Finite Group Databasebeta

← #13373155

Subgroups — #13373155

Order 1792.

866 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×#89936C211yesmaximal, characteristic
2#11739779C211yesmaximal, characteristic
3#11733934C211yesmaximal, characteristic, complemented
4C7×#1877C211yesmaximal, characteristic
5#11729801C211yesmaximal, characteristic
6#11729363C211yesmaximal, characteristic
7#11728008C211yesmaximal, characteristic, complemented
8#11727976C211yesmaximal, characteristic, complemented
9#11727907C211yesmaximal, characteristic
10Dic7×#102C211yesmaximal, characteristic
11#11727429C211yesmaximal, characteristic, complemented
12#11726603C211yesmaximal, characteristic, complemented
13#11726574C211yesmaximal, characteristic, complemented
14#11726408C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×C7×SD16C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×D4×Dic7C2211yescharacteristic
20C2×D4×Dic7C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C7×#391C2211yescharacteristic
24#90002C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29#89982—21no—
30C2×C7⋊#95—21no—
31#89936C418yescomplemented
32C2×#6392C2211yescharacteristic
33C2×#6389C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36#89749C2211yescharacteristic
37#89748C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89745C2211yescharacteristic
40#89718—21no—
41#89714—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43C4×Dic7⋊C4C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46#89426—21no—
47#89425—21no—
48#89422—21no—
49#89421—21no—
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22776—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×Dic7—21no—
65C23×Dic7—21no—
66C2×C7×SD16—22no—
67C2×C7×SD16—22no—
68C2×C7×SD16C2×C414yes—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C4×C7×D4—22no—
72C4×C7×D4—41no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79Q8×Dic7—22no—
80Q8×Dic7C2×C414yescomplemented
81D4×Dic7—22no—
82D4×Dic7—22no—
83D4×Dic7—41no—
84D4×Dic7C2×C414yes—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4D411yescharacteristic
95C2×Dic7⋊C4D411yescharacteristic
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C2×C4×Dic7C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C7×#103—41no—
104C7×#102—22no—
105C7×#95—21no—
106C7×#95—21no—
107#6409—41no—
108#6409C2×C414yescomplemented
109#6406—22no—
110#6406C2×C414yes—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95D411yescharacteristic
120C7⋊#95D411yescharacteristic
121#6392C2×C414yescomplemented
122#6389C2×C414yescomplemented
123Dic7⋊C8—22no—
124C8×Dic7—41no—
125#6377—41no—
126C2×#440—71no—
127#2355—71no—
128#2349—71no—
129#1935—71no—
130#1933—71no—
131#1921—71no—
132#1918—71no—
133#1877—71no—
134#1877D711yescharacteristic
135#1836—71no—
136#1805—71no—
137#1802—71no—
138#1771—71no—
139#1762—71no—
140C4×#102—71no—
141C24×C7—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8C22×C412yes—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4C22×C412yes—
152C2×C7×D4#4112yes—
153C2×C7×D4#4112yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—81no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—24no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—42no—
169C22×Dic7—42no—
170C22×Dic7—81no—
171C22×Dic7—81no—
172C22×Dic7—81no—
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C7×SD16—28no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8C22×C412yes—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C22⋊C4—42no—
189C42×C7—21no—
190C42×C7—21no—
191C22⋊Dic7—24no—
192C22⋊Dic7—42no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—24no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7C22×C412yes—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—22no—
203Dic7⋊C4—41no—
204Dic7⋊C4—42no—
205Dic7⋊C4—81no—
206Dic7⋊C4—81no—
207Dic7⋊C4—81no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—24no—
211C4×Dic7—81no—
212C4×Dic7—81no—
213C4×Dic7C22×C412yes—
214C4×Dic7#4112yes—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221C22×SD16C2×D711yescharacteristic
222C2×C4×Q8—71no—
223C2×C4×D4—71no—
224C2×C4×D4—71no—
225C2×C4×D4C2×D711yescharacteristic
226#440—78no—
227C2×#107—71no—
228C2×C4⋊C8—71no—
229C2×#103—71no—
230C2×#103—71no—
231C2×#103C2×D711yescharacteristic
232C2×#102—71no—
233C2×#102—71no—
234C2×#102C2×D711yescharacteristic
235C2×#97—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#391—71no—
239#391C2×D711yescharacteristic
240#389—71no—
241#389—71no—
242#387—141no—
243#387—141no—
244#385—141no—
245#385—141no—
246#384—71no—
247#382—71no—
248#382—141no—
249#380—141no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—141no—
254C2×#95—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340C2×D711yescharacteristic
263#336—71no—
264#336C2×D711yescharacteristic
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C22×D411yescharacteristic
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8—41no—
276C7×Q8—41no—
277C7×D4—22no—
278C7×D4—22no—
279C7×D4—22no—
280C7×D4#11714yes—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—84no—
297C2×C4×C7C23×C411yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—84no—
325C2×Dic7—84no—
326C2×Dic7C4×D412yes—
327C2×Dic7C4×D412yes—
328C2×Dic7#13712yes—
329C7×C8—22no—
330C7×C8—41no—
331C7⋊C8—22no—
332C7⋊C8—41no—
333C22×Q8C22×D711yescharacteristic
334C22×D4C22×D711yescharacteristic
335C23×C4—21no—
336C23×C4—141no—
337C23×C4—141no—
338C2×SD16—22no—
339C2×SD16—22no—
340C2×SD16C4×D714yes—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8C22×D711yescharacteristic
344C22×C8C22×D711yescharacteristic
345C4×Q8—74no—
346C4×Q8—142no—
347C4×D4—22no—
348C4×D4—41no—
349C4×D4—74no—
350C4×D4—142no—
351C4×D4—142no—
352C4×D4—281no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—141no—
368C2×C22⋊C4—141no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42C22×D711yescharacteristic
375#107—74no—
376C4⋊C8—142no—
377#103—41no—
378#103—74no—
379#103—281no—
380#102—22no—
381#102—74no—
382#102—142no—
383#97—74no—
384#97—281no—
385C4×C8—281no—
386#95—21no—
387#95—21no—
388#95—71no—
389#95—71no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398C22×C7—24no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7#54611yescharacteristic
407C22×C7#54811yescharacteristic
408C22×C7C2×#13711yescharacteristic
409C22×C7#57511yescharacteristic
410C22×C7#57611yescharacteristic
411C22×C7#57711yescharacteristic
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7C2×#11711yescharacteristic
418C4×C7#52011yescharacteristic
419C4×C7#55211yescharacteristic
420C4×C7#55311yescharacteristic
421Dic7—22no—
422Dic7—22no—
423Dic7—22no—
424Dic7—42no—
425Dic7—42no—
426Dic7—41no—
427Dic7—81no—
428Dic7—81no—
429C24—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8C2×C4×D712yes—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×C4×D712yes—
440C2×D4#115912yes—
441C2×D4#115912yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—81no—
452C22×C4—144no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—282no—
458C22×C4—282no—
459C22×C4—561no—
460C22×C4—561no—
461C22×C4—561no—
462C22×C4D4×D711yescharacteristic
463C22×C4D4×D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465SD16—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—72no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8C2×C4×D712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—72no—
483C4⋊C4—72no—
484C4⋊C4—142no—
485C4⋊C4—144no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—281no—
493C4⋊C4—282no—
494C4⋊C4—561no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C22⋊C4—42no—
498C22⋊C4—144no—
499C22⋊C4—282no—
500C42—21no—
501C42—21no—
502C42—72no—
503C42—72no—
504C42—144no—
505C42—141no—
506C42—141no—
507C42—561no—
508C42—561no—
509C2×C7—44no—
510C2×C7#235211yescharacteristic
511C2×C7C4×#13711yescharacteristic
512C2×C7#294611yescharacteristic
513C2×C7#331111yescharacteristic
514C2×C7#331411yescharacteristic
515C2×C7#337011yescharacteristic
516C2×C7#337211yescharacteristic
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C2×D4×D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4#644014yes—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—84no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—141no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—564no—
574C2×C4—564no—
575C2×C4C22×C4×D711yescharacteristic
576C2×C4C2×#115911yescharacteristic
577C2×C4C2×#115911yescharacteristic
578C2×C4#654811yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4#655411yescharacteristic
581C8—22no—
582C8—41no—
583C8—142no—
584C8—281no—
585C7#2277611yescharacteristic, complemented, Sylow
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C4×D4×D711yescharacteristic
594C22#9028211yescharacteristic
595C22#9028911yescharacteristic
596C22#9045311yescharacteristic
597C22#9046011yescharacteristic
598C22#9046411yescharacteristic
599C22D7×#13711yescharacteristic
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—142no—
606C4—142no—
607C4—142no—
608C4—282no—
609C4—282no—
610C4—281no—
611C4—561no—
612C4—561no—
613C4C2×#644011yescharacteristic
614C4#9023711yescharacteristic
615C4#9029211yescharacteristic
616C4#9034111yescharacteristic
617C2—44no—
618C2#1173211811yescharacteristic
619C2#1173709211yescharacteristic
620C2#1173721911yescharacteristic
621C2#1173760711yescharacteristic
622C2#1173769711yescharacteristic
623C2#1173786311yescharacteristic
624C2#1173796511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1877C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1877D7
hypocenterC7#22776
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
3. #11733934C2semidirect
7. #11728008C2semidirect
8. #11727976C2semidirect
11. #11727429C2semidirect
12. #11726603C2semidirect
13. #11726574C2semidirect
14. #11726408C2semidirect
15. #11726375C2semidirect
31. #89936C4semidirect
80. Q8×Dic7C2×C4semidirect
108. #6409C2×C4semidirect
121. #6392C2×C4semidirect
122. #6389C2×C4semidirect
585. C7#22776semidirect

Maximal subgroups

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

subgroupindex
C2×#899362
#117397792
#117339342
C7×#18772
#117298012
#117293632
#117280082
#117279762
#117279072
Dic7×#1022
#117274292
#117266032
#117265742
#117264082
#117263752
#227767