Finite Group Databasebeta

← #13502651

Subgroups — #13502651

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic, complemented
2C7×#2850C211yesmaximal, characteristic
3C2×#89826C212yesmaximal, complemented
4C2×#89822C211yesmaximal, characteristic, complemented
5C2×#89812C211yesmaximal, characteristic
6#11729446C211yesmaximal, characteristic, complemented
7#11729431C211yesmaximal, characteristic
8#11729284C211yesmaximal, characteristic, complemented
9#11727444C212yesmaximal, complemented
10#11727414C212yesmaximal, complemented
11#11727313C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C2×C7×#124C2211yescharacteristic
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×C7×#121C2211yescharacteristic
16C22×C42×C7C2211yescharacteristic
17C22×#1144C2212yes—
18C22×#1142C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4—21no—
21C22×C4×Dic7—21no—
22C2×C4×Dic14—21no—
23C2×C4×Dic14—21no—
24C2×C4×Dic14—28no—
25C2×C4×Dic14C2212yes—
26C2×C4×Dic14C2212yes—
27C7×#391C2211yescharacteristic
28C7×#390C2211yescharacteristic
29C7×#388—21no—
30C7×#388—21no—
31C7×#386—21no—
32C7×#386—21no—
33C7×#382C2211yescharacteristic
34C2×#6409C2212yes—
35C2×#6406C2212yes—
36C2×C22⋊(C7⋊C8)C2211yescharacteristic
37C2×C22⋊(C7⋊C8)C2211yescharacteristic
38#89826—28no—
39#89822—24no—
40#89812—24no—
41C2×#6384C2212yes—
42C2×#6384C2212yes—
43C2×C4⋊(C7⋊C8)C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#32909—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—21no—
55C23×Dic7—21no—
56C22×Dic14—21no—
57C22×Dic14—22no—
58C22×Dic14—24no—
59C22×Dic14—24no—
60C22×Dic14D414yes—
61C22×Dic14C2312yes—
62C7×#124—24no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C7×#121—22no—
66C7×#121—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—24no—
77C2×C42×C7D411yescharacteristic
78C2×C42×C7D411yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×#1144—24no—
82C2×#1144—24no—
83C2×#1144—24no—
84C2×#1142—24no—
85C2×#1142—24no—
86C2×#1142—24no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—24no—
89C2×C4⋊Dic7C2312yes—
90C2×Dic7⋊C4—28no—
91C2×Dic7⋊C4—22no—
92C2×Dic7⋊C4—22no—
93C2×C4×Dic7—28no—
94C2×C4×Dic7—22no—
95C2×C4×Dic7—22no—
96C22×C7⋊C8C2312yes—
97C22×C7⋊C8C2312yes—
98C4×Dic14—28no—
99C4×Dic14—28no—
100C4×Dic14—24no—
101C4×Dic14—24no—
102C4×Dic14—24no—
103C4×Dic14D418yescomplemented
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108#6409—24no—
109#6406—24no—
110C22⋊(C7⋊C8)—22no—
111C22⋊(C7⋊C8)—22no—
112#6384—24no—
113#6384—24no—
114C4⋊(C7⋊C8)—24no—
115C22×C4×Q8—71no—
116C2×#464—72no—
117C2×#451—71no—
118C2×#448—71no—
119#2850D711yescharacteristic
120#1880—72no—
121#1878—72no—
122#1874—71no—
123#1872—71no—
124#1805—72no—
125#1797—71no—
126C24×C7—21no—
127C24×C7C2×D411yescharacteristic
128C2×C7×Q8—22no—
129C2×C7×Q8—21no—
130C2×C7×Q8—22no—
131C2×C7×Q8—21no—
132C2×C7×D4—22no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4—21no—
136C2×C7×D4—42no—
137C2×C7×D4—42no—
138C2×C7×D4—42no—
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—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—41no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7#4112yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—24no—
159C22×Dic7—24no—
160C22×Dic7—24no—
161C22×Dic7—22no—
162C22×Dic7—24no—
163C22×Dic7—22no—
164C22×Dic7—42no—
165C22×Dic7—42no—
166C2×Dic14—24no—
167C2×Dic14—24no—
168C2×Dic14—216no—
169C2×Dic14—24no—
170C2×Dic14—22no—
171C2×Dic14—22no—
172C2×Dic14—22no—
173C2×Dic14—216no—
174C2×Dic14C2×D414yes—
175C2×Dic14C2×D414yes—
176C2×Dic14#4114yes—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—42no—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—42no—
187C7×C22⋊C4—42no—
188C42×C7—22no—
189C42×C7—22no—
190C42×C7—22no—
191C42×C7C2×D414yes—
192#1144—48no—
193#1142—48no—
194C4⋊Dic7—24no—
195C4⋊Dic7—22no—
196C4⋊Dic7C2×D414yes—
197Dic7⋊C4—28no—
198Dic7⋊C4—28no—
199C4×Dic7—28no—
200C4×Dic7—28no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C23×Q8—71no—
208C22×Q16—72no—
209C22×SD16—72no—
210C2×#124C2×D711yescharacteristic
211C2×C22⋊Q8C2×D711yescharacteristic
212C2×#121C2×D711yescharacteristic
213C2×C4×Q8—72no—
214C2×C4×Q8—72no—
215C2×C4×Q8—148no—
216C2×C4×Q8—141no—
217C2×C4×Q8—141no—
218C22×C4⋊C4—71no—
219C22×C4⋊C4—141no—
220C22×C42—141no—
221C22×C42C2×D711yescharacteristic
222#464—148no—
223#451—144no—
224#448—144no—
225C2×C4⋊C8—72no—
226C2×#103—72no—
227C2×#103—72no—
228C2×#103—72no—
229C2×#102—72no—
230C2×C22⋊C8—71no—
231C2×C22⋊C8—71no—
232#391C2×D711yescharacteristic
233#390C2×D711yescharacteristic
234#388—21no—
235#388—21no—
236#386—21no—
237#386—21no—
238#382C2×D711yescharacteristic
239#340—72no—
240#340—72no—
241#336—72no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7#12112yes—
254C23×C7#13512yes—
255C23×C7#13712yes—
256C23×C7C22×D411yescharacteristic
257C7×Q8—42no—
258C7×Q8—42no—
259C7×D4—42no—
260C7×D4—42no—
261C7×D4—84no—
262C2×C4×C7—28no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—44no—
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—41no—
286C2×C4×C7#12012yes—
287C2×C4×C7#12012yes—
288C2×C4×C7#12112yes—
289C2×C4×C7#13712yes—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D412yes—
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4112yes—
294C2×Dic7—216no—
295C2×Dic7—24no—
296C2×Dic7—24no—
297C2×Dic7—24no—
298C2×Dic7—24no—
299C2×Dic7—48no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302Dic14—216no—
303Dic14—24no—
304Dic14—24no—
305Dic14—24no—
306Dic14#12118yes—
307C7⋊C8—42no—
308C7⋊C8—42no—
309C22×Q8—74no—
310C22×Q8—72no—
311C22×Q8—144no—
312C22×Q8—144no—
313C22×Q8—142no—
314C22×Q8—141no—
315C22×Q8C22×D711yescharacteristic
316C22×D4—21no—
317C22×D4C22×D711yescharacteristic
318C23×C4—141no—
319C23×C4—141no—
320C23×C4C22⋊D711yescharacteristic
321C23×C4C22⋊D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C2×Q16—144no—
324C2×Q16—144no—
325C2×Q16—144no—
326C2×SD16—144no—
327C2×SD16—144no—
328C2×SD16—144no—
329C22×C8—72no—
330C22×C8—72no—
331#124—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334#121—22no—
335#121—22no—
336C4×Q8—78no—
337C4×Q8—144no—
338C4×Q8—144no—
339C4×Q8—148no—
340C4×Q8—148no—
341C4×Q8—144no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—72no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—148no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C42—24no—
358C2×C42—148no—
359C2×C42—142no—
360C2×C42—142no—
361C2×C42C22⋊D711yescharacteristic
362C2×C42C22⋊D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364C2×C42C22×D711yescharacteristic
365C4⋊C8—144no—
366#103—144no—
367#103—144no—
368#103—144no—
369#102—144no—
370C22⋊C8—142no—
371C22⋊C8—142no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7#46414yescomplemented
390C22×C7C2×#12011yescharacteristic
391C22×C7C2×#12111yescharacteristic
392C22×C7#52511yescharacteristic
393C22×C7C2×#13511yescharacteristic
394C22×C7C2×#13711yescharacteristic
395C22×C7C2×#13711yescharacteristic
396C22×C7#57511yescharacteristic
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—41no—
402C4×C7—81no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7C2×#12112yes—
406C4×C7#52512yes—
407Dic7—28no—
408Dic7—44no—
409C24—21no—
410C24C2×C22⋊D711yescharacteristic
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—74no—
416C2×Q8—74no—
417C2×Q8—74no—
418C2×Q8—1416no—
419C2×Q8—144no—
420C2×Q8—144no—
421C2×Q8—1416no—
422C2×Q8—144no—
423C2×Q8—142no—
424C2×Q8—142no—
425C2×Q8—142no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—144no—
445C22×C4—144no—
446C22×C4—144no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—142no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4D4×D712yes—
453C22×C4#115912yes—
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C23×D711yescharacteristic
460Q16—288no—
461SD16—288no—
462C2×C8—142no—
463C2×C8—142no—
464C2×C8—142no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8—142no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—74no—
474C4⋊C4—148no—
475C4⋊C4—144no—
476C4⋊C4—148no—
477C4⋊C4—142no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—148no—
488C42—148no—
489C42D4×D714yes—
490C2×C7—24no—
491C2×C7—161no—
492C2×C7#284711yescharacteristic
493C2×C7#300211yescharacteristic
494C2×C7#300411yescharacteristic
495C2×C7C2×#46411yescharacteristic
496C2×C7#303711yescharacteristic
497C2×C7#310811yescharacteristic
498C2×C7#311711yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23#650112yes—
511C23#651312yes—
512C23#651412yes—
513C23C22×C22⋊D711yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—78no—
517Q8—1416no—
518Q8—144no—
519Q8—144no—
520Q8—144no—
521D4—42no—
522D4—42no—
523D4—84no—
524C2×C4—28no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
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—1416no—
549C2×C4—144no—
550C2×C4—144no—
551C2×C4—144no—
552C2×C4—144no—
553C2×C4—288no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4#650012yes—
557C2×C4#650012yes—
558C2×C4#650112yes—
559C2×C4#651412yes—
560C2×C4C2×D4×D711yescharacteristic
561C2×C4C2×D4×D712yes—
562C2×C4C2×#115911yescharacteristic
563C2×C4D7×#4112yes—
564C8—282no—
565C8—282no—
566C7#3290911yescharacteristic, complemented, Sylow
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22#8982614yescomplemented
581C22C2×#650011yescharacteristic
582C22C2×#650111yescharacteristic
583C22C2×#651311yescharacteristic
584C22#9051311yescharacteristic
585C22C2×#651411yescharacteristic
586C22C2×#651411yescharacteristic
587C22#9051911yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—148no—
597C4—284no—
598C4D7×#12112yes—
599C4#9029712yes—
600C2—24no—
601C2—161no—
602C2#1174016811yescharacteristic
603C2#1174037111yescharacteristic
604C2#1174037811yescharacteristic
605C2C2×#8982611yescharacteristic
606C2#1174059711yescharacteristic
607C2#1174059811yescharacteristic
608C2#1174075911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2850C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2850D7
hypocenterC7#32909
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic14C2semidirect
3. C2×#89826C2semidirect
4. C2×#89822C2semidirect
6. #11729446C2semidirect
8. #11729284C2semidirect
9. #11727444C2semidirect
10. #11727414C2semidirect
11. #11727313C2semidirect
103. C4×Dic14D4semidirect
389. C22×C7#464semidirect
566. C7#32909semidirect
580. C22#89826semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C7×#28502
C2×#898262
C2×#898222
C2×#898122
#117294462
#117294312
#117292842
#117274442
#117274142
#117273132
#329097