Finite Group Databasebeta

← #13387727

Subgroups — #13387727

Order 1792.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic, complemented
3#11739821C211yesmaximal, characteristic, complemented
4C2×C4×#1142C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11729439C212yesmaximal, complemented
7#11729434C212yesmaximal, complemented
8#11729368C211yesmaximal, characteristic, complemented
9#11729358C211yesmaximal, characteristic, complemented
10#11729313C212yesmaximal, complemented
11#11729310C212yesmaximal, complemented
12C22×C7×#41—21no—
13C7×#547—24no—
14C7×#547—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C22×#1142C2211yescharacteristic
23C2×#6487C2212yes—
24C2×C4×Dic14C2211yescharacteristic
25#89993—21no—
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31#89866—24no—
32C4×#1142—24no—
33#89745—21no—
34C2×#6384C2212yes—
35C2×#6384C2211yescharacteristic
36C2×#6382C2212yes—
37C2×#6382C2211yescharacteristic
38#89720—22no—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24120—71nomaximal, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×D4—22no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—22no—
54C23×C4×C7—21no—
55C23×C4×C7C2312yes—
56C22×Dic14C2311yescharacteristic
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8D414yes—
59C7×#123—216no—
60C7×C22⋊Q8—28no—
61C7×#121—24no—
62C7×#121—24no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C4×C7×D4—22no—
66C4×C7×D4D414yes—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4D412yes—
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—22no—
79C2×C22⋊Dic7—22no—
80C2×#1142—22no—
81C2×#1142—22no—
82C2×#1142—22no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2312yes—
87#6487—24no—
88#6487—42no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—22no—
91C2×C4×Dic7—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2312yes—
94C22×C7⋊C8C2311yescharacteristic
95C4×Dic14—22no—
96C4×Dic14—22no—
97#6406—22no—
98#6406—42no—
99C7⋊#95—22no—
100C22⋊(C7⋊C8)—24no—
101C22⋊(C7⋊C8)—42no—
102#6384—24no—
103#6384—22no—
104#6382—24no—
105#6382—22no—
106C4⋊(C7⋊C8)—22no—
107C4×C7⋊C8—22no—
108C2×#547D711yescharacteristic
109C2×#475—71no—
110C2×C4×SD16—71no—
111#2538—71no—
112#2034—72no—
113#2000—72no—
114#1990—71no—
115#1954—71no—
116#1872—72no—
117#1853—72no—
118#1802—71no—
119C24×C7—21no—
120C24×C7C2×D412yes—
121C7×#41—432no—
122C2×C7×Q8—24no—
123C2×C7×Q8—42no—
124C2×C7×Q8—42no—
125C2×C7×D4—28no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4SD1614yescomplemented
136C2×C7×D4C2×D412yes—
137C22×C4×C7—28no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—24no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—44no—
148C22×C4×C7—44no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C2×Dic14—21no—
159C2×Dic14—21no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14#4112yescomplemented
164C7×C4⋊C4—28no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4C2×D412yes—
170C7×C4⋊C4C2×D414yes—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—24no—
174C42×C7C2×D412yes—
175C42×C7C2×D412yes—
176C22⋊Dic7—44no—
177C22⋊Dic7—82no—
178#1142—28no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7—42no—
184C4⋊Dic7—81no—
185Dic7⋊C4—44no—
186Dic7⋊C4—42no—
187C4×Dic7—42no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8#4112yescomplemented
197C22×#41—21no—
198C22×SD16—71no—
199#547—24no—
200#547—24no—
201C2×C4⋊Q8C2×D711yescharacteristic
202C2×#123—22no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8—72no—
205C2×#121C2×D712yes—
206C2×C4×Q8—71no—
207C2×C4×D4C2×D711yescharacteristic
208C2×C4×D4C2×D711yescharacteristic
209C22×C4⋊C4C2×D712yes—
210#475—144no—
211C4×SD16—144no—
212C2×#106—72no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×#103—72no—
216C2×#103—71no—
217C2×#102—71no—
218C2×#102—72no—
219C2×C22⋊C8—72no—
220C2×C22⋊C8—72no—
221C2×C4×C8—71no—
222#396—142no—
223#390—141no—
224#389—141no—
225#384—71no—
226#380—72no—
227#340—72no—
228#340—72no—
229#340—72no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7C2×SD1614yes—
244C23×C7C22×D411yescharacteristic
245C7×Q8—48no—
246C7×D4—22no—
247C7×D4—22no—
248C7×D4—28no—
249C7×D4—416no—
250C7×D4—41no—
251C7×D4—41no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—416no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—48no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12012yes—
271C2×C4×C7C2×SD1612yes—
272C2×C4×C7AGL(1,ℤ/8)12yes—
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×C4×C72−1+411yescharacteristic
279C2×Dic7—22no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—42no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288Dic14—22no—
289Dic14—22no—
290Dic14—41no—
291Dic14—41no—
292C7⋊C8—22no—
293C7⋊C8—41no—
294C7⋊C8—42no—
295C2×#41—28no—
296C2×#41—44no—
297C2×#41—44no—
298C22×Q8—21no—
299C22×Q8—71no—
300C22×D4—22no—
301C22×D4C22⋊D711yescharacteristic
302C22×D4C22×D711yescharacteristic
303C23×C4—22no—
304C23×C4—21no—
305C23×C4C22×D712yes—
306C2×SD16—142no—
307C2×SD16—142no—
308C2×SD16—142no—
309C22×C8—71no—
310C22×C8—71no—
311C22×C8—72no—
312C4⋊Q8—22no—
313C4⋊Q8C22⋊D714yes—
314#123—216no—
315C22⋊Q8—28no—
316C22⋊Q8—144no—
317C22⋊Q8—282no—
318#121—24no—
319#121—24no—
320C4×Q8—142no—
321C4×Q8—142no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4C22⋊D714yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D712yes—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4C22⋊D712yes—
344C2×C42—141no—
345C2×C42C22×D711yescharacteristic
346#106—142no—
347#106—144no—
348C4⋊C8—142no—
349#103—142no—
350#103—144no—
351#102—142no—
352#102—282no—
353C22⋊C8—144no—
354C22⋊C8—282no—
355C4×C8—142no—
356#95—142no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—81no—
368C22×C7—81no—
369C22×C7—81no—
370C22×C7—81no—
371C22×C7C2×#12011yescharacteristic
372C22×C7#54611yescharacteristic
373C22×C7#54711yescharacteristic
374C22×C7C22×SD1611yescharacteristic
375C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
376C22×C7#57711yescharacteristic
377C22×C7#57811yescharacteristic
378C4×C7—22no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—44no—
382C4×C7#45014yes—
383C4×C7#52511yescharacteristic
384C4×C7#53711yescharacteristic
385C4×C7#54811yescharacteristic
386C4×C7#55011yescharacteristic
387Dic7—42no—
388Dic7—82no—
389Dic7—81no—
390C24—21no—
391C24C2×C22⋊D712yes—
392#41—432no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—72no—
397C2×Q8—141no—
398C2×Q8—141no—
399C2×Q8—141no—
400C2×Q8—141no—
401C2×Q8—141no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4#114214yescomplemented
413C2×D4C2×C22⋊D712yes—
414C22×C4—28no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—141no—
429C22×C4—141no—
430C22×C4—142no—
431C22×C4C2×C22⋊D712yes—
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435SD16—148no—
436C2×C8—72no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—142no—
440C2×C8—141no—
441C2×C8—142no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—142no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—142no—
451C4⋊C4—142no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—284no—
455C4⋊C4—282no—
456C4⋊C4—282no—
457C4⋊C4—561no—
458C4⋊C4C2×C22⋊D712yes—
459C4⋊C4C2×C22⋊D714yes—
460C22⋊C4—28no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—284no—
464C22⋊C4—562no—
465C42—282no—
466C42C2×C22⋊D712yes—
467C42C2×C22⋊D712yes—
468C2×C7—44no—
469C2×C7—82no—
470C2×C7#240711yescharacteristic
471C2×C7C2×#45011yescharacteristic
472C2×C7#301311yescharacteristic
473C2×C7#329811yescharacteristic
474C2×C7#333211yescharacteristic
475C2×C7#333511yescharacteristic
476C2×C7#335311yescharacteristic
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×#114214yes—
491C23C22×C22⋊D711yescharacteristic
492Q8—48no—
493Q8—142no—
494Q8—142no—
495Q8—281no—
496Q8—281no—
497D4—28no—
498D4—22no—
499D4—22no—
500D4—416no—
501D4—41no—
502D4—41no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—416no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—142no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—281no—
526C2×C4—282no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4#649512yes—
530C2×C4C2×#114212yes—
531C2×C4#651612yes—
532C2×C4#651612yes—
533C2×C4D7×#4111yescharacteristic
534C2×C4D7×#4111yescharacteristic
535C2×C4#655311yescharacteristic
536C2×C4#655411yescharacteristic
537C2×C4C22×C22⋊D711yescharacteristic
538C2×C4C22×C22⋊D711yescharacteristic
539C8—142no—
540C8—281no—
541C8—282no—
542C7#2412011yescharacteristic, complemented, Sylow
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#649511yescharacteristic
558C22C22×#114211yescharacteristic
559C22#9051511yescharacteristic
560C22#9052011yescharacteristic
561C22#9052811yescharacteristic
562C22#9052911yescharacteristic
563C22C2×#651611yescharacteristic
564C4—22no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—282no—
569C4—562no—
570C4—561no—
571C4#8999014yes—
572C4#9051911yescharacteristic
573C4#9052211yescharacteristic
574C4#9052411yescharacteristic
575C4#9052511yescharacteristic
576C2—44no—
577C2—82no—
578C2#1174096211yescharacteristic
579C2#1174096511yescharacteristic
580C2#1174097611yescharacteristic
581C2#1174097811yescharacteristic
582C2C2×#8999011yescharacteristic
583C2#1174154211yescharacteristic
584C2#1174155911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC23×C7C22×D4
hypercenterC2×#547D7
hypocenterC7#24120
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C2×#89866C2semidirect
3. #11739821C2semidirect
4. C2×C4×#1142C2semidirect
5. #11729541C2semidirect
6. #11729439C2semidirect
7. #11729434C2semidirect
8. #11729368C2semidirect
9. #11729358C2semidirect
10. #11729313C2semidirect
11. #11729310C2semidirect
135. C2×C7×D4SD16semidirect
163. C2×Dic14#41semidirect
196. C2×C7⋊C8#41semidirect
412. C2×D4#1142semidirect
542. C7#24120semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×#898662
#117398212
C2×C4×#11422
#117295412
#117294392
#117294342
#117293682
#117293582
#117293132
#117293102
#241207