Finite Group Databasebeta

← #13387768

Subgroups — #13387768

Order 1792.

1090 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90423C211yesmaximal, characteristic
2C2×C7×#475C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11728336C212yesmaximal, complemented
7#11728334C212yesmaximal, complemented
8#11728239C211yesmaximal, characteristic, complemented
9#11728224C211yesmaximal, characteristic
10#11728027C212yesmaximal, complemented
11#11728023C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×#6510—21no—
15C2×#6509C2212yes—
16C2×Q8×Dic7—21no—
17C22×#1144C2211yescharacteristic
18#90423—24no—
19#90423—24no—
20C2×#6461—22no—
21C2×#6460C2212yes—
22C2×#6455C2212yes—
23C2×C4×D28C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#475—24no—
26C2×C7×#106C2212yes—
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#103C2212yes—
29C2×#6409C2211yescharacteristic
30C2×(C2×D7)⋊C8C2212yes—
31C2×(C2×D7)⋊C8C2212yes—
32C2×#6390C2212yes—
33C4×#1144—24no—
34#89766—22no—
35#89759C2212yes—
36C2×#6384C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38#89711—21no—
39#89707C2211yescharacteristic
40#89700—21no—
41C2×C4×C7⋊C8C2211yescharacteristic
42#89346C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24123—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×D28D411yescharacteristic
49C22×C4×D7C2312yes—
50C22×Dic14C2311yescharacteristic
51C22×C7×C8C2312yes—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8D414yes—
54C4×C7×Q8—22no—
55C4×C7×Q8—22no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2312yes—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62#6510—28no—
63#6509—24no—
64#6509—42no—
65Q8×Dic7—28no—
66C2×#1144—22no—
67C2×#1144—22no—
68C2×#1144—22no—
69C2×(C2×D7)⋊C4—22no—
70C2×(C2×D7)⋊C4—22no—
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4D412yes—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7D411yescharacteristic
78C2×C4⋊Dic7C2312yes—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C22×C7⋊C8C2311yescharacteristic
85C22×C7⋊C8C2311yescharacteristic
86#6461—216no—
87#6460—24no—
88#6460—24no—
89#6455—24no—
90#6455—24no—
91C4×D28—22no—
92C4×D28—22no—
93C4×Dic14—22no—
94C4×Dic14D414yes—
95C7×#106—24no—
96C7×C4⋊C8—22no—
97C7×#103—24no—
98#6409—22no—
99C7⋊#95—22no—
100(C2×D7)⋊C8—24no—
101(C2×D7)⋊C8—42no—
102#6390—42no—
103#6384—22no—
104#6381—22no—
105C4×C7⋊C8—22no—
106C2×#556—71no—
107C2×#475D711yescharacteristic, complemented
108C2×C4×Q16—71no—
109#2538—71no—
110#2035—72no—
111#2001—72no—
112#1990—71no—
113#1954—71no—
114#1873—72no—
115#1854—72no—
116#1803—71no—
117C23×D7C2×D412yes—
118C2×C7×Q8—24no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—42no—
125C2×C7×Q8—42no—
126C2×C7×Q8#4112yes—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2411yescharacteristic
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7—22no—
136C22×Dic7C2×D412yes—
137C2×D28—22no—
138C2×D28—22no—
139C2×D28—22no—
140C2×C4×D7—22no—
141C2×C4×D7—22no—
142C2×C4×D7—22no—
143C2×C4×D7—22no—
144C2×C4×D7—22no—
145C2×C4×D7—22no—
146C2×Dic14—21no—
147C2×Dic14—21no—
148C2×Dic14—21no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14C2×D412yes—
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—42no—
160C7×C4⋊C4C2×D414yes—
161C42×C7—42no—
162C42×C7C2×D412yes—
163C42×C7C2×D412yes—
164#1144—28no—
165(C2×D7)⋊C4—28no—
166(C2×D7)⋊C4—28no—
167(C2×D7)⋊C4—24no—
168(C2×D7)⋊C4—44no—
169(C2×D7)⋊C4—82no—
170C4⋊Dic7—28no—
171C4⋊Dic7—22no—
172C4⋊Dic7—21no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—42no—
176C4⋊Dic7—81no—
177C4⋊Dic7C2×D412yes—
178Dic7⋊C4—24no—
179Dic7⋊C4—44no—
180C4×Dic7—28no—
181C4×Dic7—24no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8#4112yes—
188C22×Q16—71no—
189#556—144no—
190#556—144no—
191C2×C4⋊Q8C2×D711yescharacteristic
192C2×#126—142no—
193C2×#124—141no—
194C2×C22⋊Q8—72no—
195C2×C22⋊Q8—72no—
196C2×C4×Q8—71no—
197C2×C4×Q8—141no—
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×D4—71no—
200C2×#117—72no—
201#475—24no—
202C4×Q16—144no—
203C2×#107—71no—
204C2×#106C2×D712yes—
205C2×C4⋊C8C2×D711yescharacteristic
206C2×#103—71no—
207C2×#103—71no—
208C2×#103—72no—
209C2×#103C2×D712yes—
210C2×C22⋊C8—72no—
211C2×C22⋊C8—72no—
212C2×C4×C8—71no—
213#396—142no—
214#390—141no—
215#389—141no—
216#384—71no—
217#380—72no—
218#340—72no—
219#340—72no—
220#340—72no—
221C23×C7C22×D411yescharacteristic
222C22×D7—22no—
223C22×D7—22no—
224C22×D7—22no—
225C22×D7—22no—
226C22×D7—22no—
227C22×D7—22no—
228C22×D7—22no—
229C7×Q8—22no—
230C7×Q8—22no—
231C7×Q8—41no—
232C7×Q8—41no—
233C7×Q8—48no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—24no—
237C2×C4×C7—22no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7#12012yes—
246C2×C4×C7#12012yes—
247C2×C4×C7#13512yes—
248C2×C4×C7#13712yes—
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7H(5,𝔽2)11yescharacteristic
254C2×C4×C72−1+411yescharacteristic
255C2×Dic7—22no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267D28—44no—
268C4×D7—42no—
269C4×D7—42no—
270C4×D7—42no—
271C4×D7—42no—
272Dic14—22no—
273Dic14—22no—
274Dic14—41no—
275Dic14—41no—
276C7×C8—42no—
277C7⋊C8—22no—
278C7⋊C8—41no—
279C22×Q8—21no—
280C22×Q8—71no—
281C22×Q8C22×D711yescharacteristic
282C22×D4—71no—
283C23×C4—72no—
284C2×Q16—142no—
285C2×Q16—142no—
286C2×Q16—142no—
287C22×C8—71no—
288C22×C8—71no—
289C22×C8C22×D712yes—
290C4⋊Q8—22no—
291C4⋊Q8C22⋊D714yes—
292#126—1416no—
293#124—148no—
294C22⋊Q8—144no—
295C22⋊Q8—144no—
296C22⋊Q8—144no—
297C22⋊Q8—282no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—74no—
301C4×Q8—148no—
302C4×Q8—142no—
303C4×D4—142no—
304C4×D4—142no—
305#117—144no—
306#117—144no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C4⋊C4C22×D712yes—
319C2×C22⋊C4—72no—
320C2×C22⋊C4—142no—
321C2×C22⋊C4—142no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4—142no—
324C2×C42—21no—
325C2×C42—142no—
326C2×C42—141no—
327C2×C42C22×D711yescharacteristic
328#107—142no—
329#106—24no—
330C4⋊C8—22no—
331#103—24no—
332#103—142no—
333#103—142no—
334#103—282no—
335C22⋊C8—144no—
336C22⋊C8—282no—
337C4×C8—142no—
338#95—142no—
339C22×C7C2×#12011yescharacteristic
340C22×C7#54611yescharacteristic
341C22×C7#54711yescharacteristic
342C22×C7C2×#13511yescharacteristic
343C22×C7C2×#13711yescharacteristic
344C22×C7#57711yescharacteristic
345C22×C7#57811yescharacteristic
346C2×D7—42no—
347C2×D7—42no—
348C2×D7—42no—
349C2×D7—42no—
350C2×D7—42no—
351C2×D7—42no—
352C2×D7—42no—
353C4×C7—22no—
354C4×C7—42no—
355C4×C7—44no—
356C4×C7—81no—
357C4×C7#45214yes—
358C4×C7#52511yescharacteristic
359C4×C7#53711yescharacteristic
360C4×C7#54811yescharacteristic
361C4×C7#55011yescharacteristic
362Dic7—42no—
363Dic7—42no—
364Dic7—42no—
365Dic7—82no—
366C24—72no—
367C2×Q8—24no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—72no—
376C2×Q8—141no—
377C2×Q8—141no—
378C2×Q8—141no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8#116112yes—
382C2×D4—142no—
383C2×D4—142no—
384C2×D4—142no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—72no—
389C22×C4—141no—
390C22×C4—141no—
391C22×C4—142no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4D4×D711yescharacteristic
399C22×C4D4×D711yescharacteristic
400C22×C4C23×D711yescharacteristic
401Q16—148no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—72no—
406C2×C8—141no—
407C2×C8—141no—
408C2×C8—141no—
409C2×C8—141no—
410C2×C8—141no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—42no—
416C4⋊C4—72no—
417C4⋊C4—148no—
418C4⋊C4—144no—
419C4⋊C4—142no—
420C4⋊C4—141no—
421C4⋊C4—142no—
422C4⋊C4—142no—
423C4⋊C4—284no—
424C4⋊C4—282no—
425C4⋊C4—561no—
426C4⋊C4C2×C22⋊D714yes—
427C22⋊C4—148no—
428C22⋊C4—148no—
429C22⋊C4—144no—
430C22⋊C4—284no—
431C22⋊C4—562no—
432C42—42no—
433C42—148no—
434C42—144no—
435C42D4×D712yes—
436C42C2×C22⋊D712yes—
437C2×C7#240711yescharacteristic
438C2×C7C2×#45211yescharacteristic
439C2×C7#301311yescharacteristic
440C2×C7#330411yescharacteristic
441C2×C7#332811yescharacteristic
442C2×C7#334211yescharacteristic
443C2×C7#335611yescharacteristic
444D7—82no—
445C23—142no—
446C23—142no—
447C23—142no—
448C23—142no—
449C23—142no—
450C23—142no—
451C23—142no—
452C23C2×D4×D711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—48no—
456Q8—41no—
457Q8—41no—
458Q8—142no—
459Q8—142no—
460Q8—281no—
461Q8—281no—
462D4—284no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—142no—
475C2×C4—142no—
476C2×C4—142no—
477C2×C4—142no—
478C2×C4—142no—
479C2×C4—281no—
480C2×C4—281no—
481C2×C4—281no—
482C2×C4—281no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4#647812yes—
491C2×C4#647912yes—
492C2×C4#650012yes—
493C2×C4#650012yes—
494C2×C4C2×D4×D711yescharacteristic
495C2×C4C2×#116111yescharacteristic
496C2×C4#655111yescharacteristic
497C2×C4D7×#4111yescharacteristic
498C2×C4#655311yescharacteristic
499C2×C4C22×C22⋊D711yescharacteristic
500C8—42no—
501C8—142no—
502C8—281no—
503C7#2412311yescharacteristic, complemented, Sylow
504C22—282no—
505C22—282no—
506C22—282no—
507C22—282no—
508C22—282no—
509C22—282no—
510C22—282no—
511C22#9032111yescharacteristic
512C22#9032211yescharacteristic
513C22C2×#647811yescharacteristic
514C22C2×#647911yescharacteristic
515C22#9046911yescharacteristic
516C22#9047011yescharacteristic
517C22C2×#650011yescharacteristic
518C4—22no—
519C4—42no—
520C4—44no—
521C4—81no—
522C4—282no—
523C4—282no—
524C4—282no—
525C4—562no—
526C4#8994514yes—
527C4#9031811yescharacteristic
528C4#9032011yescharacteristic
529C4#9050811yescharacteristic
530C4#9052511yescharacteristic
531C2—562no—
532C2#1173846811yescharacteristic
533C2#1173847111yescharacteristic
534C2#1173847411yescharacteristic
535C2#1173847811yescharacteristic
536C2#1174025711yescharacteristic
537C2C2×#8994511yescharacteristic
538C2#1174142911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#475C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenterC2×#475D7
hypocenterC7#24123
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×C7×#475C2semidirect
4. C2×C4×#1144C2semidirect
5. #11729541C2semidirect
6. #11728336C2semidirect
7. #11728334C2semidirect
8. #11728239C2semidirect
10. #11728027C2semidirect
11. #11728023C2semidirect
107. C2×#475D7semidirect
503. C7#24123semidirect

Maximal subgroups

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

subgroupindex
C2×#904232
C2×C7×#4752
#117399452
C2×C4×#11442
#117295412
#117283362
#117283342
#117282392
#117282242
#117280272
#117280232
#241237