Finite Group Databasebeta

← #13388114

Subgroups — #13388114

Order 1792.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90161C211yesmaximal, characteristic, complemented
2C4×C7×#121C211yesmaximal, characteristic
3C2×#89791C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729439C212yesmaximal
6#11729284C211yesmaximal, characteristic, complemented
7#11727414C212yesmaximal, complemented
8#11727386C212yesmaximal, complemented
9#11727280C212yesmaximal, complemented
10#11727243C211yesmaximal, characteristic, complemented
11#11727240C211yesmaximal, characteristic, complemented
12C2×C7×#121C2211yescharacteristic
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×#1142C2212yes—
17C2×#6489—21no—
18C2×#6487—21no—
19C2×#6487C2212yes—
20#90161—24no—
21#90161—24no—
22C2×#6433—21no—
23C2×#6432C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C2×C4×Dic14—21no—
26C2×C4×Dic14C2212yes—
27C7×#390C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#388—21no—
30C7×#386—21no—
31C7×#380C2211yescharacteristic
32C4×C7×C4⋊C4C2211yescharacteristic
33C4×C7×C22⋊C4—21no—
34C2×#6406C2212yes—
35C2×C22⋊(C7⋊C8)C2212yes—
36#89791—24no—
37#89780—24no—
38C2×#6384C2212yes—
39C2×#6384C2212yes—
40C2×#6384C2212yes—
41C2×#6382C2212yes—
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24151—71nomaximal, Sylow
48C22×C7×D4—21no—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C22×Dic14—21no—
55C22×Dic14C2312yes—
56C7×#121—22no—
57C7×#121—22no—
58C4×C7×D4—22no—
59C4×C7×D4—22no—
60C4×C7×D4—44no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C42×C7—22no—
71C2×C42×C7—22no—
72C2×C42×C7—21no—
73C2×C42×C7D411yescharacteristic
74C2×C42×C7D411yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—22no—
78C2×#1142—24no—
79C2×#1142—24no—
80C2×#1142—42no—
81#6489—28no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2312yes—
86#6487—28no—
87#6487—24no—
88#6487—24no—
89C2×Dic7⋊C4—22no—
90C2×Dic7⋊C4—22no—
91C2×C4×Dic7—22no—
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2312yes—
94#6433—28no—
95#6432—22no—
96#6432—22no—
97#6432D414yescomplemented
98#6432D414yescomplemented
99C4×Dic14—28no—
100C4×Dic14—24no—
101C4×Dic14—24no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105#6406—24no—
106C22⋊(C7⋊C8)—42no—
107#6384—24no—
108#6384—24no—
109#6384—42no—
110#6382—24no—
111C4⋊(C7⋊C8)—22no—
112C4⋊(C7⋊C8)—22no—
113C2×#533—71no—
114C2×#476—71no—
115C2×#461—71no—
116C4×#121D711yescharacteristic
117#2035—72no—
118#2000—72no—
119#1987—71no—
120#1981—71no—
121#1880—72no—
122#1865—72no—
123#1797—71no—
124C24×C7—21no—
125C24×C7C2×D411yescharacteristic
126C2×C7×D4—24no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4#4112yes—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7SD1614yescomplemented
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—22no—
160C22×Dic7—22no—
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—24no—
166C2×Dic14—22no—
167C2×Dic14—42no—
168C2×Dic14—42no—
169C2×Dic14C2×D414yes—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4#4112yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C42×C7—22no—
183C42×C7—22no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—42no—
187C42×C7C2×D412yes—
188C42×C7C2×D412yes—
189C22⋊Dic7—28no—
190#1142—48no—
191C4⋊Dic7—24no—
192C4⋊Dic7—24no—
193C4⋊Dic7—24no—
194C4⋊Dic7—22no—
195C4⋊Dic7C2×D414yes—
196Dic7⋊C4—28no—
197Dic7⋊C4—28no—
198C4×Dic7—28no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C22×SD16—72no—
206#533—144no—
207#533—144no—
208C2×C4⋊Q8—71no—
209C2×C4⋊Q8—71no—
210C2×#125—141no—
211C2×C22⋊Q8—72no—
212C2×C22⋊Q8—141no—
213C2×#121C2×D711yescharacteristic
214C2×C4×Q8—72no—
215C2×C4×Q8—141no—
216C2×C4×D4—21no—
217C2×C4×D4C2×D711yescharacteristic
218C2×#117—141no—
219C22×C42C2×D711yescharacteristic
220#476—144no—
221#461—144no—
222C2×#106—72no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×#103—72no—
226C2×#103—72no—
227C2×#103—72no—
228C2×#102—72no—
229C2×C22⋊C8—72no—
230#390C2×D711yescharacteristic
231#389C2×D711yescharacteristic
232#388—21no—
233#386—21no—
234#380C2×D711yescharacteristic
235C4×C4⋊C4C2×D711yescharacteristic
236C4×C22⋊C4—21no—
237#340—72no—
238#340—72no—
239#336—72no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7—22no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7C2×SD1612yes—
254C23×C7C22×D411yescharacteristic
255C7×D4—22no—
256C7×D4—22no—
257C7×D4—48no—
258C7×D4—41no—
259C7×D4—41no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—28no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—44no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—84no—
286C2×C4×C7#12012yes—
287C2×C4×C7#12012yes—
288C2×C4×C7C2×SD1612yes—
289C2×C4×C7C2×SD1612yes—
290C2×C4×C7#13712yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303Dic14—24no—
304Dic14—24no—
305Dic14—48no—
306Dic14—42no—
307Dic14—42no—
308C7⋊C8—42no—
309C7⋊C8—42no—
310C22×Q8—72no—
311C22×Q8—141no—
312C22×D4—21no—
313C22×D4C22×D711yescharacteristic
314C23×C4—21no—
315C23×C4C22⋊D711yescharacteristic
316C23×C4C22⋊D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C2×SD16—144no—
319C2×SD16—144no—
320C2×SD16—282no—
321C22×C8—72no—
322C22×C8—72no—
323C4⋊Q8—74no—
324C4⋊Q8—74no—
325C4⋊Q8—142no—
326C4⋊Q8—142no—
327#125—148no—
328C22⋊Q8—144no—
329C22⋊Q8—144no—
330C22⋊Q8—148no—
331#121—22no—
332#121—22no—
333C4×Q8—144no—
334C4×Q8—144no—
335C4×Q8—148no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—44no—
339#117—148no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—72no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—142no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—142no—
360C2×C42C22⋊D711yescharacteristic
361C2×C42C22⋊D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364#106—144no—
365C4⋊C8—142no—
366C4⋊C8—142no—
367#103—144no—
368#103—144no—
369#103—282no—
370#102—144no—
371C22⋊C8—282no—
372#95—21no—
373#95—21no—
374#95—21no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7—41no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7C2×#12011yescharacteristic
390C22×C7#52511yescharacteristic
391C22×C7#53711yescharacteristic
392C22×C7C22×SD1611yescharacteristic
393C22×C7C2×#13711yescharacteristic
394C22×C7#57511yescharacteristic
395C22×C7#57511yescharacteristic
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—42no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—81no—
402C4×C7—81no—
403C4×C7#44814yes—
404C4×C7#54611yescharacteristic
405C4×C7#54711yescharacteristic
406C4×C7#54811yescharacteristic
407C4×C7#55011yescharacteristic
408Dic7—44no—
409Dic7—44no—
410C24—21no—
411C24C2×C22⋊D711yescharacteristic
412C2×Q8—74no—
413C2×Q8—144no—
414C2×Q8—142no—
415C2×Q8—142no—
416C2×Q8—142no—
417C2×Q8—142no—
418C2×Q8—142no—
419C2×Q8—282no—
420C2×Q8—282no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4#115712yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4#114214yescomplemented
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C23×D711yescharacteristic
456SD16—288no—
457C2×C8—142no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—142no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—74no—
470C4⋊C4—148no—
471C4⋊C4—148no—
472C4⋊C4—144no—
473C4⋊C4—144no—
474C4⋊C4—144no—
475C4⋊C4—142no—
476C4⋊C4#115712yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—148no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—148no—
489C42D4×D712yes—
490C42D4×D712yes—
491C2×C7—42no—
492C2×C7—82no—
493C2×C7#240711yescharacteristic
494C2×C7C2×#44811yescharacteristic
495C2×C7#300511yescharacteristic
496C2×C7#310111yescharacteristic
497C2×C7#311611yescharacteristic
498C2×C7#324611yescharacteristic
499C2×C7#325211yescharacteristic
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C2×#114212yes—
514C23C22×C22⋊D711yescharacteristic
515Q8—144no—
516Q8—144no—
517Q8—288no—
518Q8—282no—
519Q8—282no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
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—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4C2×#114212yes—
558C2×C4C2×#114212yes—
559C2×C4#650012yes—
560C2×C4#650012yes—
561C2×C4#651412yes—
562C2×C4C2×#115711yescharacteristic
563C2×C4C2×#115711yescharacteristic
564C2×C4C2×D4×D711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4#655311yescharacteristic
567C2×C4#655411yescharacteristic
568C8—282no—
569C8—282no—
570C7#2415111yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#9048211yescharacteristic
586C22C22×#114211yescharacteristic
587C22C2×#650011yescharacteristic
588C22#9051311yescharacteristic
589C22#9051311yescharacteristic
590C22C2×#651411yescharacteristic
591C22#9052411yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—284no—
600C4—284no—
601C4#8981214yes—
602C4#9024111yescharacteristic
603C4#9024211yescharacteristic
604C4#9024311yescharacteristic
605C4#9024411yescharacteristic
606C2—42no—
607C2—82no—
608C2#1173975911yescharacteristic
609C2#1173984711yescharacteristic
610C2#1173998011yescharacteristic
611C2C2×#8981211yescharacteristic
612C2#1174037311yescharacteristic
613C2#1174060111yescharacteristic
614C2#1174076511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#121C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenterC4×#121D7
hypocenterC7#24151
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#90161C2semidirect
3. C2×#89791C2semidirect
4. C2×#89780C2semidirect
6. #11729284C2semidirect
7. #11727414C2semidirect
8. #11727386C2semidirect
9. #11727280C2semidirect
10. #11727243C2semidirect
11. #11727240C2semidirect
97. #6432D4semidirect
98. #6432D4semidirect
152. C22×C4×C7SD16semidirect
449. C22×C4#1142semidirect
570. C7#24151semidirect

Maximal subgroups

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

subgroupindex
C2×#901612
C4×C7×#1212
C2×#897912
C2×#897802
#117294392
#117292842
#117274142
#117273862
#117272802
#117272432
#117272402
#241517