Finite Group Databasebeta

← #13405984

Subgroups — #13405984

Order 1792.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90168C211yesmaximal, characteristic
2C7×#2855C211yesmaximal, characteristic, complemented
3C2×#89820C212yesmaximal, complemented
4C2×#89319C211yesmaximal, characteristic, complemented
5#11727442C211yesmaximal, characteristic, complemented
6#11727441C212yesmaximal, complemented
7#11727422C211yesmaximal, characteristic, complemented
8#11727381C211yesmaximal, characteristic, complemented
9#11727380C212yesmaximal, complemented
10#11727359C211yesmaximal, characteristic, complemented
11#11727269C212yesmaximal, complemented
12C22×#1157—21no—
13C2×C7×C22⋊Q8C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×#1143C2212yes—
16C2×#6493—21no—
17C2×#6493C2212yes—
18C23×C7⋊C8C2212yes—
19#90168—28no—
20C2×#6438—21no—
21C2×#6437C2211yescharacteristic
22C2×C4×D28—21no—
23C2×C4×D28C2212yes—
24C2×#6432C2211yescharacteristic
25C7×#391C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#389C2211yescharacteristic
29C7×#387—21no—
30C7×#387—21no—
31C7×#385—21no—
32C7×#381—21no—
33C7×#380C2211yescharacteristic
34C2×#6409C2212yes—
35#89820—28no—
36C2×#6383C2212yes—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39C2×#6382C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#89319C2211yescharacteristic
44#89319C2211yescharacteristic
45#25495—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C22×C22⋊D7—22no—
51C2×#1157—28no—
52C2×#1157—44no—
53C2×#1157—44no—
54C22×D28—21no—
55C22×D28—21no—
56C22×D28C2312yes—
57C22×C4×D7—22no—
58C22×Dic14—21no—
59C7×C22⋊Q8—24no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C42×C7—24no—
69C2×C42×C7D411yescharacteristic
70C2×C42×C7D411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×#1143—24no—
74C2×#1143—24no—
75C2×#1143—42no—
76#6493—28no—
77#6493—24no—
78#6493D418yes—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2312yes—
83C22×C7⋊C8—24no—
84C22×C7⋊C8—24no—
85C22×C7⋊C8—42no—
86C22×C7⋊C8C2312yes—
87C22×C7⋊C8C2312yes—
88#6438—28no—
89#6437—24no—
90C4×D28—28no—
91C4×D28—24no—
92C4×D28—24no—
93#6432—24no—
94C7×#95—21no—
95C7×#95—21no—
96C7×#95—21no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101#6409—24no—
102#6383—24no—
103#6383—24no—
104#6383—42no—
105#6382—24no—
106C2×#532—71no—
107C2×#465—72no—
108#2855D711yescharacteristic, complemented
109#2058—71no—
110#2041—71no—
111#2024—71no—
112#2014—71no—
113#1893—72no—
114#1877—72no—
115#1864—72no—
116C2×#336—71no—
117C24×C7C2×D411yescharacteristic
118C23×D7—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—41no—
135C22×C4×C7SD1614yescomplemented
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D414yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C2×C22⋊D7—28no—
144C2×C22⋊D7—44no—
145C2×C22⋊D7—44no—
146C22×Dic7—22no—
147#1157—432no—
148C2×D28—22no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—24no—
153C2×D28—24no—
154C2×D28—22no—
155C2×D28—42no—
156C2×D28—42no—
157C2×D28—42no—
158C2×D28—42no—
159C2×D28C2×D414yes—
160C2×C4×D7—28no—
161C2×C4×D7—44no—
162C2×C4×D7—44no—
163C2×Dic14—24no—
164C2×Dic14—42no—
165C2×Dic14—42no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—42no—
173C7×C22⋊C4—42no—
174C7×C22⋊C4—42no—
175C42×C7—22no—
176C42×C7—22no—
177C42×C7—22no—
178C42×C7—22no—
179#1143—48no—
180(C2×D7)⋊C4—28no—
181(C2×D7)⋊C4—28no—
182C4⋊Dic7—24no—
183C4⋊Dic7—22no—
184C4⋊Dic7C2×D414yes—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—48no—
191C2×C7⋊C8#4114yescomplemented
192C22×#41—141no—
193C22×SD16—72no—
194C23×C8—72no—
195#532—148no—
196C2×C4⋊Q8—71no—
197C2×#127—71no—
198C2×#124—141no—
199C2×C22⋊Q8C2×D711yescharacteristic
200C2×#121—72no—
201C2×#121—141no—
202C2×C4×D4—72no—
203C2×C4×D4—141no—
204C22×C42C2×D711yescharacteristic
205#465—148no—
206C2×#106—72no—
207C2×#103—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—72no—
211#391C2×D711yescharacteristic
212#391C2×D711yescharacteristic
213#389C2×D711yescharacteristic
214#389C2×D711yescharacteristic
215#387—21no—
216#387—21no—
217#385—21no—
218#381—21no—
219#380C2×D711yescharacteristic
220#340—72no—
221#340—72no—
222#336—71no—
223#336—72no—
224#336—71no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7#12012yes—
230C23×C7C2×SD1612yes—
231C23×C7AGL(1,ℤ/8)12yes—
232C23×C7C22×D411yescharacteristic
233C22×D7—24no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—42no—
240C7×Q8—42no—
241C7×Q8—42no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—28no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—44no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7#12012yes—
270C2×C4×C7C2×SD1612yes—
271C2×C4×C7C2×SD1612yes—
272C2×C4×C7#13512yes—
273C2×C4×C7C22×D412yes—
274C2×C4×C7C2×#4112yes—
275C2×C4×C7H(5,𝔽2)11yescharacteristic
276C2×C4×C72−1+411yescharacteristic
277C22⋊D7—416no—
278C2×Dic7—24no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281D28—24no—
282D28—24no—
283D28—48no—
284D28—48no—
285D28—42no—
286D28—42no—
287C4×D7—416no—
288Dic14—48no—
289C7⋊C8—24no—
290C7⋊C8—42no—
291C2×#41—148no—
292C2×#41—284no—
293C2×#41—284no—
294C22×Q8—141no—
295C22×Q8C22×D711yescharacteristic
296C22×D4—72no—
297C22×D4—142no—
298C22×D4—141no—
299C22×D4—141no—
300C23×C4—142no—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22×D711yescharacteristic
304C2×SD16—144no—
305C2×SD16—144no—
306C2×SD16—282no—
307C22×C8—72no—
308C22×C8—72no—
309C22×C8—144no—
310C22×C8—144no—
311C22×C8—282no—
312C4⋊Q8—144no—
313#127—144no—
314#124—148no—
315C22⋊Q8—24no—
316#121—78no—
317#121—148no—
318#121—144no—
319C4×D4—144no—
320C4×D4—144no—
321C4×D4—148no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—72no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4C22×D711yescharacteristic
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—142no—
333C2×C22⋊C4—142no—
334C2×C42—24no—
335C2×C42C22⋊D711yescharacteristic
336C2×C42C22⋊D711yescharacteristic
337C2×C42C22×D711yescharacteristic
338C2×C42C22×D711yescharacteristic
339#106—144no—
340#103—144no—
341#102—144no—
342#102—144no—
343#102—282no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7#45014yes—
358C22×C7C2×#12011yescharacteristic
359C22×C7#53211yescharacteristic
360C22×C7#53711yescharacteristic
361C22×C7C22×SD1611yescharacteristic
362C22×C7C2×#13511yescharacteristic
363C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
364C22×C7#57511yescharacteristic
365C2×D7—44no—
366C2×D7—44no—
367C2×D7—44no—
368C2×D7—82no—
369C2×D7—82no—
370C2×D7—82no—
371C2×D7—82no—
372C4×C7—22no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—22no—
376C4×C7—41no—
377C4×C7—81no—
378C4×C7—81no—
379C4×C7—81no—
380C4×C7—81no—
381C4×C7#54612yes—
382C4×C7#54712yes—
383Dic7—44no—
384C24—142no—
385C24C2×C22⋊D711yescharacteristic
386#41—2832no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—144no—
392C2×Q8—282no—
393C2×Q8—282no—
394C2×D4—74no—
395C2×D4—144no—
396C2×D4—144no—
397C2×D4—148no—
398C2×D4—142no—
399C2×D4—142no—
400C2×D4—142no—
401C2×D4—142no—
402C2×D4—142no—
403C2×D4—282no—
404C2×D4—282no—
405C2×D4—284no—
406C2×D4—284no—
407C2×D4—282no—
408C2×D4—282no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—148no—
422C22×C4—142no—
423C22×C4—284no—
424C22×C4—284no—
425C22×C4#114314yescomplemented
426C22×C4D4×D714yes—
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C23×D711yescharacteristic
433SD16—288no—
434C2×C8—74no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C2×C8—142no—
439C2×C8—142no—
440C2×C8—288no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—74no—
449C4⋊C4—144no—
450C4⋊C4—142no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—148no—
454C22⋊C4—148no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C2×C7—24no—
460C2×C7#243011yescharacteristic
461C2×C7C2×#45011yescharacteristic
462C2×C7#300211yescharacteristic
463C2×C7#316711yescharacteristic
464C2×C7#318011yescharacteristic
465C2×C7#324611yescharacteristic
466C2×C7#324711yescharacteristic
467D7—84no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—144no—
473C23—282no—
474C23—282no—
475C23—282no—
476C23—282no—
477C23—282no—
478C23—282no—
479C23#650012yes—
480C23C2×#114312yes—
481C23#651212yes—
482C23C22×C22⋊D711yescharacteristic
483Q8—42no—
484Q8—42no—
485Q8—288no—
486D4—144no—
487D4—144no—
488D4—288no—
489D4—288no—
490D4—2816no—
491D4—282no—
492D4—282no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
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—144no—
521C2×C4—2816no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4#650012yes—
525C2×C4C2×#114312yes—
526C2×C4C2×#114312yes—
527C2×C4#651312yes—
528C2×C4C2×D4×D712yes—
529C2×C4D7×#4112yes—
530C2×C4#655311yescharacteristic
531C2×C4#655411yescharacteristic
532C8—144no—
533C8—282no—
534C7#2549511yescharacteristic, complemented, Sylow
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—284no—
542C22—284no—
543C22—284no—
544C22—562no—
545C22—562no—
546C22—562no—
547C22—562no—
548C22#8981814yes—
549C22C2×#650011yescharacteristic
550C22C22×#114311yescharacteristic
551C22C2×#651211yescharacteristic
552C22C2×#651311yescharacteristic
553C22#9051311yescharacteristic
554C22#9052311yescharacteristic
555C22#9052411yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—284no—
566C4#9034512yes—
567C4#9034612yes—
568C2—24no—
569C2—564no—
570C2C2×#8981811yescharacteristic
571C2#1174029311yescharacteristic
572C2#1174037111yescharacteristic
573C2#1174058211yescharacteristic
574C2#1174059311yescharacteristic
575C2#1174078811yescharacteristic
576C2#1174080811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2855C2
socleC23×C7C22×D4
prime radicalC2×#90168C2
hypercenter#2855D7
hypocenterC7#25495
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
2. C7×#2855C2semidirect
3. C2×#89820C2semidirect
4. C2×#89319C2semidirect
5. #11727442C2semidirect
6. #11727441C2semidirect
7. #11727422C2semidirect
8. #11727381C2semidirect
9. #11727380C2semidirect
10. #11727359C2semidirect
11. #11727269C2semidirect
108. #2855D7semidirect
135. C22×C4×C7SD16semidirect
191. C2×C7⋊C8#41semidirect
425. C22×C4#1143semidirect
534. C7#25495semidirect

Maximal subgroups

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

subgroupindex
C2×#901682
C7×#28552
C2×#898202
C2×#893192
#117274422
#117274412
#117274222
#117273812
#117273802
#117273592
#117272692
#254957