Finite Group Databasebeta

← #13387757

Subgroups — #13387757

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90421C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11728389C212yesmaximal, complemented
7#11728386C212yesmaximal, complemented
8#11728239C211yesmaximal, characteristic, complemented
9#11728224C211yesmaximal, characteristic
10#11728027C212yesmaximal, complemented
11#11728022C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×#6509—22no—
15C2×#6509C2212yes—
16C22×#1144C2211yescharacteristic
17#90421—24no—
18#90421—24no—
19C2×#6460C2212yes—
20C2×#6456—21no—
21C2×D7×C4⋊C4C2212yes—
22C2×#6453—21no—
23C2×C4×D28C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#477—24no—
26C2×C7×#107C2212yes—
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#24122—71nomaximal, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8C2311yescharacteristic
48C22×D28D411yescharacteristic
49C22×C4×D7—22no—
50C22×C4×D7C2312yes—
51C22×Dic14C2311yescharacteristic
52C22×C7×C8C2312yes—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yes—
55C4×C7×Q8—22no—
56C4×C7×Q8—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2312yes—
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63#6509—216no—
64#6509—24no—
65#6509—42no—
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)⋊C4D412yes—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7D411yescharacteristic
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C22×C7⋊C8C2311yescharacteristic
84C22×C7⋊C8C2311yescharacteristic
85#6460—24no—
86#6460—24no—
87#6456—28no—
88D7×C4⋊C4—24no—
89D7×C4⋊C4—24no—
90#6453—28no—
91C4×D28—22no—
92C4×D28—22no—
93C4×Dic14—22no—
94C4×Dic14D414yes—
95C7×#107—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×#554—71no—
107C2×#477D711yescharacteristic, complemented
108C2×C4×Q16—71no—
109#2538—71no—
110#2036—72no—
111#2001—72no—
112#1990—71no—
113#1954—71no—
114#1872—72no—
115#1852—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—81no—
125C2×C7×Q8—81no—
126C2×C7×Q8#4112yes—
127C22×C4×C7—21no—
128C22×C4×C7—22no—
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×D28Q1614yescomplemented
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×C4×D7—28no—
147C2×C4×D7—82no—
148C2×C4×D7—82no—
149C2×Dic14—21no—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14C2×D412yes—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—42no—
163C7×C4⋊C4C2×D414yes—
164C42×C7—42no—
165C42×C7C2×D412yes—
166C42×C7C2×D412yes—
167#1144—28no—
168(C2×D7)⋊C4—28no—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—44no—
171(C2×D7)⋊C4—82no—
172C4⋊Dic7—28no—
173C4⋊Dic7—22no—
174C4⋊Dic7—21no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—42no—
178C4⋊Dic7—81no—
179C4⋊Dic7C2×D412yes—
180Dic7⋊C4—28no—
181Dic7⋊C4—24no—
182Dic7⋊C4—44no—
183C4×Dic7—24no—
184C2×C7⋊C8—21no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8#4112yes—
190C22×Q16—71no—
191#554—144no—
192#554—144no—
193C2×C4⋊Q8C2×D711yescharacteristic
194C2×#125—141no—
195C2×C22⋊Q8—72no—
196C2×C22⋊Q8—72no—
197C2×C22⋊Q8—142no—
198C2×C4×Q8—71no—
199C2×C4×Q8C2×D711yescharacteristic
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C22×C4⋊C4—72no—
203#477—24no—
204C4×Q16—144no—
205C2×#107—71no—
206C2×#107C2×D712yes—
207C2×C4⋊C8C2×D711yescharacteristic
208C2×#103—71no—
209C2×#103—71no—
210C2×#103—72no—
211C2×#103C2×D712yes—
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×C4×C8—71no—
215#396—142no—
216#390—141no—
217#389—141no—
218#384—71no—
219#380—72no—
220#340—72no—
221#340—72no—
222#340—72no—
223C23×C7C22×D411yescharacteristic
224C22×D7—22no—
225C22×D7—22no—
226C22×D7—22no—
227C22×D7—22no—
228C22×D7—22no—
229C22×D7—22no—
230C22×D7C2×Q1614yes—
231C7×Q8—22no—
232C7×Q8—22no—
233C7×Q8—41no—
234C7×Q8—41no—
235C7×Q8—84no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—24no—
239C2×C4×C7—22no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7#12012yes—
248C2×C4×C7#12012yes—
249C2×C4×C7C2×Q1612yes—
250C2×C4×C7#13712yes—
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C2×C4×C7H(5,𝔽2)11yescharacteristic
256C2×C4×C72−1+411yescharacteristic
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269D28—28no—
270C4×D7—42no—
271C4×D7—42no—
272C4×D7—42no—
273C4×D7—42no—
274C4×D7—88no—
275Dic14—22no—
276Dic14—22no—
277Dic14—41no—
278Dic14—41no—
279C7×C8—42no—
280C7⋊C8—22no—
281C7⋊C8—41no—
282C22×Q8—21no—
283C22×Q8—71no—
284C22×Q8C22×D711yescharacteristic
285C22×D4—71no—
286C23×C4—72no—
287C23×C4—142no—
288C2×Q16—142no—
289C2×Q16—142no—
290C2×Q16—142no—
291C22×C8—71no—
292C22×C8—71no—
293C22×C8C22×D712yes—
294C4⋊Q8—22no—
295C4⋊Q8C22⋊D714yes—
296#125—148no—
297C22⋊Q8—144no—
298C22⋊Q8—1416no—
299C22⋊Q8—144no—
300C22⋊Q8—144no—
301C22⋊Q8—282no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—74no—
305C4×Q8—142no—
306C4×D4—148no—
307C4×D4—142no—
308C4×D4—142no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C4⋊C4C22×D712yes—
324C2×C22⋊C4—72no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C42—21no—
329C2×C42—141no—
330C2×C42C22×D711yescharacteristic
331#107—24no—
332#107—142no—
333C4⋊C8—22no—
334#103—24no—
335#103—142no—
336#103—142no—
337#103—282no—
338C22⋊C8—144no—
339C22⋊C8—282no—
340C4×C8—142no—
341#95—142no—
342C22×C7C2×#12011yescharacteristic
343C22×C7#54611yescharacteristic
344C22×C7#54711yescharacteristic
345C22×C7C22×Q1611yescharacteristic
346C22×C7C2×#13711yescharacteristic
347C22×C7#57611yescharacteristic
348C22×C7#57711yescharacteristic
349C2×D7—24no—
350C2×D7—24no—
351C2×D7—24no—
352C2×D7—42no—
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—42no—
356C4×C7—22no—
357C4×C7—42no—
358C4×C7—44no—
359C4×C7—81no—
360C4×C7#45114yes—
361C4×C7#52511yescharacteristic
362C4×C7#53711yescharacteristic
363C4×C7#54811yescharacteristic
364C4×C7#55011yescharacteristic
365Dic7—42no—
366Dic7—42no—
367Dic7—42no—
368Dic7—82no—
369C24—72no—
370C2×Q8—24no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—72no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8—141no—
384C2×Q8#116112yes—
385C2×D4—74no—
386C2×D4—142no—
387C2×D4—142no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—72no—
392C22×C4—148no—
393C22×C4—141no—
394C22×C4—141no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—562no—
403C22×C4—562no—
404C22×C4D4×D711yescharacteristic
405C22×C4D4×D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407Q16—148no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—72no—
412C2×C8—141no—
413C2×C8—141no—
414C2×C8—141no—
415C2×C8—141no—
416C2×C8—141no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—42no—
422C4⋊C4—72no—
423C4⋊C4—148no—
424C4⋊C4—148no—
425C4⋊C4—144no—
426C4⋊C4—142no—
427C4⋊C4—141no—
428C4⋊C4—142no—
429C4⋊C4—142no—
430C4⋊C4—284no—
431C4⋊C4—282no—
432C4⋊C4—561no—
433C4⋊C4C2×C22⋊D714yes—
434C22⋊C4—148no—
435C22⋊C4—144no—
436C22⋊C4—284no—
437C22⋊C4—562no—
438C42—42no—
439C42—144no—
440C42D4×D712yes—
441C42C2×C22⋊D712yes—
442C2×C7#240711yescharacteristic
443C2×C7C2×#45111yescharacteristic
444C2×C7#301211yescharacteristic
445C2×C7#329911yescharacteristic
446C2×C7#333311yescharacteristic
447C2×C7#333811yescharacteristic
448C2×C7#335611yescharacteristic
449D7—44no—
450C23—74no—
451C23—142no—
452C23—142no—
453C23—142no—
454C23—142no—
455C23—142no—
456C23—142no—
457C23C2×D4×D711yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—41no—
461Q8—41no—
462Q8—84no—
463Q8—142no—
464Q8—142no—
465Q8—281no—
466Q8—281no—
467D4—148no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—142no—
480C2×C4—142no—
481C2×C4—142no—
482C2×C4—142no—
483C2×C4—142no—
484C2×C4—281no—
485C2×C4—281no—
486C2×C4—281no—
487C2×C4—281no—
488C2×C4—282no—
489C2×C4—282no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—568no—
496C2×C4D7×Q1612yes—
497C2×C4#648112yes—
498C2×C4#650012yes—
499C2×C4#650012yes—
500C2×C4C2×D4×D711yescharacteristic
501C2×C4C2×#116111yescharacteristic
502C2×C4#655111yescharacteristic
503C2×C4D7×#4111yescharacteristic
504C2×C4#655311yescharacteristic
505C2×C4C22×C22⋊D711yescharacteristic
506C8—42no—
507C8—142no—
508C8—281no—
509C7#2412211yescharacteristic, complemented, Sylow
510C22—144no—
511C22—144no—
512C22—144no—
513C22—282no—
514C22—282no—
515C22—282no—
516C22—282no—
517C22#9032111yescharacteristic
518C22#9032211yescharacteristic
519C22C2×D7×Q1611yescharacteristic
520C22C2×#648111yescharacteristic
521C22#9046511yescharacteristic
522C22#9046611yescharacteristic
523C22C2×#650011yescharacteristic
524C4—22no—
525C4—42no—
526C4—44no—
527C4—81no—
528C4—282no—
529C4—282no—
530C4—282no—
531C4—562no—
532C4#8995814yes—
533C4#9031811yescharacteristic
534C4#9032011yescharacteristic
535C4#9050811yescharacteristic
536C4#9052511yescharacteristic
537C2—284no—
538C2#1173856411yescharacteristic
539C2#1173856911yescharacteristic
540C2#1173857011yescharacteristic
541C2#1173857411yescharacteristic
542C2#1174025711yescharacteristic
543C2C2×#8995811yescharacteristic
544C2#1174139411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#477C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenterC2×#477D7
hypocenterC7#24122
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
4. C2×C4×#1144C2semidirect
5. #11729541C2semidirect
6. #11728389C2semidirect
7. #11728386C2semidirect
8. #11728239C2semidirect
10. #11728027C2semidirect
11. #11728022C2semidirect
107. C2×#477D7semidirect
139. C2×D28Q16semidirect
509. C7#24122semidirect

Maximal subgroups

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

subgroupindex
C2×#904212
C2×C7×#4772
#117399452
C2×C4×#11442
#117295412
#117283892
#117283862
#117282392
#117282242
#117280272
#117280222
#241227