Finite Group Databasebeta

← #13405714

Subgroups — #13405714

Order 1792.

754 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739833C211yesmaximal, characteristic
2C2×#89781C211yesmaximal, characteristic
3C2×#89451C211yesmaximal, characteristic, complemented
4#11731172C211yesmaximal, characteristic, complemented
5C4×C7×#102C211yesmaximal, characteristic
6#11727948C211yesmaximal, characteristic
7#11727947C211yesmaximal, characteristic
8#11727941C211yesmaximal, characteristic
9#11727941C211yesmaximal, characteristic
10#11727532C211yesmaximal, characteristic, complemented
11#11727222C211yesmaximal, characteristic, complemented
12#11726615C211yesmaximal, characteristic, complemented
13#11726614C211yesmaximal, characteristic, complemented
14#11726563C211yesmaximal, characteristic, complemented
15#11726561C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×#6433C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C2×C4×C7×C8C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23#89993C2211yescharacteristic
24#89993C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27#89982—21no—
28#89982—21no—
29C2×#6389C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33#89781—24no—
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89745C2211yescharacteristic
39#89745C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89721—21no—
45#89719—21no—
46#89708C2211yescharacteristic
47#89700—21no—
48#89700—21no—
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50#89451—24no—
51#89433—21no—
52#89428—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25475—71nomaximal, Sylow
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×Dic14D411yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7D411yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4D411yescharacteristic
93C2×Dic7⋊C4D411yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98#6433—22no—
99#6433—22no—
100C4×Dic14—22no—
101C4×Dic14—22no—
102C7×#102—22no—
103C7×#102Q814yescomplemented
104C4×C7×C8—22no—
105C7×#95—21no—
106#6406—22no—
107#6406—22no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6389—22no—
117#6389—22no—
118Dic7⋊C8—22no—
119Dic7⋊C8—22no—
120#6382—22no—
121#6382—22no—
122#6381—22no—
123#6381—22no—
124C4⋊(C7⋊C8)D414yes—
125C2×#478—71no—
126C2×#446—71no—
127#2584—71no—
128#2580—71no—
129#2088—71no—
130#2086—71no—
131#2067—71no—
132#2065—71no—
133#1991—71no—
134#1949—71no—
135#1944—71no—
136#1943—71no—
137#1936—71no—
138#1936—71no—
139C4×#102D711yescharacteristic
140C24×C7—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4C2×Q812yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—42no—
151C22×C4×C7—81no—
152C22×C4×C7—81no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C2×Dic14—22no—
165C2×Dic14—41no—
166C2×Dic14—41no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8C2×Q812yes—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4C2×Q812yes—
178C7×C4⋊C4#4112yes—
179C7×C22⋊C4—42no—
180C42×C7—81no—
181C42×C7C2×D412yes—
182C42×C7#4112yes—
183C22⋊Dic7—42no—
184C22⋊Dic7—42no—
185C22⋊Dic7—81no—
186C22⋊Dic7—81no—
187C4⋊Dic7—22no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—42no—
197C4⋊Dic7—42no—
198C4⋊Dic7—81no—
199C4⋊Dic7—81no—
200Dic7⋊C4—22no—
201Dic7⋊C4—22no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—81no—
205Dic7⋊C4—81no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C2×D412yes—
215C2×C7⋊C8C2×D412yes—
216C2×#125—71no—
217C2×C4×Q8—71no—
218C2×C4×D4C2×D711yescharacteristic
219#478—144no—
220#446—144no—
221C2×#107—71no—
222C2×#107—71no—
223C2×#106—71no—
224C2×#106—71no—
225C2×C4⋊C8—71no—
226C2×C4⋊C8—71no—
227C2×C4⋊C8—71no—
228C2×#102—71no—
229C2×#102—71no—
230C2×#102C2×D711yescharacteristic
231C2×#102C2×D711yescharacteristic
232C2×#97—71no—
233C2×#97—71no—
234C2×C4×C8C2×D711yescharacteristic
235#399—141no—
236#398—141no—
237#395—141no—
238#393—141no—
239#390—71no—
240#390—71no—
241#389—71no—
242#389—71no—
243#389—71no—
244#389—71no—
245#389—71no—
246#389—71no—
247#389—141no—
248#389—141no—
249#385—141no—
250#385—141no—
251#382—71no—
252C4×C4⋊C4C2×D711yescharacteristic
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261C23×C7—22no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7C22×D411yescharacteristic
269C7×D4—22no—
270C7×D4—22no—
271C7×D4—41no—
272C7×D4—41no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—84no—
287C2×C4×C7#12112yes—
288C2×C4×C7#12112yes—
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×Q811yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C72−1+411yescharacteristic
294C2×C4×C72−1+411yescharacteristic
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319Dic14—44no—
320C7×C8—22no—
321C7×C8—41no—
322C7⋊C8—22no—
323C7⋊C8—22no—
324C22×Q8—71no—
325C22×D4C22×D711yescharacteristic
326C23×C4—21no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8C22×D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331#125—142no—
332#125—142no—
333C4×Q8—142no—
334C4×Q8—142no—
335C4×D4—22no—
336C4×D4—22no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C42—21no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42C22×D711yescharacteristic
364#107—142no—
365#107—142no—
366#106—142no—
367#106—142no—
368C4⋊C8—74no—
369C4⋊C8—142no—
370C4⋊C8—142no—
371#102—22no—
372#102—142no—
373#102—142no—
374#102Dic1414yescomplemented
375#97—142no—
376#97—142no—
377C4×C8—22no—
378#95—21no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—81no—
391C22×C7—81no—
392C22×C7—81no—
393C22×C7—81no—
394C22×C7C2×#12111yescharacteristic
395C22×C7#54811yescharacteristic
396C22×C7D4×Q811yescharacteristic
397C22×C7#57511yescharacteristic
398C22×C7#57511yescharacteristic
399C22×C7#57611yescharacteristic
400C22×C7#57711yescharacteristic
401C4×C7—22no—
402C4×C7—22no—
403C4×C7—42no—
404C4×C7—42no—
405C4×C7—81no—
406C4×C7#53311yescharacteristic
407C4×C7#54111yescharacteristic
408C4×C7#54711yescharacteristic
409C4×C7D4×Q811yescharacteristic
410Dic7—41no—
411Dic7—41no—
412Dic7—41no—
413Dic7—41no—
414Dic7—81no—
415Dic7—81no—
416Dic7—81no—
417Dic7—81no—
418C24—21no—
419C2×Q8—142no—
420C2×Q8—281no—
421C2×Q8—281no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C2×Dic1412yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4D4×D711yescharacteristic
443C22×C4D4×D711yescharacteristic
444C22×C4C23×D711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8C2×Dic1412yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—142no—
462C4⋊C4—142no—
463C4⋊C4—142no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—282no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—561no—
477C4⋊C4—561no—
478C4⋊C4—561no—
479C4⋊C4—561no—
480C4⋊C4C2×Dic1412yes—
481C4⋊C4#115712yes—
482C22⋊C4—42no—
483C22⋊C4—282no—
484C22⋊C4—282no—
485C22⋊C4—561no—
486C22⋊C4—561no—
487C42—81no—
488C42—141no—
489C42—141no—
490C42—141no—
491C42—141no—
492C42#115712yes—
493C42D4×D712yes—
494C2×C7—82no—
495C2×C7#242911yescharacteristic
496C2×C7#305411yescharacteristic
497C2×C7#306211yescharacteristic
498C2×C7#337011yescharacteristic
499C2×C7#337111yescharacteristic
500C2×C7#338811yescharacteristic
501C2×C7#338911yescharacteristic
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D711yescharacteristic
510Q8—284no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4#644712yes—
554C2×C4#644712yes—
555C2×C4C22×Dic1411yescharacteristic
556C2×C4C2×#115711yescharacteristic
557C2×C4C2×#115711yescharacteristic
558C2×C4C2×D4×D711yescharacteristic
559C2×C4#655411yescharacteristic
560C2×C4#655411yescharacteristic
561C8—22no—
562C8—41no—
563C8—142no—
564C8—142no—
565C7#2547511yescharacteristic, complemented, Sylow
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#644711yescharacteristic
574C22D4×Dic1411yescharacteristic
575C22#9024311yescharacteristic
576C22#9044811yescharacteristic
577C22#9045311yescharacteristic
578C22#9045911yescharacteristic
579C22#9045911yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—281no—
586C4—281no—
587C4—281no—
588C4—281no—
589C4—561no—
590C4—561no—
591C4—561no—
592C4—561no—
593C4#9016111yescharacteristic
594C4#9020911yescharacteristic
595C4D4×Dic1411yescharacteristic
596C4#9024211yescharacteristic
597C2—82no—
598C2#1173093311yescharacteristic
599C2#1173619711yescharacteristic
600C2#1173628411yescharacteristic
601C2#1173701711yescharacteristic
602C2#1173702211yescharacteristic
603C2#1173707211yescharacteristic
604C2#1173708111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#102C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC4×#102D7
hypocenterC7#25475
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
3. C2×#89451C2semidirect
4. #11731172C2semidirect
10. #11727532C2semidirect
11. #11727222C2semidirect
12. #11726615C2semidirect
13. #11726614C2semidirect
14. #11726563C2semidirect
15. #11726561C2semidirect
103. C7×#102Q8semidirect
374. #102Dic14semidirect
565. C7#25475semidirect

Maximal subgroups

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

subgroupindex
#117398332
C2×#897812
C2×#894512
#117311722
C4×C7×#1022
#117279482
#117279472
#117279412
#117275322
#117272222
#117266152
#117266142
#117265632
#117265612
#254757