Finite Group Databasebeta

← #13336632

Subgroups — #13336632

Order 1792.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#476C211yesmaximal, characteristic, complemented
2#11740644C211yesmaximal, characteristic, complemented
3#11739753C211yesmaximal, characteristic
4D4×#1136C211yesmaximal, characteristic, complemented
5#11729528C211yesmaximal, characteristic, complemented
6#11728893C212yesmaximal, complemented
7#11728892C212yesmaximal, complemented
8#11728889C212yesmaximal, complemented
9#11728888C212yesmaximal, complemented
10#11728186C211yesmaximal, characteristic, complemented
11#11728174C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C22×#1136—21no—
15#90168C2211yescharacteristic
16C7×#476—21no—
17C7×#476—21no—
18C7×#476C2212yes—
19C7×#476C2212yes—
20C2×C7×#106—21no—
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×#102—21no—
23#89989—21no—
24#89979—21no—
25#89865C2212yescomplemented
26#89861—22no—
27#89860C2212yescomplemented
28#89787C2211yescharacteristic
29#89787C2211yescharacteristic
30C4×#1141—21no—
31C4×#1141—21no—
32#89784C2212yes—
33#89783—22no—
34D4×C7⋊C8C2212yes—
35#89779C2211yescharacteristic
36#89779C2211yescharacteristic
37#89739—21no—
38#89738—21no—
39#89738—21no—
40#89697C2211yescharacteristic
41#89696C2211yescharacteristic
42C4×#1136C2211yescharacteristic
43#89341C2212yescomplemented
44#89341C2212yescomplemented
45#89281C2212yescomplemented
46#89280C2212yescomplemented
47#89275C2212yescomplemented
48#89274C2212yescomplemented
49#19168—71nomaximal, Sylow
50C22×C7×Q8—21no—
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—21no—
53C2×#1157—21no—
54C22×C7×C8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8C2312yes—
59C4×C7×D4—21no—
60C4×C7×D4—21no—
61C4×C7×D4D411yescharacteristic
62C4×C7×D4D411yescharacteristic
63C4×C7×D4C2311yescharacteristic
64C4×C7×D4C2311yescharacteristic
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C22⋊C4—21no—
69C2×C42×C7C2311yescharacteristic
70C2×#1141—21no—
71C2×#1141—21no—
72#6493—21no—
73#6493—21no—
74C2×#1136—21no—
75C2×#1136—22no—
76C2×#1136—44no—
77C2×#1136D411yescharacteristic
78C2×#1136D411yescharacteristic
79C22×C7⋊C8—22no—
80#6438—21no—
81#6437C2311yescharacteristic
82C4×D28—21no—
83C4×D28—21no—
84#6432C2311yescharacteristic
85C7×#106—22no—
86C7×#106—22no—
87C7×C4⋊C8—21no—
88C7×C4⋊C8C2312yes—
89C7×#102—22no—
90C7×#102—22no—
91#6406—21no—
92#6406—21no—
93C22⋊(C7⋊C8)—21no—
94C22⋊(C7⋊C8)—22no—
95C22⋊(C7⋊C8)—22no—
96#6384—22no—
97#6384—22no—
98#6383—21no—
99#6383—21no—
100#6383—22no—
101#6383—22no—
102#6381—21no—
103#6381—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)C2312yes—
106C4⋊(C7⋊C8)C2311yescharacteristic
107C4⋊(C7⋊C8)C2311yescharacteristic
108#6377D412yes—
109C4×C7⋊C8—21no—
110C4×C7⋊C8C2312yes—
111#3141—71no—
112#3100—71no—
113C2×#476D711yescharacteristic, complemented
114D4×M16—71no—
115#1703—72no—
116#1697—72no—
117#1683—72no—
118#1677—72no—
119#1565—71no—
120#1548—71no—
121#1547—71no—
122C24×C7—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—41no—
125C2×C7×Q8—82no—
126C2×C7×D4—22no—
127C2×C7×D4—21no—
128C2×C7×D4—41no—
129C2×C7×D4C2×D411yescharacteristic
130C2×C7×D4C2×D411yescharacteristic
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7—41no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7#4111yescharacteristic
142C2×C22⋊D7—41no—
143#1157—42no—
144#1157—42no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28—21no—
148C2×C4×D7—41no—
149C2×Dic14—21no—
150C2×C7×C8—22no—
151C2×C7×C8—42no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—41no—
156C7×C4⋊C4—41no—
157C7×C4⋊C4C2×D411yescharacteristic
158C7×C4⋊C4C2×D411yescharacteristic
159C7×C22⋊C4—21no—
160C7×C22⋊C4—21no—
161C7×C22⋊C4—21no—
162C7×C22⋊C4—21no—
163C42×C7C2×D411yescharacteristic
164C42×C7C2×D411yescharacteristic
165C42×C7#4111yescharacteristic
166C42×C7C2411yescharacteristic
167#1141—42no—
168#1141—42no—
169(C2×D7)⋊C4—41no—
170(C2×D7)⋊C4—41no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173#1136—22no—
174#1136—22no—
175#1136—44no—
176#1136—42no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—44no—
181C2×C7⋊C8C2×D412yes—
182C2×C7⋊C8C2×D412yes—
183C22×M16—141no—
184#532—71no—
185C2×C4⋊Q8C2×D711yescharacteristic
186C2×C4×D4C2×D711yescharacteristic
187#489—142no—
188#487—72no—
189#486—72no—
190#480—141no—
191#476—21no—
192#476—21no—
193#476C2×D712yes—
194#476C2×D712yes—
195#474—71no—
196#474—71no—
197#459—71no—
198#459—71no—
199#447—141no—
200C4×D8—141no—
201C4×D8—141no—
202#436—72no—
203#435—142no—
204C8×D4—72no—
205#431—71no—
206C2×#106—21no—
207#423—71no—
208C2×C4⋊C8C2×D711yescharacteristic
209#417—141no—
210#417—141no—
211C2×#102—21no—
212#407—141no—
213C4×M16—71no—
214#335—72no—
215#334—72no—
216#333—72no—
217#333—72no—
218#332—72no—
219#331—72no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—81no—
225C23×C7#12111yescharacteristic
226C22×D7—41no—
227C7×Q8—82no—
228C7×Q8—82no—
229C7×D4—21no—
230C7×D4—21no—
231C7×D4—21no—
232C7×D4—42no—
233C7×D4#13712yes—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—82no—
248C2×C4×C7#12111yescharacteristic
249C2×C4×C7#12111yescharacteristic
250C2×C4×C7#12111yescharacteristic
251C2×C4×C7#13512yes—
252C2×C4×C7AGL(1,ℤ/8)12yes—
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C2×#4111yescharacteristic
256C22⋊D7—82no—
257C2×Dic7—41no—
258D28—41no—
259D28—41no—
260D28—41no—
261D28—41no—
262D28—81no—
263C4×D7—82no—
264Dic14—41no—
265Dic14—41no—
266C7×C8—42no—
267C7⋊C8—22no—
268C7⋊C8—22no—
269C7⋊C8—42no—
270C7⋊C8—41no—
271C7⋊C8—41no—
272C2×#41—141no—
273C22×Q8—21no—
274C22×D4C22⋊D711yescharacteristic
275C23×C4—21no—
276C2×D8—141no—
277C2×D8—141no—
278C2×M16—71no—
279C2×M16—71no—
280C2×M16—142no—
281C2×M16—141no—
282C2×M16—284no—
283C22×C8—21no—
284C22×C8—142no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—71no—
289C4⋊Q8C22×D712yes—
290#127—71no—
291#124—141no—
292#121—141no—
293#121—141no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—141no—
297C4×D4—141no—
298C4×D4C22⋊D711yescharacteristic
299C4×D4C22⋊D711yescharacteristic
300C4×D4C22×D711yescharacteristic
301C4×D4C22×D711yescharacteristic
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C22⋊D711yescharacteristic
305C2×C22⋊C4—21no—
306C2×C42C22×D711yescharacteristic
307#107—141no—
308#107—141no—
309#106—22no—
310#106—22no—
311C4⋊C8—21no—
312C4⋊C8—72no—
313C4⋊C8—71no—
314C4⋊C8—71no—
315C4⋊C8—141no—
316C4⋊C8C22×D712yes—
317#103—142no—
318#103—142no—
319#102—22no—
320#102—22no—
321#102—142no—
322#102—142no—
323#102—141no—
324#102—141no—
325#102—141no—
326#102—141no—
327C22⋊C8—142no—
328C22⋊C8—142no—
329C22⋊C8—141no—
330#97—72no—
331C4×C8—72no—
332C4×C8—141no—
333C22×C7—21no—
334C22×C7—21no—
335C22×C7—22no—
336C22×C7—41no—
337C22×C7—41no—
338C22×C7—41no—
339C22×C7—41no—
340C22×C7—41no—
341C22×C7—84no—
342C22×C7#46312yes—
343C22×C7C2×#12111yescharacteristic
344C2×D7—81no—
345C2×D7—81no—
346C2×D7—81no—
347C4×C7—21no—
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—41no—
351C4×C7—41no—
352C4×C7—82no—
353C4×C7#46312yes—
354C4×C7C2×#13511yescharacteristic
355C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
356C4×C7C2×#13711yescharacteristic
357C4×C7#57511yescharacteristic
358Dic7—81no—
359C24—21no—
360#41—282no—
361#41—282no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—82no—
365C2×Q8—141no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—141no—
370C2×D4—141no—
371C2×D4—141no—
372C2×D4—281no—
373C2×D4C2×C22⋊D711yescharacteristic
374C2×D4C2×C22⋊D711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—281no—
384C22×C4D4×D711yescharacteristic
385C22×C4#115911yescharacteristic
386C22×C4C2×C22⋊D711yescharacteristic
387D8—282no—
388D8—282no—
389M16—142no—
390M16—142no—
391M16—284no—
392M16—282no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—72no—
396C2×C8—72no—
397C2×C8—142no—
398C2×C8—141no—
399C2×C8—141no—
400C2×C8—284no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—141no—
407C4⋊C4—141no—
408C4⋊C4C2×C22⋊D711yescharacteristic
409C4⋊C4C2×C22⋊D711yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—281no—
415C22⋊C4—281no—
416C42D4×D711yescharacteristic
417C42#115911yescharacteristic
418C42C2×C22⋊D711yescharacteristic
419C42C23×D711yescharacteristic
420C2×C7—22no—
421C2×C7—42no—
422C2×C7—81no—
423C2×C7C2×#46311yescharacteristic
424C2×C7#305611yescharacteristic
425C2×C7#306811yescharacteristic
426D7—161no—
427C23—21no—
428C23—21no—
429C23—42no—
430C23—42no—
431C23—81no—
432C23—281no—
433C23#650211yescharacteristic
434Q8—82no—
435Q8—82no—
436Q8—281no—
437Q8—281no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—42no—
442D4—281no—
443D4—281no—
444D4—281no—
445D4—281no—
446D4—561no—
447D4—562no—
448D4#650512yes—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—281no—
464C2×C4—562no—
465C2×C4#647712yes—
466C2×C4#647912yes—
467C2×C4#650211yescharacteristic
468C2×C4#650211yescharacteristic
469C2×C4#650211yescharacteristic
470C2×C4C2×D4×D711yescharacteristic
471C2×C4C2×#115911yescharacteristic
472C2×C4C22×C22⋊D711yescharacteristic
473C8—42no—
474C8—142no—
475C8—142no—
476C8—282no—
477C8—281no—
478C8—281no—
479C7#1916811yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—561no—
490C22—561no—
491C22—561no—
492C22#8993712yes—
493C22C2×#650211yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—82no—
500C4—561no—
501C4#8993712yes—
502C4C2×#647711yescharacteristic
503C4C2×#647911yescharacteristic
504C4C2×#650511yescharacteristic
505C4#9051311yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—1121no—
510C2#1174017711yescharacteristic
511C2C2×#8993711yescharacteristic
512C2#1174138811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#476C2
socleC22×C7C2×#121
prime radical#11739753C2
hypercenterC2×#476D7
hypocenterC7#19168
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×C7×#476C2semidirect
2. #11740644C2semidirect
4. D4×#1136C2semidirect
5. #11729528C2semidirect
6. #11728893C2semidirect
7. #11728892C2semidirect
8. #11728889C2semidirect
9. #11728888C2semidirect
10. #11728186C2semidirect
11. #11728174C2semidirect
25. #89865C22semidirect
27. #89860C22semidirect
43. #89341C22semidirect
44. #89341C22semidirect
45. #89281C22semidirect
46. #89280C22semidirect
47. #89275C22semidirect
48. #89274C22semidirect
113. C2×#476D7semidirect
479. C7#19168semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4762
#117406442
#117397532
D4×#11362
#117295282
#117288932
#117288922
#117288892
#117288882
#117281862
#117281742
#191687