Finite Group Databasebeta

← #13336571

Subgroups — #13336571

Order 1792.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#474C211yesmaximal, characteristic, complemented
2#11740648C211yesmaximal, characteristic, complemented
3#11739753C211yesmaximal, characteristic
4D4×#1136C211yesmaximal, characteristic, complemented
5#11729527C211yesmaximal, characteristic, complemented
6#11728859C212yesmaximal, complemented
7#11728858C212yesmaximal, complemented
8#11728855C212yesmaximal, complemented
9#11728854C212yesmaximal, complemented
10#11728186C211yesmaximal, characteristic, complemented
11#11728174C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6505—21no—
15C2×#6505—21no—
16C22×#1136—21no—
17#90168C2211yescharacteristic
18C7×#474—21no—
19C7×#474—21no—
20C7×#474C2212yes—
21C7×#474C2212yes—
22C2×C7×#107—21no—
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×#102—21no—
25#89989—21no—
26#89979—21no—
27#89864C2212yescomplemented
28#89858—22no—
29#89857C2212yescomplemented
30#89787C2211yescharacteristic
31#89787C2211yescharacteristic
32C4×#1141—21no—
33C4×#1141—21no—
34#89784C2212yes—
35#89783—22no—
36D4×C7⋊C8C2212yes—
37#89779C2211yescharacteristic
38#89779C2211yescharacteristic
39#89739—21no—
40#89697C2211yescharacteristic
41#89696C2211yescharacteristic
42C4×#1136C2211yescharacteristic
43#89342C2212yescomplemented
44#89342C2212yescomplemented
45#89281C2212yescomplemented
46#89280C2212yescomplemented
47#89275C2212yescomplemented
48#89274C2212yescomplemented
49#19163—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×#1144—22no—
71C2×#1144—22no—
72#6505—44no—
73#6505—44no—
74C2×#1143—22no—
75C2×#1143—22no—
76C2×#1141—21no—
77C2×#1141—21no—
78#6493—21no—
79#6493—21no—
80C2×#1136—21no—
81C2×#1136—22no—
82C2×#1136—44no—
83C2×#1136D411yescharacteristic
84C2×#1136D411yescharacteristic
85C22×C7⋊C8—22no—
86#6438—21no—
87#6437C2311yescharacteristic
88C4×D28—21no—
89C4×D28—21no—
90#6432C2311yescharacteristic
91C7×#107—22no—
92C7×#107—22no—
93C7×C4⋊C8—21no—
94C7×C4⋊C8C2312yes—
95C7×#102—22no—
96C7×#102—22no—
97#6406—21no—
98#6406—21no—
99C22⋊(C7⋊C8)—21no—
100C22⋊(C7⋊C8)—22no—
101C22⋊(C7⋊C8)—22no—
102#6383—21no—
103#6383—21no—
104#6381—21no—
105#6381—21no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2311yescharacteristic
109C4⋊(C7⋊C8)C2311yescharacteristic
110#6377D412yes—
111C4×C7⋊C8—21no—
112C4×C7⋊C8C2312yes—
113#3152—71no—
114#3100—71no—
115C2×#474D711yescharacteristic, complemented
116D4×M16—71no—
117#1689—72no—
118#1685—72no—
119#1669—72no—
120#1665—72no—
121#1565—71no—
122#1547—71no—
123#1546—71no—
124C24×C7—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8—41no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×D4—22no—
130C2×C7×D4—21no—
131C2×C7×D4—41no—
132C2×C7×D4C2×D411yescharacteristic
133C2×C7×D4C2×D411yescharacteristic
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7—41no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7#4111yescharacteristic
145C2×C22⋊D7—41no—
146#1157—42no—
147#1157—42no—
148C2×D28—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×C4×D7—41no—
152C2×Dic14—21no—
153C2×C7×C8—22no—
154C2×C7×C8—42no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—41no—
159C7×C4⋊C4—41no—
160C7×C4⋊C4C2×D411yescharacteristic
161C7×C4⋊C4C2×D411yescharacteristic
162C7×C22⋊C4—21no—
163C7×C22⋊C4—21no—
164C7×C22⋊C4—21no—
165C7×C22⋊C4—21no—
166C42×C7C2×D411yescharacteristic
167C42×C7C2×D411yescharacteristic
168C42×C7#4111yescharacteristic
169C42×C7C2411yescharacteristic
170#1144—44no—
171#1144—44no—
172#1143—44no—
173#1143—44no—
174#1141—42no—
175#1141—42no—
176(C2×D7)⋊C4—41no—
177(C2×D7)⋊C4—41no—
178C4⋊Dic7—21no—
179C4⋊Dic7—21no—
180#1136—22no—
181#1136—42no—
182#1136—44no—
183#1136D814yescomplemented
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—44no—
188C2×C7⋊C8C2×D412yes—
189C2×C7⋊C8C2×D412yes—
190C2×#137—141no—
191C2×#137—141no—
192C22×M16—141no—
193#532—71no—
194C2×C4⋊Q8C2×D711yescharacteristic
195C2×C4×D4C2×D711yescharacteristic
196#497—142no—
197#494—72no—
198#492—72no—
199#480—141no—
200#474—21no—
201#474—21no—
202#474—71no—
203#474—71no—
204#474C2×D712yes—
205#474C2×D712yes—
206#459—71no—
207#459—71no—
208#447—141no—
209C4×D8—141no—
210C4×D8—141no—
211#436—72no—
212#435—142no—
213C8×D4—72no—
214#431—71no—
215C2×#107—21no—
216#423—71no—
217C2×C4⋊C8C2×D711yescharacteristic
218C2×#102—21no—
219#407—141no—
220C4×M16—71no—
221#335—72no—
222#334—72no—
223#332—72no—
224#332—72no—
225#332—72no—
226#331—72no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—81no—
232C23×C7#12111yescharacteristic
233C22×D7—41no—
234C7×Q8—42no—
235C7×Q8—42no—
236C7×Q8—82no—
237C7×D4—21no—
238C7×D4—21no—
239C7×D4—21no—
240C7×D4—42no—
241C7×D4#13712yescomplemented
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—41no—
247C2×C4×C7—42no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—82no—
256C2×C4×C7#12111yescharacteristic
257C2×C4×C7#12111yescharacteristic
258C2×C4×C7#12111yescharacteristic
259C2×C4×C7C2×D812yes—
260C2×C4×C7AGL(1,ℤ/8)12yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C2×#4111yescharacteristic
264C22⋊D7—82no—
265C2×Dic7—41no—
266D28—41no—
267D28—41no—
268D28—41no—
269D28—41no—
270D28—81no—
271C4×D7—82no—
272Dic14—41no—
273Dic14—41no—
274C7×C8—42no—
275C7⋊C8—22no—
276C7⋊C8—42no—
277C7⋊C8—41no—
278C7⋊C8—41no—
279C7⋊C8C2×D814yes—
280C2×#41—141no—
281C22×Q8—21no—
282C22×D4C22⋊D711yescharacteristic
283C23×C4—21no—
284#137—284no—
285#137—284no—
286C2×Q16—142no—
287C2×Q16—142no—
288C2×SD16—142no—
289C2×SD16—142no—
290C2×D8—141no—
291C2×D8—141no—
292C2×M16—71no—
293C2×M16—71no—
294C2×M16—142no—
295C2×M16—141no—
296C2×M16—284no—
297C22×C8—21no—
298C22×C8—142no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—71no—
303C4⋊Q8C22×D712yes—
304#127—71no—
305#124—141no—
306#121—141no—
307#121—141no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—141no—
311C4×D4—141no—
312C4×D4C22⋊D711yescharacteristic
313C4×D4C22⋊D711yescharacteristic
314C4×D4C22×D711yescharacteristic
315C4×D4C22×D711yescharacteristic
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C22⋊D711yescharacteristic
319C2×C22⋊C4—21no—
320C2×C42C22×D711yescharacteristic
321#107—22no—
322#107—22no—
323#107—141no—
324#107—141no—
325C4⋊C8—21no—
326C4⋊C8—72no—
327C4⋊C8—71no—
328C4⋊C8—71no—
329C4⋊C8—141no—
330C4⋊C8C22×D712yes—
331#102—22no—
332#102—22no—
333#102—141no—
334#102—141no—
335#102—141no—
336#102—141no—
337C22⋊C8—142no—
338C22⋊C8—142no—
339C22⋊C8—141no—
340#97—72no—
341C4×C8—72no—
342C4×C8—141no—
343C22×C7—21no—
344C22×C7—21no—
345C22×C7—22no—
346C22×C7—41no—
347C22×C7—41no—
348C22×C7—41no—
349C22×C7—41no—
350C22×C7—41no—
351C22×C7—84no—
352C22×C7#45912yes—
353C22×C7C2×#12111yescharacteristic
354C2×D7—81no—
355C2×D7—81no—
356C2×D7—81no—
357C4×C7—21no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—41no—
361C4×C7—41no—
362C4×C7—82no—
363C4×C7#45912yes—
364C4×C7C22×D811yescharacteristic
365C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
366C4×C7C2×#13711yescharacteristic
367C4×C7#57511yescharacteristic
368Dic7—81no—
369C24—21no—
370#41—282no—
371#41—282no—
372C2×Q8—22no—
373C2×Q8—41no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—141no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—141no—
381C2×D4—141no—
382C2×D4—141no—
383C2×D4—281no—
384C2×D4C2×C22⋊D711yescharacteristic
385C2×D4C2×C22⋊D711yescharacteristic
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—281no—
395C22×C4D4×D711yescharacteristic
396C22×C4#115911yescharacteristic
397C22×C4C2×C22⋊D711yescharacteristic
398Q16—284no—
399Q16—284no—
400SD16—284no—
401SD16—284no—
402D8—282no—
403D8—282no—
404M16—74no—
405M16—142no—
406M16—284no—
407M16—282no—
408C2×C8—22no—
409C2×C8—42no—
410C2×C8—72no—
411C2×C8—72no—
412C2×C8—142no—
413C2×C8—141no—
414C2×C8—141no—
415C2×C8—284no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—141no—
422C4⋊C4—141no—
423C4⋊C4C2×C22⋊D711yescharacteristic
424C4⋊C4C2×C22⋊D711yescharacteristic
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—281no—
430C22⋊C4—281no—
431C42D4×D711yescharacteristic
432C42#115911yescharacteristic
433C42C2×C22⋊D711yescharacteristic
434C42C23×D711yescharacteristic
435C2×C7—22no—
436C2×C7—42no—
437C2×C7—81no—
438C2×C7C2×#45911yescharacteristic
439C2×C7#305611yescharacteristic
440C2×C7#306911yescharacteristic
441D7—161no—
442C23—21no—
443C23—21no—
444C23—42no—
445C23—42no—
446C23—81no—
447C23—281no—
448C23#650211yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—82no—
452Q8—281no—
453Q8—281no—
454D4—21no—
455D4—21no—
456D4—21no—
457D4—42no—
458D4—281no—
459D4—281no—
460D4—281no—
461D4—281no—
462D4—561no—
463D4—562no—
464D4#650512yescomplemented
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—281no—
480C2×C4—562no—
481C2×C4D7×D812yes—
482C2×C4#647412yes—
483C2×C4#650211yescharacteristic
484C2×C4#650211yescharacteristic
485C2×C4#650211yescharacteristic
486C2×C4C2×D4×D711yescharacteristic
487C2×C4C2×#115911yescharacteristic
488C2×C4C22×C22⋊D711yescharacteristic
489C8—42no—
490C8—74no—
491C8—142no—
492C8—282no—
493C8—281no—
494C8—281no—
495C7#1916311yescharacteristic, complemented, Sylow
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—561no—
506C22—561no—
507C22—561no—
508C22#8992212yes—
509C22C2×#650211yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—561no—
517C4#8992212yes—
518C4C2×D7×D811yescharacteristic
519C4C2×#647411yescharacteristic
520C4C2×#650511yescharacteristic
521C4#9051311yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—1121no—
526C2#1174017711yescharacteristic
527C2C2×#8992211yescharacteristic
528C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#474C2
socleC22×C7C2×#121
prime radical#11739753C2
hypercenterC2×#474D7
hypocenterC7#19163
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C7×#474C2semidirect
2. #11740648C2semidirect
4. D4×#1136C2semidirect
5. #11729527C2semidirect
6. #11728859C2semidirect
7. #11728858C2semidirect
8. #11728855C2semidirect
9. #11728854C2semidirect
10. #11728186C2semidirect
11. #11728174C2semidirect
27. #89864C22semidirect
29. #89857C22semidirect
43. #89342C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89280C22semidirect
47. #89275C22semidirect
48. #89274C22semidirect
115. C2×#474D7semidirect
183. #1136D8semidirect
241. C7×D4#137semidirect
464. D4#6505semidirect
495. C7#19163semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4742
#117406482
#117397532
D4×#11362
#117295272
#117288592
#117288582
#117288552
#117288542
#117281862
#117281742
#191637