Finite Group Databasebeta

← #13405995

Subgroups — #13405995

Order 1792.

1474 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90168C211yesmaximal, characteristic, complemented
2C7×#2854C211yesmaximal, characteristic, complemented
3C2×#89810C212yesmaximal, complemented
4C2×#89319C211yesmaximal, characteristic, complemented
5#11727412C211yesmaximal, characteristic, complemented
6#11727411C212yesmaximal, complemented
7#11727392C211yesmaximal, characteristic, complemented
8#11727381C211yesmaximal, characteristic, complemented
9#11727379C212yesmaximal, complemented
10#11727358C211yesmaximal, characteristic, complemented
11#11727269C212yesmaximal, complemented
12C22×#1157—21no—
13C2×C7×#123—21no—
14C2×C7×#121C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×#1141C2212yes—
17C2×#6493—21no—
18C2×#6493C2212yes—
19C23×C7⋊C8C2212yes—
20#90168—28no—
21C2×#6438—21no—
22C2×#6437C2211yescharacteristic
23C2×C4×D28—21no—
24C2×C4×D28C2212yes—
25C2×#6432C2211yescharacteristic
26C7×#390C2211yescharacteristic
27C7×#390C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#389C2211yescharacteristic
30C7×#388—21no—
31C7×#388—21no—
32C7×#386—21no—
33C7×#380C2211yescharacteristic
34C2×#6406C2212yes—
35#89810—28no—
36C2×#6383C2212yes—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39C2×#6381C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#89319C2211yescharacteristic
44#89319C2211yescharacteristic
45#25496—71nomaximal, Sylow
46C22×C7×D4—21no—
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—22no—
52C2×#1157—28no—
53C2×#1157—44no—
54C2×#1157—44no—
55C22×D28—21no—
56C22×D28—21no—
57C22×D28C2312yes—
58C22×C4×D7—22no—
59C22×Dic14—21no—
60C7×#123—44no—
61C7×#121—24no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C42×C7—24no—
74C2×C42×C7D411yescharacteristic
75C2×C42×C7D411yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×#1141—24no—
79C2×#1141—24no—
80C2×#1141—42no—
81#6493—28no—
82#6493—24no—
83#6493D418yes—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2312yes—
88C22×C7⋊C8—24no—
89C22×C7⋊C8—24no—
90C22×C7⋊C8—42no—
91C22×C7⋊C8C2312yes—
92C22×C7⋊C8C2312yes—
93#6438—28no—
94#6437—24no—
95C4×D28—28no—
96C4×D28—24no—
97C4×D28—24no—
98#6432—24no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102#6406—24no—
103#6383—24no—
104#6383—24no—
105#6383—42no—
106#6381—24no—
107C2×#532—71no—
108C2×#466—72no—
109#2854D711yescharacteristic, complemented
110#2057—71no—
111#2041—71no—
112#2023—71no—
113#2011—71no—
114#1893—72no—
115#1879—72no—
116#1862—72no—
117C2×#336—71no—
118C24×C7—21no—
119C24×C7C2×D411yescharacteristic
120C23×D7—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—42no—
126C2×C7×D4—42no—
127C2×C7×D4—42no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—41no—
139C22×C4×C7D814yescomplemented
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D414yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C2×C22⋊D7—28no—
148C2×C22⋊D7—44no—
149C2×C22⋊D7—44no—
150C22×Dic7—22no—
151#1157—432no—
152C2×D28—22no—
153C2×D28—22no—
154C2×D28—22no—
155C2×D28—22no—
156C2×D28—24no—
157C2×D28—24no—
158C2×D28—22no—
159C2×D28—42no—
160C2×D28—42no—
161C2×D28—42no—
162C2×D28—42no—
163C2×D28C2×D414yes—
164C2×C4×D7—28no—
165C2×C4×D7—44no—
166C2×C4×D7—44no—
167C2×Dic14—24no—
168C2×Dic14—42no—
169C2×Dic14—42no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C22⋊C4—42no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C42×C7—22no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7—22no—
186#1141—48no—
187(C2×D7)⋊C4—28no—
188(C2×D7)⋊C4—28no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7C2×D414yes—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—48no—
198C2×C7⋊C8#4114yescomplemented
199C22×#41—141no—
200C22×D8—72no—
201C23×C8—72no—
202#532—148no—
203C2×C4⋊Q8—71no—
204C2×#127—71no—
205C2×#124—141no—
206C2×#123—21no—
207C2×#121—72no—
208C2×#121—141no—
209C2×#121C2×D711yescharacteristic
210C2×C4×D4—72no—
211C2×C4×D4—141no—
212C22×C42C2×D711yescharacteristic
213#466—148no—
214C2×#107—72no—
215C2×#102—72no—
216C2×#102—72no—
217C2×#102—72no—
218C2×#102—72no—
219#390C2×D711yescharacteristic
220#390C2×D711yescharacteristic
221#389C2×D711yescharacteristic
222#389C2×D711yescharacteristic
223#388—21no—
224#388—21no—
225#386—21no—
226#380C2×D711yescharacteristic
227#340—72no—
228#340—72no—
229#336—71no—
230#336—72no—
231#336—71no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7#12012yes—
244C23×C7C2×D812yes—
245C23×C7AGL(1,ℤ/8)12yes—
246C23×C7C22×D411yescharacteristic
247C22×D7—24no—
248C22×D7—42no—
249C22×D7—42no—
250C22×D7—42no—
251C22×D7—42no—
252C22×D7—42no—
253C22×D7—42no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—84no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—28no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—44no—
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#12012yes—
282C2×C4×C7C2×D812yes—
283C2×C4×C7C2×D812yes—
284C2×C4×C7#13512yes—
285C2×C4×C7C22×D412yes—
286C2×C4×C7C2×#4112yes—
287C2×C4×C7H(5,𝔽2)11yescharacteristic
288C2×C4×C72−1+411yescharacteristic
289C22⋊D7—416no—
290C2×Dic7—24no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293D28—24no—
294D28—24no—
295D28—48no—
296D28—48no—
297D28—42no—
298D28—42no—
299C4×D7—416no—
300Dic14—48no—
301C7⋊C8—24no—
302C7⋊C8—42no—
303C2×#41—148no—
304C2×#41—284no—
305C2×#41—284no—
306C22×Q8—141no—
307C22×D4—21no—
308C22×D4—72no—
309C22×D4—142no—
310C22×D4—141no—
311C22×D4—141no—
312C22×D4C22×D711yescharacteristic
313C23×C4—142no—
314C23×C4C22⋊D711yescharacteristic
315C23×C4C22⋊D711yescharacteristic
316C23×C4C22×D711yescharacteristic
317C2×D8—144no—
318C2×D8—144no—
319C2×D8—282no—
320C22×C8—72no—
321C22×C8—72no—
322C22×C8—144no—
323C22×C8—144no—
324C22×C8—282no—
325C4⋊Q8—144no—
326#127—144no—
327#124—148no—
328#123—44no—
329#121—24no—
330#121—78no—
331#121—148no—
332#121—144no—
333C4×D4—144no—
334C4×D4—144no—
335C4×D4—148no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—72no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—142no—
350C2×C22⋊C4—142no—
351C2×C42—24no—
352C2×C42C22⋊D711yescharacteristic
353C2×C42C22⋊D711yescharacteristic
354C2×C42C22×D711yescharacteristic
355C2×C42C22×D711yescharacteristic
356#107—144no—
357#102—144no—
358#102—144no—
359#102—144no—
360#102—282no—
361#95—21no—
362#95—21no—
363#95—21no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—81no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7#44714yes—
378C22×C7C2×#12011yescharacteristic
379C22×C7#53211yescharacteristic
380C22×C7#53711yescharacteristic
381C22×C7C22×D811yescharacteristic
382C22×C7C2×#13511yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C7#57511yescharacteristic
385C2×D7—44no—
386C2×D7—44no—
387C2×D7—44no—
388C2×D7—82no—
389C2×D7—82no—
390C2×D7—82no—
391C2×D7—82no—
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—41no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7#54612yes—
401C4×C7#54712yes—
402Dic7—44no—
403C24—21no—
404C24—142no—
405C24C2×C22⋊D711yescharacteristic
406#41—2832no—
407C2×Q8—144no—
408C2×Q8—282no—
409C2×Q8—282no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—74no—
418C2×D4—144no—
419C2×D4—144no—
420C2×D4—148no—
421C2×D4—142no—
422C2×D4—142no—
423C2×D4—142no—
424C2×D4—142no—
425C2×D4—142no—
426C2×D4—282no—
427C2×D4—282no—
428C2×D4—284no—
429C2×D4—284no—
430C2×D4—282no—
431C2×D4—282no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—41no—
443C22×C4—148no—
444C22×C4—142no—
445C22×C4—284no—
446C22×C4—284no—
447C22×C4#114114yescomplemented
448C22×C4D4×D714yes—
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C23×D711yescharacteristic
455D8—288no—
456C2×C8—74no—
457C2×C8—142no—
458C2×C8—142no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—288no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—74no—
470C4⋊C4—144no—
471C4⋊C4—142no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—148no—
479C22⋊C4—148no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C2×C7—24no—
485C2×C7—161no—
486C2×C7#243011yescharacteristic
487C2×C7C2×#44711yescharacteristic
488C2×C7#300211yescharacteristic
489C2×C7#316911yescharacteristic
490C2×C7#318011yescharacteristic
491C2×C7#324511yescharacteristic
492C2×C7#324711yescharacteristic
493D7—84no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—144no—
506C23—282no—
507C23—282no—
508C23—282no—
509C23—282no—
510C23—282no—
511C23—282no—
512C23C2×#114112yes—
513C23#650012yes—
514C23#651212yes—
515C23C22×C22⋊D711yescharacteristic
516Q8—288no—
517D4—42no—
518D4—42no—
519D4—84no—
520D4—144no—
521D4—144no—
522D4—288no—
523D4—288no—
524D4—2816no—
525D4—282no—
526D4—282no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—144no—
552C2×C4—2816no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4C2×#114112yes—
556C2×C4C2×#114112yes—
557C2×C4#650012yes—
558C2×C4#651312yes—
559C2×C4C2×D4×D712yes—
560C2×C4D7×#4112yes—
561C2×C4#655311yescharacteristic
562C2×C4#655411yescharacteristic
563C8—144no—
564C8—282no—
565C7#2549611yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—284no—
580C22—284no—
581C22—284no—
582C22—562no—
583C22—562no—
584C22—562no—
585C22—562no—
586C22#8980814yes—
587C22C22×#114111yescharacteristic
588C22C2×#650011yescharacteristic
589C22C2×#651211yescharacteristic
590C22C2×#651311yescharacteristic
591C22#9051311yescharacteristic
592C22#9052311yescharacteristic
593C22#9052411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—284no—
603C4#9034512yes—
604C4#9034612yes—
605C2—24no—
606C2—161no—
607C2—564no—
608C2C2×#8980811yescharacteristic
609C2#1174029311yescharacteristic
610C2#1174037111yescharacteristic
611C2#1174058211yescharacteristic
612C2#1174059311yescharacteristic
613C2#1174073611yescharacteristic
614C2#1174075511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2854C2
socleC23×C7C22×D4
hypercenter#2854D7
hypocenterC7#25496
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#90168C2semidirect
2. C7×#2854C2semidirect
3. C2×#89810C2semidirect
4. C2×#89319C2semidirect
5. #11727412C2semidirect
6. #11727411C2semidirect
7. #11727392C2semidirect
8. #11727381C2semidirect
9. #11727379C2semidirect
10. #11727358C2semidirect
11. #11727269C2semidirect
109. #2854D7semidirect
139. C22×C4×C7D8semidirect
198. C2×C7⋊C8#41semidirect
447. C22×C4#1141semidirect
565. C7#25496semidirect

Maximal subgroups

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

subgroupindex
C2×#901682
C7×#28542
C2×#898102
C2×#893192
#117274122
#117274112
#117273922
#117273812
#117273792
#117273582
#117272692
#254967