no—
203C22×C22⋊C4—531no—
204C22×C22⋊C4—1061no—
205C2×#100—536no—
206C2×#100—1063no—
207C2×#99—23no—
208C2×#99C2×D5316yes—
209#352—5316no—
210C53C22×#35211yescharacteristic, complemented, Sylow
211C25C4×D5311yescharacteristic
212C25C22⋊D5311yescharacteristic
213C22×D4—43no—
214C22×D4—41no—
215C22×D4C4×D5313yes—
216C22×D4D21213yes—
217C22×D4C22⋊D5313yes—
218C22×D4C22×D5313yes—
219C23×C4—41no—
220C23×C4—536no—
221C23×C4—1063no—
222C23×C4—1061no—
223C23×C4—1061no—
224C23×C4—2123no—
225C23×C4—2121no—
226C23×C4D21211yescharacteristic
227C2×M16—1066no—
228C2×M16—2123no—
229C22×C8—2121no—
230#123—5312no—
231#123—534no—
232#123—1066no—
233#123—1066no—
234#123—10612no—
235C2×C4⋊C4—1066no—
236C2×C4⋊C4—1066no—
237C2×C22⋊C4—26no—
238C2×C22⋊C4—43no—
239C2×C22⋊C4—536no—
240C2×C22⋊C4—1066no—
241C2×C22⋊C4—1063no—
242C2×C22⋊C4—1066no—
243#100—534no—
244#100—1066no—
245#99—26no—
246#99C22×D5314yes—
247C24—23no—
248C24—21no—
249C24—21no—
250C24—23no—
251C24—44no—
252C24—44no—
253C24#1171992113yes—
254C24#1171992113yes—
255C24#1171992111yescharacteristic
256C24C2×C4×D5313yes—
257C24C2×C22⋊D5313yes—
258C2×D4—43no—
259C2×D4—43no—
260C2×D4—43no—
261C2×D4—43no—
262C2×D4—43no—
263C2×D4—43no—
264C2×D4—43no—
265C2×D4—43no—
266C2×D4#1171992116yes—
267C2×D4C2×C4×D5313yes—
268C2×D4C2×D21213yes—
269C2×D4C2×C22⋊D5313yes—
270C2×D4C23×D5311yescharacteristic
271C22×C4—23no—
272C22×C4—21no—
273C22×C4—46no—
274C22×C4—84no—
275C22×C4—534no—
276C22×C4—1066no—
277C22×C4—1066no—
278C22×C4—1066no—
279C22×C4—2124no—
280C22×C4—2126no—
281C22×C4—2126no—
282C22×C4—2123no—
283C22×C4—2121no—
284C22×C4—2126no—
285C22×C4—2126no—
286C22×C4#1171992113yes—
287C22×C4C2×D21213yes—
288M16—1064no—
289M16—2126no—
290C2×C8—2126no—
291C4⋊C4—10612no—
292C4⋊C4—1064no—
293C22⋊C4—24no—
294C22⋊C4—46no—
295C22⋊C4—534no—
296C22⋊C4—1066no—
297C22⋊C4—10612no—
298C22⋊C4—1064no—
299C23—24no—
300C23—23no—
301C23—23no—
302C23—23no—
303C23—23no—
304C23—23no—
305C23—23no—
306C23—23no—
307C23—23no—
308C23—46no—
309C23—46no—
310C23—412no—
311C23—46no—
312C23—46no—
313C23—44no—
314C23—44no—
315C23—86no—
316C23#1313313113yes—
317C23#1313313111yescharacteristic
318C23C2×#1171992113yes—
319C23C2×#1171992113yes—
320C23C2×#1171992113yes—
321C23C22×C4×D5311yescharacteristic
322C23C22×C22⋊D5311yescharacteristic
323D4—43no—
324D4—43no—
325D4—43no—
326D4—41no—
327D4—46no—
328C2×C4—23no—
329C2×C4—23no—
330C2×C4—23no—
331C2×C4—23no—
332C2×C4—44no—
333C2×C4—812no—
334C2×C4—1064no—
335C2×C4—1064no—
336C2×C4—21212no—
337C2×C4—21212no—
338C2×C4—2126no—
339C2×C4—2126no—
340C2×C4—2124no—
341C2×C4C2×#1171992113yes—
342C2×C4C22×D21211yescharacteristic
343C8—2124no—
344C22—26no—
345C22—26no—
346C22—21no—
347C22—23no—
348C22—21no—
349C22—23no—
350C22—43no—
351C22—43no—
352C22—41no—
353C22—41no—
354C22—46no—
355C22—46no—
356C22—46no—
357C22—46no—
358C22—46no—
359C22—46no—
360C22—83no—
361C22—83no—
362C22#1681041214yesdirect product factor
363C22C2×#1313313113yes—
364C22C2×#1313313113yes—
365C22C22×#1171992111yescharacteristic
366C4—21no—
367C4—23no—
368C4—84no—
369C4—2124no—
370C4—2124no—
371C2—21no—
372C2—23no—
373C2—41no—
374C2—43no—
375C2—41no—
376C2—43no—
377C2C2×#1681041216yesdirect product factor
378C2C22×#1313313111yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC23×C53C23×C4
FrattiniC2×D4C23×D53
FittingC22×#99×C53C2
socleC23×C53#99
prime radicalC23×D4×C53C22
cosocleC2×D4×C53C24
hypercenterC22×#99D53
hypocenterC53C22×#352
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. C2×#16810412C2direct (N × H)
5. C24×Dic53C4semidirect
9. C2×#13133198C4semidirect
13. C22×C22⋊Dic53C4semidirect
16. #16810412C22direct (N × H)
26. C23×Dic53C2×C4semidirect
33. #13133198C2×C4semidirect
40. C2×C22⋊Dic53C2×C4semidirect
97. C22×Dic53C22×C4semidirect
103. C22⋊Dic53C22×C4semidirect
173. C22×C53#352semidirect
192. C2×C53C2×#352semidirect
210. C53C22×#352semidirect
362. C22#16810412direct (N × H)
377. C2C2×#16810412direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331982
C22×#131331342
C2×#168104122
C22×#99×C532
C22×#35253