Finite Group Databasebeta

← #13358334

Subgroups — #13358334

Order 1792.

1142 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90380C211yesmaximal, characteristic
2C2×#89611C211yesmaximal, characteristic
3#11731195C211yesmaximal, characteristic
4C7×#1989C211yesmaximal, characteristic, complemented
5#11729492C211yesmaximal, characteristic, complemented
6#11728213C211yesmaximal, characteristic, complemented
7#11727244C211yesmaximal, characteristic, complemented
8#11726621C211yesmaximal, characteristic
9#11726621C211yesmaximal, characteristic
10#11726620C211yesmaximal, characteristic
11D7×#340C211yesmaximal, characteristic
12#11726556C211yesmaximal, characteristic, complemented
13#11726555C211yesmaximal, characteristic, complemented
14#11726554C211yesmaximal, characteristic, complemented
15#11726553C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C22×C8×D7C2211yescharacteristic
19#90380—24no—
20#90380C418yescomplemented
21C2×#6460—21no—
22C2×#6459—21no—
23C2×#6455C2211yescharacteristic
24C2×#6455C2211yescharacteristic
25C2×D7×C4⋊C4C2211yescharacteristic
26C2×D7×C4⋊C4C2211yescharacteristic
27C2×#6453C2211yescharacteristic
28C2×#6452—21no—
29C2×#6451—21no—
30C2×#6451C2211yescharacteristic
31C2×#6435C2211yescharacteristic
32C2×#6432C2211yescharacteristic
33C2×C7×C4⋊C8C2211yescharacteristic
34C4×C7×C4⋊C4C2211yescharacteristic
35C2×Dic7⋊C8C2211yescharacteristic
36C2×Dic7⋊C8C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4×(C2×D7)⋊C4—21no—
40C4×C4⋊Dic7C2211yescharacteristic
41C4×Dic7⋊C4—21no—
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89611—24no—
44#89438—21no—
45#89421—21no—
46C7×#340C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21315—71nomaximal, Sylow
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C4×D7C2311yescharacteristic
66C22×Dic14—21no—
67C22×Dic14—21no—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C7×#125—22no—
71C7×#125C2×C414yescomplemented
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4D411yescharacteristic
83C2×(C2×D7)⋊C4D411yescharacteristic
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4D411yescharacteristic
95C2×Dic7⋊C4D411yescharacteristic
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C2×#1131—22no—
105C2×#1131—22no—
106C2×#1131—22no—
107C2×C8×D7—22no—
108C2×C8×D7—22no—
109C2×C8×D7—22no—
110#6460—28no—
111#6459—28no—
112#6455—22no—
113#6455—22no—
114#6455C2×C414yes—
115#6455Q814yescomplemented
116D7×C4⋊C4—22no—
117D7×C4⋊C4—22no—
118D7×C4⋊C4C2×C414yes—
119D7×C4⋊C4Q814yescomplemented
120#6453—22no—
121#6453C2×C414yescomplemented
122#6452—28no—
123#6451—28no—
124#6451—22no—
125#6451C2×C414yescomplemented
126#6435C2×C414yes—
127#6435D414yes—
128#6432—22no—
129#6432C2×C414yescomplemented
130C7×C4⋊C8—22no—
131C7×#95—21no—
132C7⋊#95—21no—
133C7⋊#95—21no—
134C7⋊#95—21no—
135Dic7⋊C8—22no—
136Dic7⋊C8—22no—
137C4⋊(C7⋊C8)—22no—
138C2×#544—71no—
139C2×#424—71no—
140#2274—71no—
141#1989—71no—
142#1989—71no—
143#1989—71no—
144#1989D711yescharacteristic, complemented
145#1988—71no—
146#1988—71no—
147#1987—71no—
148#1987—71no—
149#1736—71no—
150#1736—71no—
151#1735—71no—
152C2×#340—71no—
153C23×D7C2×D411yescharacteristic
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C2×C4×D7—24no—
169C2×C4×D7—24no—
170C2×C4×D7—81no—
171C2×C4×D7—81no—
172C2×C4×D7—81no—
173C2×C4×D7—81no—
174C2×C4×D7C22×C412yes—
175C2×C4×D7C2×D412yes—
176C2×C4×D7C2×Q812yes—
177C2×C4×D7C2×Q812yes—
178C2×C4×D7#4112yes—
179C2×C4×D7#4112yes—
180C2×Dic14—24no—
181C2×Dic14—24no—
182C2×Dic14—81no—
183C2×Dic14—81no—
184C2×Dic14—81no—
185C2×Dic14—81no—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8—21no—
189C2×C7×C8—21no—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4C22×C412yes—
199C7×C4⋊C4C22×C412yes—
200C7×C4⋊C4C2×Q812yes—
201C7×C4⋊C4C2×Q812yes—
202C42×C7—81no—
203C42×C7C22×C412yes—
204C42×C7C2×D412yes—
205(C2×D7)⋊C4—24no—
206(C2×D7)⋊C4—24no—
207(C2×D7)⋊C4C22⋊C414yes—
208(C2×D7)⋊C4C22⋊C414yes—
209C4⋊Dic7—24no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—24no—
215C4⋊Dic7C22×C412yes—
216C4⋊Dic7C22×C412yes—
217C4⋊Dic7C2×Q812yes—
218C4⋊Dic7C2×Q812yes—
219Dic7⋊C4—24no—
220Dic7⋊C4—24no—
221Dic7⋊C4—24no—
222Dic7⋊C4—24no—
223Dic7⋊C4C22⋊C414yes—
224Dic7⋊C4C22⋊C414yes—
225C4×Dic7—24no—
226C4×Dic7—24no—
227C4×Dic7—81no—
228C4×Dic7—81no—
229C4×Dic7—81no—
230C4×Dic7C22×C412yes—
231C4×Dic7C2×D412yes—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238#1131—44no—
239C8×D7—44no—
240C22×M16—71no—
241C23×C8—71no—
242#544—78no—
243#544—144no—
244C2×C4⋊Q8—71no—
245C2×C4⋊Q8—71no—
246C2×C4⋊Q8—141no—
247C2×#125—71no—
248C2×#125—141no—
249C2×#125C2×D711yescharacteristic
250C2×C22⋊Q8—141no—
251C2×C22⋊Q8—141no—
252C2×#117—71no—
253C2×#117—71no—
254C2×#117—71no—
255C22×C4⋊C4—71no—
256C22×C4⋊C4—71no—
257#424—144no—
258C2×C4⋊C8—71no—
259C2×C4⋊C8—71no—
260C2×C4⋊C8—71no—
261C2×C4⋊C8C2×D711yescharacteristic
262C4×C4⋊C4—71no—
263C4×C4⋊C4—71no—
264C4×C4⋊C4—71no—
265C4×C4⋊C4—141no—
266C4×C4⋊C4—141no—
267C4×C4⋊C4C2×D711yescharacteristic
268C4×C22⋊C4—141no—
269C4×C22⋊C4—141no—
270#340—71no—
271#340—71no—
272#340—71no—
273#340—71no—
274#340—71no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#340—71no—
282#340C2×D711yescharacteristic
283#340C2×D711yescharacteristic
284#340C2×D711yescharacteristic
285#340C2×D711yescharacteristic
286C23×C7C22×D411yescharacteristic
287C22×D7—21no—
288C22×D7—21no—
289C22×D7—21no—
290C22×D7—21no—
291C22×D7C2×C22⋊C412yes—
292C22×D7C22⋊Q812yes—
293C22×D7C22⋊Q812yes—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
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×C7—41no—
310C2×C4×C7C2×C22⋊C412yes—
311C2×C4×C7C2×C22⋊C412yes—
312C2×C4×C7C23×C411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×Q811yescharacteristic
315C2×C4×C7C22×Q811yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—22no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7C2×C22⋊C412yes—
335C2×Dic7C2×C22⋊C412yes—
336C2×Dic7C2×C22⋊C412yes—
337C2×Dic7C22⋊Q812yes—
338C2×Dic7C22⋊Q812yes—
339C4×D7—84no—
340C4×D7—84no—
341C4×D7C4×Q814yes—
342C4×D7C4×Q814yes—
343C4×D7#12414yes—
344C4×D7C4⋊Q814yes—
345Dic14—84no—
346Dic14—84no—
347C7×C8—41no—
348C7×C8—41no—
349C7⋊C8—41no—
350C7⋊C8—41no—
351C22×Q8—141no—
352C22×Q8—141no—
353C23×C4—71no—
354C23×C4—141no—
355C23×C4—141no—
356C2×M16—142no—
357C2×M16—142no—
358C2×M16—142no—
359C22×C8—71no—
360C22×C8—71no—
361C22×C8—142no—
362C22×C8—142no—
363C22×C8—142no—
364C22×C8C22×D711yescharacteristic
365C22×C8C22×D711yescharacteristic
366C4⋊Q8—74no—
367C4⋊Q8—74no—
368C4⋊Q8—142no—
369C4⋊Q8—148no—
370C4⋊Q8—142no—
371#125—22no—
372#125—74no—
373#125—142no—
374#125—148no—
375#125C4×D714yescomplemented
376C22⋊Q8—148no—
377C22⋊Q8—148no—
378#117—74no—
379#117—74no—
380#117—74no—
381#117—74no—
382#117—142no—
383#117—142no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—71no—
388C2×C4⋊C4—71no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
392C2×C4⋊C4—74no—
393C2×C4⋊C4—74no—
394C2×C4⋊C4—142no—
395C2×C4⋊C4—142no—
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⋊C4C22×D711yescharacteristic
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C22⋊C4—71no—
407C2×C22⋊C4—71no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C42—21no—
411C2×C42—71no—
412C2×C42—141no—
413C2×C42—141no—
414C2×C42—141no—
415C2×C42—141no—
416C2×C42—141no—
417C2×C42C22×D711yescharacteristic
418C4⋊C8—22no—
419C4⋊C8—142no—
420C4⋊C8—142no—
421C4⋊C8—142no—
422#95—21no—
423#95—141no—
424#95—141no—
425#95—141no—
426C22×C7C22×C22⋊C411yescharacteristic
427C22×C7C2×C22⋊Q811yescharacteristic
428C22×C7C2×C22⋊Q811yescharacteristic
429C22×C7#57611yescharacteristic
430C22×C7#57711yescharacteristic
431C22×C7#57711yescharacteristic
432C22×C7#57811yescharacteristic
433C2×D7—22no—
434C2×D7—22no—
435C2×D7—22no—
436C2×D7—22no—
437C2×D7—22no—
438C2×D7—22no—
439C2×D7#39114yes—
440C4×C7—22no—
441C4×C7—22no—
442C4×C7—42no—
443C4×C7—42no—
444C4×C7—42no—
445C4×C7—42no—
446C4×C7C2×C4×Q811yescharacteristic
447C4×C7C2×C4×Q811yescharacteristic
448C4×C7C2×#12411yescharacteristic
449C4×C7C2×C4⋊Q811yescharacteristic
450Dic7—22no—
451Dic7—22no—
452Dic7—42no—
453Dic7—42no—
454Dic7—42no—
455Dic7—42no—
456Dic7#39114yes—
457C24—71no—
458C2×Q8—144no—
459C2×Q8—144no—
460C2×Q8—561no—
461C2×Q8—561no—
462C2×Q8—561no—
463C2×Q8—561no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—71no—
469C22×C4—72no—
470C22×C4—72no—
471C22×C4—72no—
472C22×C4—72no—
473C22×C4—72no—
474C22×C4—72no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—144no—
478C22×C4—144no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—141no—
483C22×C4—561no—
484C22×C4—561no—
485C22×C4—561no—
486C22×C4—561no—
487C22×C4D4×D711yescharacteristic
488C22×C4D4×D711yescharacteristic
489C22×C4C23×D711yescharacteristic
490M16—284no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—284no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—74no—
511C4⋊C4—74no—
512C4⋊C4—72no—
513C4⋊C4—72no—
514C4⋊C4—72no—
515C4⋊C4—72no—
516C4⋊C4—144no—
517C4⋊C4—144no—
518C4⋊C4—144no—
519C4⋊C4—144no—
520C4⋊C4—144no—
521C4⋊C4—144no—
522C4⋊C4—141no—
523C4⋊C4—141no—
524C4⋊C4—141no—
525C4⋊C4—141no—
526C4⋊C4C2×C4×D712yes—
527C4⋊C4C2×C4×D712yes—
528C4⋊C4Q8×D712yes—
529C4⋊C4Q8×D712yes—
530C22⋊C4—74no—
531C22⋊C4—74no—
532C22⋊C4—144no—
533C22⋊C4—144no—
534C42—72no—
535C42—72no—
536C42—81no—
537C42—144no—
538C42—144no—
539C42—561no—
540C42—561no—
541C42—561no—
542C42C2×C4×D712yes—
543C42D4×D712yes—
544C2×C7C2×#39111yescharacteristic
545C2×C7#289711yescharacteristic
546C2×C7#289811yescharacteristic
547C2×C7#308111yescharacteristic
548C2×C7#308211yescharacteristic
549C2×C7#308311yescharacteristic
550C2×C7#308411yescharacteristic
551D7—24no—
552C23—72no—
553C23—72no—
554C23—72no—
555C23—141no—
556C23—141no—
557C23—141no—
558C23—141no—
559C23C2×D4×D711yescharacteristic
560Q8—564no—
561Q8—564no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—72no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—74no—
582C2×C4—74no—
583C2×C4—74no—
584C2×C4—74no—
585C2×C4—72no—
586C2×C4—72no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—564no—
604C2×C4—564no—
605C2×C4D7×C22⋊C412yes—
606C2×C4D7×C22⋊C412yes—
607C2×C4C22×C4×D711yescharacteristic
608C2×C4C2×D4×D711yescharacteristic
609C2×C4C2×Q8×D711yescharacteristic
610C2×C4C2×Q8×D711yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C2×C4D7×#4111yescharacteristic
613C8—41no—
614C8—41no—
615C8—281no—
616C8—281no—
617C7#2131511yescharacteristic, complemented, Sylow
618C22—74no—
619C22—142no—
620C22—142no—
621C22—142no—
622C22—142no—
623C22—142no—
624C22—142no—
625C22C2×D7×C22⋊C411yescharacteristic
626C22D7×C22⋊Q811yescharacteristic
627C22D7×C22⋊Q811yescharacteristic
628C22#9046111yescharacteristic
629C22#9046211yescharacteristic
630C22#9046511yescharacteristic
631C22#9046911yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—74no—
639C4—142no—
640C4—142no—
641C4—282no—
642C4—282no—
643C4—282no—
644C4—282no—
645C4C4×Q8×D711yescharacteristic
646C4C4×Q8×D711yescharacteristic
647C4D7×#12411yescharacteristic
648C4D7×C4⋊Q811yescharacteristic
649C2—144no—
650C2D7×#39111yescharacteristic
651C2#1173627711yescharacteristic
652C2#1173632711yescharacteristic
653C2#1173861311yescharacteristic
654C2#1173861511yescharacteristic
655C2#1173867711yescharacteristic
656C2#1173868011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1989C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1989D7
hypocenterC7#21315
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. C7×#1989C2semidirect
5. #11729492C2semidirect
6. #11728213C2semidirect
7. #11727244C2semidirect
12. #11726556C2semidirect
13. #11726555C2semidirect
14. #11726554C2semidirect
15. #11726553C2semidirect
20. #90380C4semidirect
71. C7×#125C2×C4semidirect
115. #6455Q8semidirect
119. D7×C4⋊C4Q8semidirect
121. #6453C2×C4semidirect
125. #6451C2×C4semidirect
129. #6432C2×C4semidirect
144. #1989D7semidirect
375. #125C4×D7semidirect
617. C7#21315semidirect

Maximal subgroups

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

subgroupindex
C2×#903802
C2×#896112
#117311952
C7×#19892
#117294922
#117282132
#117272442
#117266212
#117266202
D7×#3402
#117265562
#117265552
#117265542
#117265532
#213157