Finite Group Databasebeta

← #13356227

Subgroups — #13356227

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2311C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4#11739668C211yesmaximal, characteristic, complemented
5C2×#89736C211yesmaximal, characteristic, complemented
6#11729333C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8#11727421C212yesmaximal, complemented
9#11727420C212yesmaximal, complemented
10#11727391C212yesmaximal, complemented
11#11727390C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×#1136C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C7×#391C2212yes—
20C7×#390C2212yes—
21C7×#386—22no—
22C7×#380—21no—
23C7×#380—21no—
24#90005—24no—
25#90004C418yescomplemented
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2212yes—
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2212yes—
32C4⋊C4×Dic7—21no—
33C4⋊C4×Dic7—21no—
34#89736—24no—
35C2×#6382C2212yes—
36C2×#6381C2212yes—
37C22⋊C4×Dic7—22no—
38C4×C4⋊Dic7C2212yes—
39C4×C4⋊Dic7C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#89319C2212yes—
44#21062—71nomaximal, Sylow
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C22×C7×Q8C2311yescharacteristic
52C22×C7×D4—21no—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7C2311yescharacteristic
57C23×Dic7—21no—
58C7×#117—22no—
59C7×#117—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4C2312yes—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4D412yes—
69C2×C42×C7C2312yes—
70C2×C22⋊Dic7—21no—
71#6489—22no—
72#6489D414yes—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7C2×C414yes—
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4×Dic7—21no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—22no—
82C2×#1136—22no—
83C2×#1136—22no—
84C2×#1136C2×C414yes—
85C22×C7⋊C8—22no—
86C22×C7⋊C8—22no—
87C22×C7⋊C8—22no—
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C7×#95—22no—
92C7×#95—22no—
93#6409—22no—
94#6409—22no—
95#6409C2×C418yescomplemented
96#6406—22no—
97#6406—22no—
98#6406C2×C418yescomplemented
99C7⋊#95—22no—
100C7⋊#95—22no—
101#6382—24no—
102#6381—24no—
103C2×#428—71no—
104C2×#417—71no—
105C2×#416—71no—
106#2311D711yescharacteristic
107#2274—71no—
108#1928—72no—
109#1927—72no—
110#1926—72no—
111#1925—72no—
112C2×#340—71no—
113#1729—71no—
114C24×C7—21no—
115C24×C7C2×D411yescharacteristic
116C7×#41—28no—
117C7×#41—28no—
118C7×#41—44no—
119C7×#41—44no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8C22×C412yes—
126C2×C7×D4—24no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4C22×C412yes—
135C22×C4×C7—21no—
136C22×C4×C7—24no—
137C22×C4×C7—22no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7—81no—
142C22×C4×C7—81no—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D412yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×C4×C7#4112yes—
151C22×C4×C7#4112yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—24no—
157C22×Dic7—81no—
158C22×Dic7—81no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—42no—
164C7×C22⋊C4—24no—
165C7×C22⋊C4—82no—
166C7×C22⋊C4—82no—
167C42×C7—22no—
168C42×C7—22no—
169C22⋊Dic7—24no—
170C4⋊Dic7—24no—
171C4⋊Dic7—22no—
172C4⋊Dic7—21no—
173C4⋊Dic7—21no—
174C4⋊Dic7C22×C414yes—
175C4⋊Dic7C2×D412yes—
176C4⋊Dic7C2×D412yes—
177C4×Dic7—24no—
178C4×Dic7—82no—
179C4×Dic7—82no—
180#1136—28no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—44no—
190C2×C7⋊C8C22×C414yes—
191C22×#41C2×D711yescharacteristic
192C22×M16—71no—
193C23×C8—71no—
194C2×#117—71no—
195C2×#117C2×D711yescharacteristic
196C22×C4⋊C4—71no—
197C22×C4⋊C4C2×D711yescharacteristic
198#428—144no—
199C2×#107—72no—
200C2×#106—72no—
201#417—78no—
202#416—144no—
203C2×#103—72no—
204C2×#103—71no—
205C2×#103—71no—
206C2×#102—72no—
207C2×#102—71no—
208C2×#102—71no—
209#391C2×D712yes—
210#390C2×D712yes—
211#386—22no—
212#380—21no—
213#380—21no—
214C4×C4⋊C4—72no—
215C4×C4⋊C4—72no—
216C4×C4⋊C4—141no—
217C4×C4⋊C4—141no—
218C4×C22⋊C4—142no—
219#340—72no—
220#340—72no—
221#336—72no—
222#336—72no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—22no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7C4×D412yes—
235C23×C7#12112yes—
236C23×C7#12312yes—
237C23×C7C22×D411yescharacteristic
238C7×Q8—22no—
239C7×Q8—22no—
240C7×Q8—41no—
241C7×Q8—41no—
242C7×D4—22no—
243C7×D4—22no—
244C7×D4—48no—
245C7×D4—41no—
246C7×D4—41no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—48no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—84no—
261C2×C4×C7#11714yes—
262C2×C4×C7C4×D414yes—
263C2×C4×C7C4×D412yes—
264C2×C4×C7#12112yes—
265C2×C4×C7#12312yes—
266C2×C4×C7#12414yes—
267C2×C4×C7#12614yes—
268C2×C4×C7C23×C411yescharacteristic
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×Dic7—24no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—41no—
280C2×Dic7—42no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—84no—
284C7⋊C8—24no—
285C7⋊C8—41no—
286C7⋊C8—41no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41C4×D714yes—
293C22×Q8C22×D711yescharacteristic
294C22×D4—21no—
295C22×D4C22×D711yescharacteristic
296C23×C4—21no—
297C23×C4—21no—
298C23×C4—141no—
299C23×C4C22×D711yescharacteristic
300C2×M16—74no—
301C2×M16—142no—
302C2×M16—142no—
303C22×C8—72no—
304C22×C8—71no—
305C22×C8—71no—
306C22×C8—142no—
307C22×C8—142no—
308C22×C8—142no—
309#117—22no—
310#117—22no—
311#117—74no—
312#117—142no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—74no—
317C2×C4⋊C4—72no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22⋊D711yescharacteristic
323C2×C4⋊C4C22⋊D711yescharacteristic
324C2×C4⋊C4C22×D712yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4C22⋊D712yes—
329C2×C42—141no—
330C2×C42—142no—
331C2×C42—142no—
332C2×C42C22×D712yes—
333#107—144no—
334#106—144no—
335#103—78no—
336#103—142no—
337#103—142no—
338#102—78no—
339#102—142no—
340#102—142no—
341#95—22no—
342#95—22no—
343#95—142no—
344#95—142no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—81no—
355C22×C7—81no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7#38814yes—
359C22×C7C2×C4×D411yescharacteristic
360C22×C7C2×#12111yescharacteristic
361C22×C7C2×#12311yescharacteristic
362C22×C7#57611yescharacteristic
363C22×C7#57711yescharacteristic
364C22×C7#57711yescharacteristic
365C22×C7#57811yescharacteristic
366C4×C7—42no—
367C4×C7—42no—
368C4×C7—42no—
369C4×C7—82no—
370C4×C7#38814yes—
371C4×C7C2×#11711yescharacteristic
372C4×C7C2×C4×D411yescharacteristic
373C4×C7C2×#12411yescharacteristic
374C4×C7C2×#12611yescharacteristic
375Dic7—44no—
376Dic7—42no—
377Dic7—42no—
378C24—21no—
379C24C2×C22⋊D711yescharacteristic
380#41—28no—
381#41—28no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C2×C4×D712yes—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4C2×C4×D712yes—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—141no—
408C22×C4—141no—
409C22×C4—142no—
410C22×C4—144no—
411C22×C4—561no—
412C22×C4—561no—
413C22×C4C2×C4×D712yes—
414C22×C4#115712yes—
415C22×C4D4×D712yes—
416C22×C4#115912yes—
417C22×C4#115912yes—
418C22×C4#115912yes—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D712yes—
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423M16—148no—
424C2×C8—74no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—284no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—74no—
440C4⋊C4—72no—
441C4⋊C4—72no—
442C4⋊C4—144no—
443C4⋊C4—142no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C22⋊C4—24no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—144no—
450C42—22no—
451C42—22no—
452C42—144no—
453C42—562no—
454C42—562no—
455C2×C7—24no—
456C2×C7—82no—
457C2×C7C2×#38811yescharacteristic
458C2×C7#294611yescharacteristic
459C2×C7#294711yescharacteristic
460C2×C7#304511yescharacteristic
461C2×C7#304611yescharacteristic
462C2×C7#309811yescharacteristic
463C2×C7#309911yescharacteristic
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23C4×C22⋊D712yes—
476C23#649812yes—
477C23#650112yes—
478C23C22×C22⋊D711yescharacteristic
479Q8—22no—
480Q8—22no—
481Q8—41no—
482Q8—41no—
483D4—22no—
484D4—22no—
485D4—48no—
486D4—41no—
487D4—41no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—144no—
503C2×C4—142no—
504C2×C4—142no—
505C2×C4—281no—
506C2×C4—282no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—282no—
510C2×C4—281no—
511C2×C4—564no—
512C2×C4#644014yes—
513C2×C4#644214yes—
514C2×C4(C22⋊D7)⋊C414yes—
515C2×C4#644814yes—
516C2×C4C4×C22⋊D712yes—
517C2×C4#649812yes—
518C2×C4#650112yes—
519C2×C4C22×C4×D711yescharacteristic
520C2×C4C2×#115711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×#115911yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4C2×#115911yescharacteristic
525C8—144no—
526C8—281no—
527C8—281no—
528C7#2106211yescharacteristic, complemented, Sylow
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#8971614yes—
543C22C2×C4×C22⋊D711yescharacteristic
544C22C2×#649811yescharacteristic
545C22C2×#650111yescharacteristic
546C22#9052611yescharacteristic
547C22#9052711yescharacteristic
548C22#9052811yescharacteristic
549C22#9052911yescharacteristic
550C4—42no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—284no—
555C4—282no—
556C4—282no—
557C4#8971614yes—
558C4C2×#644011yescharacteristic
559C4C2×#644211yescharacteristic
560C4C2×(C22⋊D7)⋊C411yescharacteristic
561C4C2×#644811yescharacteristic
562C2—24no—
563C2—82no—
564C2C2×#8971611yescharacteristic
565C2#1174000911yescharacteristic
566C2#1174001611yescharacteristic
567C2#1174083011yescharacteristic
568C2#1174085111yescharacteristic
569C2#1174086311yescharacteristic
570C2#1174086411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2311C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenter#2311D7
hypocenterC7#21062
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. #11739668C2semidirect
5. C2×#89736C2semidirect
6. #11729333C2semidirect
7. C2×#89324C2semidirect
8. #11727421C2semidirect
9. #11727420C2semidirect
10. #11727391C2semidirect
11. #11727390C2semidirect
25. #90004C4semidirect
95. #6409C2×C4semidirect
98. #6406C2×C4semidirect
528. C7#21062semidirect

Maximal subgroups

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

subgroupindex
C7×#23112
C2×#900042
C2×#900052
#117396682
C2×#897362
#117293332
C2×#893242
#117274212
#117274202
#117273912
#117273902
#210627