Finite Group Databasebeta

← #13360002

Subgroups — #13360002

Order 1792.

984 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89954C211yesmaximal, characteristic
2C2×Q16×Dic7C211yesmaximal, characteristic
3C4×Q8×Dic7C211yesmaximal, characteristic
4C2×#89562C211yesmaximal, characteristic
5#11733926C211yesmaximal, characteristic
6C7×#1878C211yesmaximal, characteristic
7#11729386C211yesmaximal, characteristic
8#11727983C211yesmaximal, characteristic
9#11727975C211yesmaximal, characteristic
10Dic7×#103C211yesmaximal, characteristic
11#11727421C211yesmaximal, characteristic
12#11726549C211yesmaximal, characteristic
13#11726535C211yesmaximal, characteristic
14#11726382C211yesmaximal, characteristic
15#11726356C211yesmaximal, characteristic
16C22×C7×Q16C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×Q8×Dic7C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×#6507C2211yescharacteristic
22C22×#1144C2211yescharacteristic
23C2×#6450C2211yescharacteristic
24C2×C7×#103C2211yescharacteristic
25C2×C7×#103C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30Q16×Dic7—24no—
31#89954C418yescomplemented
32C2×#6392C2211yescharacteristic
33C2×#6390C2211yescharacteristic
34C2×#6390C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36C2×Dic7⋊C8C2211yescharacteristic
37C2×C8×Dic7C2211yescharacteristic
38C2×C8×Dic7C2211yescharacteristic
39#89748C2211yescharacteristic
40C4⋊C4×Dic7—21no—
41C4⋊C4×Dic7—21no—
42C4⋊C4×Dic7C2211yescharacteristic
43C4⋊C4×Dic7C2211yescharacteristic
44C2×#6384C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C4×C4⋊Dic7—21no—
47C4×C4⋊Dic7C2211yescharacteristic
48C42×Dic7—21no—
49C42×Dic7C2211yescharacteristic
50#89701C2211yescharacteristic
51#89562—24no—
52#89421—21no—
53#89421—21no—
54#89419—21no—
55#89419—21no—
56C7×#340C2211yescharacteristic
57C7×#336C2211yescharacteristic
58#89346C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89319C2211yescharacteristic
62C4×#462—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C2×C7×Q16—22no—
67C2×C7×Q16—22no—
68C2×C7×Q16C2×C414yes—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78Q8×Dic7—22no—
79Q8×Dic7—22no—
80Q8×Dic7—22no—
81Q8×Dic7—22no—
82Q8×Dic7C2×C414yes—
83Q8×Dic7D414yes—
84#6507—22no—
85#6507C2×C414yes—
86C2×#1144—22no—
87C2×#1144—22no—
88C2×#1144C2×C414yes—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7D411yescharacteristic
108C2×C4×Dic7D411yescharacteristic
109C2×C4×Dic7D411yescharacteristic
110C2×C4×Dic7D411yescharacteristic
111C2×C4×Dic7C2311yescharacteristic
112C2×C4×Dic7C2311yescharacteristic
113C22×C7⋊C8C2311yescharacteristic
114C22×C7⋊C8C2311yescharacteristic
115#6450—22no—
116#6450—22no—
117C7×#103—22no—
118C7×#103—22no—
119C7×#95—21no—
120C7×#95—21no—
121#6409—22no—
122#6409—22no—
123#6409C2×C414yes—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129#6392—22no—
130#6390—22no—
131#6390C2×C414yes—
132Dic7⋊C8—22no—
133Dic7⋊C8C2×C414yes—
134C8×Dic7—22no—
135C8×Dic7—22no—
136#6384—22no—
137#6381—22no—
138C2×#462—71no—
139C2×C4×Q16—71no—
140C2×C4×Q16—71no—
141C4×C4⋊Q8—71no—
142C42×Q8—71no—
143#1981—71no—
144#1947—71no—
145#1928—71no—
146#1928—71no—
147#1878—71no—
148#1878D711yescharacteristic
149#1803—71no—
150C4×C4⋊C8—71no—
151C4×#103—71no—
152C4×#103—71no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8C22×C412yes—
162C2×C7×Q8C22×C412yes—
163C2×C7×Q8C2×D412yes—
164C2×C7×Q8#4112yes—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14C22×C412yes—
186C7×Q16—28no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8C22×C412yes—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4#4112yes—
198C42×C7—24no—
199C42×C7—21no—
200C42×C7—21no—
201#1144—28no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—42no—
209C4⋊Dic7C22×C412yes—
210C4⋊Dic7C2×D412yes—
211Dic7⋊C4—24no—
212Dic7⋊C4—24no—
213Dic7⋊C4—42no—
214Dic7⋊C4—42no—
215Dic7⋊C4—42no—
216C4×Dic7—24no—
217C4×Dic7—24no—
218C4×Dic7—22no—
219C4×Dic7—22no—
220C4×Dic7—41no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7—42no—
224C4×Dic7—42no—
225C4×Dic7—42no—
226C4×Dic7Q1614yescomplemented
227C4×Dic7C22×C412yes—
228C4×Dic7C2×D412yes—
229C4×Dic7#4112yes—
230C4×Dic7#4112yes—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8C22×C412yes—
237C22×Q16—71no—
238C22×Q16C2×D711yescharacteristic
239C2×C4⋊Q8—71no—
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×Q8—71no—
243C2×C4×Q8—71no—
244C2×C4×Q8C2×D711yescharacteristic
245#462—78no—
246C4×Q16—144no—
247C4×Q16—144no—
248C2×#107—71no—
249C2×#107—71no—
250C2×C4⋊C8—71no—
251C2×C4⋊C8—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103—71no—
257C2×#103—71no—
258C2×#103C2×D711yescharacteristic
259C2×#103C2×D711yescharacteristic
260C2×C4×C8—71no—
261C2×C4×C8—71no—
262#391—71no—
263#391C2×D711yescharacteristic
264#389—141no—
265#389—141no—
266#384—71no—
267C4×C4⋊C4—71no—
268C4×C4⋊C4—71no—
269C4×C4⋊C4—71no—
270C4×C4⋊C4—141no—
271C4×C4⋊C4—141no—
272C4×C4⋊C4—141no—
273C4×C4⋊C4—141no—
274C4×C4⋊C4—141no—
275C43—71no—
276C43—141no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340C2×D711yescharacteristic
281#336—71no—
282#336C2×D711yescharacteristic
283C23×C7C22×D411yescharacteristic
284C7×Q8—22no—
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8—22no—
289C7×Q8—41no—
290C7×Q8—41no—
291C7×Q8C4×D414yes—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7C2×Q1612yes—
310C2×C4×C7C23×C411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—22no—
321C2×Dic7—22no—
322C2×Dic7—22no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7C4×D412yes—
333C2×Dic7C4×D412yes—
334C2×Dic7#12112yes—
335C2×Dic7#12112yes—
336C2×Dic7C2×Q1612yes—
337C2×Dic7C2×Q1612yes—
338C2×Dic7#13512yes—
339C2×Dic7#13712yes—
340Dic14—22no—
341Dic14—22no—
342Dic14—41no—
343Dic14—41no—
344C7×C8—22no—
345C7×C8—41no—
346C7⋊C8—22no—
347C7⋊C8—41no—
348C22×Q8—71no—
349C22×Q8C22×D711yescharacteristic
350C22×Q8C22×D711yescharacteristic
351C2×Q16—22no—
352C2×Q16—22no—
353C2×Q16—74no—
354C2×Q16—142no—
355C2×Q16—142no—
356C2×Q16C4×D714yes—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8C22×D711yescharacteristic
360C22×C8C22×D711yescharacteristic
361C4⋊Q8—74no—
362C4⋊Q8—142no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8—74no—
366C4×Q8—74no—
367C4×Q8—142no—
368C4×Q8—142no—
369C4×Q8—142no—
370C4×Q8—142no—
371C4×Q8—142no—
372C4×Q8—142no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—71no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C42—21no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—141no—
395C2×C42—141no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42—141no—
399C2×C42—141no—
400C2×C42—141no—
401C2×C42C22×D711yescharacteristic
402#107—142no—
403#107—142no—
404C4⋊C8—74no—
405C4⋊C8—142no—
406#103—22no—
407#103—22no—
408#103—74no—
409#103—74no—
410#103—142no—
411#103—142no—
412#103—142no—
413#103—142no—
414C4×C8—142no—
415C4×C8—142no—
416#95—21no—
417#95—21no—
418#95—141no—
419#95—141no—
420#95—141no—
421#95—141no—
422#95—141no—
423C22×C7C2×C4×D411yescharacteristic
424C22×C7C2×#12111yescharacteristic
425C22×C7#52511yescharacteristic
426C22×C7C22×Q1611yescharacteristic
427C22×C7C2×#13511yescharacteristic
428C22×C7C2×#13711yescharacteristic
429C22×C7#57511yescharacteristic
430C4×C7—22no—
431C4×C7—24no—
432C4×C7—41no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7—81no—
436C4×C7C2×C4×D411yescharacteristic
437C4×C7C4×#4111yescharacteristic
438C4×C7#52911yescharacteristic
439C4×C7#53211yescharacteristic
440Dic7—22no—
441Dic7—22no—
442Dic7—42no—
443Dic7—42no—
444Dic7—42no—
445Dic7—81no—
446Dic7C4×Q1614yescomplemented
447Dic7#46214yescomplemented
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—72no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×Q8—141no—
461C2×Q8—141no—
462C2×Q8C2×C4×D712yes—
463C2×Q8C2×C4×D712yes—
464C2×Q8D4×D712yes—
465C2×Q8#115912yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4D4×D711yescharacteristic
479C22×C4D4×D711yescharacteristic
480C22×C4C23×D711yescharacteristic
481Q16—28no—
482Q16—148no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—72no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8C2×C4×D712yes—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—72no—
500C4⋊C4—72no—
501C4⋊C4—144no—
502C4⋊C4—144no—
503C4⋊C4—144no—
504C4⋊C4—144no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—282no—
510C4⋊C4—282no—
511C4⋊C4—282no—
512C4⋊C4—282no—
513C4⋊C4#115912yes—
514C42—24no—
515C42—21no—
516C42—21no—
517C42—74no—
518C42—72no—
519C42—72no—
520C42—72no—
521C42—72no—
522C42—142no—
523C42—142no—
524C42—144no—
525C42—144no—
526C42—281no—
527C42—282no—
528C42—282no—
529C42—282no—
530C42—282no—
531C42—282no—
532C2×C7C4×#12111yescharacteristic
533C2×C7C2×C4×Q1611yescharacteristic
534C2×C7C4×#13711yescharacteristic
535C2×C7C2×#46211yescharacteristic
536C2×C7#303511yescharacteristic
537C2×C7#310411yescharacteristic
538C2×C7#310711yescharacteristic
539C23C2×D4×D711yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—142no—
548Q8—142no—
549Q8—281no—
550Q8—281no—
551Q8(C22⋊D7)⋊C414yes—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—72no—
570C2×C4—72no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—72no—
574C2×C4—72no—
575C2×C4—72no—
576C2×C4—72no—
577C2×C4—141no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—142no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4D7×Q1612yes—
594C2×C4C22×C4×D711yescharacteristic
595C2×C4C2×D4×D711yescharacteristic
596C2×C4C2×#115911yescharacteristic
597C2×C4C2×#115911yescharacteristic
598C2×C4D7×#4111yescharacteristic
599C2×C4D7×#4111yescharacteristic
600C8—22no—
601C8—41no—
602C8—142no—
603C8—281no—
604C7C4×#46211yescharacteristic, complemented, Sylow
605C22C4×D4×D711yescharacteristic
606C22#9027711yescharacteristic
607C22D7×#12111yescharacteristic
608C22C2×D7×Q1611yescharacteristic
609C22D7×#13511yescharacteristic
610C22D7×#13711yescharacteristic
611C22#9046811yescharacteristic
612C4—22no—
613C4—24no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—74no—
618C4—74no—
619C4—81no—
620C4—142no—
621C4—142no—
622C4—282no—
623C4—282no—
624C4—282no—
625C4—561no—
626C4C2×(C22⋊D7)⋊C411yescharacteristic
627C4C4×#115911yescharacteristic
628C4#9028711yescharacteristic
629C4#9034011yescharacteristic
630C2#1173210511yescharacteristic
631C2C4×D7×Q1611yescharacteristic
632C2#1173727411yescharacteristic
633C2#1173761711yescharacteristic
634C2#1173765811yescharacteristic
635C2D7×#46211yescharacteristic
636C2#1173804211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1878C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#1878D7
hypocenterC7C4×#462
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
31. #89954C4semidirect
226. C4×Dic7Q16semidirect
446. Dic7C4×Q16semidirect
447. Dic7#462semidirect
604. C7C4×#462semidirect

Maximal subgroups

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

subgroupindex
C2×#899542
C2×Q16×Dic72
C4×Q8×Dic72
C2×#895622
#117339262
C7×#18782
#117293862
#117279832
#117279752
Dic7×#1032
#117274212
#117265492
#117265352
#117263822
#117263562
C4×#4627