Finite Group Databasebeta

← #13280974

Subgroups — #13280974

Order 1792.

1926 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6440C211yesmaximal, characteristic
2C2×#89748C211yesmaximal, characteristic, complemented
3C2×#89714C212yesmaximal
4C2×#89422C211yesmaximal, characteristic, complemented
5C7×#1439C211yesmaximal, characteristic
6#11729320C211yesmaximal, characteristic, complemented
7#11729301C212yesmaximal
8#11729300C211yesmaximal, characteristic
9#11727164C211yesmaximal, characteristic, complemented
10#11727163C212yesmaximal, complemented
11Dic7⋊#95C212yesmaximal, complemented
12C24×Dic7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2212yes—
20C22×Dic7⋊C4C2212yes—
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C22×Dic7⋊C4C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C22×C4×Dic7C2211yescharacteristic
26C2×#6440—24no—
27C2×#6440—24no—
28C2×#6440C418yescomplemented
29C2×C7×#95C2211yescharacteristic
30C2×C7×#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89748—24no—
39#89714—28no—
40#89422—24no—
41#14340—71nomaximal, Sylow
42C25×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—22no—
48C23×Dic7—22no—
49C23×Dic7—22no—
50C23×Dic7—22no—
51C23×Dic7C2×C414yes—
52C23×Dic7D414yes—
53C23×Dic7Q814yes—
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2311yescharacteristic
57C23×Dic7C2311yescharacteristic
58C23×Dic7C2311yescharacteristic
59C23×Dic7C2311yescharacteristic
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—42no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—42no—
66C2×C7×C22⋊C4—42no—
67C2×C7×C22⋊C4C2×C414yes—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—24no—
71C2×C22⋊Dic7—42no—
72C2×C22⋊Dic7—42no—
73C2×C22⋊Dic7C2×C414yes—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—42no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—24no—
81C2×Dic7⋊C4—24no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—24no—
84C2×Dic7⋊C4—24no—
85C2×Dic7⋊C4—24no—
86C2×Dic7⋊C4—24no—
87C2×Dic7⋊C4—42no—
88C2×Dic7⋊C4—42no—
89C2×Dic7⋊C4C2×C414yes—
90C2×Dic7⋊C4C2×C414yes—
91C2×C4×Dic7—22no—
92C2×C4×Dic7—22no—
93C2×C4×Dic7C2×C414yes—
94C2×C4×Dic7C2×C414yes—
95C2×C4×Dic7D414yes—
96C2×C4×Dic7D414yes—
97#6440—232no—
98C7×#95—22no—
99C7×#95—42no—
100C7⋊#95—22no—
101C7⋊#95—24no—
102C7⋊#95—24no—
103C7⋊#95—24no—
104C7⋊#95—42no—
105C7⋊#95—42no—
106C7⋊#95—42no—
107C22×#117—71no—
108C2×#384—71no—
109C2×#382—71no—
110C2×#380—72no—
111#1441—72no—
112#1441—71no—
113#1441—71no—
114#1439—71no—
115#1439—72no—
116#1439D711yescharacteristic
117C4⋊#95—72no—
118C24×C7—21no—
119C24×C7—21no—
120C24×C7—21no—
121C24×C7—21no—
122C24×C7—22no—
123C24×C7—22no—
124C24×C7—22no—
125C24×C7—22no—
126C24×C7C22×C412yes—
127C24×C7C2×D412yes—
128C24×C7C2×Q812yes—
129C24×C7C2411yescharacteristic
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—28no—
161C22×Dic7—28no—
162C22×Dic7—28no—
163C22×Dic7—28no—
164C22×Dic7—28no—
165C22×Dic7—28no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—22no—
183C22×Dic7—22no—
184C22×Dic7C4⋊C4116yescomplemented
185C22×Dic7C22×C412yes—
186C22×Dic7C22×C412yes—
187C22×Dic7C22×C412yes—
188C22×Dic7C22×C412yes—
189C22×Dic7C2×D412yes—
190C22×Dic7C2×D412yes—
191C22×Dic7C2×D412yes—
192C22×Dic7C2×D412yes—
193C22×Dic7C2×D412yes—
194C22×Dic7C2×D412yes—
195C22×Dic7C2×Q812yes—
196C22×Dic7C2×Q812yes—
197C22×Dic7#4112yes—
198C22×Dic7#4112yes—
199C7×C4⋊C4—44no—
200C7×C22⋊C4—28no—
201C7×C22⋊C4—48no—
202C22⋊Dic7—28no—
203C22⋊Dic7—48no—
204C4⋊Dic7—44no—
205Dic7⋊C4—28no—
206Dic7⋊C4—28no—
207Dic7⋊C4—48no—
208Dic7⋊C4—48no—
209Dic7⋊C4—48no—
210C4×Dic7—28no—
211C4×Dic7C22⋊C4116yescomplemented
212C24×C4—71no—
213C2×#117—78no—
214C2×#117—144no—
215C2×#117—144no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—71no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—72no—
220C22×C4⋊C4—72no—
221C22×C4⋊C4—71no—
222C22×C4⋊C4C2×D711yescharacteristic
223C22×C22⋊C4—71no—
224C22×C22⋊C4—72no—
225C22×C22⋊C4C2×D711yescharacteristic
226C22×C22⋊C4C2×D712yes—
227C22×C42—71no—
228C22×C42—71no—
229#384—144no—
230#382—144no—
231#380—148no—
232C2×#95—72no—
233C2×#95—71no—
234C2×#95—72no—
235C2×#95—72no—
236C2×#95—72no—
237C2×#95—72no—
238C2×#95—72no—
239C2×#95C2×D711yescharacteristic
240C2×#95C2×D712yes—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—24no—
244C23×C7—24no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—22no—
249C23×C7—24no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7C2×C4⋊C414yes—
264C23×C7C23×C411yescharacteristic
265C23×C7C22×D411yescharacteristic
266C23×C7C22×D411yescharacteristic
267C23×C7C22×D411yescharacteristic
268C23×C7C22×Q811yescharacteristic
269C23×C7C2×#4111yescharacteristic
270C23×C7C2×#4111yescharacteristic
271C23×C7H(5,𝔽2)12yes—
272C23×C7H(5,𝔽2)12yes—
273C23×C72−1+412yes—
274C23×C72−1+412yes—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7C2×C22⋊C414yes—
299C2×Dic7—22no—
300C2×Dic7—22no—
301C2×Dic7—24no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7—232no—
310C2×Dic7—22no—
311C2×Dic7—24no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7C2×C22⋊C414yes—
330C2×Dic7C2×C22⋊C414yes—
331C2×Dic7C2×C4⋊C414yes—
332C2×Dic7C2×C4⋊C414yes—
333C2×Dic7#11714yes—
334C2×Dic7#12114yes—
335C2×Dic7#12114yes—
336C2×Dic7C22⋊Q814yes—
337C2×Dic7C22⋊Q814yes—
338C25C22×D711yescharacteristic
339C23×C4—71no—
340C23×C4—74no—
341C23×C4—74no—
342C23×C4—74no—
343C23×C4—71no—
344C23×C4—71no—
345C23×C4—71no—
346C23×C4—72no—
347C23×C4—72no—
348C23×C4—142no—
349C23×C4—142no—
350C23×C4—142no—
351C23×C4—142no—
352C23×C4C22×D712yes—
353C23×C4C22×D712yes—
354C23×C4C22×D711yescharacteristic
355C23×C4C22×D711yescharacteristic
356#117—1432no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—74no—
360C2×C4⋊C4—74no—
361C2×C4⋊C4—142no—
362C2×C4⋊C4—142no—
363C2×C4⋊C4—142no—
364C2×C4⋊C4—142no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—144no—
367C2×C4⋊C4—144no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—144no—
370C2×C4⋊C4—144no—
371C2×C4⋊C4—144no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—282no—
374C2×C4⋊C4—282no—
375C2×C4⋊C4—282no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—74no—
382C2×C22⋊C4—142no—
383C2×C22⋊C4—142no—
384C2×C22⋊C4—144no—
385C2×C22⋊C4—282no—
386C2×C22⋊C4—282no—
387C2×C22⋊C4C4×D714yes—
388C2×C42—74no—
389C2×C42—74no—
390C2×C42—74no—
391C2×C42—74no—
392C2×C42—142no—
393C2×C42—142no—
394#95—22no—
395#95—42no—
396#95—144no—
397#95—144no—
398#95—144no—
399#95—142no—
400#95—282no—
401#95—282no—
402#95—282no—
403C22×C7—28no—
404C22×C7—24no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—44no—
408C22×C7—42no—
409C22×C7—44no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—42no—
413C22×C7—44no—
414C22×C7C22×C22⋊C411yescharacteristic
415C22×C7C22×C4⋊C411yescharacteristic
416C22×C7C2×#11711yescharacteristic
417C22×C7#52012yes—
418C22×C7#52012yes—
419C22×C7C2×#12111yescharacteristic
420C22×C7C2×#12111yescharacteristic
421C22×C7C2×C22⋊Q811yescharacteristic
422C22×C7C2×C22⋊Q811yescharacteristic
423C22×C7#53511yescharacteristic
424C22×C7#53511yescharacteristic
425C22×C7#53512yes—
426C22×C7#53611yescharacteristic
427C22×C7#53612yes—
428C22×C7#53611yescharacteristic
429C22×C7#53712yes—
430C22×C7#53712yes—
431C22×C7#54012yes—
432C22×C7#54112yes—
433C22×C7#54212yes—
434C22×C7#54212yes—
435C22×C7#54412yes—
436C22×C7#54412yes—
437C4×C7—24no—
438C4×C7—42no—
439C4×C7—82no—
440C4×C7—82no—
441Dic7—24no—
442Dic7—24no—
443Dic7—42no—
444Dic7—82no—
445Dic7—82no—
446Dic7#38018yes—
447C24—21no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—21no—
452C24—21no—
453C24—22no—
454C24—22no—
455C24C2×C4×D712yes—
456C24D4×D712yes—
457C24Q8×D712yes—
458C24C23×D711yescharacteristic
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—72no—
480C22×C4—72no—
481C22×C4—72no—
482C22×C4—72no—
483C22×C4—72no—
484C22×C4—72no—
485C22×C4—72no—
486C22×C4—716no—
487C22×C4—72no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—72no—
491C22×C4—72no—
492C22×C4—72no—
493C22×C4—72no—
494C22×C4—148no—
495C22×C4—148no—
496C22×C4—148no—
497C22×C4—148no—
498C22×C4—148no—
499C22×C4—148no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—142no—
503C22×C4—141no—
504C22×C4—141no—
505C22×C4—142no—
506C22×C4—141no—
507C22×C4—141no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—142no—
511C22×C4—142no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4—142no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4—142no—
520C22×C4—142no—
521C22×C4—142no—
522C22×C4—142no—
523C22×C4—142no—
524C22×C4C2×C4×D712yes—
525C22×C4C2×C4×D712yes—
526C22×C4D4×D712yes—
527C22×C4D4×D712yes—
528C4⋊C4—44no—
529C4⋊C4—148no—
530C4⋊C4—148no—
531C4⋊C4—284no—
532C4⋊C4—288no—
533C4⋊C4—288no—
534C4⋊C4—288no—
535C22⋊C4—28no—
536C22⋊C4—48no—
537C22⋊C4—148no—
538C22⋊C4—288no—
539C42—716no—
540C42—148no—
541C2×C7—44no—
542C2×C7C2×#38011yescharacteristic
543C2×C7#231111yescharacteristic
544C2×C7#231111yescharacteristic
545C2×C7#231412yes—
546C2×C7#231712yes—
547C2×C7#261911yescharacteristic
548C2×C7#262012yes—
549C2×C7#262111yescharacteristic
550C2×C7#262811yescharacteristic
551C2×C7#262811yescharacteristic
552C2×C7#262912yes—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23D7×C4⋊C414yes—
576C23C22×C4×D711yescharacteristic
577C23C2×D4×D711yescharacteristic
578C23C2×D4×D711yescharacteristic
579C23C2×D4×D711yescharacteristic
580C23#654812yes—
581C23#654812yes—
582C23C2×Q8×D711yescharacteristic
583C23D7×#4111yescharacteristic
584C23D7×#4111yescharacteristic
585C23#655412yes—
586C23#655412yes—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—74no—
611C2×C4—74no—
612C2×C4—74no—
613C2×C4—74no—
614C2×C4—74no—
615C2×C4—74no—
616C2×C4—74no—
617C2×C4—74no—
618C2×C4—74no—
619C2×C4—1432no—
620C2×C4—142no—
621C2×C4—142no—
622C2×C4—142no—
623C2×C4—144no—
624C2×C4—142no—
625C2×C4—142no—
626C2×C4—144no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—142no—
631C2×C4—144no—
632C2×C4—281no—
633C2×C4—282no—
634C2×C4—281no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4—282no—
640C2×C4—282no—
641C2×C4—282no—
642C2×C4—282no—
643C2×C4—282no—
644C2×C4—282no—
645C2×C4—282no—
646C2×C4—282no—
647C2×C4—282no—
648C2×C4—282no—
649C2×C4D7×C22⋊C414yes—
650C7#1434011yescharacteristic, complemented, Sylow
651C22—28no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22C2×D7×C22⋊C411yescharacteristic
663C22C2×D7×C4⋊C411yescharacteristic
664C22D7×#11711yescharacteristic
665C22#9023712yes—
666C22#9023712yes—
667C22#9028312yes—
668C22#9028311yescharacteristic
669C22#9028812yes—
670C22D7×#12111yescharacteristic
671C22D7×#12111yescharacteristic
672C22#9029912yes—
673C22#9030612yes—
674C22#9031412yes—
675C22#9031412yes—
676C22D7×C22⋊Q811yescharacteristic
677C22D7×C22⋊Q811yescharacteristic
678C22#9033912yes—
679C22#9033911yescharacteristic
680C22#9035312yes—
681C22#9036012yes—
682C22#9036611yescharacteristic
683C22#9039212yes—
684C22#9040611yescharacteristic
685C4—24no—
686C4—42no—
687C4—78no—
688C4—82no—
689C4—82no—
690C4—144no—
691C4—144no—
692C4—282no—
693C4—562no—
694C4—562no—
695C2—44no—
696C2D7×#38011yescharacteristic
697C2#1173182712yes—
698C2#1173210711yescharacteristic
699C2#1173290112yes—
700C2#1173373911yescharacteristic
701C2#1173414311yescharacteristic
702C2#1173425511yescharacteristic
703C2#1173441712yes—
704C2#1173477712yes—
705C2#1173498311yescharacteristic
706C2#1173549411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7#14340
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#89748C2semidirect
4. C2×#89422C2semidirect
6. #11729320C2semidirect
9. #11727164C2semidirect
10. #11727163C2semidirect
11. Dic7⋊#95C2semidirect
28. C2×#6440C4semidirect
184. C22×Dic7C4⋊C4semidirect
211. C4×Dic7C22⋊C4semidirect
650. C7#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#64402
C2×#897482
C2×#897142
C2×#894222
C7×#14392
#117293202
#117293012
#117293002
#117271642
#117271632
Dic7⋊#952
#143407