Finite Group Databasebeta

← #13321505

Subgroups — #13321505

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743054C211yesmaximal, characteristic
2#11736046C211yesmaximal, characteristic
3#11730814C211yesmaximal, characteristic
4C2×C7×#328C211yesmaximal, characteristic, complemented
5#11729444C211yesmaximal, characteristic, complemented
6#11727700C211yesmaximal, characteristic, complemented
7C2×#89247C211yesmaximal, characteristic, complemented
8#11726093C212yesmaximal, complemented
9D7×#328C212yesmaximal, complemented
10#11726068C212yesmaximal, complemented
11#11726063C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2211yescharacteristic
13Q8×C22⋊D7C412yescomplemented
14C2×C4×C22⋊D7—21no—
15C2×C4×C22⋊D7C2211yescharacteristic
16C2×#6487C2211yescharacteristic
17#90327—22no—
18#90326C412yescomplemented
19#90325—21no—
20#90323C412yescomplemented
21D7×C22⋊Q8C2212yes—
22#90314—21no—
23#90313C2212yes—
24#90202—22no—
25#90202C412yescomplemented
26#90177—21no—
27#90174—21no—
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30#89984C2211yescharacteristic
31#89982—21no—
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C8×C22⋊D7—22no—
34(C22⋊D7)⋊C8—22no—
35D7×C22⋊C8C2212yes—
36#89490C2212yes—
37#89440—21no—
38#89439C2211yescharacteristic
39D7×#95C2211yescharacteristic
40#89422C2211yescharacteristic
41#89421C2211yescharacteristic
42#89419—21no—
43C7×#328C2212yes—
44C7×#328C2212yes—
45#89333C2212yes—
46#89333C2212yes—
47#89262C2212yes—
48#89262C2212yes—
49#89247C2212yes—
50#89247C2212yes—
51#17614—71nomaximal, Sylow
52C22×C7×Q8C2×C411yescharacteristic, complemented
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C22×C22⋊D7D411yescharacteristic
56C2×Q8×D7C2×C412yescomplemented
57C22×C4×D7—21no—
58C22×C4×D7C2311yescharacteristic
59C22×Dic14C2×C411yescharacteristic, complemented
60C22×C7×C8—21no—
61C7×C22⋊Q8—21no—
62C7×C22⋊Q8—21no—
63C7×C22⋊Q8C2×C412yescomplemented
64C7×C22⋊Q8C2312yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
67C2×C7×C22⋊C4—21no—
68C2×C22⋊Dic7—21no—
69C2×C22⋊Dic7—21no—
70C2×C22⋊Dic7D411yescharacteristic
71#6509—22no—
72#6509C2×C412yescomplemented
73Q8×Dic7C2×C412yescomplemented
74#6507—22no—
75#6507C2×C412yescomplemented
76C4×C22⋊D7—22no—
77C4×C22⋊D7—22no—
78C4×C22⋊D7—44no—
79C4×C22⋊D7C2×C412yes—
80C4×C22⋊D7C2×C412yes—
81C4×C22⋊D7D412yes—
82C4×C22⋊D7D412yes—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4D411yescharacteristic
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2×C411yescharacteristic, complemented
87#6487—21no—
88#6487—21no—
89#6487—22no—
90#6487—22no—
91#6487C2×C412yescomplemented
92#6487C2312yes—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4D411yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8—21no—
101C2×C8×D7—22no—
102#6460—22no—
103#6460C2×C412yescomplemented
104#6459—22no—
105#6459—22no—
106#6459—22no—
107D7×C4⋊C4—22no—
108D7×C4⋊C4C2×C412yescomplemented
109#6452—22no—
110#6452—22no—
111#6452—22no—
112#6451—22no—
113#6451C2×C412yescomplemented
114#6450—22no—
115#6450C2×C412yescomplemented
116(C22⋊D7)⋊C4—22no—
117(C22⋊D7)⋊C4—22no—
118D7×C22⋊C4—22no—
119#6441—22no—
120#6441—22no—
121#6441—22no—
122#6441—22no—
123#6440—22no—
124C7×C22⋊C8—21no—
125C7×C22⋊C8C2312yes—
126C7×#95—21no—
127C7×#95C2311yescharacteristic
128C7×#95C2311yescharacteristic
129C7⋊#95—21no—
130C7⋊#95—21no—
131C7⋊#95C2311yescharacteristic
132C7⋊#95C2311yescharacteristic
133C22⋊(C7⋊C8)—21no—
134C22⋊(C7⋊C8)C2312yes—
135(C2×D7)⋊C8—22no—
136Dic7⋊C8—22no—
137C8×Dic7—22no—
138#3501—71no—
139#2972—71no—
140#2349—71no—
141#1520—71no—
142#1520—72no—
143#1520—71no—
144#1518—72no—
145#1503—72no—
146C2×#328—71no—
147C2×#328—72no—
148C2×#328D711yescharacteristic, complemented
149C24×C7C2×D411yescharacteristic
150C23×D7C2×D411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8—81no—
153C2×C7×Q8C22×C412yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—81no—
161C22×C4×C7—81no—
162C22×C4×C7C22×C411yescharacteristic
163C22×C4×C7C22×C411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C2×C22⋊D7C22⋊C412yes—
169C2×C22⋊D7C22⋊C412yes—
170C2×C22⋊D7SD1614yes—
171C2×C22⋊D7Q1614yes—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179Q8×D7—82no—
180C2×C4×D7—22no—
181C2×C4×D7—22no—
182C2×C4×D7—42no—
183C2×C4×D7—42no—
184C2×C4×D7—81no—
185C2×C4×D7—81no—
186C2×C4×D7C22×C412yes—
187C2×C4×D7C2×D412yes—
188C2×Dic14—21no—
189C2×Dic14—22no—
190C2×Dic14—22no—
191C2×Dic14—81no—
192C2×Dic14C22×C412yes—
193C2×C7×C8—22no—
194C2×C7×C8—81no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C22×C412yes—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C22⋊Dic7—22no—
202C22⋊Dic7—22no—
203C22⋊Dic7—42no—
204C22⋊Dic7C22⋊C412yes—
205C22⋊Dic7C22⋊C412yes—
206(C2×D7)⋊C4—22no—
207(C2×D7)⋊C4—22no—
208(C2×D7)⋊C4—42no—
209(C2×D7)⋊C4C22⋊C412yes—
210(C2×D7)⋊C4C22⋊C412yes—
211C4⋊Dic7—21no—
212C4⋊Dic7—22no—
213C4⋊Dic7—22no—
214C4⋊Dic7—81no—
215C4⋊Dic7C22×C412yes—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4—22no—
220Dic7⋊C4—22no—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—42no—
225Dic7⋊C4—42no—
226Dic7⋊C4—81no—
227Dic7⋊C4C22⋊C412yes—
228Dic7⋊C4C22⋊C412yes—
229C4×Dic7—22no—
230C4×Dic7—22no—
231C4×Dic7—42no—
232C4×Dic7C22×C412yes—
233C4×Dic7C2×D412yes—
234C2×C7⋊C8—22no—
235C2×C7⋊C8—81no—
236C8×D7—82no—
237#554—72no—
238#554—72no—
239#554—142no—
240#554—142no—
241D4×Q8—72no—
242D4×Q8—72no—
243#544—141no—
244#543—141no—
245#543—141no—
246#533—72no—
247C2×C22⋊Q8—71no—
248C2×C22⋊Q8—72no—
249C2×C22⋊Q8C2×D711yescharacteristic
250#518—141no—
251C2×C4×D4—71no—
252C2×C4×D4—141no—
253C8×D4—142no—
254C8×D4—142no—
255#431—72no—
256C2×C22⋊C8—71no—
257C2×C22⋊C8—72no—
258C2×C22⋊C8C2×D711yescharacteristic
259#389—141no—
260#385—141no—
261#385—141no—
262#382—71no—
263#380—71no—
264#380—71no—
265C4×C4⋊C4—71no—
266C2×#95—71no—
267C2×#95C2×D711yescharacteristic
268#328—72no—
269#328—72no—
270#328—72no—
271#328—72no—
272#328—72no—
273#328—72no—
274#328C2×D712yes—
275#328C2×D712yes—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7C2×C22⋊C411yescharacteristic
279C23×C7C2×C22⋊C411yescharacteristic
280C23×C7C2×SD1612yes—
281C23×C7C2×Q1612yes—
282C23×C7C22×D411yescharacteristic
283C22×D7—41no—
284C22×D7—41no—
285C22×D7C2×C22⋊C412yes—
286C22×D7C2×SD1612yes—
287C22×D7C2×Q1612yes—
288C7×Q8—42no—
289C7×Q8—81no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—81no—
305C2×C4×C7—81no—
306C2×C4×C7—84no—
307C2×C4×C7C2×C22⋊C411yescharacteristic
308C2×C4×C7C2×C22⋊C411yescharacteristic
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C22⋊D7#10318yes—
312C22⋊D7#10318yes—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—81no—
324C2×Dic7—81no—
325C2×Dic7C2×C22⋊C412yes—
326C2×Dic7C2×C22⋊C412yes—
327C2×Dic7C2×C22⋊C412yes—
328C2×Dic7C2×SD1612yes—
329C2×Dic7C2×Q1612yes—
330C4×D7—42no—
331C4×D7—42no—
332C4×D7—82no—
333C4×D7—82no—
334C4×D7—84no—
335Dic14—42no—
336Dic14—82no—
337Dic14—82no—
338Dic14—81no—
339C7×C8—42no—
340C7⋊C8—42no—
341C22×Q8—72no—
342C22×Q8—71no—
343C22×Q8C4×D711yescharacteristic, complemented
344C22×D4—71no—
345C23×C4—21no—
346C23×C4—71no—
347C23×C4—141no—
348C23×C4C22×D711yescharacteristic
349C22×C8—21no—
350C22×C8—142no—
351C22×C8—141no—
352C4⋊Q8—72no—
353C4⋊Q8—72no—
354C4⋊Q8—142no—
355C4⋊Q8—142no—
356#125—142no—
357#125—142no—
358#125—142no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—72no—
362C22⋊Q8—72no—
363C22⋊Q8—72no—
364C22⋊Q8—72no—
365C22⋊Q8—142no—
366C22⋊Q8—142no—
367C22⋊Q8—142no—
368C22⋊Q8—142no—
369C22⋊Q8—142no—
370C22⋊Q8—142no—
371C22⋊Q8—142no—
372C22⋊Q8—142no—
373C22⋊Q8—142no—
374C22⋊Q8—142no—
375C22⋊Q8—142no—
376C22⋊Q8—141no—
377C22⋊Q8—141no—
378C22⋊Q8C4×D712yescomplemented
379C22⋊Q8C22×D712yes—
380C4×Q8—72no—
381C4×Q8—72no—
382C4×Q8—142no—
383C4×D4—72no—
384C4×D4—72no—
385C4×D4—72no—
386C4×D4—72no—
387C4×D4—142no—
388C4×D4—142no—
389C4×D4—142no—
390C4×D4—142no—
391C4×D4—284no—
392#117—142no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—72no—
397C2×C4⋊C4—142no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4C4×D711yescharacteristic, complemented
404C2×C22⋊C4—21no—
405C2×C22⋊C4—71no—
406C2×C22⋊C4—71no—
407C2×C22⋊C4—142no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C42—71no—
412C2×C42—141no—
413C4⋊C8—142no—
414C22⋊C8—21no—
415C22⋊C8—72no—
416C22⋊C8—142no—
417C22⋊C8—141no—
418C22⋊C8C22×D712yes—
419C4×C8—142no—
420#95—21no—
421#95—71no—
422#95—71no—
423#95—141no—
424#95—141no—
425#95C22×D711yescharacteristic
426#95C22×D711yescharacteristic
427C22×C7—21no—
428C22×C7—21no—
429C22×C7—21no—
430C22×C7—21no—
431C22×C7—21no—
432C22×C7—21no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7C2×#10312yes—
436C22×C7C2×#10312yes—
437C22×C7C22×C22⋊C411yescharacteristic
438C22×C7C22×SD1611yescharacteristic
439C22×C7C22×Q1611yescharacteristic
440C2×D7—22no—
441C2×D7—22no—
442C2×D7—42no—
443C2×D7—42no—
444C2×D7C2×#10314yes—
445C4×C7—21no—
446C4×C7—21no—
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—42no—
450C4×C7—42no—
451C4×C7—42no—
452C4×C7—81no—
453Dic7—22no—
454Dic7—22no—
455Dic7—42no—
456Dic7—42no—
457Dic7—81no—
458Dic7—81no—
459Dic7C2×#10314yes—
460C24—71no—
461C24D4×D711yescharacteristic
462C2×Q8—21no—
463C2×Q8—72no—
464C2×Q8—81no—
465C2×Q8—142no—
466C2×Q8—142no—
467C2×Q8—141no—
468C2×Q8—561no—
469C2×Q8—562no—
470C2×Q8C2×C4×D712yes—
471C2×D4—72no—
472C2×D4—72no—
473C2×D4—74no—
474C2×D4—74no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—72no—
485C22×C4—72no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—142no—
489C22×C4—142no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—282no—
495C22×C4—282no—
496C22×C4—561no—
497C22×C4—561no—
498C22×C4C2×C4×D711yescharacteristic
499C22×C4C2×C4×D711yescharacteristic
500C22×C4D4×D711yescharacteristic
501C22×C4D4×D711yescharacteristic
502C22×C4D4×D711yescharacteristic
503C22×C4C23×D711yescharacteristic
504C2×C8—22no—
505C2×C8—81no—
506C2×C8—142no—
507C2×C8—562no—
508C2×C8—561no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—72no—
513C4⋊C4—72no—
514C4⋊C4—72no—
515C4⋊C4—142no—
516C4⋊C4—142no—
517C4⋊C4—142no—
518C4⋊C4—142no—
519C4⋊C4—142no—
520C4⋊C4—142no—
521C4⋊C4—142no—
522C4⋊C4—142no—
523C4⋊C4—142no—
524C4⋊C4—141no—
525C4⋊C4—142no—
526C4⋊C4—282no—
527C4⋊C4—282no—
528C4⋊C4—561no—
529C4⋊C4—561no—
530C4⋊C4C2×C4×D712yes—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—72no—
534C22⋊C4—72no—
535C22⋊C4—72no—
536C22⋊C4—72no—
537C22⋊C4—142no—
538C22⋊C4—142no—
539C22⋊C4—142no—
540C22⋊C4—142no—
541C22⋊C4—282no—
542C22⋊C4—282no—
543C42—72no—
544C42—72no—
545C42—142no—
546C42—142no—
547C42—282no—
548C2×C7—21no—
549C2×C7—21no—
550C2×C7—22no—
551C2×C7—22no—
552C2×C7#288311yescharacteristic
553C2×C7C22×#10311yescharacteristic
554C2×C7#290111yescharacteristic
555D7—24no—
556C23—41no—
557C23—41no—
558C23—72no—
559C23—72no—
560C23—72no—
561C23—281no—
562C23—281no—
563C23D7×C22⋊C411yescharacteristic
564C23D7×C22⋊C411yescharacteristic
565C23D7×SD1612yes—
566C23D7×Q1612yes—
567C23C2×D4×D711yescharacteristic
568Q8—42no—
569Q8—81no—
570Q8—282no—
571Q8—561no—
572Q8—562no—
573Q8—562no—
574D4—78no—
575D4—78no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—72no—
591C2×C4—72no—
592C2×C4—72no—
593C2×C4—72no—
594C2×C4—72no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—84no—
598C2×C4—142no—
599C2×C4—142no—
600C2×C4—282no—
601C2×C4—282no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—562no—
611C2×C4—562no—
612C2×C4—564no—
613C2×C4—561no—
614C2×C4—561no—
615C2×C4D7×C22⋊C411yescharacteristic
616C2×C4D7×C22⋊C411yescharacteristic
617C2×C4C22×C4×D711yescharacteristic
618C2×C4C2×D4×D711yescharacteristic
619C8—42no—
620C8—282no—
621C7#1761411yescharacteristic, complemented, Sylow
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—74no—
631C22—142no—
632C22—142no—
633C22—282no—
634C22—282no—
635C22D7×#10312yes—
636C22D7×#10312yes—
637C22C2×D7×C22⋊C411yescharacteristic
638C22C2×D7×SD1611yescharacteristic
639C22C2×D7×Q1611yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—74no—
648C4—81no—
649C4—142no—
650C4—142no—
651C4—282no—
652C4—282no—
653C4—561no—
654C4—561no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—144no—
660C2#1173613311yescharacteristic
661C2C2×D7×#10311yescharacteristic
662C2#1173642811yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#328C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C22⋊D7D4
cosocleC22×C4×C7C24
hypercenterC2×#328D7
hypocenterC7#17614
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
4. C2×C7×#328C2semidirect
5. #11729444C2semidirect
6. #11727700C2semidirect
7. C2×#89247C2semidirect
8. #11726093C2semidirect
9. D7×#328C2semidirect
10. #11726068C2semidirect
11. #11726063C2semidirect
13. Q8×C22⋊D7C4semidirect
18. #90326C4semidirect
20. #90323C4semidirect
25. #90202C4semidirect
52. C22×C7×Q8C2×C4semidirect
56. C2×Q8×D7C2×C4semidirect
59. C22×Dic14C2×C4semidirect
63. C7×C22⋊Q8C2×C4semidirect
66. C2×C7×C4⋊C4C2×C4semidirect
72. #6509C2×C4semidirect
73. Q8×Dic7C2×C4semidirect
75. #6507C2×C4semidirect
86. C2×C4⋊Dic7C2×C4semidirect
91. #6487C2×C4semidirect
103. #6460C2×C4semidirect
108. D7×C4⋊C4C2×C4semidirect
113. #6451C2×C4semidirect
115. #6450C2×C4semidirect
148. C2×#328D7semidirect
343. C22×Q8C4×D7semidirect
378. C22⋊Q8C4×D7semidirect
403. C2×C4⋊C4C4×D7semidirect
621. C7#17614semidirect

Maximal subgroups

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

subgroupindex
#117430542
#117360462
#117308142
C2×C7×#3282
#117294442
#117277002
C2×#892472
#117260932
D7×#3282
#117260682
#117260632
#176147