Finite Group Databasebeta

← #13256099

Subgroups — #13256099

Order 1792.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2538C211yesmaximal, characteristic, complemented
2C2×#89799C211yesmaximal, characteristic, complemented
3C2×#89787C211yesmaximal, characteristic
4C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5C2×D4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11727334C211yesmaximal, characteristic, complemented
7#11725440C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8—21no—
9C2×C4×C7×Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C22×#1143—21no—
12C22×#1141—21no—
13C2×#6437C2211yescharacteristic
14C7×#396—22no—
15C7×#390—21no—
16C7×#389—21no—
17C7×#384C2211yescharacteristic
18C7×#380—21no—
19#89799—22no—
20#89799—22no—
21#89787—22no—
22#89787—22no—
23C2×#6383—21no—
24C2×#6383—21no—
25C2×#6383—22no—
26C2×C4⋊(C7⋊C8)C2211yescharacteristic
27C2×C4⋊(C7⋊C8)C2211yescharacteristic
28C2×C4×C7⋊C8C2211yescharacteristic
29Q8⋊(C7⋊C8)—22no—
30D4⋊(C7⋊C8)—22no—
31#10229—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C22×D28—21no—
36C22×D28—21no—
37C22×D28—21no—
38C7×C22⋊Q8—42no—
39C7×C22⋊Q8—42no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C4×C7×D4—21no—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C4⋊C4D412yes—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—22no—
58C2×C42×C7—21no—
59C2×C42×C7C2311yescharacteristic
60C2×#1143—24no—
61C2×#1143—42no—
62C2×#1143—42no—
63C2×#1141—24no—
64C2×#1141—42no—
65C2×#1141—42no—
66C22×C7⋊C8—21no—
67C22×C7⋊C8—21no—
68C22×C7⋊C8—21no—
69#6437—21no—
70#6437—21no—
71#6437—21no—
72#6437D412yes—
73C7×#95—22no—
74#6383—24no—
75#6383—24no—
76#6383—44no—
77C4⋊(C7⋊C8)—21no—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)D412yes—
80C4⋊(C7⋊C8)D412yes—
81C4×C7⋊C8—21no—
82C4×C7⋊C8—21no—
83C2×#460—71no—
84C2×#459—71no—
85#2538D711yescharacteristic, complemented
86#1957—71no—
87C2×Q8⋊C8—71no—
88C2×D4⋊C8—71no—
89#1296—71no—
90C24×C7—21no—
91C23×D7—41no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8D812yescomplemented
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4SD1612yescomplemented
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7—41no—
113C22×C4×C7—41no—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C2×D28—22no—
118C2×D28—22no—
119C2×D28—22no—
120C2×D28—42no—
121C2×D28—42no—
122C2×D28—41no—
123C2×D28—41no—
124C2×D28—41no—
125C2×D28—41no—
126C2×D28—41no—
127C2×D28—41no—
128C2×D28—41no—
129C2×D28—41no—
130C2×D28—41no—
131C2×D28—41no—
132C2×D28—41no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—42no—
138C7×C4⋊C4—44no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4—41no—
141C7×C4⋊C4—41no—
142C7×C4⋊C4—41no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C22⋊C4—44no—
146C7×C22⋊C4—41no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—82no—
149C42×C7—41no—
150C42×C7—41no—
151C42×C7C2×D411yescharacteristic
152C42×C7C2×D411yescharacteristic
153C42×C7C2×D411yescharacteristic
154C42×C7#4111yescharacteristic
155#1143—48no—
156#1141—48no—
157C2×C7⋊C8—22no—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—41no—
161C2×C7⋊C8—41no—
162C2×C7⋊C8—41no—
163C2×C7⋊C8—41no—
164C2×C7⋊C8—41no—
165C2×C7⋊C8—41no—
166C22×SD16—141no—
167C22×D8—141no—
168C2×#127—71no—
169C2×C22⋊Q8—21no—
170C2×C4×Q8C2×D711yescharacteristic
171C2×C4×D4C2×D711yescharacteristic
172#460—142no—
173#460—142no—
174#459—142no—
175#459—142no—
176C2×C4⋊C8—71no—
177C2×C4⋊C8—71no—
178C2×#102—141no—
179C2×#102—141no—
180C2×#102—142no—
181C2×C4×C8—71no—
182#396—22no—
183#390—21no—
184#389—21no—
185#384C2×D711yescharacteristic
186#380—21no—
187Q8⋊C8—142no—
188D4⋊C8—142no—
189C23×C7—22no—
190C23×C7—41no—
191C23×C7—41no—
192C23×C7—41no—
193C23×C7—41no—
194C23×C7—41no—
195C23×C7—41no—
196C23×C7#12011yescharacteristic
197C22×D7—42no—
198C22×D7—81no—
199C22×D7—81no—
200C22×D7—81no—
201C22×D7—81no—
202C22×D7—81no—
203C22×D7—81no—
204C7×Q8—22no—
205C7×Q8—22no—
206C7×Q8—41no—
207C7×Q8—41no—
208C7×D4—22no—
209C7×D4—22no—
210C7×D4—41no—
211C7×D4—41no—
212C2×C4×C7—21no—
213C2×C4×C7—21no—
214C2×C4×C7—22no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—42no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—42no—
225C2×C4×C7—41no—
226C2×C4×C7—81no—
227C2×C4×C7—81no—
228C2×C4×C7—81no—
229C2×C4×C7—81no—
230C2×C4×C7#12011yescharacteristic
231C2×C4×C7#12011yescharacteristic
232C2×C4×C7#12111yescharacteristic
233C2×C4×C7#12111yescharacteristic
234C2×C4×C7#12111yescharacteristic
235C2×C4×C7#12711yescharacteristic
236C2×C4×C7C2×D811yescharacteristic
237C2×C4×C7C2×SD1611yescharacteristic
238C2×C4×C7#13511yescharacteristic
239C2×C4×C7#13511yescharacteristic
240C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
241C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
242C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
243C2×C4×C7#13711yescharacteristic
244D28—44no—
245D28—42no—
246D28—42no—
247D28—42no—
248D28—42no—
249D28—82no—
250D28—81no—
251D28—81no—
252D28—81no—
253D28—81no—
254C7⋊C8—42no—
255C7⋊C8—42no—
256C7⋊C8—42no—
257C22×Q8C22⋊D711yescharacteristic
258C22×D4—141no—
259C22×D4—141no—
260C22×D4—141no—
261C22×D4C22⋊D711yescharacteristic
262C23×C4—21no—
263C2×SD16—144no—
264C2×SD16—282no—
265C2×SD16—282no—
266C2×D8—144no—
267C2×D8—282no—
268C2×D8—282no—
269C22×C8—141no—
270C22×C8—141no—
271C22×C8—141no—
272#127—72no—
273#127—141no—
274#127—141no—
275#127—141no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C22⋊D711yescharacteristic
291C2×C4⋊C4C22⋊D711yescharacteristic
292C2×C4⋊C4C22⋊D712yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C42—21no—
297C2×C42C22×D711yescharacteristic
298C4⋊C8—72no—
299C4⋊C8—72no—
300C4⋊C8—141no—
301C4⋊C8—141no—
302#102—144no—
303#102—144no—
304#102—284no—
305C4×C8—141no—
306C4×C8—141no—
307#95—22no—
308C22×C7—42no—
309C22×C7—42no—
310C22×C7—42no—
311C22×C7—81no—
312C22×C7—81no—
313C22×C7—81no—
314C22×C7—81no—
315C22×C7#39211yescharacteristic
316C22×C7#44711yescharacteristic
317C22×C7#44811yescharacteristic
318C22×C7#44911yescharacteristic
319C22×C7#44911yescharacteristic
320C22×C7#45311yescharacteristic
321C22×C7#45411yescharacteristic
322C2×D7—82no—
323C2×D7—82no—
324C2×D7—82no—
325C2×D7—161no—
326C2×D7—161no—
327C2×D7—161no—
328C2×D7—161no—
329C4×C7—21no—
330C4×C7—21no—
331C4×C7—42no—
332C4×C7—82no—
333C4×C7—81no—
334C4×C7—81no—
335C4×C7#44711yescharacteristic
336C4×C7#44911yescharacteristic
337C4×C7#45011yescharacteristic
338C4×C7#45211yescharacteristic
339C4×C7#46511yescharacteristic
340C4×C7#46611yescharacteristic
341C4×C7#46811yescharacteristic
342C4×C7#46911yescharacteristic
343C24—21no—
344C24—281no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8#114112yescomplemented
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—142no—
357C2×D4—142no—
358C2×D4—142no—
359C2×D4—282no—
360C2×D4—282no—
361C2×D4—281no—
362C2×D4—281no—
363C2×D4—281no—
364C2×D4—281no—
365C2×D4—281no—
366C2×D4—281no—
367C2×D4—281no—
368C2×D4—281no—
369C2×D4—281no—
370C2×D4—281no—
371C2×D4—281no—
372C2×D4#114312yescomplemented
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4C2×C22⋊D711yescharacteristic
384C22×C4C2×C22⋊D711yescharacteristic
385C22×C4C2×C22⋊D711yescharacteristic
386SD16—288no—
387D8—288no—
388C2×C8—142no—
389C2×C8—142no—
390C2×C8—142no—
391C2×C8—281no—
392C2×C8—281no—
393C2×C8—281no—
394C2×C8—281no—
395C2×C8—281no—
396C2×C8—281no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C22⋊C4—44no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—82no—
413C42—41no—
414C42—41no—
415C42D4×D711yescharacteristic
416C42D4×D711yescharacteristic
417C42D4×D711yescharacteristic
418C42#115911yescharacteristic
419C2×C7—82no—
420C2×C7#162111yescharacteristic
421C2×C7#162211yescharacteristic
422C2×C7#162711yescharacteristic
423C2×C7#162811yescharacteristic
424C2×C7#199911yescharacteristic
425C2×C7#200011yescharacteristic
426C2×C7#200611yescharacteristic
427D7—162no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—282no—
436C23—561no—
437C23—561no—
438C23—561no—
439C23—561no—
440C23—561no—
441C23—561no—
442C23#651611yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—41no—
446Q8—41no—
447D4—22no—
448D4—22no—
449D4—41no—
450D4—41no—
451D4—284no—
452D4—282no—
453D4—282no—
454D4—282no—
455D4—282no—
456D4—562no—
457D4—561no—
458D4—561no—
459D4—561no—
460D4—561no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4C2×#114111yescharacteristic
480C2×C4#649511yescharacteristic
481C2×C4#650011yescharacteristic
482C2×C4#650011yescharacteristic
483C2×C4#650111yescharacteristic
484C2×C4#650211yescharacteristic
485C2×C4#650211yescharacteristic
486C2×C4#650311yescharacteristic
487C2×C4C2×#114311yescharacteristic
488C2×C4#650511yescharacteristic
489C2×C4#651211yescharacteristic
490C2×C4#651211yescharacteristic
491C2×C4#651311yescharacteristic
492C2×C4#651311yescharacteristic
493C8—282no—
494C8—282no—
495C8—282no—
496C7#1022911yescharacteristic, complemented, Sylow
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—562no—
505C22—562no—
506C22—562no—
507C22—1121no—
508C22—1121no—
509C22—1121no—
510C22—1121no—
511C22#8998911yescharacteristic
512C22#8999511yescharacteristic
513C22#8999911yescharacteristic
514C22#9001011yescharacteristic
515C22#9001011yescharacteristic
516C22#9001311yescharacteristic
517C22#9001711yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4#8980811yescharacteristic
525C4#8980911yescharacteristic
526C4#8981011yescharacteristic
527C4#8981111yescharacteristic
528C4#8981811yescharacteristic
529C4#8981911yescharacteristic
530C4#8982011yescharacteristic
531C4#8982111yescharacteristic
532C2—82no—
533C2—1122no—
534C2#1172943811yescharacteristic
535C2#1172945311yescharacteristic
536C2#1172963311yescharacteristic
537C2#1172966811yescharacteristic
538C2#1172967211yescharacteristic
539C2#1172967711yescharacteristic
540C2#1172968511yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2538C2
socleC23×C7#120
prime radicalC2×#89787C2
hypercenter#2538D7
hypocenterC7#10229
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C7×#2538C2semidirect
2. C2×#89799C2semidirect
4. C2×Q8⋊(C7⋊C8)C2semidirect
5. C2×D4⋊(C7⋊C8)C2semidirect
6. #11727334C2semidirect
7. #11725440C2semidirect
85. #2538D7semidirect
97. C2×C7×Q8D8semidirect
103. C2×C7×D4SD16semidirect
350. C2×Q8#1141semidirect
372. C2×D4#1143semidirect
496. C7#10229semidirect

Maximal subgroups

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

subgroupindex
C7×#25382
C2×#897992
C2×#897872
C2×Q8⋊(C7⋊C8)2
C2×D4⋊(C7⋊C8)2
#117273342
#117254402
#102297