Finite Group Databasebeta

← C2×#20276409 #27536934

Subgroups — C2×#20276409 #27536934

Order 13568.

1198 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811194C211yesmaximal, characteristic
2C2×#16810607C211yesmaximal, characteristic
3#20276409C218yesmaximal, direct product factor
4C2×#16810421C214yesmaximal, complemented
5C2×C53×#413C211yesmaximal, characteristic
6C23×Dic106C411yescharacteristic
7C22×#11719917—21no—
8#16811194—22no—
9#16811194—22no—
10#16811194C414yes—
11#16811194C2214yes—
12C2×#13133215C2212yes—
13C2×#13133204C412yes—
14C2×#13133198—21no—
15C2×#13133198—21no—
16C2×#13133196—21no—
17C2×#13133196—21no—
18C2×#13133176C411yescharacteristic
19#16810607—22no—
20#16810607C2214yes—
21C2×#13133136C2212yes—
22C2×#13133134—21no—
23#16810421C22116yescomplemented
24C22×#41×C53C2211yescharacteristic
25C22×M16×C53—21no—
26C53×#413—22no—
27C53×#413C2214yes—
28C2×#101×C53C2212yes—
29C2×#100×C53—21no—
30C22×Dic106—21no—
31C22×Dic106—22no—
32C22×Dic106—42no—
33C22×Dic106—41no—
34C22×Dic106—41no—
35C22×Dic106C2×C412yes—
36C22×Dic106C2×C414yes—
37C23×Dic53—21no—
38C2×#11719917—22no—
39C2×#11719917—24no—
40C2×#11719917—41no—
41C2×#11719917—42no—
42C22×#88322—41no—
43#13133215—22no—
44#13133215—22no—
45#13133215C2×C414yes—
46#13133215C2314yes—
47#13133204—24no—
48#13133204C2×C418yes—
49#13133198—24no—
50#13133198—24no—
51#13133198—24no—
52#13133198—24no—
53#13133196—24no—
54#13133196—24no—
55#13133196—24no—
56#13133196—24no—
57#13133176—22no—
58#13133176C2×C414yes—
59C2×C4⋊Dic53—21no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×C22⋊Dic53—21no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C4×Dic53C2×C412yes—
66#13133136—22no—
67#13133136C2314yes—
68#13133134—24no—
69C2×#41×C53—41no—
70C2×#41×C53—41no—
71C2×#41×C53—42no—
72C2×#41×C53C2×C412yes—
73C2×#41×C53D412yes—
74C2×#41×C53D412yes—
75C2×#41×C53C2312yes—
76C22×Q8×C53C2311yescharacteristic
77C22×D4×C53C2×C411yescharacteristic
78C22×D4×C53D411yescharacteristic
79C22×D4×C53D411yescharacteristic
80C23×C4×C53C2×C411yescharacteristic
81C23×C4×C53D411yescharacteristic
82C23×C4×C53D411yescharacteristic
83C2×M16×C53—22no—
84C2×M16×C53—24no—
85C2×M16×C53—41no—
86C2×M16×C53—42no—
87C22×C8×C53—41no—
88#101×C53—22no—
89#101×C53C2314yes—
90#100×C53—24no—
91C24×C53C22⋊C411yescharacteristic
92C24×C53C22⋊C411yescharacteristic
93C24×C53C22⋊C411yescharacteristic
94#41×C53—42no—
95#41×C53—42no—
96#41×C53—42no—
97#41×C53—42no—
98#41×C53—42no—
99#41×C53—42no—
100#41×C53—42no—
101#41×C53—42no—
102C2×Q8×C53—41no—
103C2×Q8×C53—41no—
104C2×Q8×C53C22×C411yescharacteristic
105C2×Q8×C53C2×D411yescharacteristic
106C2×Q8×C53C2×D411yescharacteristic
107C2×Q8×C53C2411yescharacteristic
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—41no—
113C2×D4×C53—41no—
114C2×D4×C53C22⋊C412yes—
115C2×D4×C53C22⋊C412yes—
116C2×D4×C53C22⋊C412yes—
117C2×D4×C53C22×C412yes—
118C2×D4×C53C2×D412yes—
119C2×D4×C53C2×D412yes—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53C22⋊C412yes—
130C22×C4×C53C22⋊C412yes—
131C22×C4×C53C22⋊C412yes—
132C22×C4×C53C22⋊C411yescharacteristic
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C2×Dic106—24no—
145C2×Dic106—22no—
146C2×Dic106—44no—
147C2×Dic106—44no—
148C2×Dic106—42no—
149C2×Dic106—41no—
150C2×Dic106—41no—
151C2×Dic106—42no—
152C2×Dic106—42no—
153C2×Dic106—44no—
154C2×Dic106—44no—
155C2×Dic106C22×C414yes—
156M16×C53—24no—
157M16×C53—42no—
158M16×C53—44no—
159C2×C8×C53—42no—
160C2×C8×C53—44no—
161C22⋊Dic53—24no—
162C22⋊Dic53—28no—
163C22⋊Dic53—28no—
164C4⋊Dic53—24no—
165#11719919—24no—
166#11719919—24no—
167#11719919—24no—
168#11719919—24no—
169C4×Dic53—22no—
170C4×Dic53C22×C414yes—
171#11719917—24no—
172#11719917—42no—
173#11719917—44no—
174C2×#88322—42no—
175C2×#88322—44no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53C2×C22⋊C411yescharacteristic
186C23×C53C2×C22⋊C412yes—
187C23×C53C2×C22⋊C412yes—
188C23×C53C2×C22⋊C412yes—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53—41no—
192Q8×C53—41no—
193D4×C53—42no—
194D4×C53—42no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53#9912yes—
218C2×C4×C53#9911yescharacteristic
219C2×C4×C53#9911yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C412yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—48no—
230C2×Dic53—44no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233Dic106—44no—
234Dic106—44no—
235Dic106—44no—
236Dic106—44no—
237C8×C53—44no—
238#88322—44no—
239C2×#2135—531nomaximal, Sylow
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—81no—
252C22×C53—81no—
253C22×C53C2×#9911yescharacteristic
254C22×C53C2×#9911yescharacteristic
255C22×C53C22×C22⋊C411yescharacteristic
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53C2×#9912yes—
263C4×C53C2×#9912yes—
264Dic53—44no—
265Dic53—44no—
266C2×#536—531no—
267C2×#413—531no—
268C2×#413D5311yescharacteristic
269#2135—538no—
270C2×#356—534no—
271C2×C53—21no—
272C2×C53—21no—
273C2×C53—42no—
274C2×C53—42no—
275C2×C53—42no—
276C2×C53#213512yescomplemented
277C2×C53C22×#9911yescharacteristic
278C22×#41C2×D5311yescharacteristic
279C23×Q8—531no—
280C22×M16—21no—
281C22×M16—1061no—
282#536—534no—
283#536—534no—
284#536—1062no—
285#536—1062no—
286C2×C4⋊Q8—532no—
287C2×#124—532no—
288C2×#123—1061no—
289C2×#123—1061no—
290C2×C22⋊Q8—1061no—
291C2×C22⋊Q8—1061no—
292C2×#117—531no—
293#413—22no—
294#413—534no—
295#413—1062no—
296#413C2×D5314yes—
297C2×#101—532no—
298C2×#101C2×D5312yes—
299C2×#100—21no—
300C2×#100—1061no—
301#356—5316no—
302C53C2×#213511yescharacteristic, complemented, Sylow
303C2×#41—41no—
304C2×#41—41no—
305C2×#41—42no—
306C2×#41C4×D5312yes—
307C2×#41D21212yes—
308C2×#41C22⋊D5312yes—
309C2×#41C22×D5312yes—
310C22×Q8—532no—
311C22×Q8—534no—
312C22×Q8—1062no—
313C22×Q8—1061no—
314C22×Q8—2122no—
315C22×Q8—2121no—
316C22×Q8—2121no—
317C22×Q8C22×D5311yescharacteristic
318C22×D4C4×D5311yescharacteristic
319C22×D4D21211yescharacteristic
320C22×D4C22⋊D5311yescharacteristic
321C23×C4—1061no—
322C23×C4C4×D5311yescharacteristic
323C23×C4D21211yescharacteristic
324C23×C4C22⋊D5311yescharacteristic
325C2×M16—22no—
326C2×M16—24no—
327C2×M16—41no—
328C2×M16—42no—
329C2×M16—1062no—
330C2×M16—1064no—
331C2×M16—2121no—
332C2×M16—2122no—
333C22×C8—41no—
334C22×C8—2121no—
335C4⋊Q8—534no—
336C4⋊Q8—534no—
337C4⋊Q8—1062no—
338C4⋊Q8—1062no—
339#124—538no—
340#124—1064no—
341#123—1064no—
342#123—1064no—
343#123—1064no—
344#123—1064no—
345C22⋊Q8—1064no—
346C22⋊Q8—1064no—
347C22⋊Q8—1064no—
348C22⋊Q8—1064no—
349#117—534no—
350#117—1062no—
351C2×C4⋊C4—1062no—
352C2×C4⋊C4—1062no—
353C2×C4⋊C4—1061no—
354C2×C22⋊C4—1061no—
355C2×C22⋊C4—1062no—
356C2×C22⋊C4—1062no—
357C2×C42—532no—
358#101—22no—
359#101—534no—
360#101—1062no—
361#101C22×D5314yes—
362#100—24no—
363#100—1064no—
364C24#1171992111yescharacteristic
365C24#1171992111yescharacteristic
366C24#1171992111yescharacteristic
367#41—42no—
368#41—42no—
369#41—42no—
370#41—42no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—42no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—534no—
378C2×Q8—1064no—
379C2×Q8—1062no—
380C2×Q8—2124no—
381C2×Q8—2124no—
382C2×Q8—2124no—
383C2×Q8—2124no—
384C2×Q8—2122no—
385C2×Q8—2122no—
386C2×Q8—2122no—
387C2×Q8—2121no—
388C2×Q8—2121no—
389C2×Q8C2×C4×D5311yescharacteristic
390C2×Q8C2×D21211yescharacteristic
391C2×Q8C2×C22⋊D5311yescharacteristic
392C2×Q8C23×D5311yescharacteristic
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4#1171992112yes—
400C2×D4#1171992112yes—
401C2×D4#1171992112yes—
402C2×D4C2×C4×D5312yes—
403C2×D4C2×D21212yes—
404C2×D4C2×C22⋊D5312yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—1062no—
415C22×C4—1062no—
416C22×C4—1064no—
417C22×C4—2122no—
418C22×C4—2122no—
419C22×C4#1171992112yes—
420C22×C4#1171992112yes—
421C22×C4#1171992112yes—
422C22×C4#1171992111yescharacteristic
423C22×C4C2×C4×D5311yescharacteristic
424C22×C4C2×C4×D5312yes—
425C22×C4C2×D21211yescharacteristic
426C22×C4C2×D21212yes—
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5312yes—
429M16—24no—
430M16—42no—
431M16—44no—
432M16—1064no—
433M16—2122no—
434M16—2124no—
435C2×C8—42no—
436C2×C8—44no—
437C2×C8—2122no—
438C2×C8—2124no—
439C4⋊C4—1064no—
440C4⋊C4—1064no—
441C4⋊C4—1064no—
442C4⋊C4—1064no—
443C4⋊C4—1064no—
444C22⋊C4—1064no—
445C22⋊C4—1068no—
446C22⋊C4—1068no—
447C42—534no—
448C42—1062no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23C2×#1171992111yescharacteristic
459C23C2×#1171992112yes—
460C23C2×#1171992112yes—
461C23C2×#1171992112yes—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—2124no—
467Q8—2124no—
468Q8—2124no—
469Q8—2124no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—1064no—
495C2×C4—1064no—
496C2×C4—2124no—
497C2×C4—2128no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4#1313313112yes—
501C2×C4#1313313111yescharacteristic
502C2×C4#1313313111yescharacteristic
503C2×C4C2×#1171992111yescharacteristic
504C2×C4C2×#1171992111yescharacteristic
505C2×C4C2×#1171992111yescharacteristic
506C2×C4C2×#1171992112yes—
507C2×C4C22×C4×D5311yescharacteristic
508C2×C4C22×C22⋊D5311yescharacteristic
509C2×C4C22×D21211yescharacteristic
510C8—44no—
511C8—2124no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—81no—
524C22—81no—
525C22C2×#1313313111yescharacteristic
526C22C2×#1313313111yescharacteristic
527C22C22×#1171992111yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—2124no—
535C4—2124no—
536C4C2×#1313313112yes—
537C4C2×#1313313112yes—
538C2—21no—
539C2—21no—
540C2—42no—
541C2—42no—
542C2—42no—
543C2#2027640912yesdirect product factor
544C2C22×#1313313111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C23×C4
FrattiniC2×Q8C23×D53
FittingC2×C53×#413C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC2×Q8×C53C24
hypercenterC2×#413D53
hypocenterC53C2×#2135
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #20276409C2direct (N × H)
4. C2×#16810421C2semidirect
23. #16810421C22semidirect
276. C2×C53#2135semidirect
302. C53C2×#2135semidirect
543. C2#20276409direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111942
C2×#168106072
#202764092
C2×#168104212
C2×C53×#4132
C2×#213553