Finite Group Databasebeta

← #13361001

Subgroups — #13361001

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×#6493C211yesmaximal, characteristic
2C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
3C2×#89507C211yesmaximal, characteristic
4C7×#1749C211yesmaximal, characteristic, complemented
5#11727686C211yesmaximal, characteristic
6#11727681C211yesmaximal, characteristic, complemented
7#11727679C211yesmaximal, characteristic, complemented
8#11727568C212yesmaximal, complemented
9#11727566C212yesmaximal, complemented
10#11726379C212yesmaximal, complemented
11#11726377C212yesmaximal, complemented
12C22×C42×C7C2211yescharacteristic
13C2×#6493C2211yescharacteristic
14C2×C4×C22⋊D7—21no—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7—21no—
18C2×C4×D28C2211yescharacteristic
19C2×C7×C22⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2211yescharacteristic
21C2×C7×#97C2212yes—
22#89985—21no—
23#89984C2211yescharacteristic
24C2×C7⋊#95—21no—
25C2×#6395C2212yes—
26C2×#6395C2212yes—
27C2×#6392C2212yes—
28C2×#6391C2212yes—
29#89712—21no—
30#89711C2211yescharacteristic
31C4×(C2×D7)⋊C4—21no—
32C4×C4⋊Dic7—21no—
33C4×C4⋊Dic7—21no—
34C4×C4⋊Dic7C2212yes—
35C4×C4⋊Dic7C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37#89700C2211yescharacteristic
38#89507C418yescomplemented
39C7×#336C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#21580—71nomaximal, Sylow
45C23×C4×C7D411yescharacteristic
46C23×C4×C7D411yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C22×C22⋊D7—21no—
49C23×Dic7—21no—
50C23×Dic7—21no—
51C22×D28C2311yescharacteristic
52C22×C4×D7—21no—
53C22×C7×C8C2312yes—
54C22×C7×C8C2312yes—
55C2×C42×C7—24no—
56C2×C42×C7D411yescharacteristic
57C2×C42×C7D411yescharacteristic
58C2×C42×C7C2311yescharacteristic
59C2×C42×C7C2311yescharacteristic
60C2×C22⋊Dic7—21no—
61#6493—22no—
62#6493C2×C414yes—
63C4×C22⋊D7—44no—
64C2×(C2×D7)⋊C4—21no—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7—24no—
72C2×C4⋊Dic7C2×C414yescomplemented
73C2×C4⋊Dic7C2311yescharacteristic
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2312yes—
78C2×Dic7⋊C4—21no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—44no—
83C4×D28—24no—
84C7×C22⋊C8—22no—
85C7×C22⋊C8C2×C414yescomplemented
86C7×#97—24no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92#6395—24no—
93#6395C2×C418yescomplemented
94#6392—24no—
95#6391C2×C418yescomplemented
96C2×#481—71no—
97C4×#121—71no—
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#1749D711yescharacteristic, complemented
107C24×C7C2×D411yescharacteristic
108C23×D7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—41no—
117C22×C4×C7C22×C412yes—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7#4112yes—
124C22×C4×C7C2411yescharacteristic
125C2×C22⋊D7—24no—
126C2×C22⋊D7—42no—
127C2×C22⋊D7—42no—
128C22×Dic7—24no—
129C22×Dic7—24no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7—42no—
136C22×Dic7—42no—
137C22×Dic7—42no—
138C22×Dic7—42no—
139C2×D28—22no—
140C2×D28—22no—
141C2×D28—21no—
142C2×D28C22×C412yes—
143C2×C4×D7—42no—
144C2×C4×D7—42no—
145C2×C4×D7—42no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C2×C7×C8—22no—
151C2×C7×C8C22×C414yes—
152C42×C7—22no—
153C42×C7—22no—
154C42×C7—22no—
155C42×C7#4114yes—
156C22⋊Dic7—42no—
157(C2×D7)⋊C4—24no—
158(C2×D7)⋊C4—42no—
159(C2×D7)⋊C4—42no—
160(C2×D7)⋊C4—42no—
161C4⋊Dic7—24no—
162C4⋊Dic7—24no—
163C4⋊Dic7—22no—
164C4⋊Dic7—22no—
165C4⋊Dic7—21no—
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4⋊Dic7C22×C414yes—
169C4⋊Dic7C22×C412yes—
170C4⋊Dic7#4114yes—
171Dic7⋊C4—42no—
172C4×Dic7—44no—
173C4×Dic7—42no—
174C4×Dic7—44no—
175C2×#121—71no—
176C2×C4×D4—71no—
177C2×C4×D4—141no—
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⋊C8C2×D711yescharacteristic
188C2×C22⋊C8C2×D711yescharacteristic
189C2×#97C2×D712yes—
190#390—71no—
191#389—71no—
192#388—141no—
193#386—141no—
194#380—71no—
195C4×C4⋊C4—72no—
196C4×C4⋊C4—72no—
197C4×C4⋊C4—71no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4—141no—
200C4×C22⋊C4—141no—
201C2×#95—141no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340—72no—
206#336C2×D712yes—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—21no—
211C23×C7C4×D412yes—
212C23×C7AGL(1,ℤ/8)12yes—
213C23×C7#13712yes—
214C23×C7C22×D411yescharacteristic
215C22×D7—22no—
216C22×D7—41no—
217C22×D7—41no—
218C22×D7—41no—
219C22×D7—41no—
220C22×D7—41no—
221C22×D7—41no—
222C2×C4×C7—28no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—44no—
237C2×C4×C7C4×D412yes—
238C2×C4×C7#12312yes—
239C2×C4×C7#12312yes—
240C2×C4×C7AGL(1,ℤ/8)12yes—
241C2×C4×C7C23×C411yescharacteristic
242C2×C4×C7C2×#4111yescharacteristic
243C2×C4×C7C2×#4112yes—
244C2×C4×C7C2×#4112yes—
245C22⋊D7—48no—
246C2×Dic7—24no—
247C2×Dic7—22no—
248C2×Dic7—24no—
249C2×Dic7—48no—
250C2×Dic7—48no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—41no—
254C2×Dic7—42no—
255C2×Dic7—41no—
256C2×Dic7—42no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263D28—24no—
264D28—42no—
265C4×D7—84no—
266C7×C8—24no—
267C7×C8—42no—
268C22×D4—71no—
269C22×D4—141no—
270C23×C4—141no—
271C23×C4—141no—
272C23×C4—141no—
273C23×C4D2811yescharacteristic
274C23×C4D2811yescharacteristic
275C23×C4C22×D711yescharacteristic
276C22×C8C22×D712yes—
277C22×C8C22×D712yes—
278#121—74no—
279#121—142no—
280C4×D4—144no—
281C4×D4—284no—
282C2×C4⋊C4—74no—
283C2×C4⋊C4—72no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—71no—
286C2×C4⋊C4—71no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—144no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—142no—
292C2×C4⋊C4—141no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C22⋊C4—141no—
297C2×C22⋊C4—141no—
298C2×C42—24no—
299C2×C42—141no—
300C2×C42—142no—
301C2×C42—142no—
302C2×C42—284no—
303C2×C42D2811yescharacteristic
304C2×C42D2811yescharacteristic
305C2×C42C22×D711yescharacteristic
306C2×C42C22×D711yescharacteristic
307#107—144no—
308#106—78no—
309#102—78no—
310#102—144no—
311C22⋊C8—22no—
312C22⋊C8C4×D714yescomplemented
313#97—24no—
314#95—142no—
315#95—142no—
316#95—141no—
317#95—141no—
318#95—141no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7#44014yes—
326C22×C7C2×C4×D411yescharacteristic
327C22×C7C2×#12311yescharacteristic
328C22×C7#52511yescharacteristic
329C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
330C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C7C2×#13711yescharacteristic
332C22×C7#57511yescharacteristic
333C2×D7—42no—
334C2×D7—42no—
335C2×D7—42no—
336C2×D7—81no—
337C2×D7—81no—
338C2×D7—81no—
339C2×D7—81no—
340C4×C7—22no—
341C4×C7—22no—
342C4×C7—22no—
343C4×C7—22no—
344C4×C7—41no—
345C4×C7C4×#4112yes—
346C4×C7#52912yes—
347Dic7—44no—
348Dic7—42no—
349Dic7—44no—
350Dic7—81no—
351Dic7—81no—
352C24—141no—
353C24C2×D2811yescharacteristic
354C2×D4—72no—
355C2×D4—144no—
356C2×D4—141no—
357C2×D4—142no—
358C2×D4—142no—
359C2×D4—282no—
360C2×D4—282no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—144no—
370C22×C4—144no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—141no—
374C22×C4—141no—
375C22×C4—141no—
376C22×C4—282no—
377C22×C4—282no—
378C22×C4—282no—
379C22×C4—282no—
380C22×C4—282no—
381C22×C4—282no—
382C22×C4—282no—
383C22×C4C2×C4×D712yes—
384C22×C4C2×D2811yescharacteristic
385C22×C4C2×D2811yescharacteristic
386C22×C4C2×D2811yescharacteristic
387C22×C4C2×D2811yescharacteristic
388C22×C4C2×D2811yescharacteristic
389C22×C4#115712yes—
390C22×C4C23×D711yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8C2×C4×D714yes—
397C4⋊C4—74no—
398C4⋊C4—72no—
399C4⋊C4—74no—
400C4⋊C4—144no—
401C4⋊C4—144no—
402C4⋊C4—142no—
403C4⋊C4—141no—
404C4⋊C4—142no—
405C4⋊C4—142no—
406C4⋊C4—142no—
407C4⋊C4—282no—
408C22⋊C4—144no—
409C22⋊C4—282no—
410C22⋊C4—282no—
411C22⋊C4—282no—
412C22⋊C4—282no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—284no—
417C42—282no—
418C42—284no—
419C42#115914yes—
420C2×C7—24no—
421C2×C7C4×#12311yescharacteristic
422C2×C7C2×#44011yescharacteristic
423C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
424C2×C7#309211yescharacteristic
425C2×C7#309411yescharacteristic
426C2×C7#310911yescharacteristic
427C2×C7#311711yescharacteristic
428D7—82no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—142no—
434C23—281no—
435C23—281no—
436C23—281no—
437C23—281no—
438C23—281no—
439C23—281no—
440C23C4×D2812yes—
441C23#647112yes—
442C23#647212yes—
443C23C22×D2811yescharacteristic
444D4—144no—
445D4—288no—
446D4—282no—
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—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—44no—
462C2×C4—142no—
463C2×C4—144no—
464C2×C4—144no—
465C2×C4—288no—
466C2×C4—288no—
467C2×C4—282no—
468C2×C4—282no—
469C2×C4—281no—
470C2×C4—282no—
471C2×C4—281no—
472C2×C4—282no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—281no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4—564no—
480C2×C4C4×D2812yes—
481C2×C4#644912yes—
482C2×C4#644912yes—
483C2×C4#647112yes—
484C2×C4C22×C4×D711yescharacteristic
485C2×C4C2×#115711yescharacteristic
486C2×C4C2×#115912yes—
487C2×C4D7×#4112yes—
488C8—24no—
489C8—42no—
490C7#2158011yescharacteristic, complemented, Sylow
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—282no—
498C22—282no—
499C22—282no—
500C22—561no—
501C22—561no—
502C22—561no—
503C22—561no—
504C22#8948214yes—
505C22C2×C4×D2811yescharacteristic
506C22C2×#644911yescharacteristic
507C22#9021511yescharacteristic
508C22C2×#647111yescharacteristic
509C22C2×#647111yescharacteristic
510C22C2×#647211yescharacteristic
511C22#9043911yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—41no—
517C4—282no—
518C4—284no—
519C4—284no—
520C4—561no—
521C4—561no—
522C4C4×#115912yes—
523C4#9022912yes—
524C2—24no—
525C2—562no—
526C2C4×#644911yescharacteristic
527C2C2×#8948211yescharacteristic
528C2C4×#647111yescharacteristic
529C2#1173606711yescharacteristic
530C2#1173606911yescharacteristic
531C2#1173678811yescharacteristic
532C2#1173680211yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1749C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenter#1749D7
hypocenterC7#21580
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic7C2semidirect
4. C7×#1749C2semidirect
6. #11727681C2semidirect
7. #11727679C2semidirect
8. #11727568C2semidirect
9. #11727566C2semidirect
10. #11726379C2semidirect
11. #11726377C2semidirect
38. #89507C4semidirect
72. C2×C4⋊Dic7C2×C4semidirect
85. C7×C22⋊C8C2×C4semidirect
93. #6395C2×C4semidirect
95. #6391C2×C4semidirect
106. #1749D7semidirect
312. C22⋊C8C4×D7semidirect
490. C7#21580semidirect

Maximal subgroups

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

subgroupindex
C4×#64932
C2×C4×C4⋊Dic72
C2×#895072
C7×#17492
#117276862
#117276812
#117276792
#117275682
#117275662
#117263792
#117263772
#215807