Finite Group Databasebeta

← #13276680

Subgroups — #13276680

Order 1792.

2558 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic7C211yesmaximal, characteristic
2C2×C7×#390C211yesmaximal, characteristic
3C2×#89993C211yesmaximal, characteristic
4C2×#89757C211yesmaximal, characteristic, complemented
5C2×#89722C214yesmaximal, complemented
6C2×#89707C211yesmaximal, characteristic, complemented
7#11729317C211yesmaximal, characteristic, complemented
8#11729300C214yesmaximal, complemented
9C4×C7⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C24×Dic7C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2214yes—
14C22×C42×C7C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2214yes—
17C2×D4×Dic7—24no—
18C2×D4×Dic7—24no—
19C2×D4×Dic7—24no—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2212yes—
22C22×C4⋊Dic7C2214yes—
23C22×C4⋊Dic7C2211yescharacteristic
24C22×C4×Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26C7×#390C418yescomplemented
27#89993—24no—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2214yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32#89757—24no—
33#89722—216no—
34#89707—24no—
35#13977—71nomaximal, Sylow
36C25×C7C2312yes—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4C2×C414yes—
42C22×C7×D4D414yes—
43C22×C7×D4Q814yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—24no—
48C23×Dic7—24no—
49C23×Dic7—24no—
50C23×Dic7—28no—
51C23×Dic7—44no—
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2311yescharacteristic
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4C2×C414yescomplemented
60C2×C7×C22⋊C4—28no—
61C2×C7×C22⋊C4—28no—
62C2×C7×C22⋊C4C2×C4116yescomplemented
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7C2×C414yescomplemented
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—28no—
70C2×C22⋊Dic7—28no—
71C2×C22⋊Dic7—44no—
72D4×Dic7—232no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—28no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—28no—
80C2×C4⋊Dic7—28no—
81C2×C4⋊Dic7—24no—
82C2×C4⋊Dic7—44no—
83C2×C4⋊Dic7D414yes—
84C2×C4×Dic7—22no—
85C2×C4×Dic7—22no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—22no—
89C2×C4×Dic7D414yes—
90C7⋊#95—24no—
91C7⋊#95—28no—
92C7⋊#95—24no—
93C7⋊#95D418yescomplemented
94C22×C4×D4—71no—
95C2×#390—71no—
96C2×#390D711yescharacteristic
97C2×#389—71no—
98C2×#385—74no—
99C2×#384—71no—
100#1439—74no—
101C4⋊#95—71no—
102C4×#95—71no—
103C24×C7—24no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7—24no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—24no—
110C24×C7—24no—
111C24×C7C22×C414yes—
112C24×C7C2×D414yes—
113C24×C7C2×Q814yes—
114C24×C7C2411yescharacteristic
115C2×C7×D4—28no—
116C2×C7×D4—28no—
117C2×C7×D4—28no—
118C2×C7×D4—28no—
119C2×C7×D4—28no—
120C2×C7×D4—28no—
121C2×C7×D4C4⋊C4116yescomplemented
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—24no—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—22no—
130C22×C4×C7C22×C414yes—
131C22×C4×C7C22×C414yes—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C2×D414yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×Q812yes—
138C22×C4×C7#4114yes—
139C22×C4×C7#4112yes—
140C22×C4×C7#4112yes—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—22no—
144C22×Dic7—216no—
145C22×Dic7—24no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—24no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—24no—
164C22×Dic7—416no—
165C22×Dic7—416no—
166C22×Dic7—48no—
167C22×Dic7—48no—
168C22×Dic7C2×D414yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D414yes—
171C22×Dic7C2×D414yes—
172C22×Dic7C2×D412yes—
173C7×C4⋊C4—28no—
174C7×C22⋊C4—232no—
175C42×C7—28no—
176C22⋊Dic7—216no—
177C22⋊Dic7—416no—
178C4⋊Dic7—28no—
179C4⋊Dic7—416no—
180C4⋊Dic7—44no—
181C4⋊Dic7—48no—
182C4×Dic7—28no—
183C4×Dic7—44no—
184C23×D4C2×D711yescharacteristic
185C24×C4—72no—
186C2×C4×D4—144no—
187C2×C4×D4—144no—
188C2×C4×D4—144no—
189C22×C4⋊C4—71no—
190C22×C4⋊C4—71no—
191C22×C4⋊C4—72no—
192C22×C4⋊C4—74no—
193C22×C4⋊C4C2×D711yescharacteristic
194C22×C22⋊C4—72no—
195C22×C22⋊C4—74no—
196C22×C22⋊C4C2×D714yes—
197C22×C42—71no—
198C22×C42—71no—
199C22×C42C2×D711yescharacteristic
200#390—144no—
201#390Dic718yescomplemented
202#389—144no—
203#385—1416no—
204#384—144no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—74no—
208C2×#95—72no—
209C23×C7—28no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—28no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—28no—
218C23×C7—28no—
219C23×C7—28no—
220C23×C7—28no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—44no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—44no—
227C23×C7—44no—
228C23×C7—44no—
229C23×C7—44no—
230C23×C7—44no—
231C23×C7C2×C4⋊C418yes—
232C23×C7C23×C411yescharacteristic
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D411yescharacteristic
235C23×C7C22×D411yescharacteristic
236C23×C7C22×D412yes—
237C23×C7C22×Q811yescharacteristic
238C23×C7C2×#4111yescharacteristic
239C23×C7C2×#4111yescharacteristic
240C23×C7C2×#4112yes—
241C23×C7H(5,𝔽2)12yes—
242C23×C72−1+412yes—
243C7×D4—232no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—28no—
251C2×C4×C7—24no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—44no—
255C2×C4×C7C2×C4⋊C414yes—
256C2×C4×C7C4×D414yes—
257C2×C4×C7C4×D414yes—
258C2×C4×C7C4×D418yescomplemented
259C2×C4×C7#12014yes—
260C2×C4×C7C22⋊Q814yes—
261C2×C4×C7C22⋊Q814yes—
262C2×C4×C7#12314yes—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—22no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—28no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—432no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—44no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—44no—
289C2×Dic7—44no—
290C2×Dic7#12714yes—
291C25C22×D712yes—
292C22×D4—22no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4Dic1414yes—
297C22×D4D2814yes—
298C22×D4C2×Dic714yes—
299C23×C4—71no—
300C23×C4—72no—
301C23×C4—71no—
302C23×C4—72no—
303C23×C4—72no—
304C23×C4—144no—
305C23×C4—144no—
306C23×C4—148no—
307C23×C4—144no—
308C23×C4—284no—
309C23×C4C22×D712yes—
310C23×C4C22×D712yes—
311C23×C4C22×D711yescharacteristic
312C4×D4—1432no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—74no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—142no—
318C2×C4⋊C4—148no—
319C2×C4⋊C4—148no—
320C2×C4⋊C4—144no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—148no—
324C2×C4⋊C4—142no—
325C2×C4⋊C4—284no—
326C2×C4⋊C4C2×Dic714yescomplemented
327C2×C22⋊C4—28no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—144no—
330C2×C22⋊C4—144no—
331C2×C22⋊C4—144no—
332C2×C22⋊C4—148no—
333C2×C22⋊C4—148no—
334C2×C22⋊C4—284no—
335C2×C22⋊C4C2×Dic7116yescomplemented
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—74no—
339C2×C42—142no—
340C2×C42—142no—
341C2×C42—142no—
342C2×C42—142no—
343C2×C42—142no—
344C2×C42C2×Dic714yescomplemented
345#95—78no—
346#95—148no—
347#95—144no—
348#95—144no—
349C22×C7—216no—
350C22×C7—28no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7—44no—
354C22×C7—48no—
355C22×C7—48no—
356C22×C7—44no—
357C22×C7—44no—
358C22×C7—48no—
359C22×C7—44no—
360C22×C7C22×C4⋊C411yescharacteristic
361C22×C7C2×C4×D411yescharacteristic
362C22×C7C2×C4×D412yes—
363C22×C7C2×C4×D411yescharacteristic
364C22×C7#52012yes—
365C22×C7C2×#12011yescharacteristic
366C22×C7C2×C22⋊Q811yescharacteristic
367C22×C7C2×C22⋊Q811yescharacteristic
368C22×C7C2×#12311yescharacteristic
369C22×C7C2×#12711yescharacteristic
370C22×C7#53211yescharacteristic
371C22×C7#53212yes—
372C22×C7#53511yescharacteristic
373C22×C7#53611yescharacteristic
374C22×C7#54212yes—
375C22×C7D4212yes—
376C22×C7D4212yes—
377C22×C7#54712yes—
378C22×C7#54712yes—
379C22×C7#54812yes—
380C22×C7D4×Q812yes—
381C22×C7#55312yes—
382C22×C7#55412yes—
383C4×C7—28no—
384C4×C7—44no—
385C4×C7#38518yescomplemented
386Dic7—24no—
387Dic7—44no—
388Dic7—42no—
389Dic7—44no—
390Dic7—82no—
391C24—24no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—24no—
398C24—24no—
399C24C2×Dic1414yes—
400C24C2×D2814yes—
401C24C22×Dic714yes—
402C24C23×D711yescharacteristic
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4C4⋊Dic7116yescomplemented
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—74no—
419C22×C4—72no—
420C22×C4—74no—
421C22×C4—72no—
422C22×C4—74no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—142no—
426C22×C4—1416no—
427C22×C4—141no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—144no—
431C22×C4—141no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—144no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—144no—
444C22×C4—142no—
445C22×C4—144no—
446C22×C4—288no—
447C22×C4—2816no—
448C22×C4—2816no—
449C22×C4—288no—
450C22×C4C2×Dic1412yes—
451C22×C4C2×D2812yes—
452C22×C4D4×D714yes—
453C22×C4D4×D712yes—
454C22×C4D4×D712yes—
455C22×C4#115914yes—
456C22×C4#115912yes—
457C22×C4#115912yes—
458C22×C4C22×Dic714yes—
459C22×C4C22×Dic714yes—
460C22×C4C22×Dic712yes—
461C4⋊C4—28no—
462C4⋊C4—148no—
463C4⋊C4—2816no—
464C4⋊C4—288no—
465C4⋊C4—284no—
466C22⋊C4—232no—
467C22⋊C4—1416no—
468C22⋊C4—2816no—
469C42—28no—
470C42—148no—
471C42—284no—
472C2×C7—48no—
473C2×C7C2×#38511yescharacteristic
474C2×C7C4×#12711yescharacteristic
475C2×C7#232611yescharacteristic
476C2×C7D4×C4⋊C412yes—
477C2×C7#236112yes—
478C2×C7#242212yes—
479C2×C7#243211yescharacteristic
480C2×C7#243311yescharacteristic
481C2×C7#245212yes—
482C2×C7#254211yescharacteristic
483C2×C7#256611yescharacteristic
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23C2×C4⋊Dic718yes—
507C23C22×Dic1411yescharacteristic
508C23C22×D2811yescharacteristic
509C23C2×D4×D711yescharacteristic
510C23C2×D4×D712yes—
511C23C2×D4×D711yescharacteristic
512C23C2×#115911yescharacteristic
513C23C2×#115912yes—
514C23C2×#115911yescharacteristic
515C23#655312yes—
516C23#655412yes—
517C23C23×Dic711yescharacteristic
518D4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—74no—
531C2×C4—142no—
532C2×C4—144no—
533C2×C4—144no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—148no—
541C2×C4—144no—
542C2×C4—142no—
543C2×C4—2832no—
544C2×C4—282no—
545C2×C4—282no—
546C2×C4—282no—
547C2×C4—284no—
548C2×C4—282no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—284no—
556C2×C4—284no—
557C2×C4—282no—
558C2×C4#644114yes—
559C2×C4#644114yes—
560C2×C4#644514yes—
561C2×C4#644914yes—
562C2×C4C2×C4⋊Dic714yes—
563C2×C4D4×Dic714yes—
564C2×C4D4×Dic714yes—
565C2×C4D4×Dic718yescomplemented
566C7#1397711yescharacteristic, complemented, Sylow
567C22—216no—
568C22—28no—
569C22—48no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22C2×#644111yescharacteristic
579C22C2×#644111yescharacteristic
580C22C2×#644511yescharacteristic
581C22C2×#644911yescharacteristic
582C22D4×Dic1412yes—
583C22#9023412yes—
584C22D4×D2812yes—
585C22#9024612yes—
586C22#9028912yes—
587C22#9030012yes—
588C22#9030212yes—
589C22#9030912yes—
590C22#9036511yescharacteristic
591C22#9036611yescharacteristic
592C22#9037312yes—
593C22D7×#12711yescharacteristic
594C22#9040712yes—
595C22#9040711yescharacteristic
596C22C22×C4⋊Dic711yescharacteristic
597C22C2×D4×Dic712yes—
598C22C2×D4×Dic711yescharacteristic
599C22C2×D4×Dic711yescharacteristic
600C22#9051812yes—
601C4—28no—
602C4—44no—
603C4—144no—
604C4—284no—
605C4—282no—
606C4—284no—
607C4—562no—
608C4#8972218yescomplemented
609C2—48no—
610C2#1173375911yescharacteristic
611C2#1173376712yes—
612C2#1173379411yescharacteristic
613C2#1173380711yescharacteristic
614C2#1173388612yes—
615C2#1173389011yescharacteristic
616C2C2×#8972211yescharacteristic
617C2D4×C4⋊Dic712yes—
618C2#1174008112yes—
619C2#1174039811yescharacteristic
620C2Dic7×#12711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#13977
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#89757C2semidirect
5. C2×#89722C2semidirect
6. C2×#89707C2semidirect
7. #11729317C2semidirect
8. #11729300C2semidirect
9. C4×C7⋊#95C2semidirect
26. C7×#390C4semidirect
59. C2×C7×C4⋊C4C2×C4semidirect
62. C2×C7×C22⋊C4C2×C4semidirect
65. C2×C42×C7C2×C4semidirect
93. C7⋊#95D4semidirect
121. C2×C7×D4C4⋊C4semidirect
201. #390Dic7semidirect
258. C2×C4×C7C4×D4semidirect
326. C2×C4⋊C4C2×Dic7semidirect
335. C2×C22⋊C4C2×Dic7semidirect
344. C2×C42C2×Dic7semidirect
385. C4×C7#385semidirect
409. C2×D4C4⋊Dic7semidirect
565. C2×C4D4×Dic7semidirect
566. C7#13977semidirect
608. C4#89722semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic72
C2×C7×#3902
C2×#899932
C2×#897572
C2×#897222
C2×#897072
#117293172
#117293002
C4×C7⋊#952
#139777