Finite Group Databasebeta

← #13383723

Subgroups — #13383723

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11733408C211yesmaximal, characteristic
2D7×#382C211yesmaximal, characteristic
3C7×#2101C211yesmaximal, characteristic, complemented
4#11727367C211yesmaximal, characteristic, complemented
5#11727296C211yesmaximal, characteristic, complemented
6#11726621C211yesmaximal, characteristic
7#11726621C211yesmaximal, characteristic
8#11726621C211yesmaximal, characteristic
9#11726621C211yesmaximal, characteristic
10#11726600C211yesmaximal, characteristic, complemented
11#11726599C211yesmaximal, characteristic, complemented
12#11726598C211yesmaximal, characteristic, complemented
13#11726566C211yesmaximal, characteristic, complemented
14#11726414C211yesmaximal, characteristic
15#11726398C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×C42×D7C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#382C2211yescharacteristic
25#89761—21no—
26#89757C2211yescharacteristic
27#89749C2211yescharacteristic
28#89749C2211yescharacteristic
29C4⋊C4×Dic7—21no—
30#89745C2211yescharacteristic
31#89708C2211yescharacteristic
32#89700C2211yescharacteristic
33#89440—21no—
34D7×#95—21no—
35D7×#95—21no—
36#89422—21no—
37#89422—21no—
38#89421—21no—
39C7×#340C2211yescharacteristic
40C7×#340C2211yescharacteristic
41C7×#340C2211yescharacteristic
42C7×#340C2211yescharacteristic
43C7×#336C2211yescharacteristic
44#89346C2211yescharacteristic
45#89346C2211yescharacteristic
46#89346C2211yescharacteristic
47#89346C2211yescharacteristic
48#89345C2211yescharacteristic
49#89345C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#23746—71nomaximal, Sylow
60C22×C4×D7—21no—
61C22×C4×D7—21no—
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D411yescharacteristic
64C22×C4×D7C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C2×#1131—22no—
92C2×#1131—22no—
93C2×#1131—22no—
94C2×#1131—22no—
95C2×#1131—22no—
96C2×#1131—22no—
97#6455—22no—
98#6455—22no—
99#6455—22no—
100#6455Q814yes—
101D7×C4⋊C4—22no—
102D7×C4⋊C4—22no—
103D7×C4⋊C4—22no—
104D7×C4⋊C4—22no—
105C42×D7—22no—
106C42×D7—22no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119#2296—71no—
120C2×#382—71no—
121#2101—71no—
122#2101—71no—
123#2101—71no—
124#2101—71no—
125#2101—71no—
126#2101—71no—
127#2101—71no—
128#2101D711yescharacteristic, complemented
129#1736—71no—
130#1736—71no—
131#1736—71no—
132#1736—71no—
133#1728—71no—
134C23×D7C2×D411yescharacteristic
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—41no—
152C2×C4×D7—41no—
153C2×C4×D7—41no—
154C2×C4×D7—41no—
155C2×C4×D7—42no—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×C4×D7C2×Q812yes—
162C2×C4×D7#4112yes—
163C2×C4×D7#4112yes—
164C2×C4×D7#4112yes—
165C2×C4×D7#4112yes—
166C2×C4×D7#4112yes—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4C2×Q812yes—
182C42×C7—21no—
183C42×C7—21no—
184(C2×D7)⋊C4—24no—
185(C2×D7)⋊C4—42no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—42no—
194C4⋊Dic7C2×Q812yes—
195Dic7⋊C4—42no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—24no—
202C4×Dic7—22no—
203C4×Dic7—22no—
204C4×Dic7—42no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211#1131—44no—
212#1131—44no—
213C22×M16—71no—
214C22×M16—71no—
215C2×#117—71no—
216C2×#117—71no—
217C22×C4⋊C4—71no—
218C22×C4⋊C4—71no—
219C22×C42—71no—
220#389—71no—
221#389—71no—
222#389—71no—
223#389C2×D711yescharacteristic
224#385—141no—
225#385—141no—
226#382—71no—
227#382—71no—
228#382—71no—
229#382—141no—
230#382—141no—
231#382C2×D711yescharacteristic
232C4×C4⋊C4—141no—
233C4×C4⋊C4—141no—
234C2×#95—141no—
235C2×#95—141no—
236#340—71no—
237#340—71no—
238#340—71no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340C2×D711yescharacteristic
249#340C2×D711yescharacteristic
250#340C2×D711yescharacteristic
251#340C2×D711yescharacteristic
252#336—71no—
253#336—71no—
254#336—71no—
255#336C2×D711yescharacteristic
256C23×C7C22×D411yescharacteristic
257C22×D7—21no—
258C22×D7—21no—
259C22×D7—21no—
260C22×D7—21no—
261C22×D7C22⋊Q812yes—
262C22×D7#12312yes—
263C22×D7#12412yes—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7C22×Q811yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×Dic7—22no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7C22⋊Q812yes—
307C2×Dic7#12312yes—
308C2×Dic7#12412yes—
309C4×D7—44no—
310C4×D7—44no—
311C4×D7—84no—
312C4×D7—84no—
313C4×D7#12514yes—
314C4×D7#12514yes—
315C4×D7#12614yes—
316C4×D7#12614yes—
317C7×C8—41no—
318C7×C8—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—71no—
324C23×C4—141no—
325C23×C4—141no—
326C2×M16—142no—
327C2×M16—142no—
328C2×M16—142no—
329C2×M16—142no—
330C2×M16—142no—
331C2×M16—142no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8C22×D711yescharacteristic
335C22×C8C22×D711yescharacteristic
336#117—74no—
337#117—142no—
338#117—142no—
339#117—142no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C42—71no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—142no—
364C2×C42—142no—
365C2×C42—141no—
366C2×C42—141no—
367C2×C42C22×D711yescharacteristic
368#95—21no—
369#95—21no—
370#95—21no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380C22×C7C2×C22⋊Q811yescharacteristic
381C22×C7C2×#12311yescharacteristic
382C22×C7C2×#12411yescharacteristic
383C22×C7#57511yescharacteristic
384C22×C7#57511yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57511yescharacteristic
387C2×D7—22no—
388C2×D7—22no—
389C2×D7—22no—
390C2×D7—22no—
391C2×D7—22no—
392C2×D7—22no—
393C2×D7#40014yes—
394C4×C7—42no—
395C4×C7—41no—
396C4×C7—41no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7C2×#12511yescharacteristic
401C4×C7C2×#12511yescharacteristic
402C4×C7C2×#12611yescharacteristic
403C4×C7C2×#12611yescharacteristic
404Dic7—42no—
405Dic7—41no—
406Dic7—41no—
407Dic7—81no—
408Dic7—81no—
409Dic7—81no—
410Dic7#40014yes—
411C24—71no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—71no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—72no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—72no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—141no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—281no—
432C22×C4—281no—
433C22×C4—281no—
434C22×C4—281no—
435C22×C4—282no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—282no—
439C22×C4—282no—
440C22×C4—282no—
441C22×C4D4×D711yescharacteristic
442C22×C4D4×D711yescharacteristic
443C22×C4C23×D711yescharacteristic
444M16—284no—
445M16—284no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—72no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—282no—
479C4⋊C4Q8×D712yes—
480C22⋊C4—144no—
481C22⋊C4—282no—
482C42—21no—
483C42—21no—
484C42—142no—
485C42—142no—
486C42—144no—
487C42—141no—
488C42—141no—
489C42—282no—
490C2×C7C2×#40011yescharacteristic
491C2×C7#308011yescharacteristic
492C2×C7#308011yescharacteristic
493C2×C7#309311yescharacteristic
494C2×C7#309311yescharacteristic
495C2×C7#314011yescharacteristic
496C2×C7#314011yescharacteristic
497D7—24no—
498C23—72no—
499C23—72no—
500C23—72no—
501C23—141no—
502C23—141no—
503C23—141no—
504C23—141no—
505C23C2×D4×D711yescharacteristic
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—72no—
525C2×C4—72no—
526C2×C4—72no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—74no—
530C2×C4—74no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—142no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—284no—
539C2×C4—284no—
540C2×C4—281no—
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—564no—
552C2×C4—564no—
553C2×C4C2×Q8×D711yescharacteristic
554C2×C4D7×#4111yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4D7×#4111yescharacteristic
557C2×C4D7×#4111yescharacteristic
558C2×C4D7×#4111yescharacteristic
559C8—41no—
560C8—41no—
561C8—281no—
562C8—281no—
563C7#2374611yescharacteristic, complemented, Sylow
564C22—74no—
565C22—142no—
566C22—142no—
567C22—142no—
568C22—142no—
569C22—142no—
570C22—142no—
571C22D7×C22⋊Q811yescharacteristic
572C22D7×#12311yescharacteristic
573C22D7×#12411yescharacteristic
574C22#9045911yescharacteristic
575C22#9045911yescharacteristic
576C22#9046411yescharacteristic
577C22#9046811yescharacteristic
578C4—41no—
579C4—41no—
580C4—42no—
581C4—74no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—282no—
586C4—281no—
587C4—281no—
588C4—561no—
589C4—561no—
590C4—561no—
591C4D7×#12511yescharacteristic
592C4D7×#12511yescharacteristic
593C4D7×#12611yescharacteristic
594C4D7×#12611yescharacteristic
595C2—144no—
596C2D7×#40011yescharacteristic
597C2#1173861211yescharacteristic
598C2#1173867611yescharacteristic
599C2#1173881711yescharacteristic
600C2#1173896111yescharacteristic
601C2#1173908511yescharacteristic
602C2#1173912711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2101C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#2101D7
hypocenterC7#23746
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. C7×#2101C2semidirect
4. #11727367C2semidirect
5. #11727296C2semidirect
10. #11726600C2semidirect
11. #11726599C2semidirect
12. #11726598C2semidirect
13. #11726566C2semidirect
15. #11726398C2semidirect
128. #2101D7semidirect
563. C7#23746semidirect

Maximal subgroups

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

subgroupindex
#117334082
D7×#3822
C7×#21012
#117273672
#117272962
#117266212
#117266002
#117265992
#117265982
#117265662
#117264142
#117263982
#237467