Finite Group Databasebeta

← #13411916

Subgroups — #13411916

Order 1792.

786 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740687C211yesmaximal, characteristic, complemented
2#11733747C211yesmaximal, characteristic
3C7×#2023C211yesmaximal, characteristic
4#11729533C211yesmaximal, characteristic, complemented
5#11727933C211yesmaximal, characteristic
6#11727917C211yesmaximal, characteristic
7#11727907C211yesmaximal, characteristic
8Dic7×#102C211yesmaximal, characteristic
9#11727406C211yesmaximal, characteristic
10#11726601C211yesmaximal, characteristic, complemented
11#11726553C211yesmaximal, characteristic, complemented
12#11726550C211yesmaximal, characteristic, complemented
13#11726543C211yesmaximal, characteristic, complemented
14#11726377C211yesmaximal, characteristic, complemented
15#11726371C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×#6507—21no—
18C2×D4×Dic7C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×#6440—21no—
21C2×C7×#102C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C7×#390C2211yescharacteristic
24#90001—21no—
25#89993C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89751—21no—
33#89748C2211yescharacteristic
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745C2211yescharacteristic
38#89744—21no—
39C2×#6382C2211yescharacteristic
40#89717—21no—
41#89716—21no—
42#89714—21no—
43C4×C4⋊Dic7C2211yescharacteristic
44C4×Dic7⋊C4C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×C4⋊(C7⋊C8)C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#336C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26024—71nomaximal, Sylow
60C22×C7×Q8—21no—
61C22×C7×D4C2311yescharacteristic
62C23×Dic7—21no—
63C22×Dic14—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—41no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78#6507—44no—
79D4×Dic7—22no—
80D4×Dic7Q814yescomplemented
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4D411yescharacteristic
93C2×Dic7⋊C4D411yescharacteristic
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7C2311yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6451—22no—
102#6451—41no—
103#6440—44no—
104C7×#102—22no—
105C7×#102—41no—
106#6406—22no—
107#6406—41no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115#6391—22no—
116Dic7⋊C8—22no—
117Dic7⋊C8—41no—
118C8×Dic7—22no—
119#6382—22no—
120C4⋊(C7⋊C8)—41no—
121#2551—71no—
122#2354—71no—
123#2065—71no—
124#2065—71no—
125#2023—71no—
126#2023D711yescharacteristic
127#1987—71no—
128#1987—71no—
129#1948—71no—
130#1939—71no—
131#1920—71no—
132#1920—71no—
133#1834—71no—
134#1802—71no—
135C4×#102—71no—
136C24×C7—21no—
137C2×C7×Q8—42no—
138C2×C7×Q8—42no—
139C2×C7×Q8—81no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4C2×Q812yes—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—24no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C2×Dic14—42no—
164C2×Dic14—42no—
165C2×Dic14—81no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4#4112yes—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—81no—
181C42×C7—21no—
182C42×C7—21no—
183C22⋊Dic7—24no—
184C22⋊Dic7—42no—
185C22⋊Dic7—81no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7C2×Q812yes—
193C4⋊Dic7#4112yes—
194Dic7⋊C4—22no—
195Dic7⋊C4—42no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—81no—
201Dic7⋊C4—81no—
202Dic7⋊C4SD1614yes—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—42no—
206C4×Dic7—42no—
207C4×Dic7—42no—
208C4×Dic7C2×Q812yes—
209C4×Dic7#4112yes—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C4⋊Q8—71no—
217C2×C4⋊Q8—141no—
218C2×C4⋊Q8C2×D711yescharacteristic
219C2×C4×D4—71no—
220C2×#117—141no—
221C2×#106—71no—
222C2×#106—71no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102C2×D711yescharacteristic
229C2×#102C2×D711yescharacteristic
230C2×C4×C8—71no—
231#395—141no—
232#395—141no—
233#390—71no—
234#390C2×D711yescharacteristic
235#389—71no—
236#389—71no—
237#389—141no—
238#388—141no—
239#388—141no—
240#384—71no—
241#384—71no—
242#380—141no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247#340—71no—
248#340—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#336C2×D711yescharacteristic
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×Q8—84no—
268C7×D4—22no—
269C7×D4—22no—
270C7×D4—41no—
271C7×D4—41no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
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×C7C2×SD1612yes—
287C2×C4×C7C22×Q811yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—48no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7C22⋊Q812yes—
314C2×Dic7C22⋊Q812yes—
315C2×Dic7C2×SD1612yes—
316C2×Dic7C2×SD1612yes—
317C2×Dic7AGL(1,ℤ/8)12yes—
318Dic14—84no—
319C7×C8—41no—
320C7×C8—41no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C22×Q8—21no—
324C22×Q8—141no—
325C22×D4C22×D711yescharacteristic
326C23×C4—141no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8C22×D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331C4⋊Q8—22no—
332C4⋊Q8—41no—
333C4⋊Q8—142no—
334C4⋊Q8—281no—
335C4⋊Q8—284no—
336C4×D4—74no—
337C4×D4—142no—
338#117—284no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—141no—
364C2×C42—141no—
365C2×C42—141no—
366C2×C42C22×D711yescharacteristic
367#106—142no—
368#106—142no—
369C4⋊C8—142no—
370C4⋊C8—281no—
371C4⋊C8—281no—
372#102—22no—
373#102—41no—
374#102—142no—
375#102—281no—
376C4×C8—142no—
377#95—71no—
378#95—71no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—81no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7—81no—
391C22×C7C2×C22⋊Q811yescharacteristic
392C22×C7#54611yescharacteristic
393C22×C7#54811yescharacteristic
394C22×C7C22×SD1611yescharacteristic
395C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C7#57611yescharacteristic
397C22×C7#57811yescharacteristic
398C4×C7—22no—
399C4×C7—42no—
400C4×C7—41no—
401C4×C7—81no—
402C4×C7—81no—
403C4×C7#53311yescharacteristic
404C4×C7#54411yescharacteristic
405C4×C7#55511yescharacteristic
406C4×C7#55611yescharacteristic
407Dic7—22no—
408Dic7—22no—
409Dic7—42no—
410Dic7—41no—
411Dic7—42no—
412Dic7—81no—
413Dic7—81no—
414Dic7#47614yes—
415C24—21no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—282no—
420C2×Q8—282no—
421C2×Q8—561no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4Q8×D712yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—144no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4D4×D711yescharacteristic
443C22×C4D4×D711yescharacteristic
444C22×C4C23×D711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—74no—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—142no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—561no—
479C4⋊C4—561no—
480C4⋊C4#115912yes—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—144no—
484C22⋊C4—282no—
485C22⋊C4—561no—
486C42—21no—
487C42—21no—
488C42—72no—
489C42—72no—
490C42—141no—
491C42—141no—
492C42—282no—
493C42—282no—
494C42—282no—
495C2×C7—82no—
496C2×C7#243911yescharacteristic
497C2×C7C2×#47611yescharacteristic
498C2×C7#308311yescharacteristic
499C2×C7#329811yescharacteristic
500C2×C7#332511yescharacteristic
501C2×C7#336211yescharacteristic
502C2×C7#336711yescharacteristic
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D711yescharacteristic
511Q8—84no—
512Q8—564no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—72no—
534C2×C4—72no—
535C2×C4—72no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—142no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—142no—
545C2×C4—288no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4D7×SD1612yes—
557C2×C4C2×#115911yescharacteristic
558C2×C4#654811yescharacteristic
559C2×C4C2×Q8×D711yescharacteristic
560C2×C4D7×#4111yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655411yescharacteristic
563C8—41no—
564C8—41no—
565C8—281no—
566C8—281no—
567C7#2602411yescharacteristic, complemented, Sylow
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#9028211yescharacteristic
576C22D7×C22⋊Q811yescharacteristic
577C22#9036811yescharacteristic
578C22#9044811yescharacteristic
579C22C2×D7×SD1611yescharacteristic
580C22D7×AGL(1,ℤ/8)11yescharacteristic
581C22#9047011yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—74no—
586C4—81no—
587C4—81no—
588C4—142no—
589C4—142no—
590C4—281no—
591C4—282no—
592C4—282no—
593C4—561no—
594C4—561no—
595C4#9031311yescharacteristic
596C4#9031411yescharacteristic
597C4#9036111yescharacteristic
598C4#9036211yescharacteristic
599C2—82no—
600C2#1173422711yescharacteristic
601C2#1173746611yescharacteristic
602C2#1173749411yescharacteristic
603C2#1173841311yescharacteristic
604C2D7×#47611yescharacteristic
605C2#1173903311yescharacteristic
606C2#1173911511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2023C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2023D7
hypocenterC7#26024
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #11740687C2semidirect
4. #11729533C2semidirect
10. #11726601C2semidirect
11. #11726553C2semidirect
12. #11726550C2semidirect
13. #11726543C2semidirect
14. #11726377C2semidirect
15. #11726371C2semidirect
80. D4×Dic7Q8semidirect
567. C7#26024semidirect

Maximal subgroups

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

subgroupindex
#117406872
#117337472
C7×#20232
#117295332
#117279332
#117279172
#117279072
Dic7×#1022
#117274062
#117266012
#117265532
#117265502
#117265432
#117263772
#117263712
#260247