Finite Group Databasebeta

← #13381676

Subgroups — #13381676

Order 1792.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#575C211yesmaximal, characteristic
2C2×#90009C211yesmaximal, characteristic, complemented
3C2×#90006C211yesmaximal, characteristic, complemented
4#11741517C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6C22×#6396C211yesmaximal, characteristic, complemented
7#11729859C214yesmaximal, complemented
8#11729847C214yesmaximal, complemented
9#11729741C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C7×#575—24no—
13C7×#575—24no—
14C2×C7×#137—21no—
15C2×C7×#137C2212yes—
16C2×C7×AGL(1,ℤ/8)—21no—
17C2×C7×AGL(1,ℤ/8)C2212yes—
18C2×C7×#135—22no—
19C22×C7×M16C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22C22×#1136C2211yescharacteristic
23C23×C7⋊C8C2211yescharacteristic
24#90009—24no—
25#90006—22no—
26#90006C2214yes—
27#90005—22no—
28#90005—22no—
29#90005—22no—
30#90004C2214yes—
31C2×#6410C2212yescomplemented
32#89997—21no—
33C2×#6409C2212yescomplemented
34C2×#6407C2212yescomplemented
35#89987—21no—
36C2×#6406C2212yescomplemented
37C2×C22⋊(C7⋊C8)—22no—
38C2×#6396—24no—
39C2×#6396C2214yescomplemented
40#89349C2214yescomplemented
41#23530—71nomaximal, Sylow
42C2×C7×#41—21no—
43C2×C7×#41—21no—
44C2×C7×#41—22no—
45C2×C7×#41—21no—
46C2×C7×#41—21no—
47C2×C7×#41—21no—
48C2×C7×#41—21no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41D412yes—
56C2×C7×#41D414yes—
57C2×C7×#41C2312yes—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8C2311yescharacteristic
60C22×C7×D4—21no—
61C22×C7×D4—21no—
62C22×C7×D4D411yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C7×#137—28no—
68C7×#137—24no—
69C7×#137—24no—
70C7×AGL(1,ℤ/8)—28no—
71C7×AGL(1,ℤ/8)—24no—
72C7×AGL(1,ℤ/8)—24no—
73C7×#135—216no—
74C2×C7×Q16—22no—
75C2×C7×SD16—22no—
76C2×C7×SD16—22no—
77C2×C7×D8—22no—
78C2×C7×M16—21no—
79C2×C7×M16—22no—
80C2×C7×M16D414yes—
81C2×C7×M16C2312yes—
82C22×C7×C8—21no—
83C2×C22⋊Dic7—21no—
84#6489—21no—
85#6489—21no—
86#6489—21no—
87#6489—21no—
88C2×C4⋊Dic7C2312yes—
89C2×C4×Dic7—21no—
90C2×#1136—21no—
91C2×#1136—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136—22no—
95C2×#1136—22no—
96C2×#1136C2312yes—
97C2×#1136C2312yes—
98C22×C7⋊C8—22no—
99C22×C7⋊C8—22no—
100C22×C7⋊C8—21no—
101C22×C7⋊C8—21no—
102C22×C7⋊C8D412yes—
103C22×C7⋊C8D412yes—
104C22×C7⋊C8C2312yes—
105#6410—24no—
106#6409—24no—
107#6409—24no—
108#6407—24no—
109#6406—24no—
110#6406—24no—
111C22⋊(C7⋊C8)—28no—
112#6396—28no—
113C2×#575D711yescharacteristic
114C22×#108—71no—
115#2896—71no—
116C2×#416—71no—
117C2×#413—71no—
118C2×#408—71no—
119#2052—74no—
120#2051—74no—
121#1742—71no—
122C24×C7—21no—
123C24×C7—21no—
124C24×C7#4111yescharacteristic
125C7×#41—24no—
126C7×#41—24no—
127C7×#41—24no—
128C7×#41—24no—
129C7×#41—24no—
130C7×#41—24no—
131C7×#41—22no—
132C7×#41—22no—
133C7×#41—22no—
134C7×#41—22no—
135C7×#41—42no—
136C7×#41—42no—
137C7×#41—42no—
138C7×#41—42no—
139C7×#41—42no—
140C7×#41—42no—
141C7×#41—42no—
142C7×#41—42no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—22no—
146C2×C7×Q8—22no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8C2×D412yes—
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—22no—
154C2×C7×D4—24no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—41no—
165C2×C7×D4—41no—
166C2×C7×D4—42no—
167C2×C7×D4—42no—
168C2×C7×D4—42no—
169C2×C7×D4C2×D412yes—
170C2×C7×D4C2×D412yes—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—24no—
175C22×C4×C7—22no—
176C22×C4×C7—41no—
177C22×C4×C7—41no—
178C22×C4×C7—42no—
179C22×C4×C7—42no—
180C22×C4×C7—42no—
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2×D412yes—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C22×C4×C7#4112yes—
188C22×C4×C7#4111yescharacteristic
189C22×C4×C7#4112yes—
190C22×C4×C7C2411yescharacteristic
191C22×Dic7—22no—
192C7×Q16—28no—
193C7×SD16—28no—
194C7×SD16—28no—
195C7×D8—28no—
196C7×M16—24no—
197C7×M16—22no—
198C7×M16C2×D414yes—
199C2×C7×C8—24no—
200C2×C7×C8—22no—
201C22⋊Dic7—41no—
202C22⋊Dic7—41no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4×Dic7—41no—
206C4×Dic7—41no—
207#1136—22no—
208#1136—22no—
209#1136—22no—
210#1136—22no—
211#1136—42no—
212#1136—42no—
213C2×C7⋊C8—24no—
214C2×C7⋊C8—24no—
215C2×C7⋊C8—22no—
216C2×C7⋊C8—22no—
217C2×C7⋊C8—22no—
218C2×C7⋊C8—22no—
219C2×C7⋊C8—42no—
220C2×C7⋊C8—42no—
221C2×C7⋊C8C2×D414yes—
222C22×#41C2×D711yescharacteristic
223C22×#41C2×D711yescharacteristic
224#575—24no—
225#575—24no—
226C2×#137—21no—
227C2×#137C2×D712yes—
228C2×AGL(1,ℤ/8)—21no—
229C2×AGL(1,ℤ/8)C2×D712yes—
230C2×#135—22no—
231C22×M16—71no—
232C22×M16—71no—
233C22×M16C2×D711yescharacteristic
234C23×C8—71no—
235C2×#117—71no—
236C2×#108—74no—
237C2×#108—144no—
238#419—141no—
239#418—141no—
240#417—74no—
241#416—142no—
242#416—142no—
243#416—142no—
244C2×#103—72no—
245C2×#102—72no—
246#413—144no—
247C2×#101—72no—
248C2×#100—72no—
249#408—74no—
250#408—142no—
251C2×C22⋊C8—142no—
252#343—74no—
253C23×C7—22no—
254C23×C7—22no—
255C23×C7—42no—
256C23×C7—44no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—42no—
260C23×C7—82no—
261C23×C7#12111yescharacteristic
262C23×C7#12111yescharacteristic
263C23×C7#12312yes—
264C23×C7#12412yes—
265C23×C7C2×#4111yescharacteristic
266C7×Q8—22no—
267C7×Q8—22no—
268C7×Q8—22no—
269C7×Q8—22no—
270C7×Q8—41no—
271C7×Q8—41no—
272C7×Q8—42no—
273C7×D4—22no—
274C7×D4—22no—
275C7×D4—22no—
276C7×D4—22no—
277C7×D4—42no—
278C7×D4—42no—
279C7×D4—44no—
280C7×D4—44no—
281C7×D4—44no—
282C7×D4—41no—
283C7×D4—41no—
284C7×D4—42no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—44no—
293C2×C4×C7—44no—
294C2×C4×C7—44no—
295C2×C4×C7#12012yes—
296C2×C4×C7#12012yes—
297C2×C4×C7#12112yes—
298C2×C4×C7#12112yes—
299C2×C4×C7#12112yes—
300C2×C4×C7#12112yes—
301C2×C4×C7#12111yescharacteristic
302C2×C4×C7#12111yescharacteristic
303C2×C4×C7#12312yes—
304C2×C4×C7#12312yes—
305C2×C4×C7#12312yes—
306C2×C4×C7#12412yes—
307C2×C4×C7C22×D411yescharacteristic
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×Dic7—42no—
314C2×Dic7—82no—
315C7×C8—24no—
316C7⋊C8—24no—
317C7⋊C8—42no—
318C7⋊C8—42no—
319C2×#41—21no—
320C2×#41—22no—
321C2×#41—21no—
322C2×#41—21no—
323C2×#41—21no—
324C2×#41—22no—
325C2×#41—22no—
326C2×#41—22no—
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41C22⋊D712yes—
333C2×#41C22⋊D714yes—
334C2×#41C22×D712yes—
335C22×Q8C22⋊D711yescharacteristic
336C22×Q8C22×D711yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4C22⋊D711yescharacteristic
340C22×D4C22×D711yescharacteristic
341C23×C4—21no—
342C23×C4—21no—
343C23×C4C22×D711yescharacteristic
344#137—28no—
345#137—24no—
346#137—24no—
347AGL(1,ℤ/8)—28no—
348AGL(1,ℤ/8)—24no—
349AGL(1,ℤ/8)—24no—
350#135—216no—
351C2×Q16—22no—
352C2×SD16—22no—
353C2×SD16—22no—
354C2×D8—22no—
355C2×M16—21no—
356C2×M16—22no—
357C2×M16—72no—
358C2×M16—72no—
359C2×M16—141no—
360C2×M16—141no—
361C2×M16—142no—
362C2×M16—142no—
363C2×M16—142no—
364C2×M16—142no—
365C2×M16C22⋊D714yes—
366C2×M16C22×D712yes—
367C22×C8—21no—
368C22×C8—72no—
369C22×C8—72no—
370C22×C8—72no—
371C22×C8—142no—
372C22×C8—142no—
373C22×C8—141no—
374C22×C8—141no—
375#117—141no—
376#117—141no—
377#117—141no—
378#117—141no—
379C2×C4⋊C4—72no—
380C2×C22⋊C4—141no—
381C2×C42—141no—
382#108—148no—
383#103—144no—
384#103—144no—
385#102—144no—
386#102—144no—
387#101—144no—
388#100—144no—
389C22⋊C8—148no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—42no—
397C22×C7—44no—
398C22×C7—82no—
399C22×C7—84no—
400C22×C7—82no—
401C22×C7—82no—
402C22×C7#39412yes—
403C22×C7#39412yes—
404C22×C7C2×#12111yescharacteristic
405C22×C7C2×#12311yescharacteristic
406C22×C7C2×#12411yescharacteristic
407C4×C7—42no—
408C4×C7—42no—
409C4×C7#39412yes—
410C4×C7#39412yes—
411C4×C7C2×#12011yescharacteristic
412C4×C7C2×#12111yescharacteristic
413C4×C7C2×#12111yescharacteristic
414C4×C7C2×#12311yescharacteristic
415Dic7—82no—
416C24—21no—
417C24—21no—
418C24#115911yescharacteristic
419#41—24no—
420#41—24no—
421#41—24no—
422#41—24no—
423#41—24no—
424#41—24no—
425#41—22no—
426#41—22no—
427#41—22no—
428#41—22no—
429#41—42no—
430#41—42no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—42no—
435#41—42no—
436#41—42no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8C2×C22⋊D712yes—
446C2×Q8C2×C22⋊D712yes—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4C2×C22⋊D712yes—
464C2×D4C2×C22⋊D712yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—142no—
476C22×C4D4×D711yescharacteristic
477C22×C4D4×D711yescharacteristic
478C22×C4#115912yes—
479C22×C4#115911yescharacteristic
480C22×C4#115912yes—
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D712yes—
484C22×C4C2×C22⋊D712yes—
485C22×C4C23×D711yescharacteristic
486Q16—28no—
487SD16—28no—
488SD16—28no—
489D8—28no—
490M16—24no—
491M16—22no—
492M16—142no—
493M16—142no—
494M16—142no—
495M16—142no—
496M16—282no—
497M16—282no—
498M16C2×C22⋊D714yes—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—74no—
502C2×C8—144no—
503C2×C8—144no—
504C2×C8—142no—
505C2×C8—142no—
506C2×C8—142no—
507C2×C8—142no—
508C2×C8—282no—
509C2×C8—282no—
510C4⋊C4—142no—
511C4⋊C4—142no—
512C22⋊C4—281no—
513C22⋊C4—281no—
514C42—281no—
515C42—281no—
516C2×C7—21no—
517C2×C7—21no—
518C2×C7—22no—
519C2×C7—22no—
520C2×C7—82no—
521C2×C7—82no—
522C2×C7C2×#39411yescharacteristic
523C2×C7#306712yes—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—44no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—82no—
532C23#649812yes—
533C23#649912yes—
534C23#650211yescharacteristic
535C23#650211yescharacteristic
536C23C2×#115911yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—42no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—42no—
549D4—42no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—41no—
554D4—41no—
555D4—42no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—282no—
567C2×C4—562no—
568C2×C4#649812yes—
569C2×C4#649812yes—
570C2×C4#649812yes—
571C2×C4#649912yes—
572C2×C4#650112yes—
573C2×C4#650112yes—
574C2×C4#650212yes—
575C2×C4#650212yes—
576C2×C4#650211yescharacteristic
577C2×C4#650211yescharacteristic
578C2×C4#651612yes—
579C2×C4#651612yes—
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×#115911yescharacteristic
582C2×C4C2×#115911yescharacteristic
583C2×C4C22×C22⋊D711yescharacteristic
584C2×C4C22×C22⋊D711yescharacteristic
585C2×C4C22×C22⋊D711yescharacteristic
586C8—24no—
587C8—144no—
588C8—282no—
589C8—282no—
590C7#2353011yescharacteristic, complemented, Sylow
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—84no—
601C22—82no—
602C22—82no—
603C22#8999412yes—
604C22#8999412yes—
605C22C2×#649811yescharacteristic
606C22C2×#649911yescharacteristic
607C22C2×#650211yescharacteristic
608C4—42no—
609C4—42no—
610C4—562no—
611C4#8999412yes—
612C4#8999412yes—
613C4C2×#649811yescharacteristic
614C4C2×#650111yescharacteristic
615C4C2×#650211yescharacteristic
616C4C2×#651611yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2—82no—
622C2—82no—
623C2#1174144612yes—
624C2C2×#8999411yescharacteristic
subgroupG/H
centerC2×C4#6502
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#575C2
socleC22×C7C2×#121
hypercenterC2×#575D7
hypocenterC7#23530
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
2. C2×#90009C2semidirect
3. C2×#90006C2semidirect
4. #11741517C2semidirect
5. C2×#90005C2semidirect
6. C22×#6396C2semidirect
7. #11729859C2semidirect
8. #11729847C2semidirect
9. #11729741C2semidirect
31. C2×#6410C22semidirect
33. C2×#6409C22semidirect
34. C2×#6407C22semidirect
36. C2×#6406C22semidirect
39. C2×#6396C22semidirect
40. #89349C22semidirect
590. C7#23530semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5752
C2×#900092
C2×#900062
#117415172
C2×#900052
C22×#63962
#117298592
#117298472
#117297412
#235307