Finite Group Databasebeta

← #13285779

Subgroups — #13285779

Order 1792.

1982 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C2×C7×#385C211yesmaximal, characteristic
3C2×C4×C22⋊Dic7C211yesmaximal, characteristic
4C2×#89751C211yesmaximal, characteristic, complemented
5C2×#89744C212yesmaximal, complemented
6#11729319C211yesmaximal, characteristic, complemented
7#11729301C212yesmaximal
8#11729301C212yesmaximal
9#11727178C211yesmaximal, characteristic, complemented
10#11727174C211yesmaximal, characteristic, complemented
11#11727159C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic7C2212yes—
19C22×C22⋊Dic7C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C2×#6487—24no—
22C2×#6487—28no—
23C22×Dic7⋊C4C2212yes—
24C22×Dic7⋊C4C2212yes—
25C22×Dic7⋊C4C2212yes—
26C22×C4×Dic7C2212yes—
27C7×#385—24no—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30C4×C22⋊Dic7—24no—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2211yescharacteristic
39#89751—24no—
40#89744—28no—
41#14763—71nomaximal, Sylow
42C25×C7C2311yescharacteristic
43C23×C4×C7—24no—
44C23×C4×C7—22no—
45C23×C4×C7—22no—
46C23×C4×C7—42no—
47C23×C4×C7D414yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2312yes—
56C23×Dic7C2312yes—
57C23×Dic7C2312yes—
58C22×Dic14—22no—
59C22×Dic14—24no—
60C22×Dic14—22no—
61C22×Dic14—22no—
62C22×Dic14—42no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—42no—
73C2×C22⋊Dic7—24no—
74C2×C22⋊Dic7—24no—
75C2×C22⋊Dic7—24no—
76C2×C22⋊Dic7—24no—
77C2×C22⋊Dic7—24no—
78C2×C22⋊Dic7Q818yescomplemented
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—24no—
81#6487—232no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—24no—
84C2×Dic7⋊C4—24no—
85C2×Dic7⋊C4—24no—
86C2×Dic7⋊C4—24no—
87C2×Dic7⋊C4—24no—
88C2×Dic7⋊C4—24no—
89C2×Dic7⋊C4—42no—
90C2×Dic7⋊C4D418yes—
91C2×C4×Dic7—24no—
92C2×C4×Dic7—24no—
93C2×C4×Dic7—42no—
94C7×#95—22no—
95C7×#95—22no—
96C7⋊#95—22no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95—24no—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—42no—
103C7⋊#95D414yes—
104C22×C22⋊Q8—71no—
105C2×#395—71no—
106C2×#389—72no—
107C2×#385D711yescharacteristic
108C2×C4×C22⋊C4—71no—
109#1446—71no—
110#1444—71no—
111#1441—71no—
112#1439—72no—
113#1439—72no—
114#1436—72no—
115C24×C7—22no—
116C24×C7—21no—
117C24×C7—22no—
118C24×C7—21no—
119C24×C7—21no—
120C24×C7—21no—
121C24×C7—22no—
122C24×C7—21no—
123C24×C7—21no—
124C24×C7C2×D412yes—
125C24×C7C2×Q814yes—
126C24×C7C2411yescharacteristic
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—28no—
143C22×C4×C7—28no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—44no—
155C22×C4×C7—48no—
156C22×C4×C7—48no—
157C22×C4×C7C2×D414yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7#4112yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
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×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7C2×D414yes—
187C22×Dic7C2×D414yes—
188C22×Dic7C2×D414yes—
189C22×Dic7C2×Q814yes—
190C22×Dic7C2×Q814yes—
191C22×Dic7#4114yes—
192C2×Dic14—28no—
193C2×Dic14—48no—
194C2×Dic14—48no—
195C2×Dic14—44no—
196C2×Dic14—44no—
197C7×C4⋊C4—44no—
198C7×C4⋊C4—44no—
199C7×C22⋊C4—28no—
200C7×C22⋊C4—44no—
201C22⋊Dic7—216no—
202C22⋊Dic7—216no—
203C4⋊Dic7—28no—
204Dic7⋊C4—216no—
205Dic7⋊C4—216no—
206Dic7⋊C4—48no—
207C4×Dic7—48no—
208C23×Q8—71no—
209C24×C4C2×D711yescharacteristic
210C2×C22⋊Q8—148no—
211C2×C22⋊Q8—144no—
212C22×C4⋊C4—72no—
213C22×C4⋊C4—71no—
214C22×C4⋊C4—72no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4C2×D711yescharacteristic
217C22×C4⋊C4C2×D711yescharacteristic
218C22×C22⋊C4—72no—
219C22×C22⋊C4—72no—
220C22×C22⋊C4C2×D711yescharacteristic
221C22×C22⋊C4C2×D711yescharacteristic
222C22×C42—72no—
223#395—144no—
224#389—148no—
225#385—24no—
226C4×C22⋊C4—144no—
227C2×#95—71no—
228C2×#95—72no—
229C2×#95—72no—
230C2×#95—72no—
231C2×#95—72no—
232C2×#95—72no—
233C2×#95—71no—
234C2×#95—72no—
235C2×#95C2×D711yescharacteristic
236C2×#95C2×D711yescharacteristic
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—24no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7C4⋊Q814yes—
260C23×C7C22×D412yes—
261C23×C7C22×D411yescharacteristic
262C23×C7C22×D411yescharacteristic
263C23×C7C22×Q812yes—
264C23×C7C2×#4111yescharacteristic
265C23×C7C2×#4111yescharacteristic
266C23×C7C2×#4111yescharacteristic
267C23×C7C2×#4112yes—
268C23×C7H(5,𝔽2)12yes—
269C23×C72−1+411yescharacteristic
270C23×C72−1+411yescharacteristic
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—416no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
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—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7#12014yes—
303C2×C4×C7#12114yes—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—24no—
307C2×Dic7—24no—
308C2×Dic7—24no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—24no—
318C2×Dic7—24no—
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×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7C22⋊Q818yescomplemented
332Dic14—416no—
333C25C22×D711yescharacteristic
334C22×Q8—144no—
335C22×Q8—142no—
336C22×Q8—142no—
337C22×Q8—142no—
338C22×Q8—282no—
339C23×C4—24no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—42no—
343C23×C4—72no—
344C23×C4—72no—
345C23×C4—72no—
346C23×C4—72no—
347C23×C4C22⋊D714yes—
348C23×C4C22×D711yescharacteristic
349C23×C4C22×D711yescharacteristic
350C23×C4C22×D711yescharacteristic
351C23×C4C22×D711yescharacteristic
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D711yescharacteristic
354C22⋊Q8—1432no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—78no—
361C2×C4⋊C4—144no—
362C2×C4⋊C4—144no—
363C2×C4⋊C4—144no—
364C2×C4⋊C4—144no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—144no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—144no—
370C2×C4⋊C4—282no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—78no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—144no—
380C2×C22⋊C4—144no—
381C2×C22⋊C4—144no—
382C2×C42—144no—
383C2×C42—144no—
384C2×C42—282no—
385#95—22no—
386#95—22no—
387#95—74no—
388#95—144no—
389#95—144no—
390#95—144no—
391#95—144no—
392#95—144no—
393#95—142no—
394#95—282no—
395C22×C7—24no—
396C22×C7—24no—
397C22×C7—24no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—44no—
403C22×C7—44no—
404C22×C7—44no—
405C22×C7—44no—
406C22×C7C2×#12011yescharacteristic
407C22×C7C2×#12111yescharacteristic
408C22×C7C2×C22⋊Q812yes—
409C22×C7#52511yescharacteristic
410C22×C7#52512yes—
411C22×C7C2×C4⋊Q811yescharacteristic
412C22×C7#53212yes—
413C22×C7#53211yescharacteristic
414C22×C7#53312yes—
415C22×C7#53411yescharacteristic
416C22×C7#53611yescharacteristic
417C22×C7#53912yes—
418C22×C7#54212yes—
419C22×C7D4212yes—
420C22×C7#54612yes—
421C22×C7#54711yescharacteristic
422C22×C7#54811yescharacteristic
423C22×C7D4×Q812yes—
424C22×C7D4×Q812yes—
425C22×C7#55211yescharacteristic
426C22×C7#55412yes—
427C22×C7#55412yes—
428C22×C7#55611yescharacteristic
429C4×C7—24no—
430C4×C7—24no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—81no—
434C4×C7—81no—
435Dic7—28no—
436Dic7—44no—
437Dic7—44no—
438Dic7—44no—
439C24—21no—
440C24—21no—
441C24—22no—
442C24—21no—
443C24—21no—
444C24—22no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24Q8×D714yes—
449C24C2×C22⋊D712yes—
450C24C23×D711yescharacteristic
451C2×Q8—148no—
452C2×Q8—288no—
453C2×Q8—288no—
454C2×Q8—284no—
455C2×Q8—284no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—74no—
487C22×C4—74no—
488C22×C4—74no—
489C22×C4—74no—
490C22×C4—74no—
491C22×C4—74no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—142no—
511C22×C4—142no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4#115712yes—
515C22×C4#115712yes—
516C22×C4D4×D714yes—
517C22×C4D4×D712yes—
518C22×C4#115912yes—
519C22×C4C2×C22⋊D712yes—
520C22×C4C2×C22⋊D712yes—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—1416no—
524C4⋊C4—148no—
525C4⋊C4—1416no—
526C4⋊C4—288no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—1416no—
530C22⋊C4—1416no—
531C42—288no—
532C2×C7—44no—
533C2×C7#238811yescharacteristic
534C2×C7#241011yescharacteristic
535C2×C7#241511yescharacteristic
536C2×C7#243411yescharacteristic
537C2×C7#244912yes—
538C2×C7#245212yes—
539C2×C7#246512yes—
540C2×C7#255612yes—
541C2×C7#258311yescharacteristic
542C2×C7#266311yescharacteristic
543C2×C7#285711yescharacteristic
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23#650714yes—
567C23C2×#115711yescharacteristic
568C23C2×#115711yescharacteristic
569C23C2×D4×D712yes—
570C23C2×D4×D711yescharacteristic
571C23C2×#115911yescharacteristic
572C23#654812yes—
573C23C2×Q8×D712yes—
574C23D7×#4112yes—
575C23#655411yescharacteristic
576C23#655411yescharacteristic
577C23C22×C22⋊D711yescharacteristic
578Q8—2816no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—416no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—78no—
611C2×C4—144no—
612C2×C4—144no—
613C2×C4—144no—
614C2×C4—144no—
615C2×C4—144no—
616C2×C4—144no—
617C2×C4—144no—
618C2×C4—144no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—144no—
623C2×C4—144no—
624C2×C4—144no—
625C2×C4—144no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4#650014yes—
639C2×C4#650214yes—
640C7#1476311yescharacteristic, complemented, Sylow
641C22—24no—
642C22—24no—
643C22—24no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22#9020212yes—
653C22#9020212yes—
654C22#9023211yescharacteristic
655C22#9023311yescharacteristic
656C22#9023812yes—
657C22#9023912yes—
658C22#9024211yescharacteristic
659C22#9024311yescharacteristic
660C22#9028112yes—
661C22#9028212yes—
662C22#9031312yes—
663C22D7×C22⋊Q812yes—
664C22#9032312yes—
665C22#9032312yes—
666C22#9035112yes—
667C22#9035312yes—
668C22#9048211yescharacteristic
669C22C2×#650011yescharacteristic
670C22C2×#650211yescharacteristic
671C22#9049511yescharacteristic
672C22C2×#650711yescharacteristic
673C22#9051611yescharacteristic
674C22#9052311yescharacteristic
675C4—24no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—81no—
681C4—148no—
682C4—284no—
683C4—284no—
684C4—284no—
685C2—44no—
686C2#1173187012yes—
687C2#1173187812yes—
688C2#1173191912yes—
689C2#1173205412yes—
690C2#1173962611yescharacteristic
691C2#1173979611yescharacteristic
692C2#1173984511yescharacteristic
693C2#1174005111yescharacteristic
694C2#1174020511yescharacteristic
695C2#1174025611yescharacteristic
696C2#1174031311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC2×C7×#385C2
prime radicalC25×C7C23
hypercenterC2×#385D7
hypocenterC7#14763
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
4. C2×#89751C2semidirect
5. C2×#89744C2semidirect
6. #11729319C2semidirect
9. #11727178C2semidirect
10. #11727174C2semidirect
11. #11727159C2semidirect
78. C2×C22⋊Dic7Q8semidirect
331. C2×Dic7C22⋊Q8semidirect
640. C7#14763semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C2×C7×#3852
C2×C4×C22⋊Dic72
C2×#897512
C2×#897442
#117293192
#117293012
#117271782
#117271742
#117271592
#147637