Finite Group Databasebeta

← #13409928

Subgroups — #13409928

Order 1792.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C2×#89837C211yesmaximal, characteristic, complemented
3#11739920C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic
5#11729537C212yesmaximal, complemented
6#11729493C212yesmaximal, complemented
7#11729492C211yesmaximal, characteristic, complemented
8#11729389C211yesmaximal, characteristic, complemented
9#11729363C211yesmaximal, characteristic
10#11729361C212yesmaximal
11#11729360C212yesmaximal
12C7×#554—24no—
13C7×#554—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125C2211yescharacteristic
16C2×C7×#125C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4—21no—
25C2×#6507—21no—
26C2×#6432C2211yescharacteristic
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29#89837—24no—
30#89780—24no—
31#89755—21no—
32#89755—21no—
33#89746—21no—
34C2×#6382C2212yes—
35C2×#6382C2212yes—
36C2×#6381C2212yes—
37C4×C4⋊Dic7C2211yescharacteristic
38#89700C2212yes—
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89319C2212yes—
46#89319C2212yes—
47#25851—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C22×Dic14—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×#125—22no—
58C7×#125—22no—
59C7×#125—22no—
60C7×#125D414yes—
61C7×C22⋊Q8—28no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—28no—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C4×C7×D4D414yes—
72C2×C7×C4⋊C4—28no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4D411yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90#6507—44no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—22no—
95C2×Dic7⋊C4—22no—
96C2×C4×Dic7—21no—
97C22×C7⋊C8C2312yes—
98C22×C7⋊C8C2312yes—
99#6432—22no—
100#6432—22no—
101#6406—24no—
102#6406Q818yescomplemented
103C7⋊#95—22no—
104C7⋊#95—22no—
105C7⋊#95—21no—
106#6382—24no—
107#6382—24no—
108#6381—24no—
109C4⋊(C7⋊C8)—22no—
110#6377—22no—
111C2×#554D711yescharacteristic
112C2×#501—71no—
113C2×#476—71no—
114#2546—71no—
115#2059—72no—
116#2058—72no—
117#1989—71no—
118#1985—71no—
119#1864—72no—
120#1862—72no—
121#1762—71no—
122C24×C7—21no—
123C2×C7×Q8—24no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—41no—
127C2×C7×Q8—82no—
128C2×C7×D4—21no—
129C2×C7×D4—22no—
130C2×C7×D4C2×D412yes—
131C2×C7×D4C2×Q814yes—
132C22×C4×C7—24no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—24no—
137C22×C4×C7—24no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—82no—
141C22×C4×C7—82no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—22no—
150C22×Dic7—22no—
151C2×Dic14—42no—
152C2×Dic14—42no—
153C2×Dic14—42no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4C2×D412yes—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D412yes—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—24no—
171C42×C7—21no—
172C42×C7—21no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7C2×D412yes—
176C42×C7C2×D412yes—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7C2×Q814yes—
182Dic7⋊C4—44no—
183Dic7⋊C4—44no—
184C4×Dic7—42no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—24no—
189C2×C7⋊C8C2×Q814yes—
190#554—24no—
191#554—24no—
192C2×C4⋊Q8—71no—
193C2×C4⋊Q8—141no—
194C2×C4⋊Q8C2×D711yescharacteristic
195C2×#125C2×D711yescharacteristic
196C2×#125C2×D711yescharacteristic
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8—21no—
200C2×C4×Q8C2×D711yescharacteristic
201C2×C4×D4C2×D711yescharacteristic
202C2×C4×D4C2×D711yescharacteristic
203C2×C4×D4C2×D711yescharacteristic
204C22×C4⋊C4—21no—
205#501—144no—
206#476—144no—
207C2×#107—72no—
208C2×#106—72no—
209C2×#106—72no—
210C2×C4⋊C8—71no—
211C2×#102—72no—
212C2×#102—72no—
213C2×#97—71no—
214#398—141no—
215#398—141no—
216#391—141no—
217#389—72no—
218#389—72no—
219C4×C4⋊C4—71no—
220#340—72no—
221#340—72no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7—22no—
226C23×C7—24no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—42no—
230C23×C7C22×D411yescharacteristic
231C7×Q8—44no—
232C7×Q8—84no—
233C7×D4—22no—
234C7×D4—24no—
235C7×D4C4⋊Q814yescomplemented
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—22no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—48no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—84no—
257C2×C4×C7—84no—
258C2×C4×C7#12012yes—
259C2×C4×C7#12012yes—
260C2×C4×C7AGL(1,ℤ/8)12yes—
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C22×D411yescharacteristic
263C2×C4×C7C22×Q812yes—
264C2×C4×C72−1+412yes—
265C2×Dic7—24no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271Dic14—84no—
272C7⋊C8—24no—
273C7⋊C8—42no—
274C22×Q8—21no—
275C22×Q8—141no—
276C22×Q8C22⋊D711yescharacteristic
277C22×D4C22×D711yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C22×C8—72no—
282C22×C8—72no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—142no—
286C4⋊Q8—142no—
287C4⋊Q8—284no—
288#125—22no—
289#125—22no—
290#125—22no—
291#125C22⋊D714yes—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4C22⋊D714yes—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—72no—
310C2×C4⋊C4—72no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4C22⋊D711yescharacteristic
315C2×C4⋊C4C22⋊D711yescharacteristic
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C4⋊C4C22×D711yescharacteristic
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C42—141no—
324C2×C42C22×D711yescharacteristic
325C2×C42C22×D711yescharacteristic
326C2×C42C22×D711yescharacteristic
327#107—144no—
328#106—144no—
329#106—144no—
330C4⋊C8—142no—
331#102—78no—
332#102—144no—
333#97—142no—
334#95—141no—
335#95—142no—
336#95—142no—
337C22×C7—24no—
338C22×C7—44no—
339C22×C7—44no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7C2×#12011yescharacteristic
343C22×C7D4×Q812yes—
344C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C7#57511yescharacteristic
346C22×C7#57712yes—
347C4×C7—22no—
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—41no—
351C4×C7—42no—
352C4×C7—41no—
353C4×C7—41no—
354C4×C7—42no—
355C4×C7—42no—
356C4×C7C2×C4⋊Q811yescharacteristic
357C4×C7#53611yescharacteristic
358C4×C7D4×Q812yes—
359Dic7—44no—
360Dic7—82no—
361C24—21no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—82no—
367C2×Q8—282no—
368C2×Q8—282no—
369C2×Q8—282no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4Q8×D714yes—
373C2×D4C2×C22⋊D712yes—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—142no—
385C22×C4—142no—
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388C22×C4C2×C22⋊D711yescharacteristic
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C2×C22⋊D711yescharacteristic
391C22×C4C2×C22⋊D711yescharacteristic
392C22×C4C23×D711yescharacteristic
393C2×C8—74no—
394C2×C8—142no—
395C2×C8—142no—
396C2×C8—144no—
397C2×C8—142no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—74no—
410C4⋊C4—142no—
411C4⋊C4—142no—
412C4⋊C4—142no—
413C4⋊C4—284no—
414C4⋊C4—284no—
415C4⋊C4—282no—
416C4⋊C4C2×C22⋊D712yes—
417C4⋊C4C2×C22⋊D712yes—
418C4⋊C4C2×C22⋊D712yes—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—282no—
427C42C2×C22⋊D712yes—
428C42C2×C22⋊D712yes—
429C2×C7—44no—
430C2×C7#243411yescharacteristic
431C2×C7#300611yescharacteristic
432C2×C7#300911yescharacteristic
433C2×C7#338412yes—
434C2×C7#338812yes—
435C23—24no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23C22×C22⋊D711yescharacteristic
441Q8—44no—
442Q8—84no—
443Q8—564no—
444D4—24no—
445D4—22no—
446D4#650714yescomplemented
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—144no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4#651212yes—
476C2×C4#651612yes—
477C2×C4#651612yes—
478C2×C4C2×Q8×D712yes—
479C2×C4#655112yes—
480C2×C4C22×C22⋊D711yescharacteristic
481C2×C4C22×C22⋊D711yescharacteristic
482C8—144no—
483C8—282no—
484C7#2585111yescharacteristic, complemented, Sylow
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22Q8×C22⋊D712yes—
491C22C2×#651211yescharacteristic
492C22#9051311yescharacteristic
493C22#9052712yes—
494C22C2×#651611yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—284no—
505C4—562no—
506C4C2×#650711yescharacteristic
507C4Q8×C22⋊D712yes—
508C4#9050711yescharacteristic
509C2—44no—
510C2#1174107512yes—
511C2#1174107712yes—
512C2#1174151011yescharacteristic
513C2#1174152711yescharacteristic
514C2#1174155011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#554C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×#554D7
hypocenterC7#25851
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#89837C2semidirect
3. #11739920C2semidirect
5. #11729537C2semidirect
6. #11729493C2semidirect
7. #11729492C2semidirect
8. #11729389C2semidirect
102. #6406Q8semidirect
235. C7×D4C4⋊Q8semidirect
446. D4#6507semidirect
484. C7#25851semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×#898372
#117399202
C2×#897802
#117295372
#117294932
#117294922
#117293892
#117293632
#117293612
#117293602
#258517