Finite Group Databasebeta

← C4×#89939 #13362343

Subgroups — C4×#89939 #13362343

Order 1792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×SD16C211yesmaximal, characteristic
2C2×#89939C211yesmaximal, characteristic
3C4×#6507C211yesmaximal, characteristic, complemented
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5C4×#6499C211yesmaximal, characteristic
6C2×C4×#1142C211yesmaximal, characteristic
7C2×C4×Dic28C211yesmaximal, characteristic, complemented
8#11729804C211yesmaximal, characteristic
9#11728329C211yesmaximal, characteristic, complemented
10#11728310C211yesmaximal, characteristic, complemented
11#11728013C211yesmaximal, characteristic, complemented
12#11727982C211yesmaximal, characteristic, complemented
13#11727946C211yesmaximal, characteristic
14#11727913C211yesmaximal, characteristic
15C4×#6389C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6507C2211yescharacteristic
20C22×#1144C2211yescharacteristic
21C2×#6499C2211yescharacteristic
22C22×#1142C2211yescharacteristic
23C22×Dic28C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C4×C7×SD16—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31#90002C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C4×C22⋊Dic7—22no—
36#89939C418yesdirect product factor
37C2×#6392C2211yescharacteristic
38C2×#6390C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×#6389C2211yescharacteristic
43C2×#6389C2211yescharacteristic
44C4×#1144—24no—
45C4×#1142—24no—
46#89749C2211yescharacteristic
47#89748C2211yescharacteristic
48#89746C2211yescharacteristic
49#89746C2211yescharacteristic
50#89744—22no—
51C2×#6384C2211yescharacteristic
52C2×#6384C2211yescharacteristic
53C2×#6382C2211yescharacteristic
54C2×#6381C2211yescharacteristic
55#89717—22no—
56C4×Dic7⋊C4—22no—
57C42×Dic7C2211yescharacteristic
58C2×C4×C7⋊C8C2211yescharacteristic
59C4×Dic28—24no—
60C4×#497—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×Dic14C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C2×C7×SD16—22no—
67C2×C7×SD16—22no—
68C2×C7×SD16C2×C414yes—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—22no—
82C2×C22⋊Dic7—22no—
83#6507—22no—
84#6507C2×C414yescomplemented
85C2×#1144—22no—
86C2×#1144—22no—
87C2×#1144C2×C414yescomplemented
88#6499—22no—
89#6499C2×C414yes—
90C2×#1142—22no—
91C2×#1142—22no—
92C2×#1142C2×C414yes—
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—22no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—22no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7D412yes—
102C2×C4×Dic7D412yes—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C2×Dic28—22no—
108C2×Dic28—22no—
109C2×Dic28C2×C414yescomplemented
110C4×Dic14—22no—
111C4×Dic14—22no—
112C4×Dic14—22no—
113C4×Dic14—22no—
114C7×#106—22no—
115C7×#103—22no—
116C7×#102—22no—
117C4×C7×C8D414yes—
118#6409—22no—
119#6406—22no—
120C7⋊#95—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C7⋊#95—22no—
124#6392—22no—
125#6390—22no—
126#6390—22no—
127#6389—22no—
128#6389—22no—
129#6389—22no—
130#6389C2×C414yescomplemented
131#6384—22no—
132#6384—22no—
133#6382—22no—
134#6381—22no—
135C4×C7⋊C8D414yes—
136C2×#497—71no—
137C2×C4×Q16—71no—
138C2×C4×Q16—71no—
139C2×C4×SD16—71no—
140C2×C4×SD16D711yescharacteristic
141C4×C4⋊Q8—71no—
142C4×#124—71no—
143#1973—71no—
144#1973—71no—
145#1971—71no—
146#1971—71no—
147#1960—71no—
148#1959—71no—
149#1847—71no—
150C4×#97—71no—
151C24×C7—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4C22×C412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—42no—
168C22×C4×C7—44no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7C2×D412yes—
177C22×Dic7C2×D412yes—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14C22×C412yes—
189C2×Dic14C22×C412yes—
190C7×SD16—28no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8C22×C412yes—
194C2×C7×C8C2×D412yes—
195C2×C7×C8C2×D412yes—
196C2×C7×C8#4112yes—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—42no—
202C7×C22⋊C4—42no—
203C42×C7—42no—
204C42×C7C2×D412yes—
205C42×C7C2×D412yes—
206C22⋊Dic7—28no—
207C22⋊Dic7—44no—
208#1144—28no—
209#1142—28no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214Dic7⋊C4—28no—
215Dic7⋊C4—44no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218C4×Dic7—24no—
219C4×Dic7—21no—
220C4×Dic7—21no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7—42no—
224C4×Dic7C22×C412yes—
225C4×Dic7#4112yes—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8C22×C412yes—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×D412yes—
231C2×C7⋊C8#4112yes—
232Dic28—28no—
233C22×Q16—71no—
234C22×Q16—71no—
235C22×SD16—71no—
236C22×SD16C2×D711yescharacteristic
237C2×C4⋊Q8—71no—
238C2×#124—71no—
239C2×C4×Q8—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×D4C2×D711yescharacteristic
243#497—78no—
244C4×Q16—144no—
245C4×Q16—144no—
246C4×SD16—24no—
247C4×SD16—144no—
248C2×#107—71no—
249C2×#107—71no—
250C2×#106—71no—
251C2×#106C2×D711yescharacteristic
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103C2×D711yescharacteristic
258C2×#102—71no—
259C2×#102C2×D711yescharacteristic
260C2×#97—71no—
261C2×#97—71no—
262C2×#97—71no—
263C2×#97—71no—
264C2×C4×C8—71no—
265C2×C4×C8C2×D711yescharacteristic
266#391—71no—
267#391—71no—
268#391—71no—
269#390—71no—
270#389—142no—
271#388—142no—
272#384—71no—
273#382—71no—
274C4×C4⋊C4—142no—
275C4×C22⋊C4—142no—
276C43—71no—
277C23×C7—22no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—42no—
281C23×C7—42no—
282C23×C7C22×D411yescharacteristic
283C7×Q8—22no—
284C7×Q8—22no—
285C7×Q8—41no—
286C7×Q8—41no—
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—41no—
290C7×D4—41no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—84no—
300C2×C4×C7#12712yes—
301C2×C4×C7#12712yes—
302C2×C4×C7#13714yes—
303C2×C4×C7C23×C411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7C4×D414yes—
325Dic14—22no—
326Dic14—22no—
327Dic14—22no—
328Dic14—22no—
329Dic14—41no—
330Dic14—41no—
331Dic14—41no—
332Dic14—41no—
333C7×C8—22no—
334C7×C8C4×D414yes—
335C7⋊C8—22no—
336C7⋊C8C4×D414yes—
337C22×Q8—71no—
338C22×Q8—71no—
339C22×Q8C22×D711yescharacteristic
340C22×D4C22×D711yescharacteristic
341C23×C4—21no—
342C2×Q16—74no—
343C2×Q16—74no—
344C2×Q16—142no—
345C2×Q16—142no—
346C2×Q16—142no—
347C2×Q16—142no—
348C2×SD16—22no—
349C2×SD16—22no—
350C2×SD16—74no—
351C2×SD16—142no—
352C2×SD16—142no—
353C2×SD16C4×D714yes—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8C22×D711yescharacteristic
357C22×C8C22×D711yescharacteristic
358C4⋊Q8—74no—
359C4⋊Q8—142no—
360#124—74no—
361#124—142no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—142no—
367C4×Q8—142no—
368C4×D4—22no—
369C4×D4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—142no—
381C2×C22⋊C4—142no—
382C2×C42—21no—
383C2×C42—71no—
384C2×C42—72no—
385C2×C42—71no—
386C2×C42—72no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390#107—142no—
391#107—142no—
392#106—22no—
393#106—142no—
394#103—22no—
395#103—142no—
396#103—142no—
397#103—142no—
398#103—142no—
399#103—142no—
400#102—22no—
401#102—142no—
402#97—74no—
403#97—142no—
404#97—142no—
405#97—142no—
406C4×C8—74no—
407C4×C8C22⋊D714yes—
408#95—142no—
409#95—142no—
410#95—142no—
411#95—142no—
412C22×C7—42no—
413C22×C7—44no—
414C22×C7—81no—
415C22×C7—81no—
416C22×C7—82no—
417C22×C7C2×C4×D411yescharacteristic
418C22×C7C2×#12711yescharacteristic
419C22×C7#53211yescharacteristic
420C22×C7C2×#13712yes—
421C22×C7#57512yes—
422C4×C7—22no—
423C4×C7—42no—
424C4×C7—81no—
425C4×C7—81no—
426C4×C7#49714yescomplemented
427C4×C7C2×C4×D411yescharacteristic
428C4×C7C2×C4×D411yescharacteristic
429C4×C7#53211yescharacteristic
430C4×C7#53211yescharacteristic
431Dic7—24no—
432Dic7—42no—
433Dic7—42no—
434Dic7—42no—
435Dic7—81no—
436Dic7—81no—
437C24—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—72no—
444C2×Q8—72no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8—141no—
452C2×Q8—141no—
453C2×Q8—141no—
454C2×Q8—141no—
455C2×Q8C2×C4×D712yes—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C2×C4×D712yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—72no—
468C22×C4—72no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4D4×D711yescharacteristic
474C22×C4D4×D711yescharacteristic
475C22×C4C23×D711yescharacteristic
476Q16—148no—
477Q16—148no—
478SD16—28no—
479SD16—148no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—72no—
483C2×C8—72no—
484C2×C8—72no—
485C2×C8—72no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8C2×C4×D712yes—
489C2×C8#115712yes—
490C2×C8C2×C22⋊D712yes—
491C2×C8C2×C22⋊D712yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—148no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—141no—
502C4⋊C4—284no—
503C4⋊C4—282no—
504C4⋊C4—282no—
505C22⋊C4—42no—
506C22⋊C4—148no—
507C22⋊C4—284no—
508C42—42no—
509C42—72no—
510C42—72no—
511C42—144no—
512C42—141no—
513C42—141no—
514C42—282no—
515C42—282no—
516C42—282no—
517C42D4×D712yes—
518C42C2×C22⋊D712yes—
519C2×C7—82no—
520C2×C7C4×#12711yescharacteristic
521C2×C7C4×#13712yes—
522C2×C7C2×#49711yescharacteristic
523C2×C7#315011yescharacteristic
524C2×C7#317212yes—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23C2×D4×D711yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—142no—
536Q8—142no—
537Q8—142no—
538Q8—142no—
539Q8—281no—
540Q8—281no—
541Q8—281no—
542Q8—281no—
543D4—22no—
544D4—22no—
545D4—41no—
546D4—41no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—74no—
556C2×C4—84no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4#647814yes—
573C2×C4#650312yes—
574C2×C4#650312yes—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4C22×C22⋊D711yescharacteristic
581C8—22no—
582C8—74no—
583C8—142no—
584C8C4×C22⋊D714yes—
585C7C4×#49711yescharacteristic, complemented, Sylow
586C22—44no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C4×D4×D711yescharacteristic
592C22#9023911yescharacteristic
593C22C2×#647812yes—
594C22#9045912yes—
595C22C2×#650311yescharacteristic
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—144no—
601C4—282no—
602C4—282no—
603C4—282no—
604C4—561no—
605C4—561no—
606C4#8993914yesdirect product factor
607C4C4×D4×D711yescharacteristic
608C4#9023911yescharacteristic
609C4C2×C4×C22⋊D711yescharacteristic
610C4#9052311yescharacteristic
611C2—82no—
612C2C4×#647812yes—
613C2#1173711012yes—
614C2C4×#650311yescharacteristic
615C2C2×#8993911yescharacteristic
616C2#1174134911yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×C4×SD16D7
hypocenterC7C4×#497
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. C4×#6507C2semidirect
4. C2×C4×#1144C2semidirect
7. C2×C4×Dic28C2semidirect
9. #11728329C2semidirect
10. #11728310C2semidirect
11. #11728013C2semidirect
12. #11727982C2semidirect
15. C4×#6389C2semidirect
36. #89939C4direct (N × H)
84. #6507C2×C4semidirect
87. C2×#1144C2×C4semidirect
109. C2×Dic28C2×C4semidirect
130. #6389C2×C4semidirect
426. C4×C7#497semidirect
585. C7C4×#497semidirect
606. C4#89939direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C7×SD162
C2×#899392
C4×#65072
C2×C4×#11442
C4×#64992
C2×C4×#11422
C2×C4×Dic282
#117298042
#117283292
#117283102
#117280132
#117279822
#117279462
#117279132
C4×#63892
C4×#4977