Finite Group Databasebeta

← #13365902

Subgroups — #13365902

Order 1792.

836 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89531C211yesmaximal, characteristic
2#11733734C211yesmaximal, characteristic
3#11733412C211yesmaximal, characteristic, complemented
4C7×#1936C211yesmaximal, characteristic
5#11727928C211yesmaximal, characteristic
6#11727910C211yesmaximal, characteristic
7#11727907C211yesmaximal, characteristic
8#11727573C211yesmaximal, characteristic, complemented
9#11727563C211yesmaximal, characteristic, complemented
10#11727399C211yesmaximal, characteristic
11#11727374C211yesmaximal, characteristic, complemented
12#11726574C211yesmaximal, characteristic, complemented
13#11726554C211yesmaximal, characteristic, complemented
14#11726550C211yesmaximal, characteristic, complemented
15#11726544C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×#6451C2211yescharacteristic
18C2×#6450—21no—
19C2×#6440—21no—
20C2×C7×#102C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#389C2211yescharacteristic
25#89993C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32#89748C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89745C2211yescharacteristic
37#89744—21no—
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42C22⋊C4×Dic7—21no—
43C42×Dic7C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47#89531C418yescomplemented
48#89421—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22116—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×Dic7—21no—
63C22×Dic14—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76D4×Dic7—22no—
77D4×Dic7C2×C414yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7D411yescharacteristic
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6451—22no—
100#6451C2×C414yes—
101#6450—44no—
102#6440—44no—
103C7×#102—41no—
104C7×#102C2×C414yes—
105C7×#97—41no—
106C7×#95—21no—
107#6406—41no—
108#6406C2×C414yes—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115#6392C2×C414yes—
116#6391—41no—
117Dic7⋊C8—22no—
118Dic7⋊C8C2×C414yes—
119#6382—41no—
120#6381C2×C414yes—
121#6377—41no—
122C2×#474—71no—
123C4×C4⋊Q8—71no—
124#2293—71no—
125#1988—71no—
126#1948—71no—
127#1943—71no—
128#1943—71no—
129#1936—71no—
130#1936D711yescharacteristic
131#1926—71no—
132#1926—71no—
133#1921—71no—
134#1921—71no—
135#1802—71no—
136#1767—71no—
137C24×C7—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4C22×C412yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—24no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—42no—
156C22×Dic7—42no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C2×Dic14—24no—
162C2×Dic14—42no—
163C2×Dic14—42no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8C22×C412yes—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4C22×C412yes—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—42no—
179C42×C7—21no—
180C42×C7—21no—
181C22⋊Dic7—24no—
182C22⋊Dic7—42no—
183C22⋊Dic7—42no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7C22×C412yes—
191C4⋊Dic7C22×C412yes—
192Dic7⋊C4—24no—
193Dic7⋊C4—42no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—22no—
201C4×Dic7—22no—
202C4×Dic7—22no—
203C4×Dic7—41no—
204C4×Dic7—42no—
205C4×Dic7—42no—
206C4×Dic7—42no—
207C4×Dic7C22×C412yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C412yes—
215C2×C4⋊Q8—71no—
216C2×C4×Q8—141no—
217C2×C4×D4—71no—
218C2×#117—141no—
219#474—78no—
220C2×#107—71no—
221C2×#107—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102C2×D711yescharacteristic
229C2×#102C2×D711yescharacteristic
230C2×#97—71no—
231C2×#97C2×D711yescharacteristic
232#391—141no—
233#390—71no—
234#390C2×D711yescharacteristic
235#389—71no—
236#389—71no—
237#389—71no—
238#389—141no—
239#389C2×D711yescharacteristic
240#388—141no—
241#388—141no—
242#384—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—141no—
247C4×C22⋊C4—141no—
248C43—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340C2×D711yescharacteristic
256#340C2×D711yescharacteristic
257#336—71no—
258#336—71no—
259C23×C7—22no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7C22×D411yescharacteristic
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—41no—
270C7×D4—41no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
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×C7C23×C411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7H(5,𝔽2)11yescharacteristic
289C2×C4×C72−1+411yescharacteristic
290C2×C4×C72−1+411yescharacteristic
291C2×C4×C72−1+411yescharacteristic
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—48no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7C4×D412yes—
313C2×Dic7C4×D412yes—
314C2×Dic7AGL(1,ℤ/8)12yes—
315C2×Dic7AGL(1,ℤ/8)12yes—
316C2×Dic7#13712yes—
317Dic14—48no—
318C7×C8—22no—
319C7×C8—41no—
320C7⋊C8—22no—
321C7⋊C8—41no—
322C22×Q8—141no—
323C22×D4C22×D711yescharacteristic
324C23×C4—141no—
325C22×C8—71no—
326C22×C8—71no—
327C22×C8C22×D711yescharacteristic
328C22×C8C22×D711yescharacteristic
329C4⋊Q8—74no—
330C4⋊Q8—142no—
331C4×Q8—284no—
332C4×D4—74no—
333C4×D4—142no—
334#117—284no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—141no—
361C2×C42—141no—
362C2×C42—141no—
363C2×C42C22×D711yescharacteristic
364#107—74no—
365#107—74no—
366#106—281no—
367#106—281no—
368C4⋊C8—74no—
369C4⋊C8—142no—
370#102—41no—
371#102—74no—
372#102—281no—
373#102C4×D714yes—
374#97—41no—
375#97—281no—
376#95—21no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—81no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7C2×C4×D411yescharacteristic
391C22×C7#53611yescharacteristic
392C22×C7#53711yescharacteristic
393C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C7C2×#13711yescharacteristic
396C22×C7#57511yescharacteristic
397C4×C7—42no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—81no—
401C4×C7—81no—
402C4×C7#51911yescharacteristic
403C4×C7#52011yescharacteristic
404C4×C7#54111yescharacteristic
405C4×C7#54211yescharacteristic
406Dic7—22no—
407Dic7—22no—
408Dic7—42no—
409Dic7—42no—
410Dic7—41no—
411Dic7—81no—
412Dic7—81no—
413Dic7#44214yes—
414C24—21no—
415C2×Q8—144no—
416C2×Q8—282no—
417C2×Q8—282no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4C2×C4×D712yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—144no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4D4×D711yescharacteristic
439C22×C4D4×D711yescharacteristic
440C22×C4C23×D711yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—72no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8C2×C4×D712yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—72no—
460C4⋊C4—72no—
461C4⋊C4—144no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—282no—
470C4⋊C4—282no—
471C4⋊C4—282no—
472C4⋊C4—282no—
473C4⋊C4C2×C4×D712yes—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—144no—
477C22⋊C4—282no—
478C22⋊C4—282no—
479C42—21no—
480C42—21no—
481C42—72no—
482C42—72no—
483C42—142no—
484C42—142no—
485C42—142no—
486C42—141no—
487C42—141no—
488C42—281no—
489C42—282no—
490C42—282no—
491C42—282no—
492C2×C7—82no—
493C2×C7#232311yescharacteristic
494C2×C7C2×#44211yescharacteristic
495C2×C7C4×#13711yescharacteristic
496C2×C7#323511yescharacteristic
497C2×C7#323911yescharacteristic
498C2×C7#325111yescharacteristic
499C2×C7#325411yescharacteristic
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×D4×D711yescharacteristic
508Q8—288no—
509D4—22no—
510D4—22no—
511D4—41no—
512D4—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—141no—
542C2×C4—288no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4C22×C4×D711yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4#655411yescharacteristic
557C2×C4#655411yescharacteristic
558C2×C4#655411yescharacteristic
559C8—22no—
560C8—41no—
561C8—142no—
562C8—281no—
563C7#2211611yescharacteristic, complemented, Sylow
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C4×D4×D711yescharacteristic
572C22#9028811yescharacteristic
573C22#9036611yescharacteristic
574C22D7×AGL(1,ℤ/8)11yescharacteristic
575C22D7×AGL(1,ℤ/8)11yescharacteristic
576C22#9046411yescharacteristic
577C22D7×#13711yescharacteristic
578C4—41no—
579C4—41no—
580C4—42no—
581C4—74no—
582C4—81no—
583C4—81no—
584C4—142no—
585C4—142no—
586C4—281no—
587C4—282no—
588C4—282no—
589C4—561no—
590C4—561no—
591C4#9020711yescharacteristic
592C4#9023711yescharacteristic
593C4#9035311yescharacteristic
594C4#9039211yescharacteristic
595C2—82no—
596C2#1173290411yescharacteristic
597C2#1173721911yescharacteristic
598C2D7×#44211yescharacteristic
599C2#1173823911yescharacteristic
600C2#1173829911yescharacteristic
601C2#1173913611yescharacteristic
602C2#1173916711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1936C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1936D7
hypocenterC7#22116
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
3. #11733412C2semidirect
8. #11727573C2semidirect
9. #11727563C2semidirect
11. #11727374C2semidirect
12. #11726574C2semidirect
13. #11726554C2semidirect
14. #11726550C2semidirect
15. #11726544C2semidirect
47. #89531C4semidirect
563. C7#22116semidirect

Maximal subgroups

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

subgroupindex
C2×#895312
#117337342
#117334122
C7×#19362
#117279282
#117279102
#117279072
#117275732
#117275632
#117273992
#117273742
#117265742
#117265542
#117265502
#117265442
#221167