Finite Group Databasebeta

← #13360995

Subgroups — #13360995

Order 1792.

1048 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#121C211yesmaximal, characteristic
2C2×#89806C211yesmaximal, characteristic
3C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
4#11729431C211yesmaximal, characteristic
5#11729363C212yesmaximal, complemented
6#11729283C211yesmaximal, characteristic, complemented
7#11727390C212yesmaximal, complemented
8#11727356C212yesmaximal, complemented
9#11727258C211yesmaximal, characteristic, complemented
10#11727256C211yesmaximal, characteristic, complemented
11#11727235C212yesmaximal, complemented
12C2×C7×#121C2211yescharacteristic
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4×Dic7—21no—
19C7×#390C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#388—21no—
22C7×#386—21no—
23C7×#380C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C22⋊C4—21no—
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28C2×C7⋊#95—21no—
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31#89806C418yescomplemented
32C2×#6382C2212yes—
33C2×#6381C2212yes—
34C4×C4⋊Dic7—21no—
35C4×C4⋊Dic7—21no—
36C4×C4⋊Dic7C2212yes—
37C4×C4⋊Dic7C2212yes—
38C2×#6377C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89319C2212yes—
44#21580—71nomaximal, Sylow
45C22×C7×D4—21no—
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×Dic7—21no—
52C23×Dic7—21no—
53C7×#121—22no—
54C7×#121C2×C414yes—
55C4×C7×D4—24no—
56C4×C7×D4—44no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—21no—
66C2×C42×C7—24no—
67C2×C42×C7—21no—
68C2×C42×C7D411yescharacteristic
69C2×C42×C7D411yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—24no—
76C2×C4⋊Dic7C2×C414yescomplemented
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2312yes—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—44no—
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2312yes—
84C7×#95—21no—
85C7×#95—21no—
86C7×#95—21no—
87#6406—24no—
88#6406C2×C418yescomplemented
89C7⋊#95—22no—
90C7⋊#95—22no—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2×C414yescomplemented
93#6382C2×C418yescomplemented
94#6381—24no—
95#6377—24no—
96C2×#481—71no—
97C4×#121D711yescharacteristic
98C2×C4×C4⋊C4—71no—
99#1925—72no—
100#1920—72no—
101#1874—71no—
102#1852—71no—
103#1824—71no—
104#1776—72no—
105#1762—72no—
106#1749—71no—
107C24×C7—21no—
108C24×C7C2×D411yescharacteristic
109C2×C7×D4—24no—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4—21no—
113C2×C7×D4—42no—
114C2×C7×D4—42no—
115C2×C7×D4C22×C412yes—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—41no—
127C22×C4×C7—42no—
128C22×C4×C7—42no—
129C22×C4×C7—42no—
130C22×C4×C7C22×C412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7#4112yes—
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—24no—
139C22×Dic7—24no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—42no—
143C22×Dic7—42no—
144C22×Dic7—42no—
145C22×Dic7—42no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—42no—
150C7×C4⋊C4C22×C412yes—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—42no—
153C7×C22⋊C4—42no—
154C7×C22⋊C4—42no—
155C7×C22⋊C4—42no—
156C42×C7—22no—
157C42×C7—22no—
158C42×C7—22no—
159C42×C7—42no—
160C42×C7#4114yes—
161C4⋊Dic7—24no—
162C4⋊Dic7—24no—
163C4⋊Dic7—22no—
164C4⋊Dic7—22no—
165C4⋊Dic7C22×C414yes—
166C4⋊Dic7#4114yes—
167C4×Dic7—44no—
168C4×Dic7—44no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8C22×C414yes—
175C2×#121C2×D711yescharacteristic
176C2×C4×D4—21no—
177C2×C4×D4C2×D711yescharacteristic
178C22×C4⋊C4—71no—
179C22×C4⋊C4—71no—
180C22×C42—141no—
181C22×C42C2×D711yescharacteristic
182#481—78no—
183C2×#107—72no—
184C2×#106—72no—
185C2×#102—72no—
186C2×#102—72no—
187C2×C22⋊C8—71no—
188C2×C22⋊C8—71no—
189C2×#97—72no—
190#390C2×D711yescharacteristic
191#389C2×D711yescharacteristic
192#388—21no—
193#386—21no—
194#380C2×D711yescharacteristic
195C4×C4⋊C4—72no—
196C4×C4⋊C4—72no—
197C4×C4⋊C4—141no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4C2×D711yescharacteristic
200C4×C22⋊C4—21no—
201C2×#95—141no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340—72no—
206#336—72no—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—21no—
211C23×C7—22no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7C4×D412yes—
219C23×C7AGL(1,ℤ/8)12yes—
220C23×C7#13712yes—
221C23×C7C22×D411yescharacteristic
222C7×D4—24no—
223C7×D4—48no—
224C7×D4—42no—
225C2×C4×C7—28no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—44no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—84no—
250C2×C4×C7C4×D412yes—
251C2×C4×C7#12312yes—
252C2×C4×C7#12312yes—
253C2×C4×C7AGL(1,ℤ/8)12yes—
254C2×C4×C7C23×C411yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7C2×#4112yes—
257C2×C4×C7C2×#4112yes—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—48no—
261C2×Dic7—48no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C7⋊C8—24no—
267C7⋊C8—42no—
268C22×D4—21no—
269C22×D4C22×D711yescharacteristic
270C23×C4—21no—
271C23×C4—141no—
272C23×C4—141no—
273C23×C4C22⋊D711yescharacteristic
274C23×C4C22⋊D711yescharacteristic
275C23×C4C22×D711yescharacteristic
276C22×C8—72no—
277C22×C8—72no—
278#121—22no—
279#121C4×D714yes—
280C4×D4—24no—
281C4×D4—44no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—74no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—144no—
287C2×C4⋊C4—142no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4C22×D711yescharacteristic
291C2×C4⋊C4C22×D711yescharacteristic
292C2×C4⋊C4C22×D711yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C42—24no—
299C2×C42—21no—
300C2×C42—142no—
301C2×C42—142no—
302C2×C42—284no—
303C2×C42C22⋊D711yescharacteristic
304C2×C42C22⋊D711yescharacteristic
305C2×C42C22×D711yescharacteristic
306C2×C42C22×D711yescharacteristic
307#107—144no—
308#106—78no—
309#102—78no—
310#102—144no—
311C22⋊C8—74no—
312C22⋊C8—142no—
313#97—144no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—142no—
318#95—142no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—42no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7—81no—
329C22×C7—81no—
330C22×C7—81no—
331C22×C7—81no—
332C22×C7#44014yes—
333C22×C7C2×C4×D411yescharacteristic
334C22×C7C2×#12311yescharacteristic
335C22×C7#52511yescharacteristic
336C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
337C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
338C22×C7C2×#13711yescharacteristic
339C22×C7#57511yescharacteristic
340C4×C7—22no—
341C4×C7—22no—
342C4×C7—22no—
343C4×C7—22no—
344C4×C7—41no—
345C4×C7—42no—
346C4×C7—81no—
347C4×C7—81no—
348C4×C7C4×#4112yes—
349C4×C7#52912yes—
350Dic7—44no—
351Dic7—44no—
352C24—21no—
353C24C2×C22⋊D711yescharacteristic
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4C2×C4×D712yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—144no—
376C22×C4—144no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—282no—
380C22×C4—282no—
381C22×C4—282no—
382C22×C4—282no—
383C22×C4C2×C4×D712yes—
384C22×C4#115712yes—
385C22×C4C2×C22⋊D711yescharacteristic
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388C22×C4C2×C22⋊D711yescharacteristic
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C23×D711yescharacteristic
391C2×C8—74no—
392C2×C8—142no—
393C2×C8—142no—
394C2×C8—142no—
395C2×C8—142no—
396C2×C8—142no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—74no—
402C4⋊C4—74no—
403C4⋊C4—144no—
404C4⋊C4—144no—
405C4⋊C4—142no—
406C4⋊C4—142no—
407C4⋊C4C2×C4×D712yes—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—42no—
417C42—284no—
418C42—284no—
419C42#115914yes—
420C2×C7—24no—
421C2×C7—82no—
422C2×C7C4×#12311yescharacteristic
423C2×C7C2×#44011yescharacteristic
424C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
425C2×C7#309211yescharacteristic
426C2×C7#309411yescharacteristic
427C2×C7#310911yescharacteristic
428C2×C7#311711yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C4×C22⋊D712yes—
441C23#649512yes—
442C23#651412yes—
443C23C22×C22⋊D711yescharacteristic
444D4—24no—
445D4—48no—
446D4—42no—
447C2×C4—28no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—84no—
472C2×C4—144no—
473C2×C4—144no—
474C2×C4—288no—
475C2×C4—288no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4C4×C22⋊D712yes—
481C2×C4#649812yes—
482C2×C4#649812yes—
483C2×C4#651212yes—
484C2×C4C22×C4×D711yescharacteristic
485C2×C4C2×#115711yescharacteristic
486C2×C4C2×#115912yes—
487C2×C4D7×#4112yes—
488C8—144no—
489C8—282no—
490C7#2158011yescharacteristic, complemented, Sylow
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#8979014yes—
505C22C2×C4×C22⋊D711yescharacteristic
506C22C2×#649511yescharacteristic
507C22C2×#649811yescharacteristic
508C22C2×#651211yescharacteristic
509C22#9051311yescharacteristic
510C22C2×#651411yescharacteristic
511C22#9051911yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—41no—
518C4—81no—
519C4—81no—
520C4—284no—
521C4—284no—
522C4C4×#115912yes—
523C4#9022912yes—
524C2—24no—
525C2—82no—
526C2C2×#8979011yescharacteristic
527C2C4×#649811yescharacteristic
528C2C4×#651211yescharacteristic
529C2#1174006111yescharacteristic
530C2#1174035911yescharacteristic
531C2#1174059811yescharacteristic
532C2#1174074511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#121C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenterC4×#121D7
hypocenterC7#21580
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
3. C2×C4×C4⋊Dic7C2semidirect
5. #11729363C2semidirect
6. #11729283C2semidirect
7. #11727390C2semidirect
8. #11727356C2semidirect
9. #11727258C2semidirect
10. #11727256C2semidirect
11. #11727235C2semidirect
31. #89806C4semidirect
76. C2×C4⋊Dic7C2×C4semidirect
88. #6406C2×C4semidirect
92. C22⋊(C7⋊C8)C2×C4semidirect
93. #6382C2×C4semidirect
490. C7#21580semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1212
C2×#898062
C2×C4×C4⋊Dic72
#117294312
#117293632
#117292832
#117273902
#117273562
#117272582
#117272562
#117272352
#215807