Finite Group Databasebeta

← #13362274

Subgroups — #13362274

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#440C211yesmaximal, characteristic, complemented
2C2×#89940C211yesmaximal, characteristic
3C4×#6510C211yesmaximal, characteristic, complemented
4C2×#89802C211yesmaximal, characteristic, complemented
5C4×#6499C211yesmaximal, characteristic, complemented
6C2×#89786C211yesmaximal, characteristic
7C2×#89482C211yesmaximal, characteristic, complemented
8#11729804C211yesmaximal, characteristic, complemented
9#11728383C211yesmaximal, characteristic, complemented
10#11728361C211yesmaximal, characteristic, complemented
11#11728013C211yesmaximal, characteristic, complemented
12#11727980C211yesmaximal, characteristic, complemented
13#11727945C211yesmaximal, characteristic
14#11727912C211yesmaximal, characteristic, complemented
15Dic7×#97C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6510C2211yescharacteristic
20C22×#1144C2211yescharacteristic
21C2×#6499C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C22×#1132C2211yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C7×#440—24no—
27C2×C7×#107C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C7×#97C2211yescharacteristic
31#90002C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C4×C22⋊Dic7—22no—
36#89940C418yescomplemented
37C2×#6395C2211yescharacteristic
38C2×#6392C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89802—24no—
45#89786—24no—
46#89762—22no—
47#89760C2211yescharacteristic
48#89749C2211yescharacteristic
49#89749C2211yescharacteristic
50#89746C2211yescharacteristic
51C2×#6384C2211yescharacteristic
52C2×#6383C2211yescharacteristic
53C2×#6382C2211yescharacteristic
54C2×#6382C2211yescharacteristic
55#89717—22no—
56C4×(C2×D7)⋊C4—22no—
57C42×Dic7C2211yescharacteristic
58C2×#6377C2211yescharacteristic
59#89482—24no—
60#21693—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×D28C2311yescharacteristic
65C22×C4×D7—21no—
66C22×Dic14C2311yescharacteristic
67C2×C7×SD16—22no—
68C2×C7×SD16—22no—
69C2×C7×SD16C2×C414yescomplemented
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C4×C7×Q8—22no—
73C4×C7×Q8—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C22⋊Dic7—22no—
83C2×C22⋊Dic7—22no—
84#6510—22no—
85#6510C2×C414yescomplemented
86C2×#1144—22no—
87C2×#1144—22no—
88C2×#1144C2×C414yescomplemented
89#6499—22no—
90#6499C2×C414yescomplemented
91C2×#1141—22no—
92C2×#1141—22no—
93C2×#1141C2×C414yes—
94C2×(C2×D7)⋊C4—22no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—22no—
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7D412yes—
102C2×C4×Dic7D412yes—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C2×#1132—22no—
108C2×#1132—22no—
109C2×#1132C2×C414yescomplemented
110C4×D28—22no—
111C4×D28—22no—
112C4×Dic14—22no—
113C4×Dic14—22no—
114C7×#107—22no—
115C7×#103—22no—
116C7×#102—22no—
117C7×#97D414yes—
118#6409—22no—
119#6406—22no—
120C7⋊#95—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C7⋊#95—22no—
124#6395—22no—
125#6392—22no—
126#6390—22no—
127#6389—22no—
128#6389—22no—
129C8×Dic7—22no—
130C8×Dic7C2×C414yescomplemented
131#6384—22no—
132#6383—22no—
133#6382—22no—
134#6382—22no—
135#6377D414yes—
136C2×#495—71no—
137C2×#442—71no—
138C2×#441—71no—
139C2×#440—71no—
140C2×#440D711yescharacteristic, complemented
141C4×#124—71no—
142C4×#124—71no—
143#1973—71no—
144#1973—71no—
145#1972—71no—
146#1971—71no—
147#1958—71no—
148#1958—71no—
149#1846—71no—
150C4×#97—71no—
151C24×C7—21no—
152C23×D7—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8C22×C412yes—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4C22×C412yes—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—44no—
169C22×C4×C7—42no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7C2×D412yes—
177C22×Dic7C2×D412yes—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28—21no—
182C2×D28—21no—
183C2×D28C22×C412yes—
184C2×C4×D7—44no—
185C2×C4×D7—42no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—21no—
191C2×Dic14C22×C412yes—
192C7×SD16—28no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C2×D412yes—
197C2×C7×C8C2×D412yes—
198C2×C7×C8#4112yes—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—42no—
204C7×C22⋊C4—42no—
205C42×C7—42no—
206C42×C7C2×D412yes—
207C42×C7C2×D412yes—
208C22⋊Dic7—28no—
209C22⋊Dic7—44no—
210#1144—28no—
211#1141—28no—
212(C2×D7)⋊C4—28no—
213(C2×D7)⋊C4—44no—
214(C2×D7)⋊C4—42no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219Dic7⋊C4—42no—
220C4×Dic7—21no—
221C4×Dic7—21no—
222C4×Dic7—42no—
223C4×Dic7—42no—
224C4×Dic7—42no—
225C4×Dic7C22×C412yes—
226C4×Dic7#4112yes—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C22×C412yes—
230C2×C7⋊C8C2×D412yes—
231C2×C7⋊C8C2×D412yes—
232C2×C7⋊C8#4112yes—
233#1132—28no—
234C22×Q16—71no—
235C22×SD16—71no—
236C22×SD16C2×D711yescharacteristic
237C22×D8—71no—
238C2×#124—71no—
239C2×#124—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8C2×D711yescharacteristic
242C2×C4×D4—71no—
243C2×C4×D4C2×D711yescharacteristic
244#495—78no—
245#442—144no—
246#441—144no—
247#440—24no—
248#440—144no—
249C2×#107—71no—
250C2×#107C2×D711yescharacteristic
251C2×#106—71no—
252C2×#106—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103C2×D711yescharacteristic
257C2×#102—71no—
258C2×#102—71no—
259C2×#102—71no—
260C2×#102C2×D711yescharacteristic
261C2×#97—71no—
262C2×#97—71no—
263C2×#97—71no—
264C2×#97C2×D711yescharacteristic
265C2×C4×C8—71no—
266C2×C4×C8—71no—
267#391—71no—
268#391—71no—
269#390—71no—
270#390—71no—
271#388—142no—
272#388—142no—
273#382—71no—
274#382—71no—
275C4×C22⋊C4—142no—
276C4×C22⋊C4—142no—
277C43—71no—
278C23×C7—22no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—42no—
282C23×C7—42no—
283C23×C7C22×D411yescharacteristic
284C22×D7—22no—
285C22×D7—41no—
286C22×D7—41no—
287C22×D7—42no—
288C22×D7—42no—
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—41no—
296C7×D4—41no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—84no—
307C2×C4×C7#12712yes—
308C2×C4×C7#12712yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—22no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7C4×D414yes—
328D28—22no—
329D28—22no—
330D28—41no—
331D28—41no—
332C4×D7—84no—
333Dic14—22no—
334Dic14—22no—
335Dic14—41no—
336Dic14—41no—
337C7×C8—22no—
338C7×C8C4×D414yes—
339C7⋊C8—22no—
340C7⋊C8C4×D414yes—
341C22×Q8—71no—
342C22×Q8C22×D711yescharacteristic
343C22×D4—71no—
344C22×D4C22×D711yescharacteristic
345C23×C4—21no—
346C23×C4—141no—
347C2×Q16—74no—
348C2×Q16—142no—
349C2×Q16—142no—
350C2×SD16—22no—
351C2×SD16—22no—
352C2×SD16—74no—
353C2×SD16—142no—
354C2×SD16—142no—
355C2×SD16C4×D714yescomplemented
356C2×D8—74no—
357C2×D8—142no—
358C2×D8—142no—
359C22×C8—71no—
360C22×C8—71no—
361C22×C8C22×D711yescharacteristic
362C22×C8C22×D711yescharacteristic
363#124—74no—
364#124—74no—
365#124—142no—
366#124—142no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8—142no—
370C4×Q8—142no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—142no—
374C4×D4—142no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—142no—
383C2×C22⋊C4—142no—
384C2×C22⋊C4—142no—
385C2×C22⋊C4—142no—
386C2×C22⋊C4—141no—
387C2×C42—21no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—72no—
391C2×C42—72no—
392C2×C42—141no—
393C2×C42C22×D711yescharacteristic
394#107—22no—
395#107—142no—
396#106—142no—
397#106—142no—
398#103—22no—
399#103—142no—
400#103—142no—
401#103—142no—
402#102—22no—
403#102—142no—
404#102—142no—
405#102—142no—
406#97—74no—
407#97—142no—
408#97—142no—
409#97C22⋊D714yes—
410C4×C8—74no—
411C4×C8—142no—
412#95—142no—
413#95—142no—
414#95—142no—
415#95—142no—
416C22×C7—42no—
417C22×C7—44no—
418C22×C7—81no—
419C22×C7—81no—
420C22×C7—82no—
421C22×C7C2×C4×D411yescharacteristic
422C22×C7C2×#12711yescharacteristic
423C22×C7#53211yescharacteristic
424C22×C7#57512yes—
425C22×C7#57512yes—
426C2×D7—42no—
427C2×D7—44no—
428C2×D7—81no—
429C2×D7—81no—
430C2×D7—82no—
431C4×C7—22no—
432C4×C7—22no—
433C4×C7—42no—
434C4×C7—81no—
435C4×C7—81no—
436C4×C7C2×C4×D411yescharacteristic
437C4×C7C2×C4×D411yescharacteristic
438C4×C7#53211yescharacteristic
439C4×C7#53211yescharacteristic
440Dic7—24no—
441Dic7—42no—
442Dic7—42no—
443Dic7—81no—
444Dic7—81no—
445C24—21no—
446C24—141no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—72no—
453C2×Q8—141no—
454C2×Q8—141no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8C2×C4×D712yes—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—72no—
465C2×D4—141no—
466C2×D4—141no—
467C2×D4—141no—
468C2×D4—141no—
469C2×D4—141no—
470C2×D4C2×C4×D712yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—72no—
477C22×C4—72no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—284no—
482C22×C4—282no—
483C22×C4D4×D711yescharacteristic
484C22×C4D4×D711yescharacteristic
485C22×C4C23×D711yescharacteristic
486Q16—148no—
487SD16—28no—
488SD16—148no—
489D8—148no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—72no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—72no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8C2×C4×D712yes—
499C2×C8#115712yes—
500C2×C8C2×C22⋊D712yes—
501C2×C8C2×C22⋊D712yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—282no—
512C22⋊C4—42no—
513C22⋊C4—148no—
514C22⋊C4—148no—
515C22⋊C4—284no—
516C22⋊C4—284no—
517C22⋊C4—282no—
518C42—42no—
519C42—72no—
520C42—72no—
521C42—141no—
522C42—141no—
523C42—282no—
524C42—282no—
525C42—282no—
526C42D4×D712yes—
527C42C2×C22⋊D712yes—
528C2×C7—82no—
529C2×C7C4×#12711yescharacteristic
530C2×C7#294312yes—
531C2×C7#315311yescharacteristic
532C2×C7#315311yescharacteristic
533C2×C7#318312yes—
534D7—82no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—142no—
541C23—282no—
542C23—282no—
543C23—281no—
544C23—281no—
545C23C2×D4×D711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—142no—
551Q8—142no—
552Q8—281no—
553Q8—281no—
554D4—22no—
555D4—22no—
556D4—41no—
557D4—41no—
558D4—142no—
559D4—142no—
560D4—281no—
561D4—281no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—74no—
572C2×C4—84no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—282no—
582C2×C4—282no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—564no—
586C2×C4#650312yes—
587C2×C4#650312yes—
588C2×C4C22×C4×D711yescharacteristic
589C2×C4C2×#115711yescharacteristic
590C2×C4C2×D4×D711yescharacteristic
591C2×C4D7×#4111yescharacteristic
592C2×C4D7×#4111yescharacteristic
593C2×C4C22×C22⋊D711yescharacteristic
594C8—22no—
595C8—74no—
596C8—142no—
597C8C4×C22⋊D714yes—
598C7#2169311yescharacteristic, complemented, Sylow
599C22—42no—
600C22—44no—
601C22—82no—
602C22—81no—
603C22—81no—
604C22—282no—
605C22—284no—
606C22—562no—
607C22—561no—
608C22—561no—
609C22C4×D4×D711yescharacteristic
610C22#9023911yescharacteristic
611C22#9046412yes—
612C22#9046812yes—
613C22C2×#650311yescharacteristic
614C4—22no—
615C4—22no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—144no—
620C4—282no—
621C4—282no—
622C4—561no—
623C4—561no—
624C4C4×D4×D711yescharacteristic
625C4#9023911yescharacteristic
626C4C2×C4×C22⋊D711yescharacteristic
627C4#9052311yescharacteristic
628C2—82no—
629C2—562no—
630C2#1173721712yes—
631C2#1173723812yes—
632C2C4×#650311yescharacteristic
633C2#1174138311yescharacteristic
634C2#1174141811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#440C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
cosocleC22×C4×C7C24
hypercenterC2×#440D7
hypocenterC7#21693
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×C7×#440C2semidirect
3. C4×#6510C2semidirect
4. C2×#89802C2semidirect
5. C4×#6499C2semidirect
7. C2×#89482C2semidirect
8. #11729804C2semidirect
9. #11728383C2semidirect
10. #11728361C2semidirect
11. #11728013C2semidirect
12. #11727980C2semidirect
14. #11727912C2semidirect
15. Dic7×#97C2semidirect
36. #89940C4semidirect
69. C2×C7×SD16C2×C4semidirect
85. #6510C2×C4semidirect
88. C2×#1144C2×C4semidirect
90. #6499C2×C4semidirect
109. C2×#1132C2×C4semidirect
130. C8×Dic7C2×C4semidirect
140. C2×#440D7semidirect
355. C2×SD16C4×D7semidirect
598. C7#21693semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4402
C2×#899402
C4×#65102
C2×#898022
C4×#64992
C2×#897862
C2×#894822
#117298042
#117283832
#117283612
#117280132
#117279802
#117279452
#117279122
Dic7×#972
#216937