Finite Group Databasebeta

← #13387729

Subgroups — #13387729

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×#90246C211yesmaximal, characteristic
2C7×#2538C211yesmaximal, characteristic, complemented
3C2×#89792C211yesmaximal, characteristic, complemented
4C2×C4×#1143C211yesmaximal, characteristic, complemented
5#11729453C212yesmaximal, complemented
6#11729446C212yesmaximal, complemented
7#11729368C211yesmaximal, characteristic, complemented
8#11727338C211yesmaximal, characteristic, complemented
9#11727336C211yesmaximal, characteristic, complemented
10#11727278C212yesmaximal, complemented
11#11727259C212yesmaximal, complemented
12C22×#1159—21no—
13C2×C7×C22⋊Q8C2212yes—
14C2×C4×C7×Q8C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C22×#1143C2211yescharacteristic
17C2×#6493C2212yes—
18C22×C4⋊Dic7C2212yes—
19#90246—24no—
20#90246—24no—
21C2×#6460—21no—
22C2×#6449—22no—
23C2×C4×D28C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C7×#396—22no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#384C2211yescharacteristic
29C7×#380C2212yes—
30C2×#6409C2211yescharacteristic
31C2×#6409C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34C4×#1143—24no—
35#89792—24no—
36C2×#6383C2211yescharacteristic
37C2×#6383C2212yes—
38C2×#6382C2211yescharacteristic
39C2×#6382C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#24120—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4D411yescharacteristic
48C23×C4×C7C2312yes—
49C22×C22⋊D7—22no—
50C23×Dic7—22no—
51C2×#1159—28no—
52C2×#1159—44no—
53C2×#1159—44no—
54C22×D28C2311yescharacteristic
55C22×C4×D7—21no—
56C22×Dic14—21no—
57C7×C22⋊Q8—24no—
58C7×C22⋊Q8—42no—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4D412yes—
73C2×C42×C7—21no—
74C2×C42×C7C2311yescharacteristic
75C2×#1143—22no—
76C2×#1143—22no—
77C2×#1143—22no—
78#6493—24no—
79#6493—24no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—24no—
86C2×C4⋊Dic7C2312yes—
87C2×C4⋊Dic7C2311yescharacteristic
88C22×C7⋊C8C2311yescharacteristic
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2311yescharacteristic
91#6460—28no—
92#6449—216no—
93C4×D28—22no—
94C4×D28D414yes—
95#6432—22no—
96#6432D414yes—
97C7×#95—22no—
98#6409—24no—
99#6409—22no—
100C22⋊(C7⋊C8)—24no—
101C22⋊(C7⋊C8)—42no—
102#6383—22no—
103#6383—42no—
104#6382—24no—
105#6382—22no—
106C4⋊(C7⋊C8)—22no—
107C4×C7⋊C8—22no—
108C2×#547—71no—
109C2×#475—71no—
110C2×C4×SD16—71no—
111#2538D711yescharacteristic, complemented
112#2034—72no—
113#2000—72no—
114#1990—71no—
115#1954—71no—
116#1872—72no—
117#1853—72no—
118#1802—71no—
119C24×C7C2×D412yes—
120C23×D7—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8#4112yescomplemented
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4SD1614yescomplemented
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C2×C22⋊D7—28no—
144C2×C22⋊D7—44no—
145C2×C22⋊D7—44no—
146C22×Dic7—28no—
147C22×Dic7—22no—
148C22×Dic7—21no—
149C22×Dic7—44no—
150C22×Dic7—44no—
151#1159—432no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28C2×D412yes—
158C2×C4×D7—24no—
159C2×C4×D7—42no—
160C2×C4×D7—42no—
161C2×Dic14—24no—
162C2×Dic14—42no—
163C2×Dic14—42no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—44no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—81no—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—44no—
175C7×C22⋊C4—82no—
176C42×C7—42no—
177C42×C7C2×D412yes—
178C42×C7C2×D412yes—
179#1143—28no—
180(C2×D7)⋊C4—28no—
181(C2×D7)⋊C4—24no—
182C4⋊Dic7—28no—
183C4⋊Dic7—24no—
184C4⋊Dic7—22no—
185C4⋊Dic7—21no—
186C4⋊Dic7C2×D414yes—
187C4⋊Dic7C2×D412yes—
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—141no—
198C22×SD16—71no—
199#547—144no—
200#547—144no—
201C2×C4⋊Q8—71no—
202C2×#123—142no—
203C2×C22⋊Q8—141no—
204C2×C22⋊Q8C2×D712yes—
205C2×#121—72no—
206C2×C4×Q8C2×D711yescharacteristic
207C2×C4×D4—71no—
208C2×C4×D4C2×D711yescharacteristic
209C22×C4⋊C4—72no—
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—22no—
223#390—21no—
224#389—21no—
225#384C2×D711yescharacteristic
226#380C2×D712yes—
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×C7C2×SD1614yes—
237C23×C7C22×D411yescharacteristic
238C22×D7—22no—
239C22×D7—41no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7—41no—
243C22×D7—41no—
244C22×D7—41no—
245C7×Q8—22no—
246C7×Q8—22no—
247C7×Q8—41no—
248C7×Q8—41no—
249C7×D4—28no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7C2×SD1612yes—
271C2×C4×C7AGL(1,ℤ/8)12yes—
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C72−1+411yescharacteristic
278C22⋊D7—416no—
279C2×Dic7—24no—
280C2×Dic7—22no—
281C2×Dic7—416no—
282C2×Dic7—42no—
283C2×Dic7—41no—
284C2×Dic7—42no—
285C2×Dic7—41no—
286D28—22no—
287D28—22no—
288D28—41no—
289D28—41no—
290C4×D7—48no—
291Dic14—48no—
292C7⋊C8—22no—
293C7⋊C8—41no—
294C7⋊C8—42no—
295C2×#41—148no—
296C2×#41—284no—
297C2×#41—284no—
298C22×Q8—141no—
299C22×Q8C22×D711yescharacteristic
300C22×D4—71no—
301C22×D4—142no—
302C22×D4C22⋊D711yescharacteristic
303C23×C4—142no—
304C23×C4—141no—
305C23×C4C22×D712yes—
306C2×SD16—142no—
307C2×SD16—142no—
308C2×SD16—142no—
309C22×C8—71no—
310C22×C8—71no—
311C22×C8—72no—
312C4⋊Q8—74no—
313C4⋊Q8—142no—
314#123—1416no—
315C22⋊Q8—24no—
316C22⋊Q8—42no—
317C22⋊Q8—148no—
318#121—144no—
319#121—144no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—74no—
325C4×D4—142no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—144no—
333C2×C4⋊C4—144no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D712yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4C22⋊D712yes—
344C2×C42—21no—
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—22no—
357C22×C7—24no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7C2×#12011yescharacteristic
365C22×C7#54611yescharacteristic
366C22×C7#54711yescharacteristic
367C22×C7C22×SD1611yescharacteristic
368C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C7#57711yescharacteristic
370C22×C7#57811yescharacteristic
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—81no—
375C2×D7—81no—
376C2×D7—81no—
377C2×D7—81no—
378C4×C7—22no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—82no—
382C4×C7—81no—
383C4×C7#45014yes—
384C4×C7#52511yescharacteristic
385C4×C7#53711yescharacteristic
386C4×C7#54811yescharacteristic
387C4×C7#55011yescharacteristic
388Dic7—44no—
389Dic7—42no—
390C24—141no—
391C24C2×C22⋊D712yes—
392#41—2832no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—144no—
399C2×Q8—282no—
400C2×Q8—282no—
401C2×Q8#115912yescomplemented
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—72no—
405C2×D4—148no—
406C2×D4—141no—
407C2×D4—141no—
408C2×D4—141no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—284no—
412C2×D4—284no—
413C2×D4#114314yescomplemented
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—148no—
424C22×C4—141no—
425C22×C4—142no—
426C22×C4—144no—
427C22×C4—284no—
428C22×C4—284no—
429C22×C4—282no—
430C22×C4—282no—
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—141no—
440C2×C8—142no—
441C2×C8—141no—
442C2×C8—142no—
443C2×C8—142no—
444C2×C8—141no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—72no—
454C4⋊C4—74no—
455C4⋊C4—81no—
456C4⋊C4—148no—
457C4⋊C4—144no—
458C4⋊C4—141no—
459C4⋊C4—142no—
460C22⋊C4—24no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—148no—
464C22⋊C4—144no—
465C42—42no—
466C42D4×D712yes—
467C42D4×D712yes—
468C2×C7—44no—
469C2×C7#240711yescharacteristic
470C2×C7C2×#45011yescharacteristic
471C2×C7#301311yescharacteristic
472C2×C7#329811yescharacteristic
473C2×C7#333211yescharacteristic
474C2×C7#333511yescharacteristic
475C2×C7#335311yescharacteristic
476D7—82no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—142no—
484C23—281no—
485C23—281no—
486C23—281no—
487C23—281no—
488C23—281no—
489C23—281no—
490C23C2×#114314yes—
491C23C22×C22⋊D711yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—288no—
497D4—28no—
498D4—142no—
499D4—142no—
500D4—2816no—
501D4—281no—
502D4—281no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—142no—
522C2×C4—144no—
523C2×C4—2816no—
524C2×C4—281no—
525C2×C4—282no—
526C2×C4—281no—
527C2×C4—282no—
528C2×C4—288no—
529C2×C4#650012yes—
530C2×C4#650012yes—
531C2×C4C2×#114312yes—
532C2×C4#651212yes—
533C2×C4C2×D4×D711yescharacteristic
534C2×C4C2×D4×D711yescharacteristic
535C2×C4C2×#115911yescharacteristic
536C2×C4#654811yescharacteristic
537C2×C4D7×#4111yescharacteristic
538C2×C4#655411yescharacteristic
539C8—142no—
540C8—282no—
541C8—281no—
542C7#2412011yescharacteristic, complemented, Sylow
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—282no—
551C22—282no—
552C22—282no—
553C22—561no—
554C22—561no—
555C22—561no—
556C22—561no—
557C22C2×#650011yescharacteristic
558C22#9049711yescharacteristic
559C22C22×#114311yescharacteristic
560C22C2×#651211yescharacteristic
561C22#9051511yescharacteristic
562C22#9052711yescharacteristic
563C22#9052911yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—81no—
569C4—282no—
570C4—284no—
571C4#8981814yes—
572C4#9028911yescharacteristic
573C4#9029011yescharacteristic
574C4#9029711yescharacteristic
575C4#9029911yescharacteristic
576C2—44no—
577C2—562no—
578C2#1174014611yescharacteristic
579C2C2×#8981811yescharacteristic
580C2#1174087211yescharacteristic
581C2#1174099511yescharacteristic
582C2#1174101911yescharacteristic
583C2#1174106711yescharacteristic
584C2#1174110711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2538C2
socleC23×C7C22×D4
prime radicalC2×#90246C2
hypercenter#2538D7
hypocenterC7#24120
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C7×#2538C2semidirect
3. C2×#89792C2semidirect
4. C2×C4×#1143C2semidirect
5. #11729453C2semidirect
6. #11729446C2semidirect
7. #11729368C2semidirect
8. #11727338C2semidirect
9. #11727336C2semidirect
10. #11727278C2semidirect
11. #11727259C2semidirect
111. #2538D7semidirect
126. C2×C7×Q8#41semidirect
129. C2×C7×D4SD16semidirect
196. C2×C7⋊C8#41semidirect
401. C2×Q8#1159semidirect
413. C2×D4#1143semidirect
542. C7#24120semidirect

Maximal subgroups

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

subgroupindex
C2×#902462
C7×#25382
C2×#897922
C2×C4×#11432
#117294532
#117294462
#117293682
#117273382
#117273362
#117272782
#117272592
#241207