Finite Group Databasebeta

← #13359077

Subgroups — #13359077

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90316C211yesmaximal, characteristic
2C2×#89894C211yesmaximal, characteristic
3#11731313C211yesmaximal, characteristic
4C7×#1894C211yesmaximal, characteristic
5#11729743C211yesmaximal, characteristic
6#11729450C211yesmaximal, characteristic
7#11727982C211yesmaximal, characteristic, complemented
8#11727979C211yesmaximal, characteristic, complemented
9#11727974C211yesmaximal, characteristic, complemented
10Dic7×#103C211yesmaximal, characteristic, complemented
11#11727268C211yesmaximal, characteristic
12#11726559C211yesmaximal, characteristic, complemented
13#11726558C211yesmaximal, characteristic, complemented
14#11726537C211yesmaximal, characteristic, complemented
15#11726535C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×C7×C22⋊Q8C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×#6507C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C2×#6487C2211yescharacteristic
22C23×C7⋊C8C2211yescharacteristic
23#90316—24no—
24#90316C418yescomplemented
25C2×#6453—21no—
26C2×#6453C2211yescharacteristic
27C2×#6452—21no—
28C2×#6450C2211yescharacteristic
29C2×#6442—21no—
30C2×#6442—21no—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2211yescharacteristic
33C7×#380C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36#89984C2211yescharacteristic
37#89894—24no—
38C2×#6390C2211yescharacteristic
39C2×#6390C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×#6389C2211yescharacteristic
42C2×C8×Dic7C2211yescharacteristic
43C2×C8×Dic7C2211yescharacteristic
44#89749C2211yescharacteristic
45#89748C2211yescharacteristic
46C4⋊C4×Dic7C2211yescharacteristic
47C4⋊C4×Dic7C2211yescharacteristic
48C2×#6384C2211yescharacteristic
49C2×#6384C2211yescharacteristic
50#89717—21no—
51#89716—21no—
52#89426—21no—
53#89425—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21413—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C23×C4×C7C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8C2×C414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7D411yescharacteristic
82C2×C22⋊Dic7D411yescharacteristic
83Q8×Dic7—22no—
84Q8×Dic7C2×C414yes—
85#6507—22no—
86#6507C2×C414yes—
87#6489C2×C414yes—
88#6489D414yes—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7D411yescharacteristic
92C2×C4⋊Dic7D411yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96#6487—22no—
97#6487C2×C414yes—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×Dic7⋊C4—21no—
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2311yescharacteristic
111C22×C7⋊C8—22no—
112C22×C7⋊C8—22no—
113C22×C7⋊C8—22no—
114C22×C7⋊C8C2311yescharacteristic
115C22×C7⋊C8C2311yescharacteristic
116#6453—28no—
117#6453—22no—
118#6453C2×C414yes—
119#6452—28no—
120#6450—22no—
121#6450C2×C414yes—
122#6442—28no—
123#6442—28no—
124C7×#103—22no—
125C7×#103—22no—
126C7×#95—21no—
127#6409—22no—
128#6409—22no—
129C7⋊#95—21no—
130C7⋊#95—21no—
131C7⋊#95—21no—
132C7⋊#95—21no—
133C7⋊#95—21no—
134#6390—22no—
135#6390—22no—
136#6389—22no—
137#6389—22no—
138C8×Dic7—22no—
139C8×Dic7—22no—
140#6384—22no—
141#6384—22no—
142C2×#541—71no—
143C2×#405—71no—
144#2314—71no—
145#1992—71no—
146#1991—71no—
147#1960—71no—
148#1956—71no—
149#1894—71no—
150#1894D711yescharacteristic
151#1892—71no—
152#1892—71no—
153#1763—71no—
154#1763—71no—
155C4×#103—71no—
156C4×#103—71no—
157C24×C7C2×D411yescharacteristic
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8C22×C412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7C22×C412yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7C2×D411yescharacteristic
181C22×Dic7C2×D411yescharacteristic
182C22×Dic7C2×D411yescharacteristic
183C22×Dic7C2×D411yescharacteristic
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14C22×C412yes—
190C7×M16—28no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8#4112yes—
196C2×C7×C8#4112yes—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4C22×C412yes—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—81no—
206C22⋊Dic7—24no—
207C22⋊Dic7—81no—
208C22⋊Dic7C22⋊C414yes—
209C22⋊Dic7C22⋊C414yes—
210C4⋊Dic7—21no—
211C4⋊Dic7—24no—
212C4⋊Dic7—24no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217C4⋊Dic7C22⋊C414yes—
218C4⋊Dic7C22⋊C414yes—
219C4⋊Dic7C22×C412yes—
220Dic7⋊C4—24no—
221Dic7⋊C4—24no—
222Dic7⋊C4—24no—
223Dic7⋊C4—24no—
224Dic7⋊C4—24no—
225Dic7⋊C4—81no—
226Dic7⋊C4—81no—
227C4×Dic7—24no—
228C4×Dic7—24no—
229C4×Dic7—81no—
230C4×Dic7—81no—
231C4×Dic7C22×C412yes—
232C4×Dic7C22×C412yes—
233C4×Dic7C2×D412yes—
234C4×Dic7C2×D412yes—
235C2×C7⋊C8—28no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8—21no—
239C2×C7⋊C8—21no—
240C2×C7⋊C8#4112yes—
241C2×C7⋊C8#4112yes—
242C22×M16C2×D711yescharacteristic
243C23×C8—71no—
244#541—78no—
245#541—144no—
246C2×C4⋊Q8—71no—
247C2×#126—141no—
248C2×#126—141no—
249C2×#125—71no—
250C2×#125—141no—
251C2×#125—141no—
252C2×C22⋊Q8—71no—
253C2×C22⋊Q8C2×D711yescharacteristic
254C2×C4×Q8—71no—
255C2×C4×Q8—71no—
256C2×#117—71no—
257C2×#103—71no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#103—71no—
263C2×#103C2×D711yescharacteristic
264C2×#103C2×D711yescharacteristic
265#405—144no—
266C2×#97—71no—
267C2×#97—71no—
268C2×C4×C8—71no—
269C2×C4×C8—71no—
270#388—141no—
271#388—141no—
272#387—141no—
273#387—141no—
274#384—71no—
275#382—71no—
276#380—71no—
277#380C2×D711yescharacteristic
278C4×C4⋊C4—71no—
279C4×C4⋊C4—71no—
280#340—71no—
281#340—71no—
282#340—71no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340C2×D711yescharacteristic
287#340C2×D711yescharacteristic
288C23×C7—21no—
289C23×C7—21no—
290C23×C7—21no—
291C23×C7—21no—
292C23×C7—21no—
293C23×C7—21no—
294C23×C7C2×C22⋊C412yes—
295C23×C7C22×D411yescharacteristic
296C7×Q8—22no—
297C7×Q8—22no—
298C7×Q8—41no—
299C7×Q8—41no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7C2×C22⋊C412yes—
319C2×C4×C7C23×C411yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—21no—
332C2×Dic7—21no—
333C2×Dic7—22no—
334C2×Dic7—22no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7C2×C22⋊C412yes—
346C2×Dic7C2×C22⋊C412yes—
347C2×Dic7C2×C22⋊C412yes—
348C2×Dic7C2×C22⋊C412yes—
349Dic14—22no—
350Dic14—22no—
351Dic14—41no—
352Dic14—41no—
353C7×C8—22no—
354C7×C8—22no—
355C7⋊C8—22no—
356C7⋊C8—22no—
357C22×Q8—71no—
358C22×Q8C22×D711yescharacteristic
359C23×C4C22×D711yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C22×C8—71no—
364C22×C8—71no—
365C22×C8—142no—
366C22×C8—142no—
367C22×C8—142no—
368C22×C8C22×D711yescharacteristic
369C22×C8C22×D711yescharacteristic
370C4⋊Q8—74no—
371C4⋊Q8—142no—
372#126—148no—
373#126—148no—
374#125—74no—
375#125—142no—
376#125—148no—
377#125—148no—
378C22⋊Q8—22no—
379C22⋊Q8—74no—
380C22⋊Q8—142no—
381C22⋊Q8C4×D714yes—
382C4×Q8—74no—
383C4×Q8—74no—
384C4×Q8—142no—
385C4×Q8—142no—
386#117—74no—
387#117—74no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
392C2×C4⋊C4—71no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4—141no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—71no—
409C2×C22⋊C4—71no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4—141no—
412C2×C42—71no—
413C2×C42—71no—
414C2×C42—141no—
415C2×C42—141no—
416C2×C42—141no—
417C2×C42—141no—
418#103—22no—
419#103—22no—
420#103—142no—
421#103—142no—
422#103—142no—
423#103—142no—
424#103—142no—
425#103—142no—
426#97—142no—
427#97—142no—
428C4×C8—142no—
429C4×C8—142no—
430#95—21no—
431#95—141no—
432#95—141no—
433#95—141no—
434#95—141no—
435#95—141no—
436C22×C7—22no—
437C22×C7—22no—
438C22×C7—22no—
439C22×C7—41no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—41no—
443C22×C7C22×C22⋊C411yescharacteristic
444C22×C7#53211yescharacteristic
445C22×C7#53211yescharacteristic
446C22×C7#57711yescharacteristic
447C22×C7#57711yescharacteristic
448C22×C7#57811yescharacteristic
449C22×C7#57811yescharacteristic
450C4×C7—22no—
451C4×C7—42no—
452C4×C7—42no—
453C4×C7—81no—
454C4×C7—81no—
455C4×C7C4×#4111yescharacteristic
456C4×C7C4×#4111yescharacteristic
457C4×C7#52511yescharacteristic
458C4×C7#52511yescharacteristic
459Dic7—22no—
460Dic7—22no—
461Dic7—22no—
462Dic7—22no—
463Dic7—42no—
464Dic7—42no—
465Dic7—81no—
466Dic7—81no—
467C24D4×D711yescharacteristic
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—72no—
474C2×Q8—141no—
475C2×Q8—141no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8C2×C4×D712yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—71no—
489C22×C4—71no—
490C22×C4—71no—
491C22×C4—71no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4C2×C4×D712yes—
497C22×C4D4×D711yescharacteristic
498C22×C4D4×D712yes—
499C22×C4C23×D711yescharacteristic
500M16—28no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—72no—
506C2×C8—72no—
507C2×C8—148no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8#115912yes—
513C2×C8#115912yes—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—74no—
521C4⋊C4—74no—
522C4⋊C4—72no—
523C4⋊C4—144no—
524C4⋊C4—144no—
525C4⋊C4—144no—
526C4⋊C4—144no—
527C4⋊C4—144no—
528C4⋊C4—144no—
529C4⋊C4—144no—
530C4⋊C4—141no—
531C4⋊C4—141no—
532C4⋊C4—141no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—561no—
536C4⋊C4—561no—
537C4⋊C4C2×C4×D712yes—
538C22⋊C4—24no—
539C22⋊C4—74no—
540C22⋊C4—74no—
541C22⋊C4—81no—
542C22⋊C4—144no—
543C22⋊C4—561no—
544C42—72no—
545C42—72no—
546C42—72no—
547C42—72no—
548C42—144no—
549C42—144no—
550C42—561no—
551C42—561no—
552C2×C7—42no—
553C2×C7#229711yescharacteristic
554C2×C7#289811yescharacteristic
555C2×C7#289811yescharacteristic
556C2×C7#317411yescharacteristic
557C2×C7#317711yescharacteristic
558C2×C7#317811yescharacteristic
559C2×C7#317811yescharacteristic
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23D7×C22⋊C412yes—
567C23C2×D4×D711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—142no—
573Q8—142no—
574Q8—281no—
575Q8—281no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—72no—
595C2×C4—72no—
596C2×C4—72no—
597C2×C4—72no—
598C2×C4—141no—
599C2×C4—141no—
600C2×C4—141no—
601C2×C4—141no—
602C2×C4—141no—
603C2×C4—141no—
604C2×C4—141no—
605C2×C4—141no—
606C2×C4—142no—
607C2×C4—142no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4D7×C22⋊C412yes—
619C2×C4C22×C4×D711yescharacteristic
620C2×C4C2×D4×D711yescharacteristic
621C2×C4C2×#115911yescharacteristic
622C2×C4C2×#115911yescharacteristic
623C2×C4D7×#4111yescharacteristic
624C2×C4D7×#4111yescharacteristic
625C8—22no—
626C8—22no—
627C8—142no—
628C8—142no—
629C7#2141311yescharacteristic, complemented, Sylow
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×D7×C22⋊C411yescharacteristic
638C22#9028711yescharacteristic
639C22#9028711yescharacteristic
640C22#9046611yescharacteristic
641C22#9046911yescharacteristic
642C22#9047011yescharacteristic
643C22#9047011yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—142no—
650C4—142no—
651C4—142no—
652C4—142no—
653C4—282no—
654C4—282no—
655C4—561no—
656C4—561no—
657C4C4×#115911yescharacteristic
658C4C4×#115911yescharacteristic
659C4#9027711yescharacteristic
660C4#9034711yescharacteristic
661C2—42no—
662C2#1173210311yescharacteristic
663C2#1173632811yescharacteristic
664C2#1173640211yescharacteristic
665C2#1173823111yescharacteristic
666C2#1173823311yescharacteristic
667C2#1173833511yescharacteristic
668C2#1173833611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1894C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1894D7
hypocenterC7#21413
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
7. #11727982C2semidirect
8. #11727979C2semidirect
9. #11727974C2semidirect
10. Dic7×#103C2semidirect
12. #11726559C2semidirect
13. #11726558C2semidirect
14. #11726537C2semidirect
15. #11726535C2semidirect
24. #90316C4semidirect
629. C7#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#903162
C2×#898942
#117313132
C7×#18942
#117297432
#117294502
#117279822
#117279792
#117279742
Dic7×#1032
#117272682
#117265592
#117265582
#117265372
#117265352
#214137