Finite Group Databasebeta

← #13363721

Subgroups — #13363721

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90371C211yesmaximal, characteristic
2C2×#89609C211yesmaximal, characteristic, complemented
3#11731198C211yesmaximal, characteristic, complemented
4C7×#1983C211yesmaximal, characteristic, complemented
5#11729475C211yesmaximal, characteristic, complemented
6#11728215C211yesmaximal, characteristic, complemented
7#11728025C212yesmaximal, complemented
8#11727975C212yesmaximal, complemented
9#11727957C212yesmaximal
10#11727907C212yesmaximal, complemented
11#11727245C211yesmaximal, characteristic, complemented
12C2×C7×#124C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C2×#6501C2211yescharacteristic
16C2×D4×Dic7C2211yescharacteristic
17C22×#1131C2212yes—
18#90371—24no—
19#90371C418yescomplemented
20C2×#6446—21no—
21C2×(C22⋊D7)⋊C4—21no—
22C2×#6442—21no—
23C2×#6440—21no—
24C2×#6435—21no—
25C2×C42×D7C2211yescharacteristic
26C2×#6433C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#103C2212yes—
29C2×C7×#102C2212yes—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33C2×Dic7⋊C8C2212yes—
34#89759C2212yes—
35#89748C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37C4×Dic7⋊C4—21no—
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89609—24no—
40#89439—21no—
41#89422—21no—
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21890—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⋊Dic7C2312yes—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C2×#1131—24no—
90C2×#1131—42no—
91C2×#1131—42no—
92#6446—28no—
93(C22⋊D7)⋊C4—28no—
94#6442—28no—
95#6440—28no—
96#6435—44no—
97C42×D7C2×C414yes—
98C42×D7D414yes—
99#6433—22no—
100#6433C2×C414yes—
101C7×C4⋊C8—22no—
102C7×#103—42no—
103C7×#102—42no—
104C7×#95—21no—
105#6409—42no—
106#6406—42no—
107C7⋊#95—21no—
108C7⋊#95—22no—
109Dic7⋊C8—24no—
110C4⋊(C7⋊C8)—22no—
111C2×#529—71no—
112C2×#423—71no—
113#2285—71no—
114#1989—71no—
115#1989—71no—
116#1983—71no—
117#1983D711yescharacteristic, complemented
118#1895—72no—
119#1894—72no—
120#1803—72no—
121#1802—72no—
122C24×C7—21no—
123C23×D7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8C22×C412yes—
128C2×C7×D4—21no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4C22×C412yes—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—24no—
138C2×C22⋊D7—44no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—24no—
142C22×Dic7—22no—
143C22×Dic7—42no—
144C22×Dic7—42no—
145C22×Dic7C2×D411yescharacteristic
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C2×C4×D7—22no—
149C2×C4×D7—22no—
150C2×C4×D7—22no—
151C2×C4×D7—22no—
152C2×C4×D7—42no—
153C2×C4×D7C22⋊C414yes—
154C2×C4×D7C22⋊C414yes—
155C2×C4×D7C22×C412yes—
156C2×C4×D7C2×D412yes—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—24no—
164C42×C7—42no—
165C42×C7C22×C412yes—
166C42×C7C2×D412yes—
167C22⋊Dic7—24no—
168C22⋊Dic7—24no—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—44no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—24no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7C22×C412yes—
177C4⋊Dic7C22×C412yes—
178Dic7⋊C4—24no—
179Dic7⋊C4—24no—
180Dic7⋊C4—24no—
181Dic7⋊C4—44no—
182Dic7⋊C4—44no—
183C4×Dic7—24no—
184C4×Dic7—42no—
185C4×Dic7C22⋊C414yes—
186C4×Dic7C22⋊C414yes—
187C4×Dic7C22×C412yes—
188C4×Dic7C2×D412yes—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192#1131—48no—
193C22×M16—72no—
194#529—78no—
195#529—144no—
196C2×#126—141no—
197C2×#125—71no—
198C2×#124C2×D711yescharacteristic
199C2×#123—141no—
200C2×C22⋊Q8—71no—
201C2×#121—71no—
202C2×C4×Q8—71no—
203C2×C4×D4—71no—
204C2×C4×D4—141no—
205C2×#117—141no—
206C2×#117—141no—
207C22×C42—71no—
208#423—144no—
209C2×C4⋊C8—72no—
210C2×C4⋊C8—71no—
211C2×C4⋊C8C2×D711yescharacteristic
212C2×#103—72no—
213C2×#103C2×D712yes—
214C2×#102—72no—
215C2×#102C2×D712yes—
216#384—72no—
217#382—141no—
218#380—72no—
219#380—141no—
220C4×C4⋊C4—71no—
221C4×C4⋊C4—141no—
222C4×C4⋊C4C2×D711yescharacteristic
223#340—72no—
224#340—72no—
225#340—72no—
226#340—72no—
227C23×C7—22no—
228C23×C7—42no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—42no—
232C23×C7C22×D411yescharacteristic
233C22×D7—22no—
234C22×D7—21no—
235C22×D7—21no—
236C22×D7—22no—
237C22×D7C2×C22⋊C412yes—
238C7×Q8—24no—
239C7×Q8—41no—
240C7×Q8—41no—
241C7×D4—24no—
242C7×D4—41no—
243C7×D4—41no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7C2×C22⋊C412yes—
253C2×C4×C7C2×C22⋊C412yes—
254C2×C4×C7C23×C411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7H(5,𝔽2)12yes—
257C2×C4×C72−1+412yes—
258C22⋊D7—48no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—48no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7C2×C22⋊C412yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7C2×C22⋊C412yes—
273C2×Dic7AGL(1,ℤ/8)12yes—
274C2×Dic7#13712yes—
275C4×D7—24no—
276C4×D7—22no—
277C4×D7—22no—
278C4×D7—28no—
279C4×D7—44no—
280C7×C8—42no—
281C7⋊C8—42no—
282C22×Q8C22×D711yescharacteristic
283C22×D4—141no—
284C22×D4C22×D711yescharacteristic
285C23×C4—71no—
286C23×C4—71no—
287C23×C4—71no—
288C23×C4—141no—
289C2×M16—144no—
290C2×M16—282no—
291C2×M16—282no—
292C22×C8—72no—
293C22×C8C22×D712yes—
294#126—148no—
295#125—74no—
296#125—142no—
297#124—22no—
298#124C4×D714yes—
299#123—148no—
300C22⋊Q8—74no—
301C22⋊Q8—142no—
302#121—74no—
303#121—142no—
304C4×Q8—74no—
305C4×Q8—142no—
306C4×D4—74no—
307C4×D4—142no—
308C4×D4—148no—
309#117—148no—
310#117—284no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4C22×D712yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—141no—
324C2×C22⋊C4—141no—
325C2×C22⋊C4—141no—
326C2×C22⋊C4—142no—
327C2×C42—21no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—74no—
332C2×C42—74no—
333C2×C42—141no—
334C2×C42—141no—
335C2×C42C22×D711yescharacteristic
336C4⋊C8—22no—
337C4⋊C8—144no—
338C4⋊C8—142no—
339#103—42no—
340#103—282no—
341#102—42no—
342#102—282no—
343#95—21no—
344#95—142no—
345#95—141no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—82no—
350C22×C7—82no—
351C22×C7C22×C22⋊C411yescharacteristic
352C22×C7#53712yes—
353C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C7C2×#13711yescharacteristic
355C22×C7#57512yes—
356C2×D7—22no—
357C2×D7—22no—
358C2×D7—22no—
359C2×D7—42no—
360C2×D7—42no—
361C4×C7—22no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—82no—
365C4×C7#52012yes—
366C4×C7#53511yescharacteristic
367C4×C7#53611yescharacteristic
368Dic7—22no—
369Dic7—22no—
370Dic7—42no—
371Dic7—42no—
372Dic7—82no—
373Dic7#41714yes—
374C24—21no—
375C24—71no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×C4×D712yes—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—144no—
384C2×D4—284no—
385C2×D4C2×C4×D712yes—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—71no—
389C22×C4—71no—
390C22×C4—71no—
391C22×C4—74no—
392C22×C4—72no—
393C22×C4—72no—
394C22×C4—74no—
395C22×C4—142no—
396C22×C4—142no—
397C22×C4—144no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—142no—
403C22×C4—282no—
404C22×C4—282no—
405C22×C4—282no—
406C22×C4D4×D711yescharacteristic
407C22×C4D4×D711yescharacteristic
408C22×C4C23×D711yescharacteristic
409M16—288no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—142no—
414C2×C8—142no—
415C2×C8—142no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—72no—
419C4⋊C4—72no—
420C4⋊C4—144no—
421C4⋊C4—144no—
422C4⋊C4—144no—
423C4⋊C4—144no—
424C4⋊C4—141no—
425C4⋊C4—141no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4—284no—
429C4⋊C4—284no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—144no—
433C22⋊C4—144no—
434C22⋊C4—144no—
435C22⋊C4—284no—
436C42—42no—
437C42—74no—
438C42—74no—
439C42—72no—
440C42—72no—
441C42—144no—
442C42—282no—
443C42C2×C4×D712yes—
444C42D4×D712yes—
445C2×C7—82no—
446C2×C7#231111yescharacteristic
447C2×C7C2×#41711yescharacteristic
448C2×C7#289611yescharacteristic
449C2×C7#325112yes—
450C2×C7#325212yes—
451D7—42no—
452C23—22no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—72no—
458C23—142no—
459C23—141no—
460C23—141no—
461C23—142no—
462C23C2×D4×D711yescharacteristic
463Q8—24no—
464Q8—41no—
465Q8—41no—
466D4—24no—
467D4—41no—
468D4—41no—
469D4—288no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—72no—
479C2×C4—72no—
480C2×C4—72no—
481C2×C4—72no—
482C2×C4—72no—
483C2×C4—148no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—144no—
488C2×C4—142no—
489C2×C4—142no—
490C2×C4—142no—
491C2×C4—284no—
492C2×C4—288no—
493C2×C4—282no—
494C2×C4—281no—
495C2×C4—281no—
496C2×C4—282no—
497C2×C4—282no—
498C2×C4—282no—
499C2×C4D7×C22⋊C412yes—
500C2×C4D7×C22⋊C412yes—
501C2×C4C22×C4×D711yescharacteristic
502C2×C4C2×D4×D711yescharacteristic
503C2×C4#654812yes—
504C2×C4#655412yes—
505C8—42no—
506C8—282no—
507C7#2189011yescharacteristic, complemented, Sylow
508C22—42no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—142no—
514C22—142no—
515C22—142no—
516C22—282no—
517C22—282no—
518C22C2×D7×C22⋊C411yescharacteristic
519C22#9029912yes—
520C22#9045912yes—
521C22D7×AGL(1,ℤ/8)11yescharacteristic
522C22D7×#13711yescharacteristic
523C4—22no—
524C4—22no—
525C4—42no—
526C4—74no—
527C4—82no—
528C4—142no—
529C4—142no—
530C4—282no—
531C4—282no—
532C4—562no—
533C4#9023712yes—
534C4#9036611yescharacteristic
535C4#9040611yescharacteristic
536C2—82no—
537C2—282no—
538C2#1173373911yescharacteristic
539C2#1173627511yescharacteristic
540C2D7×#41711yescharacteristic
541C2#1173795812yes—
542C2#1173802212yes—
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#21890
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×#89609C2semidirect
3. #11731198C2semidirect
4. C7×#1983C2semidirect
5. #11729475C2semidirect
6. #11728215C2semidirect
7. #11728025C2semidirect
8. #11727975C2semidirect
10. #11727907C2semidirect
11. #11727245C2semidirect
19. #90371C4semidirect
117. #1983D7semidirect
507. C7#21890semidirect

Maximal subgroups

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

subgroupindex
C2×#903712
C2×#896092
#117311982
C7×#19832
#117294752
#117282152
#117280252
#117279752
#117279572
#117279072
#117272452
#218907