Finite Group Databasebeta

← C2×#20273734 #27356212

Subgroups — C2×#20273734 #27356212

Order 13568.

1524 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1524 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16811185C211yesmaximal, characteristic
2C2×#16810941C212yesmaximal, complemented
3#20273734C218yesmaximal, direct product factor
4C2×#16810356C212yesmaximal, complemented
5C2×#16810347C211yesmaximal, characteristic, complemented
6C2×C53×#399C211yesmaximal, characteristic, complemented
7C23×C22⋊D53—21no—
8#16811185—24no—
9#16811185—22no—
10#16811185C2214yes—
11C2×#13133201—22no—
12C2×#13133191—21no—
13C2×#13133191C2212yes—
14C2×#13133187—22no—
15#16810941—24no—
16#16810941C2218yescomplemented
17C2×#13133148—22no—
18C2×#13133137—22no—
19C2×C22⋊#88322C2212yescomplemented
20#16810356C2218yescomplemented
21#16810347C2214yescomplemented
22C22×C4⋊C4×C53—21no—
23C22×C4⋊C4×C53C2211yescharacteristic, complemented
24C22×C22⋊C4×C53—21no—
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#399—22no—
27C53×#399C2214yescomplemented
28C2×#95×C53C2211yescharacteristic, complemented
29C24×D53—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C22⋊D53—22no—
33C22×C22⋊D53—24no—
34C22×C22⋊D53—22no—
35C22×C22⋊D53—24no—
36C22×C22⋊D53—44no—
37C22×C22⋊D53—44no—
38C23×Dic53—21no—
39C22×#88322—22no—
40#13133201—28no—
41#13133201—28no—
42#13133191—24no—
43#13133191—24no—
44#13133191—22no—
45#13133191—22no—
46#13133191D414yes—
47#13133191C2314yes—
48#13133187—28no—
49#13133187—28no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53D412yes—
52C2×#11719921—22no—
53C2×#11719921—22no—
54#13133148—28no—
55#13133137—28no—
56C22⋊#88322—22no—
57C22⋊#88322C2314yescomplemented
58C25×C53D411yescharacteristic
59C23×C4×C53—21no—
60C23×C4×C53—21no—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C2×C4⋊C4×C53—24no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—44no—
68C2×C4⋊C4×C53C2312yescomplemented
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—22no—
71C2×C22⋊C4×C53—22no—
72C2×C22⋊C4×C53—44no—
73C2×C22⋊C4×C53D412yes—
74C2×C22⋊C4×C53C2312yes—
75#95×C53—21no—
76#95×C53C2312yescomplemented
77C24×C53—21no—
78C24×C53—21no—
79C24×C53—22no—
80C24×C53—42no—
81C24×C53—42no—
82C24×C53SD1614yes—
83C24×C53C2×D411yescharacteristic
84C24×C53C2×D412yes—
85C23×D53—22no—
86C23×D53—24no—
87C23×D53—44no—
88C23×D53—82no—
89C23×D53—82no—
90C22×C4×C53—24no—
91C22×C4×C53—21no—
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2411yescharacteristic
108C2×C22⋊D53—28no—
109C2×C22⋊D53—24no—
110C2×C22⋊D53—24no—
111C2×C22⋊D53—48no—
112C2×C22⋊D53—416no—
113C2×C22⋊D53—48no—
114C2×C22⋊D53—48no—
115C2×C22⋊D53—44no—
116C2×C22⋊D53—44no—
117C22×Dic53—24no—
118C22×Dic53—22no—
119C22×Dic53—44no—
120C2×D212—24no—
121C2×D212—22no—
122C2×D212—44no—
123C2×D212—82no—
124C2×D212C2×D414yes—
125C4⋊C4×C53—24no—
126C4⋊C4×C53—48no—
127C4⋊C4×C53—42no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—48no—
131#11719921—28no—
132#11719921—24no—
133#11719921—24no—
134C4⋊Dic53—24no—
135C4⋊Dic53—22no—
136C4⋊Dic53C2×D414yes—
137C2×#88322—24no—
138C2×#88322—44no—
139C23×C53—22no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53—24no—
143C23×C53—22no—
144C23×C53—24no—
145C23×C53—48no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—84no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53#12011yescharacteristic
154C23×C53#12011yescharacteristic
155C23×C53C2×SD1612yes—
156C23×C53C2×SD1614yes—
157C23×C53AGL(1,ℤ/8)12yes—
158C23×C53C22×D411yescharacteristic
159C22×D53—24no—
160C22×D53—48no—
161C22×D53—44no—
162C22×D53—44no—
163C22×D53—44no—
164C22×D53—84no—
165C22×D53—84no—
166C22×D53—816no—
167C2×C4×C53—24no—
168C2×C4×C53—24no—
169C2×C4×C53—22no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—48no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—81no—
182C2×C4×C53—82no—
183C2×C4×C53—82no—
184C2×C4×C53#12012yes—
185C2×C4×C53C22×D412yes—
186C22⋊D53—416no—
187C22⋊D53—44no—
188C22⋊D53—44no—
189C22⋊D53—48no—
190C2×Dic53—24no—
191C2×Dic53—48no—
192C2×Dic53—44no—
193D212—44no—
194D212—82no—
195D212—84no—
196#88322—44no—
197C2×#1613—531nomaximal, Sylow
198C22×C53—24no—
199C22×C53—24no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—24no—
205C22×C53—44no—
206C22×C53—48no—
207C22×C53—42no—
208C22×C53—42no—
209C22×C53—82no—
210C22×C53—84no—
211C22×C53#45012yes—
212C22×C53#45012yes—
213C22×C53#45312yes—
214C22×C53#45812yes—
215C22×C53C2×#12011yescharacteristic
216C22×C53C22×SD1611yescharacteristic
217C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
218C2×D53—44no—
219C2×D53—44no—
220C2×D53—44no—
221C2×D53—816no—
222C2×D53—88no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—44no—
226C4×C53—42no—
227C4×C53—82no—
228Dic53—44no—
229C2×#534—531no—
230C2×#481—532no—
231C2×#399D5311yescharacteristic, complemented
232#1613—538no—
233C2×#327—532no—
234C2×#323—531no—
235C2×C53—21no—
236C2×C53—21no—
237C2×C53—22no—
238C2×C53—44no—
239C2×C53#161314yescomplemented
240C2×C53C2×#45011yescharacteristic
241C2×C53C2×#45311yescharacteristic
242C2×C53C2×#45811yescharacteristic
243D53—84no—
244C23×D4—1061no—
245#534—534no—
246#534—1064no—
247#534—1062no—
248C2×#123—1062no—
249C2×#121—532no—
250C2×#121—1061no—
251C2×#120—1062no—
252C22×C4⋊C4—21no—
253C22×C4⋊C4C2×D5311yescharacteristic, complemented
254C22×C22⋊C4—21no—
255C22×C22⋊C4C2×D5311yescharacteristic
256#481—538no—
257#481—1064no—
258C2×#106—1062no—
259C2×#102—1062no—
260C2×C22⋊C8—532no—
261#399—22no—
262#399C2×D5314yescomplemented
263C2×#95C2×D5311yescharacteristic, complemented
264#327—538no—
265#323—534no—
266C53C2×#161311yescharacteristic, complemented, Sylow
267C25—1061no—
268C25C22⋊D5311yescharacteristic
269C22×D4—532no—
270C22×D4—1061no—
271C22×D4—1062no—
272C22×D4—1064no—
273C22×D4—1062no—
274C22×D4—1064no—
275C22×D4—2124no—
276C22×D4—2124no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—1061no—
280C23×C4C22⋊D5311yescharacteristic
281C23×C4C22×D5311yescharacteristic
282C22×C8—1062no—
283#123—1068no—
284#123—1068no—
285#121—534no—
286#121—534no—
287#121—1064no—
288#121—1062no—
289#121—1062no—
290#121—1064no—
291#120—1068no—
292#120—1068no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—532no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4C22×D5312yescomplemented
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—1062no—
306C2×C22⋊C4—1062no—
307C2×C22⋊C4C22⋊D5312yes—
308C2×C22⋊C4C22×D5312yes—
309#106—1068no—
310#102—1068no—
311C22⋊C8—534no—
312C22⋊C8—1062no—
313#95—21no—
314#95C22×D5312yescomplemented
315C24—21no—
316C24—21no—
317C24—22no—
318C24—42no—
319C24—42no—
320C24—1064no—
321C24—1062no—
322C24—4242no—
323C24—4242no—
324C24—2124no—
325C24#1171992414yes—
326C24C2×C22⋊D5311yescharacteristic
327C24C2×C22⋊D5312yes—
328C2×D4—534no—
329C2×D4—1064no—
330C2×D4—1068no—
331C2×D4—1064no—
332C2×D4—1064no—
333C2×D4—1062no—
334C2×D4—4242no—
335C2×D4—2124no—
336C2×D4—21216no—
337C2×D4—2128no—
338C2×D4—2128no—
339C2×D4—2128no—
340C2×D4—2124no—
341C2×D4—2124no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—1064no—
356C22×C4—1062no—
357C22×C4—2124no—
358C22×C4D4×D5312yes—
359C22×C4D4×D5312yes—
360C22×C4C2×C22⋊D5312yes—
361C22×C4C2×C22⋊D5311yescharacteristic
362C22×C4C23×D5311yescharacteristic
363C2×C8—1064no—
364C2×C8—2124no—
365C4⋊C4—24no—
366C4⋊C4—48no—
367C4⋊C4—42no—
368C4⋊C4—534no—
369C4⋊C4—1064no—
370C4⋊C4—1062no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—48no—
374C22⋊C4—1068no—
375C22⋊C4—1064no—
376C22⋊C4—1064no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—24no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23—48no—
386C23—42no—
387C23—42no—
388C23—84no—
389C23—81no—
390C23—81no—
391C23—1064no—
392C23—42416no—
393C23—4244no—
394C23—4244no—
395C23—2128no—
396C23—2124no—
397C23—2124no—
398C23—2124no—
399C23#1313318511yescharacteristic
400C23#1313318511yescharacteristic
401C23C2×#1171992412yes—
402C23C2×#1171992414yes—
403C23#1313324212yes—
404C23C22×C22⋊D5311yescharacteristic
405D4—4242no—
406D4—4244no—
407D4—2124no—
408D4—21216no—
409D4—2128no—
410D4—2124no—
411D4—2124no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—48no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—1064no—
430C2×C4—2128no—
431C2×C4—2124no—
432C2×C4#1313318512yes—
433C2×C4C2×D4×D5312yes—
434C8—2124no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—24no—
442C22—44no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22—82no—
447C22—84no—
448C22—42416no—
449C22—4248no—
450C22—2124no—
451C22—2124no—
452C22—2124no—
453C22#1681077312yes—
454C22#1681077312yes—
455C22#1681079012yes—
456C22#1681081212yes—
457C22C2×#1313318511yescharacteristic
458C22C22×#1171992411yescharacteristic
459C22C2×#1313324211yescharacteristic
460C4—22no—
461C4—22no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—2124no—
466C2—21no—
467C2—21no—
468C2—22no—
469C2—44no—
470C2—4244no—
471C2#2027373414yesdirect product factor
472C2C2×#1681077311yescharacteristic
473C2C2×#1681079011yescharacteristic
474C2C2×#1681081211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#399C2
socleC23×C53#120
prime radicalC2×#16811185C2
hypercenterC2×#399D53
hypocenterC53C2×#1613
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
2. C2×#16810941C2semidirect
3. #20273734C2direct (N × H)
4. C2×#16810356C2semidirect
5. C2×#16810347C2semidirect
6. C2×C53×#399C2semidirect
16. #16810941C22semidirect
19. C2×C22⋊#88322C22semidirect
20. #16810356C22semidirect
21. #16810347C22semidirect
23. C22×C4⋊C4×C53C22semidirect
27. C53×#399C22semidirect
28. C2×#95×C53C22semidirect
57. C22⋊#88322C23semidirect
68. C2×C4⋊C4×C53C23semidirect
76. #95×C53C23semidirect
231. C2×#399D53semidirect
239. C2×C53#1613semidirect
253. C22×C4⋊C4C2×D53semidirect
262. #399C2×D53semidirect
263. C2×#95C2×D53semidirect
266. C53C2×#1613semidirect
300. C2×C4⋊C4C22×D53semidirect
314. #95C22×D53semidirect
471. C2#20273734direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111852
C2×#168109412
#202737342
C2×#168103562
C2×#168103472
C2×C53×#3992
C2×#161353