Finite Group Databasebeta

← #13319952

Subgroups — #13319952

Order 1792.

948 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90353C211yesmaximal, characteristic
2#11736077C211yesmaximal, characteristic, complemented
3#11736047C211yesmaximal, characteristic, complemented
4C7×#1468C211yesmaximal, characteristic, complemented
5#11729456C211yesmaximal, characteristic, complemented
6#11727726C211yesmaximal, characteristic, complemented
7#11727677C211yesmaximal, characteristic, complemented
8#11725901C212yesmaximal, complemented
9#11725881C212yesmaximal, complemented
10#11725876C212yesmaximal, complemented
11#11725856C212yesmaximal, complemented
12C2×C7×#123C2211yescharacteristic
13C2×#6502C411yescharacteristic
14C2×#6499C411yescharacteristic
15C2×C4×C22⋊D7C2211yescharacteristic
16C2×#6487C2211yescharacteristic
17#90353—22no—
18#90353—22no—
19#90353—22no—
20#90353—22no—
21C2×#6461C411yescharacteristic
22C2×#6459C2211yescharacteristic
23C2×#6451C411yescharacteristic
24C2×#6450C2211yescharacteristic
25C2×#6448C411yescharacteristic
26C2×#6448C411yescharacteristic
27C2×#6446C2211yescharacteristic
28C2×#6442C411yescharacteristic
29C2×#6441C411yescharacteristic
30C2×#6440C2211yescharacteristic
31C7×#407C2211yescharacteristic
32C2×C7×C22⋊C8C2211yescharacteristic
33#89979C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35#89899—22no—
36#89883—22no—
37#89502—22no—
38#89500—22no—
39#89498C2212yes—
40D7×C22⋊C8C2212yes—
41#89491C2212yes—
42#89491C2212yes—
43C7×#324C2212yes—
44C7×#323C2212yes—
45#89323C2212yes—
46#89320C2212yes—
47#89264C2212yes—
48#89264C2212yes—
49#89264C2212yes—
50#89264C2212yes—
51#17407—71nomaximal, Sylow
52C22×C7×D4C2×C411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C22×C22⋊D7C2×C411yescharacteristic
55C22×C22⋊D7C2×C411yescharacteristic
56C23×Dic7C2×C411yescharacteristic
57C22×C4×D7C2311yescharacteristic
58C22×Dic14C2×C411yescharacteristic
59C22×Dic14C2×C411yescharacteristic
60C22×Dic14C2×C411yescharacteristic
61C2×C7×M16—21no—
62C22×C7×C8—21no—
63C7×#123—21no—
64C7×#123—21no—
65C7×#123—21no—
66C7×#123—21no—
67C2×C7×C4⋊C4C2×C411yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4C2×C411yescharacteristic
70C2×C7×C22⋊C4C2×C411yescharacteristic
71C2×C7×C22⋊C4C2311yescharacteristic
72C2×C22⋊Dic7C2×C411yescharacteristic
73C2×C22⋊Dic7C2×C411yescharacteristic
74C2×C22⋊Dic7C2×C411yescharacteristic
75C2×C22⋊Dic7C2×C411yescharacteristic
76C2×C22⋊Dic7C2×C411yescharacteristic
77#6502—22no—
78#6502—22no—
79#6499—22no—
80#6499—22no—
81C4×C22⋊D7D412yes—
82C4×C22⋊D7D412yes—
83C4×C22⋊D7D412yes—
84C4×C22⋊D7D412yes—
85C2×(C2×D7)⋊C4C2×C411yescharacteristic
86C2×(C2×D7)⋊C4C2×C411yescharacteristic
87C2×(C2×D7)⋊C4C2×C411yescharacteristic
88C2×(C2×D7)⋊C4C2×C411yescharacteristic
89C2×C4⋊Dic7C2×C411yescharacteristic
90C2×C4⋊Dic7C2×C411yescharacteristic
91#6487—21no—
92#6487—21no—
93#6487—21no—
94#6487—21no—
95C2×Dic7⋊C4C2×C411yescharacteristic
96C2×Dic7⋊C4C2×C411yescharacteristic
97C2×Dic7⋊C4C2×C411yescharacteristic
98C2×Dic7⋊C4C2×C411yescharacteristic
99C2×Dic7⋊C4C2311yescharacteristic
100C2×Dic7⋊C4C2311yescharacteristic
101C2×C4×Dic7C2×C411yescharacteristic
102C2×C4×Dic7C2×C411yescharacteristic
103C2×C4×Dic7C2×C411yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C2×#1136—21no—
106C22×C7⋊C8—21no—
107C2×#1131—22no—
108C2×C8×D7—22no—
109#6461—22no—
110#6461—22no—
111#6459—22no—
112#6459—22no—
113#6451—22no—
114#6451—22no—
115#6450—22no—
116#6450—22no—
117#6448—22no—
118#6448—22no—
119#6448—22no—
120#6448—22no—
121#6446—22no—
122#6446—22no—
123#6442—22no—
124#6442—22no—
125#6441—22no—
126#6441—22no—
127#6440—22no—
128#6440—22no—
129C7×C22⋊C8—21no—
130C7×C22⋊C8—21no—
131C7×C22⋊C8C2312yes—
132C7×C22⋊C8C2312yes—
133C22⋊(C7⋊C8)—21no—
134C22⋊(C7⋊C8)—21no—
135C22⋊(C7⋊C8)C2312yes—
136C22⋊(C7⋊C8)C2312yes—
137(C2×D7)⋊C8—22no—
138(C2×D7)⋊C8—22no—
139#6389—22no—
140#6389—22no—
141Dic7⋊C8—22no—
142Dic7⋊C8—22no—
143C2×#542—71no—
144#2973—71no—
145#2973—71no—
146#1472—71no—
147#1472—71no—
148#1472—72no—
149#1468—72no—
150#1468—71no—
151#1468D711yescharacteristic, complemented
152#1467—72no—
153#1464—72no—
154C24×C7C22×C411yescharacteristic
155C24×C7C22×C411yescharacteristic
156C23×D7C22×C411yescharacteristic
157C2×C7×D4—21no—
158C2×C7×D4—21no—
159C2×C7×D4—81no—
160C22×C4×C7—81no—
161C22×C4×C7C22×C411yescharacteristic
162C22×C4×C7C22×C411yescharacteristic
163C22×C4×C7C22×C411yescharacteristic
164C22×C4×C7C22×C411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—22no—
171C2×C22⋊D7—42no—
172C2×C22⋊D7—81no—
173C2×C22⋊D7C22⋊C412yes—
174C2×C22⋊D7C22⋊C412yes—
175C22×Dic7—22no—
176C22×Dic7—42no—
177C22×Dic7C22×C411yescharacteristic
178C22×Dic7C22×C411yescharacteristic
179C22×Dic7C22×C411yescharacteristic
180C22×Dic7C22×C411yescharacteristic
181C22×Dic7C22×C411yescharacteristic
182C22×Dic7C22×C411yescharacteristic
183C22×Dic7C22×C411yescharacteristic
184C2×C4×D7—81no—
185C2×C4×D7C2×D412yes—
186C2×C4×D7C2×D412yes—
187C2×Dic14—22no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14—22no—
191C2×Dic14—42no—
192C2×Dic14—81no—
193C2×Dic14—81no—
194C7×M16—81no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—81no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—21no—
208C22⋊Dic7—21no—
209C22⋊Dic7—21no—
210C22⋊Dic7—21no—
211C22⋊Dic7—21no—
212C22⋊Dic7—22no—
213C22⋊Dic7—22no—
214C22⋊Dic7C22⋊C412yes—
215C22⋊Dic7C22⋊C412yes—
216(C2×D7)⋊C4—22no—
217(C2×D7)⋊C4—22no—
218(C2×D7)⋊C4—22no—
219(C2×D7)⋊C4C22⋊C412yes—
220(C2×D7)⋊C4C22⋊C412yes—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7—22no—
224Dic7⋊C4—21no—
225Dic7⋊C4—21no—
226Dic7⋊C4—21no—
227Dic7⋊C4—21no—
228Dic7⋊C4—22no—
229Dic7⋊C4—22no—
230Dic7⋊C4—22no—
231Dic7⋊C4C22⋊C412yes—
232Dic7⋊C4C22⋊C412yes—
233C4×Dic7—22no—
234C4×Dic7—22no—
235C4×Dic7—22no—
236C4×Dic7C2×D412yes—
237C4×Dic7C2×D412yes—
238#1136—81no—
239C2×C7⋊C8—22no—
240C2×C7⋊C8—22no—
241C2×C7⋊C8—81no—
242#1131—82no—
243C8×D7—82no—
244#542—142no—
245#542—142no—
246#542—142no—
247#542—142no—
248C2×C4⋊Q8—71no—
249C2×#126—71no—
250C2×#126—71no—
251C2×#124—71no—
252C2×#124—71no—
253C2×#124—71no—
254C2×#123—71no—
255C2×#123C2×D711yescharacteristic
256C2×C22⋊Q8—71no—
257C2×C22⋊Q8—71no—
258C2×C22⋊Q8—71no—
259C2×#121—71no—
260C2×C4×Q8—71no—
261C2×C4×D4—71no—
262C2×#117—71no—
263#435—142no—
264#435—142no—
265#435—142no—
266#435—142no—
267#432—72no—
268#432—72no—
269#407—71no—
270#407—72no—
271#407C2×D711yescharacteristic
272C2×C22⋊C8—71no—
273C2×C22⋊C8—72no—
274C2×C22⋊C8C2×D711yescharacteristic
275#324—72no—
276#324—72no—
277#324—72no—
278#324—72no—
279#324—72no—
280#324C2×D712yes—
281#323—72no—
282#323C2×D712yes—
283C23×C7—22no—
284C23×C7—21no—
285C23×C7—22no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7C2×C22⋊C411yescharacteristic
291C23×C7C2×C22⋊C411yescharacteristic
292C23×C7C23×C411yescharacteristic
293C22×D7—22no—
294C22×D7—41no—
295C22×D7—41no—
296C22×D7C2×C22⋊C412yes—
297C7×D4—81no—
298C7×D4—81no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
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×C7—41no—
311C2×C4×C7C2×C22⋊C411yescharacteristic
312C2×C4×C7C2×C22⋊C411yescharacteristic
313C2×C4×C7C22×D411yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C22⋊D7—42no—
316C22⋊D7—42no—
317C22⋊D7—82no—
318C2×Dic7—22no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—44no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7C2×C22⋊C412yes—
331C2×Dic7C2×C22⋊C412yes—
332C2×Dic7C2×C22⋊C412yes—
333C4×D7—42no—
334C4×D7—42no—
335Dic14—44no—
336Dic14—82no—
337Dic14—81no—
338Dic14—81no—
339C7×C8—42no—
340C7×C8—42no—
341C7⋊C8—42no—
342C7⋊C8—42no—
343C22×Q8—71no—
344C22×Q8—71no—
345C22×Q8—71no—
346C22×D4—71no—
347C22×D4—71no—
348C22×D4C4×D711yescharacteristic
349C23×C4—71no—
350C23×C4—71no—
351C23×C4C22×D711yescharacteristic
352C2×M16—21no—
353C2×M16—142no—
354C2×M16—141no—
355C22×C8—21no—
356C22×C8—142no—
357C22×C8—141no—
358C4⋊Q8—142no—
359C4⋊Q8—142no—
360#126—142no—
361#126—142no—
362#126—142no—
363#126—142no—
364#124—142no—
365#124—142no—
366#124—142no—
367#124—142no—
368#124—142no—
369#124—142no—
370#123—21no—
371#123—21no—
372#123—21no—
373#123—21no—
374#123—142no—
375#123—142no—
376C22⋊Q8—142no—
377C22⋊Q8—142no—
378C22⋊Q8—142no—
379C22⋊Q8—142no—
380C22⋊Q8—141no—
381C22⋊Q8—141no—
382C22⋊Q8—141no—
383C22⋊Q8—141no—
384#121—142no—
385#121—142no—
386C4×Q8—142no—
387C4×Q8—142no—
388C4×D4—72no—
389C4×D4—72no—
390C4×D4—72no—
391C4×D4—72no—
392#117—142no—
393#117—142no—
394C2×C4⋊C4—71no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—71no—
398C2×C4⋊C4—71no—
399C2×C4⋊C4—71no—
400C2×C4⋊C4—71no—
401C2×C4⋊C4—71no—
402C2×C4⋊C4C4×D711yescharacteristic
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C22⋊C4—71no—
405C2×C22⋊C4—71no—
406C2×C22⋊C4—71no—
407C2×C22⋊C4—71no—
408C2×C22⋊C4—71no—
409C2×C22⋊C4—71no—
410C2×C22⋊C4—71no—
411C2×C22⋊C4—71no—
412C2×C22⋊C4—71no—
413C2×C22⋊C4C4×D711yescharacteristic
414C2×C22⋊C4C4×D711yescharacteristic
415C2×C22⋊C4C22×D711yescharacteristic
416C2×C42—71no—
417C2×C42—71no—
418C2×C42—71no—
419C2×C42—71no—
420C4⋊C8—142no—
421C4⋊C8—142no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—72no—
425C22⋊C8—72no—
426C22⋊C8—142no—
427C22⋊C8—142no—
428C22⋊C8—141no—
429C22⋊C8—141no—
430C22⋊C8C22×D712yes—
431C22⋊C8C22×D712yes—
432#97—142no—
433#97—142no—
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—21no—
440C22×C7—21no—
441C22×C7—21no—
442C22×C7—41no—
443C22×C7—41no—
444C22×C7—41no—
445C22×C7—82no—
446C22×C7—82no—
447C22×C7C22×C22⋊C411yescharacteristic
448C22×C7#56812yes—
449C2×D7—22no—
450C2×D7—22no—
451C2×D7—22no—
452C2×D7—82no—
453C4×C7—21no—
454C4×C7—21no—
455C4×C7—21no—
456C4×C7—21no—
457C4×C7—41no—
458C4×C7—41no—
459C4×C7—41no—
460Dic7—22no—
461Dic7—22no—
462Dic7—22no—
463Dic7—41no—
464Dic7—41no—
465Dic7—41no—
466Dic7—41no—
467C24—71no—
468C24C2×C4×D711yescharacteristic
469C24C2×C4×D711yescharacteristic
470C2×Q8—142no—
471C2×Q8—142no—
472C2×Q8—141no—
473C2×Q8—141no—
474C2×Q8—282no—
475C2×Q8—561no—
476C2×Q8—561no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—72no—
480C2×D4—72no—
481C2×D4—81no—
482C2×D4—142no—
483C2×D4—282no—
484C2×D4—561no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—71no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—71no—
491C22×C4—71no—
492C22×C4—71no—
493C22×C4—71no—
494C22×C4—81no—
495C22×C4—142no—
496C22×C4—282no—
497C22×C4—561no—
498C22×C4C2×C4×D711yescharacteristic
499C22×C4C2×C4×D711yescharacteristic
500C22×C4C2×C4×D711yescharacteristic
501C22×C4C2×C4×D711yescharacteristic
502C22×C4D4×D711yescharacteristic
503C22×C4D4×D711yescharacteristic
504C22×C4D4×D711yescharacteristic
505C22×C4D4×D711yescharacteristic
506C22×C4C23×D711yescharacteristic
507M16—81no—
508M16—562no—
509M16—561no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—81no—
513C2×C8—142no—
514C2×C8—142no—
515C2×C8—562no—
516C2×C8—561no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—72no—
522C4⋊C4—72no—
523C4⋊C4—142no—
524C4⋊C4—142no—
525C4⋊C4—142no—
526C4⋊C4—142no—
527C4⋊C4—141no—
528C4⋊C4—141no—
529C4⋊C4—141no—
530C4⋊C4—141no—
531C4⋊C4—141no—
532C4⋊C4—141no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—72no—
540C22⋊C4—72no—
541C22⋊C4—72no—
542C22⋊C4—72no—
543C22⋊C4—142no—
544C22⋊C4—142no—
545C22⋊C4—142no—
546C22⋊C4—142no—
547C22⋊C4—142no—
548C22⋊C4—141no—
549C22⋊C4—141no—
550C22⋊C4—141no—
551C22⋊C4—141no—
552C42—72no—
553C42—72no—
554C42—142no—
555C42—142no—
556C42—142no—
557C2×C7—22no—
558C2×C7—41no—
559C2×C7—41no—
560C2×C7—81no—
561C2×C7#287911yescharacteristic
562C2×C7#288311yescharacteristic
563C2×C7#288911yescharacteristic
564D7—42no—
565C23—22no—
566C23—21no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—72no—
573C23—142no—
574C23—281no—
575C23—281no—
576C23D7×C22⋊C411yescharacteristic
577C23D7×C22⋊C411yescharacteristic
578C23C22×C4×D711yescharacteristic
579Q8—284no—
580Q8—561no—
581Q8—561no—
582Q8—562no—
583D4—81no—
584D4—81no—
585D4—282no—
586D4—282no—
587D4—562no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—72no—
601C2×C4—72no—
602C2×C4—72no—
603C2×C4—142no—
604C2×C4—141no—
605C2×C4—141no—
606C2×C4—141no—
607C2×C4—141no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—282no—
611C2×C4—282no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—284no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4D7×C22⋊C411yescharacteristic
618C2×C4D7×C22⋊C411yescharacteristic
619C2×C4C2×D4×D711yescharacteristic
620C2×C4C2×D4×D711yescharacteristic
621C8—42no—
622C8—42no—
623C8—282no—
624C8—282no—
625C7#1740711yescharacteristic, complemented, Sylow
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—82no—
638C22—82no—
639C22—142no—
640C22—142no—
641C22—142no—
642C22—562no—
643C22C2×D7×C22⋊C411yescharacteristic
644C22#9044112yes—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—142no—
653C4—142no—
654C4—142no—
655C4—281no—
656C4—281no—
657C4—281no—
658C4—281no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2—81no—
663C2—282no—
664C2#1173610711yescharacteristic
665C2#1173613311yescharacteristic
666C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1468C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#6502C4
cosocleC22×C4×C7C24
hypercenter#1468D7
hypocenterC7#17407
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1468

Complements

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

NH ≅ G/Ntype
2. #11736077C2semidirect
3. #11736047C2semidirect
4. C7×#1468C2semidirect
5. #11729456C2semidirect
6. #11727726C2semidirect
7. #11727677C2semidirect
8. #11725901C2semidirect
9. #11725881C2semidirect
10. #11725876C2semidirect
11. #11725856C2semidirect
151. #1468D7semidirect
625. C7#17407semidirect

Maximal subgroups

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

subgroupindex
C2×#903532
#117360772
#117360472
C7×#14682
#117294562
#117277262
#117276772
#117259012
#117258812
#117258762
#117258562
#174077