Finite Group Databasebeta

← C2×#20273683 #27356263

Subgroups — C2×#20273683 #27356263

Order 13568.

1480 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1480 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×#16811204C211yesmaximal, characteristic
2C2×#16810762C212yesmaximal, complemented
3#20273683C218yesmaximal, direct product factor
4C2×#16810356C212yesmaximal, complemented
5C2×#16810349C211yesmaximal, characteristic, complemented
6C2×C53×#395C211yesmaximal, characteristic, complemented
7C22×#11719945—21no—
8C22×#11719924—22no—
9#16811204—24no—
10#16811204—22no—
11#16811204C2214yes—
12C2×#13133197—22no—
13C2×#13133192—22no—
14C2×#13133191—21no—
15C2×#13133191C2212yes—
16#16810762—24no—
17#16810762C2218yescomplemented
18C2×#13133141—22no—
19C2×C22⋊#88322C2212yes—
20#16810356C2218yescomplemented
21#16810349C2214yescomplemented
22C23×Q8×C53C2211yescharacteristic
23C22×C4⋊C4×C53—21no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C53×#395—22no—
26C53×#395C2214yescomplemented
27C2×#95×C53—21no—
28C2×#95×C53C2211yescharacteristic
29C2×#11719945—24no—
30C2×#11719945—24no—
31C2×#11719945—44no—
32C2×#11719945—44no—
33C22×Dic106—21no—
34C22×D212—21no—
35C22×D212—21no—
36C22×D212D412yes—
37C22×C22⋊D53—22no—
38C22×C4×D53—22no—
39C2×#11719924—28no—
40C2×#11719924—48no—
41C22×#88322—22no—
42#13133197—28no—
43#13133197—28no—
44#13133192—28no—
45#13133192—28no—
46#13133191—24no—
47#13133191—24no—
48#13133191—22no—
49#13133191—22no—
50#13133191D414yes—
51#13133191C2314yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53D412yes—
54C2×#11719921—22no—
55C2×#11719921—22no—
56#13133141—28no—
57C22⋊#88322—22no—
58C22⋊#88322C2314yes—
59C22×Q8×C53—21no—
60C22×Q8×C53—22no—
61C22×Q8×C53—22no—
62C22×Q8×C53—44no—
63C22×Q8×C53C2312yes—
64C23×C4×C53—21no—
65C23×C4×C53—21no—
66C23×C4×C53D411yescharacteristic
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2311yescharacteristic
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—24no—
72C2×C4⋊C4×C53—44no—
73C2×C4⋊C4×C53D412yes—
74C2×C4⋊C4×C53C2312yes—
75#95×C53—24no—
76#95×C53—21no—
77#95×C53C2312yes—
78C24×C53C2×D411yescharacteristic
79C23×D53—22no—
80C2×Q8×C53—28no—
81C2×Q8×C53—24no—
82C2×Q8×C53—44no—
83C2×Q8×C53—44no—
84C2×Q8×C53—44no—
85C2×Q8×C53—48no—
86C2×Q8×C53—42no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—24no—
91C22×C4×C53—24no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—21no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53SD1614yescomplemented
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2411yescharacteristic
110C2×C22⋊D53—24no—
111C2×C22⋊D53—24no—
112C2×C22⋊D53—44no—
113C2×C22⋊D53—44no—
114C22×Dic53—22no—
115#11719945—48no—
116#11719945—416no—
117#11719945—48no—
118C2×D212—24no—
119C2×D212—24no—
120C2×D212—22no—
121C2×D212—44no—
122C2×D212—44no—
123C2×D212—82no—
124C2×D212C2×D414yes—
125C2×C4×D53—28no—
126C2×C4×D53—48no—
127C2×Dic106—24no—
128C2×Dic106—44no—
129C4⋊C4×C53—24no—
130C4⋊C4×C53—24no—
131C4⋊C4×C53—48no—
132#11719924—416no—
133#11719921—28no—
134#11719921—24no—
135#11719921—24no—
136C4⋊Dic53—24no—
137C4⋊Dic53—22no—
138C4⋊Dic53C2×D414yes—
139C2×#88322—24no—
140C2×#88322—44no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53#12011yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53C2×SD1612yes—
146C23×C53AGL(1,ℤ/8)12yes—
147C23×C53C22×D411yescharacteristic
148C22×D53—24no—
149C22×D53—44no—
150C22×D53—44no—
151C22×D53—44no—
152Q8×C53—28no—
153Q8×C53—44no—
154Q8×C53—48no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—24no—
158C2×C4×C53—24no—
159C2×C4×C53—24no—
160C2×C4×C53—24no—
161C2×C4×C53—24no—
162C2×C4×C53—22no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—48no—
167C2×C4×C53—42no—
168C2×C4×C53—44no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—44no—
174C2×C4×C53—42no—
175C2×C4×C53—81no—
176C2×C4×C53#12012yes—
177C2×C4×C53C2×SD1614yescomplemented
178C2×C4×C53C22×D412yes—
179C22⋊D53—44no—
180C22⋊D53—44no—
181C22⋊D53—48no—
182C2×Dic53—24no—
183C2×Dic53—44no—
184D212—44no—
185D212—48no—
186D212—82no—
187D212—84no—
188C4×D53—416no—
189Dic106—48no—
190#88322—44no—
191C2×#1604—531nomaximal, Sylow
192C22×C53—24no—
193C22×C53—24no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53#45012yes—
199C22×C53#45012yes—
200C22×C53#45612yes—
201C22×C53B(4,𝔽2)12yes—
202C22×C53C2×#12011yescharacteristic
203C22×C53C22×SD1611yescharacteristic
204C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C2×D53—44no—
206C2×D53—44no—
207C2×D53—44no—
208C2×D53—88no—
209C4×C53—24no—
210C4×C53—22no—
211C4×C53—22no—
212C4×C53—44no—
213C4×C53—42no—
214C4×C53—44no—
215Dic53—44no—
216C2×#537—531no—
217C2×#448—532no—
218C2×#395D5311yescharacteristic, complemented
219#1604—538no—
220C2×#327—532no—
221C2×#324—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—22no—
225C2×C53#160414yescomplemented
226C2×C53C2×#45011yescharacteristic
227C2×C53C2×#45611yescharacteristic
228C2×C53C2×B(4,𝔽2)11yescharacteristic
229D53—84no—
230C22×#41—1061no—
231C23×Q8C2×D5311yescharacteristic
232C22×SD16—1062no—
233#537—534no—
234#537—1064no—
235#537—1062no—
236C2×C22⋊Q8—1062no—
237C2×#121—532no—
238C2×#121—1062no—
239C2×#121—1061no—
240C22×C4⋊C4—21no—
241C22×C4⋊C4C2×D5311yescharacteristic
242#448—538no—
243#448—1064no—
244C2×#103—1062no—
245C2×C22⋊C8—532no—
246#395—22no—
247#395C2×D5314yescomplemented
248C2×#95—21no—
249C2×#95C2×D5311yescharacteristic
250#327—538no—
251#324—534no—
252C53C2×#160411yescharacteristic, complemented, Sylow
253C2×#41—1064no—
254C2×#41—1064no—
255C2×#41—2124no—
256C2×#41—2124no—
257C22×Q8—21no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—44no—
261C22×Q8—1061no—
262C22×Q8C22×D5312yes—
263C22×D4—532no—
264C22×D4—1061no—
265C22×D4—1061no—
266C22×D4—1062no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—1062no—
270C23×C4C22⋊D5311yescharacteristic
271C23×C4C22⋊D5311yescharacteristic
272C23×C4C22×D5311yescharacteristic
273C2×SD16—1068no—
274C2×SD16—2128no—
275C22×C8—1062no—
276C22⋊Q8—1068no—
277C22⋊Q8—1068no—
278#121—534no—
279#121—534no—
280#121—1068no—
281#121—1068no—
282#121—1064no—
283#121—1062no—
284#121—1062no—
285#121—1064no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—532no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4C22⋊D5312yes—
293C2×C4⋊C4C22×D5312yes—
294C2×C22⋊C4—1062no—
295C2×C22⋊C4—1062no—
296#103—1068no—
297C22⋊C8—534no—
298C22⋊C8—1062no—
299#95—24no—
300#95—21no—
301#95C22×D5312yes—
302C24—1062no—
303C24C2×C22⋊D5311yescharacteristic
304#41—21216no—
305#41—2128no—
306#41—2128no—
307C2×Q8—28no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—42no—
314C2×Q8—1064no—
315C2×Q8—2124no—
316C2×D4—534no—
317C2×D4—1064no—
318C2×D4—1064no—
319C2×D4—1064no—
320C2×D4—1064no—
321C2×D4—1062no—
322C2×D4—4242no—
323C2×D4—2124no—
324C2×D4—2124no—
325C2×D4—2124no—
326C2×D4—2124no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—1068no—
344C22×C4—1062no—
345C22×C4—2128no—
346C22×C4#1171992414yescomplemented
347C22×C4D4×D5312yes—
348C22×C4D4×D5312yes—
349C22×C4C2×C22⋊D5312yes—
350C22×C4C2×C22⋊D5312yes—
351C22×C4C2×C22⋊D5311yescharacteristic
352C22×C4C23×D5311yescharacteristic
353SD16—21216no—
354C2×C8—1064no—
355C2×C8—2124no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C4⋊C4—534no—
360C4⋊C4—1064no—
361C4⋊C4—1062no—
362C22⋊C4—1068no—
363C22⋊C4—1064no—
364C22⋊C4—1064no—
365C23—22no—
366C23—22no—
367C23—1064no—
368C23—2124no—
369C23—2124no—
370C23—2124no—
371C23#1313318511yescharacteristic
372C23#1313318511yescharacteristic
373C23C2×#1171992412yes—
374C23#1313324212yes—
375C23C22×C22⋊D5311yescharacteristic
376Q8—28no—
377Q8—44no—
378Q8—48no—
379Q8—2128no—
380D4—4242no—
381D4—4244no—
382D4—2124no—
383D4—2128no—
384D4—2128no—
385D4—2124no—
386D4—2124no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—1064no—
409C2×C4—21216no—
410C2×C4—2124no—
411C2×C4#1313318512yes—
412C2×C4C2×#1171992414yescomplemented
413C2×C4C2×D4×D5312yes—
414C8—2124no—
415C22—24no—
416C22—24no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—4248no—
422C22—2124no—
423C22—2124no—
424C22—2124no—
425C22#1681077312yes—
426C22#1681077312yes—
427C22#1681080512yes—
428C22#1681080912yes—
429C22C2×#1313318511yescharacteristic
430C22C22×#1171992411yescharacteristic
431C22C2×#1313324211yescharacteristic
432C4—22no—
433C4—22no—
434C4—24no—
435C4—44no—
436C4—44no—
437C4—42no—
438C4—2124no—
439C2—21no—
440C2—22no—
441C2—21no—
442C2—4244no—
443C2#2027368314yesdirect product factor
444C2C2×#1681077311yescharacteristic
445C2C2×#1681080511yescharacteristic
446C2C2×#1681080911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#395C2
socleC23×C53#120
prime radicalC2×#16811204C2
hypercenterC2×#395D53
hypocenterC53C2×#1604
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
2. C2×#16810762C2semidirect
3. #20273683C2direct (N × H)
4. C2×#16810356C2semidirect
5. C2×#16810349C2semidirect
6. C2×C53×#395C2semidirect
17. #16810762C22semidirect
20. #16810356C22semidirect
21. #16810349C22semidirect
26. C53×#395C22semidirect
103. C22×C4×C53SD16semidirect
177. C2×C4×C53C2×SD16semidirect
218. C2×#395D53semidirect
225. C2×C53#1604semidirect
247. #395C2×D53semidirect
252. C53C2×#1604semidirect
346. C22×C4#11719924semidirect
412. C2×C4C2×#11719924semidirect
443. C2#20273683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112042
C2×#168107622
#202736832
C2×#168103562
C2×#168103492
C2×C53×#3952
C2×#160453