Finite Group Databasebeta

← #13275622

Subgroups — #13275622

Order 1792.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
2C2×#89721C211yesmaximal, characteristic
3C2×C22⋊C4×Dic7C211yesmaximal, characteristic
4C2×C22⋊C4×Dic7C212yesmaximal
5C2×#89421C212yesmaximal, complemented
6C7×#1439C211yesmaximal, characteristic
7#11729322C211yesmaximal, characteristic, complemented
8#11729301C212yesmaximal
9#11727176C211yesmaximal, characteristic, complemented
10#11727175C212yesmaximal, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C24×Dic7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2212yes—
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2212yes—
24C22×C4×Dic7C2211yescharacteristic
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2211yescharacteristic
35C4⋊C4×Dic7—24no—
36#89721C418yescomplemented
37C22⋊C4×Dic7—28no—
38C22⋊C4×Dic7—24no—
39#89421—28no—
40C4×#399—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—24no—
47C23×Dic7—22no—
48C23×Dic7—22no—
49C23×Dic7—24no—
50C23×Dic7Q814yes—
51C23×Dic7C2312yes—
52C23×Dic7C2311yescharacteristic
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C23×Dic7C2312yes—
56C23×Dic7C2311yescharacteristic
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4C2×C414yes—
65C2×C22⋊Dic7—22no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—22no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7C2×C418yes—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7C2×C414yescomplemented
74C2×Dic7⋊C4—24no—
75C2×Dic7⋊C4—24no—
76C2×Dic7⋊C4—24no—
77C2×Dic7⋊C4—24no—
78C2×Dic7⋊C4—24no—
79C2×Dic7⋊C4C2×C418yescomplemented
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7—22no—
86C2×C4×Dic7—24no—
87C2×C4×Dic7—24no—
88C2×C4×Dic7D418yes—
89C2×C4×Dic7D414yes—
90C7×#95—24no—
91C7×#95—22no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95C2×C414yescomplemented
100C2×#399—71no—
101C2×C4×C4⋊C4—72no—
102C2×C4×C4⋊C4—71no—
103C2×C4×C22⋊C4—72no—
104C2×C4×C22⋊C4—71no—
105#1445—72no—
106#1445—71no—
107#1444—71no—
108#1439—72no—
109#1439D711yescharacteristic
110C4×#95—71no—
111C24×C7—22no—
112C24×C7—21no—
113C24×C7—22no—
114C24×C7—21no—
115C24×C7—21no—
116C24×C7—22no—
117C24×C7—21no—
118C24×C7—22no—
119C24×C7C22×C412yes—
120C24×C7C2×Q812yes—
121C24×C7#4112yes—
122C24×C7C2411yescharacteristic
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7C22×C412yes—
142C22×C4×C7C22×C412yes—
143C22×C4×C7C2×D414yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7#4114yes—
146C22×C4×C7#4112yes—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—22no—
169C22×Dic7—28no—
170C22×Dic7—216no—
171C22×Dic7—48no—
172C22×Dic7—44no—
173C22×Dic7—44no—
174C22×Dic7C22×C414yes—
175C22×Dic7C22×C414yes—
176C22×Dic7C2×D412yes—
177C22×Dic7C2×D414yes—
178C22×Dic7C2×D414yes—
179C22×Dic7C2×D412yes—
180C22×Dic7C2×Q812yes—
181C22×Dic7C2×Q812yes—
182C22×Dic7#4112yes—
183C22×Dic7#4114yes—
184C22×Dic7#4112yes—
185C22×Dic7#4114yes—
186C7×C4⋊C4—28no—
187C7×C22⋊C4—216no—
188C7×C22⋊C4—28no—
189C22⋊Dic7—28no—
190C22⋊Dic7—216no—
191C4⋊Dic7—28no—
192Dic7⋊C4—216no—
193Dic7⋊C4—216no—
194C4×Dic7—216no—
195C4×Dic7—28no—
196C4×Dic7—44no—
197C4×Dic7—48no—
198C24×C4—71no—
199C22×C4⋊C4—72no—
200C22×C4⋊C4—71no—
201C22×C4⋊C4—72no—
202C22×C4⋊C4C2×D711yescharacteristic
203C22×C22⋊C4—72no—
204C22×C22⋊C4—71no—
205C22×C22⋊C4C2×D712yes—
206C22×C22⋊C4C2×D711yescharacteristic
207C22×C42—72no—
208C22×C42—71no—
209C22×C42—71no—
210C22×C42—72no—
211#399—78no—
212C4×C4⋊C4—148no—
213C4×C4⋊C4—144no—
214C4×C22⋊C4—148no—
215C4×C22⋊C4—144no—
216C2×#95—71no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—71no—
222C2×#95—71no—
223C2×#95—72no—
224C2×#95C2×D711yescharacteristic
225C2×#95C2×D712yes—
226C23×C7—24no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—24no—
230C23×C7—22no—
231C23×C7—24no—
232C23×C7—22no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—24no—
236C23×C7—22no—
237C23×C7—24no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—41no—
246C23×C7—42no—
247C23×C7—41no—
248C23×C7C4×Q814yes—
249C23×C7C23×C411yescharacteristic
250C23×C7C22×D411yescharacteristic
251C23×C7C22×D412yes—
252C23×C7C22×Q811yescharacteristic
253C23×C7C2×#4111yescharacteristic
254C23×C7C2×#4111yescharacteristic
255C23×C7C2×#4112yes—
256C23×C7C2×#4112yes—
257C23×C7C2×#4111yescharacteristic
258C23×C7C2×#4112yes—
259C23×C7C2×#4111yescharacteristic
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7C4×D414yes—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—22no—
294C2×Dic7—24no—
295C2×Dic7—22no—
296C2×Dic7—24no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—42no—
301C2×Dic7—41no—
302C2×Dic7—42no—
303C2×Dic7—41no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—416no—
313C2×Dic7C4×D418yes—
314C2×Dic7C22⋊Q814yes—
315C2×Dic7C22⋊Q818yes—
316C2×Dic7#12318yes—
317C2×Dic7#12314yes—
318C2×Dic7#12714yes—
319C25C22×D711yescharacteristic
320C23×C4—71no—
321C23×C4—72no—
322C23×C4—74no—
323C23×C4—72no—
324C23×C4—71no—
325C23×C4—71no—
326C23×C4—71no—
327C23×C4—142no—
328C23×C4—144no—
329C23×C4—142no—
330C23×C4—144no—
331C23×C4C22×D711yescharacteristic
332C23×C4C22×D712yes—
333C23×C4C22×D712yes—
334C23×C4C22×D711yescharacteristic
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—78no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—144no—
340C2×C4⋊C4—144no—
341C2×C4⋊C4—144no—
342C2×C4⋊C4—144no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—142no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—78no—
352C2×C22⋊C4—144no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—144no—
355C2×C22⋊C4—142no—
356C2×C22⋊C4—142no—
357C2×C22⋊C4C4×D714yes—
358C2×C42—78no—
359C2×C42—74no—
360C2×C42—144no—
361C2×C42—144no—
362C2×C42—144no—
363C2×C42—142no—
364C2×C42—144no—
365C2×C42—144no—
366C2×C42—144no—
367C2×C42—144no—
368#95—24no—
369#95—22no—
370#95—74no—
371#95—144no—
372#95—144no—
373#95—142no—
374#95—144no—
375#95—144no—
376#95—142no—
377#95—144no—
378C22×C7—24no—
379C22×C7—28no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7—44no—
383C22×C7—42no—
384C22×C7—44no—
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—44no—
389C22×C7C2×C4×D411yescharacteristic
390C22×C7C2×C4×D412yes—
391C22×C7C2×C4×Q811yescharacteristic
392C22×C7C4×#4111yescharacteristic
393C22×C7C4×#4112yes—
394C22×C7C2×C22⋊Q811yescharacteristic
395C22×C7C2×C22⋊Q812yes—
396C22×C7C2×#12312yes—
397C22×C7C2×#12311yescharacteristic
398C22×C7#52512yes—
399C22×C7#52511yescharacteristic
400C22×C7#52511yescharacteristic
401C22×C7#52512yes—
402C22×C7#52912yes—
403C22×C7#52912yes—
404C22×C7#52911yescharacteristic
405C22×C7#52912yes—
406C22×C7#52912yes—
407C22×C7C2×#12711yescharacteristic
408C22×C7#53212yes—
409C22×C7#53211yescharacteristic
410C22×C7#53311yescharacteristic
411C22×C7#53312yes—
412C4×C7—24no—
413C4×C7—42no—
414C4×C7—44no—
415C4×C7—44no—
416Dic7—28no—
417Dic7—44no—
418Dic7—42no—
419Dic7—42no—
420Dic7—42no—
421Dic7#39918yescomplemented
422C24—21no—
423C24—22no—
424C24—21no—
425C24—22no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—22no—
430C24C2×C4×D712yes—
431C24#115912yes—
432C24Q8×D712yes—
433C24C23×D711yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—72no—
453C22×C4—74no—
454C22×C4—72no—
455C22×C4—72no—
456C22×C4—74no—
457C22×C4—74no—
458C22×C4—74no—
459C22×C4—74no—
460C22×C4—74no—
461C22×C4—72no—
462C22×C4—72no—
463C22×C4—72no—
464C22×C4—148no—
465C22×C4—1416no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—141no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—141no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—142no—
486C22×C4—141no—
487C22×C4—142no—
488C22×C4—284no—
489C22×C4—288no—
490C22×C4—284no—
491C22×C4C2×C4×D712yes—
492C22×C4C2×C4×D712yes—
493C22×C4D4×D714yes—
494C22×C4D4×D712yes—
495C22×C4#115914yes—
496C22×C4#115912yes—
497C4⋊C4—28no—
498C4⋊C4—1416no—
499C4⋊C4—1416no—
500C4⋊C4—148no—
501C22⋊C4—216no—
502C22⋊C4—28no—
503C22⋊C4—1416no—
504C22⋊C4—148no—
505C42—148no—
506C42—1416no—
507C42—288no—
508C42—284no—
509C2×C7—44no—
510C2×C7C4×#12311yescharacteristic
511C2×C7C4×#12312yes—
512C2×C7C4×C22⋊Q811yescharacteristic
513C2×C7C4×C22⋊Q812yes—
514C2×C7C4×#12711yescharacteristic
515C2×C7C2×#39911yescharacteristic
516C2×C7#239512yes—
517C2×C7#239511yescharacteristic
518C2×C7#239612yes—
519C2×C7#239611yescharacteristic
520C2×C7#239711yescharacteristic
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23#645014yes—
544C23C22×C4×D711yescharacteristic
545C23C2×D4×D711yescharacteristic
546C23C2×D4×D712yes—
547C23C2×#115912yes—
548C23C2×#115911yescharacteristic
549C23C2×#115911yescharacteristic
550C23C2×Q8×D711yescharacteristic
551C23D7×#4111yescharacteristic
552C23D7×#4112yes—
553C23D7×#4111yescharacteristic
554C23D7×#4112yes—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—74no—
579C2×C4—74no—
580C2×C4—78no—
581C2×C4—78no—
582C2×C4—74no—
583C2×C4—78no—
584C2×C4—144no—
585C2×C4—144no—
586C2×C4—144no—
587C2×C4—144no—
588C2×C4—144no—
589C2×C4—144no—
590C2×C4—144no—
591C2×C4—144no—
592C2×C4—144no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—144no—
596C2×C4—142no—
597C2×C4—144no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—2816no—
601C2×C4—282no—
602C2×C4—281no—
603C2×C4—282no—
604C2×C4—281no—
605C2×C4—282no—
606C2×C4—281no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4—282no—
611C2×C4—281no—
612C2×C4—282no—
613C2×C4(C22⋊D7)⋊C414yes—
614C7C4×#39911yescharacteristic, complemented, Sylow
615C22—24no—
616C22—28no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22C2×(C22⋊D7)⋊C411yescharacteristic
627C22C2×#645011yescharacteristic
628C22#9021311yescharacteristic
629C22C4×#115912yes—
630C22C4×D4×D712yes—
631C22#9027712yes—
632C22#9027711yescharacteristic
633C22#9028712yes—
634C22#9029111yescharacteristic
635C22#9029112yes—
636C22#9029711yescharacteristic
637C22#9029712yes—
638C22#9031311yescharacteristic
639C22#9031312yes—
640C22D7×C22⋊Q811yescharacteristic
641C22D7×C22⋊Q812yes—
642C22#9034012yes—
643C22D7×#12312yes—
644C22D7×#12311yescharacteristic
645C22#9035911yescharacteristic
646C22#9037112yes—
647C22#9039712yes—
648C22D7×#12711yescharacteristic
649C4—24no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—78no—
654C4—148no—
655C4—282no—
656C4—282no—
657C4—284no—
658C4—282no—
659C2—44no—
660C2#1173131111yescharacteristic
661C2#1173182612yes—
662C2#1173210611yescharacteristic
663C2#1173289812yes—
664C2#1173373811yescharacteristic
665C2#1173411211yescharacteristic
666C2#1173422311yescharacteristic
667C2#1173433712yes—
668C2#1173472712yes—
669C2#1173490311yescharacteristic
670C2D7×#39911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1439C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1439D7
hypocenterC7C4×#399
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×Dic7C2semidirect
5. C2×#89421C2semidirect
7. #11729322C2semidirect
9. #11727176C2semidirect
10. #11727175C2semidirect
11. Dic7×#95C2semidirect
36. #89721C4semidirect
73. C2×C4⋊Dic7C2×C4semidirect
79. C2×Dic7⋊C4C2×C4semidirect
99. C7⋊#95C2×C4semidirect
421. Dic7#399semidirect
614. C7C4×#399semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic72
C2×#897212
C2×C22⋊C4×Dic72
C2×#894212
C7×#14392
#117293222
#117293012
#117271762
#117271752
Dic7×#952
C4×#3997