Finite Group Databasebeta

← #13364935

Subgroups — #13364935

Order 1792.

1092 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90002C211yesmaximal, characteristic, complemented
2C2×#89999C211yesmaximal, characteristic
3#11731310C211yesmaximal, characteristic
4C7×#1787C211yesmaximal, characteristic, complemented
5#11728030C212yesmaximal, complemented
6#11728011C212yesmaximal, complemented
7#11728008C212yesmaximal, complemented
8#11727995C212yesmaximal, complemented
9#11727686C211yesmaximal, characteristic
10Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
11#11727259C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×#1143C2212yes—
14C2×#6493C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4×Dic7C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×(C22⋊D7)⋊C4—21no—
19C2×C7×#103C2212yes—
20C2×C7×#103C2212yes—
21C2×C7×C22⋊C8C2211yescharacteristic
22C7×#380C2211yescharacteristic
23#90002—21no—
24#90002—21no—
25#90002C2212yes—
26#90002C2212yes—
27#89999C418yescomplemented
28C2×#6409C2212yes—
29C2×C7⋊#95—21no—
30C2×C7⋊#95—21no—
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6395C2212yes—
33C2×C8×Dic7C2212yes—
34#89760C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36#89745C2211yescharacteristic
37C2×#6382C2212yes—
38#89730—21no—
39#89441—21no—
40#89438—21no—
41#89346C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22041—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yescomplemented
50C22×C7×Q8C2312yes—
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C23×Dic7D411yescharacteristic
57C22×D28C2311yescharacteristic
58C22×C4×D7—21no—
59C22×C7×C8C2312yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×#1143—42no—
64C2×#1143—42no—
65C2×#1143C2×C418yescomplemented
66#6493—22no—
67#6493C2×C414yes—
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—21no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—21no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—42no—
74C2×C4⋊Dic7C2312yes—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×Dic7⋊C4—21no—
77C2×C4×Dic7—21no—
78C2×C4×Dic7—42no—
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C22×C7⋊C8C2312yes—
84#6456—42no—
85(C22⋊D7)⋊C4—44no—
86C7×#103—42no—
87C7×#103—42no—
88C7×C22⋊C8—22no—
89C7×#95—21no—
90#6409C2×C418yescomplemented
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—22no—
97C22⋊(C7⋊C8)C2×C414yescomplemented
98#6395—42no—
99C8×Dic7—42no—
100#6382—42no—
101C2×#448—71no—
102C4×#121—71no—
103C2×#391—71no—
104#1968—72no—
105#1933—72no—
106#1927—72no—
107#1877—72no—
108#1874—71no—
109#1853—71no—
110#1787D711yescharacteristic, complemented
111C4×C22⋊C8—71no—
112C24×C7C2×D411yescharacteristic
113C23×D7—21no—
114C2×C7×Q8—24no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—48no—
121C2×C7×Q8—42no—
122C2×C7×Q8—42no—
123C2×C7×Q8—48no—
124C2×C7×Q8C22×C414yes—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—24no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7#4112yes—
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—24no—
138C2×C22⋊D7—42no—
139C2×C22⋊D7—42no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—41no—
145C22×Dic7—41no—
146C22×Dic7—41no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C22×Dic7SD1614yescomplemented
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C2×D28—22no—
156C2×D28—22no—
157C2×D28—21no—
158C2×D28C22×C412yes—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×C4×D7—42no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C22⋊C4—42no—
168#1143—48no—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—42no—
171(C2×D7)⋊C4—42no—
172(C2×D7)⋊C4—42no—
173C4⋊Dic7—21no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—42no—
177C4⋊Dic7C22×C412yes—
178Dic7⋊C4—42no—
179C4×Dic7—22no—
180C4×Dic7—22no—
181C4×Dic7—41no—
182C4×Dic7—41no—
183C4×Dic7—42no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8C22×C414yes—
187C23×Q8C2×D711yescharacteristic
188C22×SD16—72no—
189C2×#121—71no—
190C2×C4×D4—71no—
191C2×C4×D4—141no—
192C22×C4⋊C4—71no—
193C22×C42—71no—
194#448—78no—
195C2×#106—72no—
196C2×#103—72no—
197C2×#103C2×D712yes—
198C2×#103C2×D712yes—
199C2×#102—72no—
200C2×C22⋊C8—71no—
201C2×C22⋊C8C2×D711yescharacteristic
202C2×C4×C8—72no—
203#391—72no—
204#391—72no—
205#391—141no—
206#391—141no—
207#390—71no—
208#389—71no—
209#388—141no—
210#386—141no—
211#380C2×D711yescharacteristic
212C4×C4⋊C4—71no—
213C4×C22⋊C4—141no—
214C2×#95—141no—
215C2×#95—141no—
216#340—72no—
217#340—72no—
218#336—72no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7C4×D412yes—
224C23×C7C2×SD1612yes—
225C23×C7#13512yes—
226C23×C7C22×D411yescharacteristic
227C22×D7—22no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C7×Q8—24no—
235C7×Q8—24no—
236C7×Q8—48no—
237C7×Q8—42no—
238C7×Q8—42no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—48no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7C4×D412yes—
254C2×C4×C7C23×C411yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7H(5,𝔽2)12yes—
257C2×C4×C7H(5,𝔽2)12yes—
258C22⋊D7—48no—
259C2×Dic7—21no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—21no—
264C2×Dic7—21no—
265C2×Dic7—22no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—44no—
270C2×Dic7—44no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—84no—
277C2×Dic7C2×SD1612yes—
278C2×Dic7C2×SD1612yes—
279C2×Dic7#13712yes—
280D28—24no—
281D28—42no—
282C4×D7—84no—
283C7×C8—42no—
284C7⋊C8—24no—
285C22×Q8—21no—
286C22×Q8—22no—
287C22×Q8—42no—
288C22×Q8—42no—
289C22×Q8C4×D714yescomplemented
290C22×Q8C22×D712yes—
291C22×D4—71no—
292C22×D4—141no—
293C23×C4—21no—
294C23×C4—71no—
295C23×C4—71no—
296C23×C4—141no—
297C23×C4—141no—
298C23×C4C22×D711yescharacteristic
299C2×SD16—78no—
300C2×SD16—282no—
301C2×SD16—282no—
302C22×C8—72no—
303C22×C8C22×D712yes—
304#121—74no—
305#121—142no—
306C4×D4—282no—
307C4×D4—284no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—72no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—282no—
313C2×C4⋊C4C22×D711yescharacteristic
314C2×C4⋊C4C22×D711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—141no—
317C2×C22⋊C4—141no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4—141no—
320C2×C42—71no—
321C2×C42—71no—
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—141no—
325C2×C42—282no—
326#106—282no—
327#103—42no—
328#103—42no—
329#103—78no—
330#102—282no—
331C22⋊C8—22no—
332C22⋊C8—74no—
333C4×C8—282no—
334#95—21no—
335#95—142no—
336#95—142no—
337#95—142no—
338#95—142no—
339#95—141no—
340#95—141no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7C4×SD1614yes—
348C22×C7C2×C4×D411yescharacteristic
349C22×C7#53411yescharacteristic
350C22×C7#53511yescharacteristic
351C22×C7C22×SD1611yescharacteristic
352C22×C7C2×#13511yescharacteristic
353C22×C7C2×#13711yescharacteristic
354C22×C7#57511yescharacteristic
355C2×D7—42no—
356C2×D7—42no—
357C2×D7—42no—
358C2×D7—81no—
359C2×D7—81no—
360C2×D7—81no—
361C2×D7—81no—
362C4×C7—22no—
363C4×C7—44no—
364C4×C7—81no—
365C4×C7—81no—
366C4×C7#51812yes—
367C4×C7#53812yes—
368Dic7—22no—
369Dic7—22no—
370Dic7—22no—
371Dic7—41no—
372Dic7—42no—
373Dic7—82no—
374C24—141no—
375C24D4×D711yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8C2×C4×D714yes—
387C2×D4—72no—
388C2×D4—144no—
389C2×D4—141no—
390C2×D4—142no—
391C2×D4—142no—
392C2×D4—282no—
393C2×D4—282no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—71no—
405C22×C4—74no—
406C22×C4—71no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—141no—
410C22×C4—142no—
411C22×C4—282no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4—281no—
415C22×C4—281no—
416C22×C4—281no—
417C22×C4—282no—
418C22×C4—282no—
419C22×C4—282no—
420C22×C4C2×C4×D712yes—
421C22×C4D4×D711yescharacteristic
422C22×C4#115912yes—
423C22×C4C23×D711yescharacteristic
424SD16—288no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—74no—
429C2×C8—142no—
430C2×C8—142no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—72no—
434C4⋊C4—141no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—282no—
438C4⋊C4—282no—
439C22⋊C4—42no—
440C22⋊C4—144no—
441C22⋊C4—282no—
442C22⋊C4—282no—
443C22⋊C4—282no—
444C42—142no—
445C42—142no—
446C42—281no—
447C42—281no—
448C42—282no—
449C2×C7—24no—
450C2×C7#232111yescharacteristic
451C2×C7C2×C4×SD1611yescharacteristic
452C2×C7C4×#13711yescharacteristic
453C2×C7#318711yescharacteristic
454C2×C7#319011yescharacteristic
455C2×C7#320611yescharacteristic
456C2×C7#321111yescharacteristic
457D7—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—142no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23—281no—
468C23—281no—
469C23(C22⋊D7)⋊C412yes—
470C23D7×SD1612yes—
471C23#648212yes—
472C23C2×D4×D711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—144no—
479D4—288no—
480D4—282no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—72no—
496C2×C4—72no—
497C2×C4—72no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—141no—
501C2×C4—141no—
502C2×C4—141no—
503C2×C4—141no—
504C2×C4—142no—
505C2×C4—141no—
506C2×C4—141no—
507C2×C4—141no—
508C2×C4—284no—
509C2×C4—284no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—564no—
516C2×C4—564no—
517C2×C4(C22⋊D7)⋊C412yes—
518C2×C4C22×C4×D711yescharacteristic
519C2×C4C2×#115911yescharacteristic
520C2×C4#654812yes—
521C2×C4#654812yes—
522C8—42no—
523C8—144no—
524C7#2204111yescharacteristic, complemented, Sylow
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—282no—
532C22—282no—
533C22—282no—
534C22—561no—
535C22—561no—
536C22—561no—
537C22—561no—
538C22#8956014yes—
539C22C2×(C22⋊D7)⋊C411yescharacteristic
540C22#9028011yescharacteristic
541C22#9028311yescharacteristic
542C22C2×D7×SD1611yescharacteristic
543C22C2×#648211yescharacteristic
544C22D7×#13711yescharacteristic
545C22#9046811yescharacteristic
546C4—22no—
547C4—44no—
548C4—81no—
549C4—81no—
550C4—142no—
551C4—142no—
552C4—142no—
553C4—281no—
554C4—282no—
555C4—562no—
556C4#9017712yes—
557C4#9028412yes—
558C2—24no—
559C2—562no—
560C2#1173126611yescharacteristic
561C2C2×#8956011yescharacteristic
562C2#1173639111yescharacteristic
563C2#1173747911yescharacteristic
564C2#1173748811yescharacteristic
565C2#1173763311yescharacteristic
566C2#1173764611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1787C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenter#1787D7
hypocenterC7#22041
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C2×#90002C2semidirect
4. C7×#1787C2semidirect
5. #11728030C2semidirect
6. #11728011C2semidirect
7. #11728008C2semidirect
8. #11727995C2semidirect
10. Dic7×C22⋊C8C2semidirect
11. #11727259C2semidirect
27. #89999C4semidirect
49. C22×C7×Q8C2×C4semidirect
65. C2×#1143C2×C4semidirect
90. #6409C2×C4semidirect
97. C22⋊(C7⋊C8)C2×C4semidirect
110. #1787D7semidirect
150. C22×Dic7SD16semidirect
289. C22×Q8C4×D7semidirect
524. C7#22041semidirect

Maximal subgroups

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

subgroupindex
C2×#900022
C2×#899992
#117313102
C7×#17872
#117280302
#117280112
#117280082
#117279952
#117276862
Dic7×C22⋊C82
#117272592
#220417