Finite Group Databasebeta

← #13412754

Subgroups — #13412754

Order 1792.

750 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731782C211yesmaximal, characteristic, complemented
2#11731084C211yesmaximal, characteristic
3C7×#1947C211yesmaximal, characteristic, complemented
4#11728395C211yesmaximal, characteristic
5#11728393C211yesmaximal, characteristic
6#11728369C211yesmaximal, characteristic, complemented
7#11728360C211yesmaximal, characteristic, complemented
8#11728238C211yesmaximal, characteristic
9#11728205C211yesmaximal, characteristic, complemented
10#11727340C211yesmaximal, characteristic
11#11727212C211yesmaximal, characteristic, complemented
12#11726646C211yesmaximal, characteristic
13#11726569C211yesmaximal, characteristic, complemented
14#11726549C211yesmaximal, characteristic, complemented
15#11726534C211yesmaximal, characteristic
16C2×#6456C2211yescharacteristic
17C2×#6452—21no—
18C2×#6435—21no—
19C2×C7×#107C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C7×#384C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89760C2211yescharacteristic
31#89757C2211yescharacteristic
32#89756—21no—
33#89752—21no—
34#89748C2211yescharacteristic
35#89748C2211yescharacteristic
36#89748C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89744—21no—
40C2×#6383C2211yescharacteristic
41C2×#6383C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89712—21no—
46#89711C2211yescharacteristic
47C4×Dic7⋊C4C2211yescharacteristic
48#89700C2211yescharacteristic
49#89441—21no—
50#89439—21no—
51C7×#340C2211yescharacteristic
52C7×#340C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26086—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4D411yescharacteristic
89C2×Dic7⋊C4D411yescharacteristic
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6456—22no—
97#6456Q814yes—
98#6452—44no—
99#6435—44no—
100C7×#107—41no—
101C7×#107—41no—
102C7×C4⋊C8—22no—
103C7×#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95D411yescharacteristic
109C7⋊#95D411yescharacteristic
110#6395—22no—
111#6395—41no—
112#6392—41no—
113Dic7⋊C8—22no—
114Dic7⋊C8—22no—
115C8×Dic7—22no—
116#6383—22no—
117#6383—41no—
118#6381—22no—
119#6381—22no—
120#6381—41no—
121#2552—71no—
122#2354—71no—
123#2093—71no—
124#2093—71no—
125#2067—71no—
126#2067—71no—
127#1947—71no—
128#1947—71no—
129#1947D711yescharacteristic, complemented
130#1939—71no—
131#1918—71no—
132#1918—71no—
133#1845—71no—
134#1802—71no—
135C4×#102—71no—
136C23×D7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2411yescharacteristic
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28C2×Q812yes—
157C2×C4×D7—24no—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8#4112yes—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4C2×Q812yes—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7—42no—
179(C2×D7)⋊C4—24no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—42no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4—81no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—42no—
191Dic7⋊C4—22no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198Dic7⋊C4—81no—
199Dic7⋊C4D814yes—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—42no—
203C4×Dic7—42no—
204C4×Dic7C2×Q812yes—
205C4×Dic7#4112yes—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8#4112yes—
212C2×#125—141no—
213C2×C4×D4—71no—
214C2×#117—141no—
215C2×#107—71no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107—71no—
219C2×#107C2×D711yescharacteristic
220C2×#107C2×D711yescharacteristic
221C2×C4⋊C8—71no—
222C2×C4⋊C8—71no—
223C2×C4⋊C8C2×D711yescharacteristic
224C2×#102—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×C4×C8—71no—
229#398—141no—
230#398—141no—
231#390—71no—
232#390—71no—
233#389—71no—
234#389—71no—
235#389—141no—
236#388—141no—
237#388—141no—
238#384—71no—
239#384—71no—
240#384—71no—
241#384C2×D711yescharacteristic
242#380—141no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4C2×D711yescharacteristic
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340C2×D711yescharacteristic
252#340C2×D711yescharacteristic
253#336—71no—
254#336—71no—
255C23×C7C22×D411yescharacteristic
256C22×D7—22no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C22×D7—41no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
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×C7C2×D812yes—
281C2×C4×C7C22×Q811yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—41no—
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×Dic7C22⋊Q812yes—
304C2×Dic7C22⋊Q812yes—
305C2×Dic7C2×D812yes—
306C2×Dic7C2×D812yes—
307C2×Dic7#13712yes—
308D28—22no—
309D28—22no—
310D28—41no—
311D28—41no—
312C4×D7—48no—
313C7×C8—22no—
314C7×C8—41no—
315C7⋊C8—22no—
316C7⋊C8—41no—
317C22×D4—71no—
318C23×C4—141no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323#125—284no—
324C4×D4—74no—
325C4×D4—142no—
326#117—284no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C22⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C42—21no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—141no—
355C2×C42—141no—
356C2×C42C22×D711yescharacteristic
357#107—41no—
358#107—41no—
359#107—142no—
360#107—142no—
361#107—281no—
362#107—281no—
363C4⋊C8—22no—
364C4⋊C8—142no—
365C4⋊C8—142no—
366#102—142no—
367#102—142no—
368#102—281no—
369#102—281no—
370C4×C8—142no—
371#95—21no—
372#95—71no—
373#95—71no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378C22×C7C2×C22⋊Q811yescharacteristic
379C22×C7#54711yescharacteristic
380C22×C7#55011yescharacteristic
381C22×C7C22×D811yescharacteristic
382C22×C7C2×#13711yescharacteristic
383C22×C7#57611yescharacteristic
384C22×C7#57811yescharacteristic
385C2×D7—42no—
386C2×D7—42no—
387C2×D7—42no—
388C2×D7—81no—
389C2×D7—81no—
390C2×D7—81no—
391C2×D7—81no—
392C4×C7—22no—
393C4×C7—41no—
394C4×C7—42no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7—81no—
398C4×C7#53311yescharacteristic
399C4×C7#54411yescharacteristic
400C4×C7#55211yescharacteristic
401C4×C7#55311yescharacteristic
402Dic7—22no—
403Dic7—22no—
404Dic7—41no—
405Dic7—81no—
406Dic7—81no—
407Dic7—81no—
408Dic7#47414yes—
409C24—141no—
410C2×D4—72no—
411C2×D4—141no—
412C2×D4—141no—
413C2×D4—141no—
414C2×D4—141no—
415C2×D4—141no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—144no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—282no—
429C22×C4—282no—
430C22×C4D4×D711yescharacteristic
431C22×C4D4×D711yescharacteristic
432C22×C4C23×D711yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—72no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8#115712yes—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—74no—
454C4⋊C4—81no—
455C4⋊C4—142no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—282no—
463C4⋊C4—282no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—282no—
467C4⋊C4—561no—
468C4⋊C4—561no—
469C4⋊C4—561no—
470C4⋊C4Q8×D712yes—
471C22⋊C4—144no—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C22⋊C4—561no—
475C22⋊C4—561no—
476C42—21no—
477C42—21no—
478C42—42no—
479C42—72no—
480C42—72no—
481C42—141no—
482C42—141no—
483C42—282no—
484C42—282no—
485C2×C7#244111yescharacteristic
486C2×C7C2×#47411yescharacteristic
487C2×C7#308311yescharacteristic
488C2×C7#333411yescharacteristic
489C2×C7#335711yescharacteristic
490C2×C7#339111yescharacteristic
491C2×C7#339811yescharacteristic
492D7—82no—
493C23—142no—
494C23—281no—
495C23—281no—
496C23—281no—
497C23—281no—
498C23—281no—
499C23—281no—
500C23C2×D4×D711yescharacteristic
501D4—142no—
502D4—142no—
503D4—281no—
504D4—281no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
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—72no—
523C2×C4—72no—
524C2×C4—72no—
525C2×C4—72no—
526C2×C4—72no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—288no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4D7×D812yes—
545C2×C4C2×#115711yescharacteristic
546C2×C4#654811yescharacteristic
547C2×C4C2×Q8×D711yescharacteristic
548C2×C4#655111yescharacteristic
549C2×C4D7×#4111yescharacteristic
550C2×C4D7×#4111yescharacteristic
551C8—22no—
552C8—41no—
553C8—142no—
554C8—281no—
555C7#2608611yescharacteristic, complemented, Sylow
556C22—282no—
557C22—282no—
558C22—282no—
559C22—561no—
560C22—561no—
561C22—561no—
562C22—561no—
563C22#9019811yescharacteristic
564C22#9029011yescharacteristic
565C22D7×C22⋊Q811yescharacteristic
566C22C2×D7×D811yescharacteristic
567C22#9045711yescharacteristic
568C22#9046511yescharacteristic
569C22D7×#13711yescharacteristic
570C4—22no—
571C4—42no—
572C4—41no—
573C4—74no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—142no—
578C4—142no—
579C4—281no—
580C4—561no—
581C4—561no—
582C4—561no—
583C4#9023111yescharacteristic
584C4#9026911yescharacteristic
585C4#9038411yescharacteristic
586C4#9042011yescharacteristic
587C2—562no—
588C2#1173344611yescharacteristic
589C2#1173659911yescharacteristic
590C2#1173668311yescharacteristic
591C2#1173776911yescharacteristic
592C2#1173791411yescharacteristic
593C2D7×#47411yescharacteristic
594C2#1173855611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1947D7
hypocenterC7#26086
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. #11731782C2semidirect
3. C7×#1947C2semidirect
6. #11728369C2semidirect
7. #11728360C2semidirect
9. #11728205C2semidirect
11. #11727212C2semidirect
13. #11726569C2semidirect
14. #11726549C2semidirect
129. #1947D7semidirect
555. C7#26086semidirect

Maximal subgroups

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

subgroupindex
#117317822
#117310842
C7×#19472
#117283952
#117283932
#117283692
#117283602
#117282382
#117282052
#117273402
#117272122
#117266462
#117265692
#117265492
#117265342
#260867