Finite Group Databasebeta

← #13358926

Subgroups — #13358926

Order 1792.

1182 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×#90371C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic, complemented
3C4×#6455C211yesmaximal, characteristic, complemented
4C7×#1983C211yesmaximal, characteristic, complemented
5#11729475C211yesmaximal, characteristic, complemented
6#11728215C211yesmaximal, characteristic, complemented
7#11728023C212yesmaximal, complemented
8#11727974C212yesmaximal, complemented
9#11727955C212yesmaximal
10#11727906C212yesmaximal, complemented
11#11727245C211yesmaximal, characteristic, complemented
12C2×C7×#124C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C2×#6501C2211yescharacteristic
16C2×D4×Dic7C2211yescharacteristic
17#90371—24no—
18#90371C418yescomplemented
19C2×#6455C2212yes—
20C2×#6446—21no—
21C2×(C22⋊D7)⋊C4—21no—
22C2×#6442—21no—
23C2×#6440—21no—
24C2×C42×D7C2211yescharacteristic
25C2×#6433C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C2×C7×#103C2212yes—
28C2×C7×#102C2212yes—
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6406C2212yes—
32C2×(C2×D7)⋊C8C2212yes—
33C2×#6389C2212yes—
34C4⋊C4×Dic7C2212yes—
35C4×(C2×D7)⋊C4—21no—
36C4×C4⋊Dic7C2211yescharacteristic
37C42×Dic7—21no—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89607—24no—
40#89438—21no—
41#89420—21no—
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21396—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C22×C22⋊D7—21no—
50C23×Dic7—21no—
51C22×C4×D7D411yescharacteristic
52C22×C4×D7D411yescharacteristic
53C22×C4×D7C2311yescharacteristic
54C22×C7×C8C2312yes—
55C7×#124—22no—
56C7×#124C2×C414yes—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—21no—
64#6509—22no—
65#6509C2×C414yes—
66Q8×Dic7—22no—
67Q8×Dic7C2×C414yes—
68#6501—22no—
69#6501C2×C414yes—
70D4×Dic7—22no—
71D4×Dic7C2×C414yes—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—22no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2312yes—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89#6455—24no—
90#6455—24no—
91#6446—28no—
92(C22⋊D7)⋊C4—28no—
93#6442—28no—
94#6440—28no—
95C42×D7C2×C414yes—
96C42×D7D414yes—
97#6433—22no—
98#6433C2×C414yes—
99C7×C4⋊C8—22no—
100C7×#103—24no—
101C7×#102—24no—
102C7×#95—21no—
103#6409—24no—
104#6406—24no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107(C2×D7)⋊C8—24no—
108#6389—24no—
109C4⋊(C7⋊C8)—22no—
110C2×#529—71no—
111C2×#432—71no—
112C4×#117—71no—
113#1989—71no—
114#1989—71no—
115#1983—71no—
116#1983D711yescharacteristic, complemented
117#1875—72no—
118#1873—72no—
119#1763—72no—
120#1762—72no—
121C24×C7—21no—
122C23×D7C2×D411yescharacteristic
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8C22×C412yes—
127C2×C7×D4—21no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4C22×C412yes—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2411yescharacteristic
136C2×C22⋊D7—24no—
137C2×C22⋊D7—44no—
138C22×Dic7—22no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—24no—
142C22×Dic7—42no—
143C22×Dic7—42no—
144C22×Dic7C2×D411yescharacteristic
145C22×Dic7C2×D411yescharacteristic
146C22×Dic7C2×D411yescharacteristic
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—42no—
150C2×C4×D7—41no—
151C2×C4×D7—41no—
152C2×C4×D7C22⋊C414yes—
153C2×C4×D7C22⋊C414yes—
154C2×C4×D7C22×C412yes—
155C2×C4×D7C2×D412yes—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4#4114yes—
161C7×C22⋊C4—24no—
162C7×C22⋊C4—24no—
163C42×C7—42no—
164C42×C7C22×C412yes—
165C42×C7C2×D412yes—
166C22⋊Dic7—24no—
167C22⋊Dic7—24no—
168(C2×D7)⋊C4—28no—
169(C2×D7)⋊C4—24no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—22no—
173C4⋊Dic7—24no—
174C4⋊Dic7C22×C412yes—
175C4⋊Dic7C22×C412yes—
176C4⋊Dic7#4114yes—
177Dic7⋊C4—24no—
178Dic7⋊C4—24no—
179Dic7⋊C4—24no—
180C4×Dic7—28no—
181C4×Dic7—24no—
182C4×Dic7—44no—
183C4×Dic7—42no—
184C4×Dic7C22⋊C414yes—
185C4×Dic7C22⋊C414yes—
186C4×Dic7C22×C412yes—
187C4×Dic7C2×D412yes—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191#529—78no—
192#529—144no—
193C2×#126—141no—
194C2×#125—71no—
195C2×#124C2×D711yescharacteristic
196C2×#123—141no—
197C2×C22⋊Q8—71no—
198C2×#121—71no—
199C2×C4×Q8—71no—
200C2×C4×D4—71no—
201C2×C4×D4—141no—
202C2×#117—72no—
203C2×#117—141no—
204C22×C42—71no—
205#432—144no—
206C2×C4⋊C8—71no—
207C2×C4⋊C8C2×D711yescharacteristic
208C2×#103—72no—
209C2×#103C2×D712yes—
210C2×#102—72no—
211C2×#102C2×D712yes—
212C2×C22⋊C8—72no—
213C2×#97—72no—
214C4×C4⋊C4—72no—
215C4×C4⋊C4—71no—
216C4×C4⋊C4C2×D711yescharacteristic
217C4×C22⋊C4—141no—
218C4×C22⋊C4—141no—
219#376—141no—
220C43—141no—
221#340—72no—
222#340—72no—
223#340—72no—
224#340—72no—
225C23×C7—22no—
226C23×C7—42no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—42no—
230C23×C7C22×D411yescharacteristic
231C22×D7—22no—
232C22×D7—22no—
233C22×D7—21no—
234C22×D7—21no—
235C22×D7C2×C22⋊C412yes—
236C7×Q8—24no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—24no—
240C7×D4—41no—
241C7×D4—41no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7C2×C22⋊C412yes—
251C2×C4×C7C2×C22⋊C412yes—
252C2×C4×C7C23×C411yescharacteristic
253C2×C4×C7C22×D411yescharacteristic
254C2×C4×C7C2×#4112yes—
255C2×C4×C7C2×#4112yes—
256C22⋊D7—48no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—21no—
260C2×Dic7—21no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—24no—
264C2×Dic7—48no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7C2×C22⋊C412yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7C2×C22⋊C412yes—
273C4×D7—24no—
274C4×D7—22no—
275C4×D7—22no—
276C4×D7—44no—
277C4×D7—44no—
278C7×C8—42no—
279C7⋊C8—42no—
280C22×Q8C22×D711yescharacteristic
281C22×D4—141no—
282C22×D4C22×D711yescharacteristic
283C23×C4—71no—
284C23×C4—71no—
285C23×C4—71no—
286C23×C4—141no—
287C22×C8—72no—
288C22×C8C22×D712yes—
289#126—148no—
290#125—74no—
291#125—142no—
292#124—22no—
293#124C4×D714yes—
294#123—148no—
295C22⋊Q8—74no—
296C22⋊Q8—142no—
297#121—74no—
298#121—142no—
299C4×Q8—74no—
300C4×Q8—142no—
301C4×D4—74no—
302C4×D4—142no—
303C4×D4—148no—
304#117—148no—
305#117—144no—
306#117—144no—
307C2×C4⋊C4—72no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4C22×D712yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—142no—
318C2×C22⋊C4—141no—
319C2×C22⋊C4—141no—
320C2×C22⋊C4—141no—
321C2×C42—21no—
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—71no—
325C2×C42—74no—
326C2×C42—74no—
327C2×C42—142no—
328C2×C42—141no—
329C2×C42—142no—
330C2×C42—141no—
331C2×C42C22×D711yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—142no—
334#103—24no—
335#103—144no—
336#102—24no—
337#102—144no—
338C22⋊C8—144no—
339#97—144no—
340#95—21no—
341#95—142no—
342#95—141no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—82no—
347C22×C7—82no—
348C22×C7C22×C22⋊C411yescharacteristic
349C22×C7#52512yes—
350C22×C7#57512yes—
351C22×C7#57511yescharacteristic
352C22×C7#57511yescharacteristic
353C2×D7—22no—
354C2×D7—22no—
355C2×D7—22no—
356C2×D7—42no—
357C2×D7—42no—
358C4×C7—22no—
359C4×C7—22no—
360C4×C7—44no—
361C4×C7—42no—
362C4×C7C4×#4112yes—
363C4×C7#53211yescharacteristic
364C4×C7#53211yescharacteristic
365Dic7—22no—
366Dic7—22no—
367Dic7—22no—
368Dic7—44no—
369Dic7—42no—
370Dic7—42no—
371C24—21no—
372C24—71no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C2×C4×D712yes—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—144no—
381C2×D4—284no—
382C2×D4C2×C4×D712yes—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—71no—
386C22×C4—71no—
387C22×C4—71no—
388C22×C4—72no—
389C22×C4—72no—
390C22×C4—74no—
391C22×C4—74no—
392C22×C4—144no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—142no—
396C22×C4—141no—
397C22×C4—141no—
398C22×C4—282no—
399C22×C4—281no—
400C22×C4—281no—
401C22×C4—282no—
402C22×C4—282no—
403C22×C4D4×D711yescharacteristic
404C22×C4D4×D711yescharacteristic
405C22×C4C23×D711yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—142no—
410C2×C8—142no—
411C2×C8—142no—
412C4⋊C4—22no—
413C4⋊C4—72no—
414C4⋊C4—72no—
415C4⋊C4—74no—
416C4⋊C4—144no—
417C4⋊C4—144no—
418C4⋊C4—144no—
419C4⋊C4—144no—
420C4⋊C4—141no—
421C4⋊C4—141no—
422C4⋊C4—142no—
423C4⋊C4#115914yes—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—144no—
427C22⋊C4—148no—
428C22⋊C4—144no—
429C22⋊C4—144no—
430C42—42no—
431C42—74no—
432C42—74no—
433C42—72no—
434C42—72no—
435C42—144no—
436C42—148no—
437C42—282no—
438C42—284no—
439C42C2×C4×D712yes—
440C42D4×D712yes—
441C2×C7—82no—
442C2×C7#229711yescharacteristic
443C2×C7#289611yescharacteristic
444C2×C7#289611yescharacteristic
445C2×C7#311112yes—
446C2×C7#311112yes—
447D7—42no—
448C23—22no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—72no—
454C23—142no—
455C23—141no—
456C23—141no—
457C23—142no—
458C23C2×D4×D711yescharacteristic
459Q8—24no—
460Q8—41no—
461Q8—41no—
462D4—24no—
463D4—41no—
464D4—41no—
465D4—288no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—72no—
475C2×C4—72no—
476C2×C4—72no—
477C2×C4—142no—
478C2×C4—141no—
479C2×C4—141no—
480C2×C4—142no—
481C2×C4—142no—
482C2×C4—144no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—144no—
487C2×C4—284no—
488C2×C4—284no—
489C2×C4—288no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—281no—
495C2×C4D7×C22⋊C412yes—
496C2×C4D7×C22⋊C412yes—
497C2×C4C22×C4×D711yescharacteristic
498C2×C4C2×D4×D711yescharacteristic
499C2×C4C2×#115912yes—
500C2×C4D7×#4112yes—
501C8—42no—
502C8—282no—
503C7#2139611yescharacteristic, complemented, Sylow
504C22—42no—
505C22—42no—
506C22—42no—
507C22—82no—
508C22—82no—
509C22—142no—
510C22—142no—
511C22—142no—
512C22—282no—
513C22—282no—
514C22C2×D7×C22⋊C411yescharacteristic
515C22#9029712yes—
516C22#9045912yes—
517C22#9046411yescharacteristic
518C22#9046811yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—142no—
524C4—142no—
525C4—142no—
526C4—282no—
527C4—284no—
528C4—282no—
529C4C4×#115912yes—
530C4#9035911yescharacteristic
531C4#9040711yescharacteristic
532C2—82no—
533C2—282no—
534C2#1173373511yescharacteristic
535C2#1173627511yescharacteristic
536C2#1173632611yescharacteristic
537C2#1173795112yes—
538C2#1173801512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1983C2
socleC23×C7C22×D4
prime radicalC2×#6501C22
cosocleC22×C4×C7C24
hypercenter#1983D7
hypocenterC7#21396
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. C2×#89607C2semidirect
3. C4×#6455C2semidirect
4. C7×#1983C2semidirect
5. #11729475C2semidirect
6. #11728215C2semidirect
7. #11728023C2semidirect
8. #11727974C2semidirect
10. #11727906C2semidirect
11. #11727245C2semidirect
18. #90371C4semidirect
116. #1983D7semidirect
503. C7#21396semidirect

Maximal subgroups

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

subgroupindex
C2×#903712
C2×#896072
C4×#64552
C7×#19832
#117294752
#117282152
#117280232
#117279742
#117279552
#117279062
#117272452
#213967