Finite Group Databasebeta

← #13363687

Subgroups — #13363687

Order 1792.

1150 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×#90315C211yesmaximal, characteristic
2C22×Dic7⋊C8C211yesmaximal, characteristic, complemented
3#11731308C211yesmaximal, characteristic, complemented
4C7×#1892C211yesmaximal, characteristic, complemented
5#11729727C211yesmaximal, characteristic
6#11729450C211yesmaximal, characteristic, complemented
7#11727984C212yesmaximal, complemented
8#11727976C212yesmaximal, complemented
9#11727269C211yesmaximal, characteristic
10#11726647C212yesmaximal, complemented
11#11726555C212yesmaximal, complemented
12C23×C7×C8C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×#6510C2211yescharacteristic
15C2×Q8×Dic7C2211yescharacteristic
16C2×#6493C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19#90315—24no—
20#90315C418yescomplemented
21C2×#6461—21no—
22C2×#6456C2211yescharacteristic
23C2×#6455—21no—
24C2×#6453C2211yescharacteristic
25C2×#6449—21no—
26C2×(C22⋊D7)⋊C4—21no—
27C2×C7×#103C2212yes—
28C7×#380C2211yescharacteristic
29C2×#6409C2212yes—
30#89984C2211yescharacteristic
31C2×#6395C2212yes—
32C2×Dic7⋊C8—24no—
33C2×Dic7⋊C8C2212yes—
34C2×Dic7⋊C8C2212yes—
35#89749C2212yes—
36C4⋊C4×Dic7C2212yes—
37C2×#6383C2212yes—
38#89715—21no—
39C22⋊C4×Dic7—21no—
40#89423—21no—
41#89420—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21886—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C22×C22⋊D7—21no—
50C23×Dic7D411yescharacteristic
51C23×Dic7D411yescharacteristic
52C22×D28C2311yescharacteristic
53C22×C4×D7—21no—
54C22×C7×C8—22no—
55C22×C7×C8—42no—
56C22×C7×C8C2312yes—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8C2×C414yes—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C22⋊Dic7—21no—
65#6510—22no—
66#6510C2×C414yes—
67Q8×Dic7—22no—
68Q8×Dic7C2×C414yes—
69#6493—22no—
70#6493C2×C414yes—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×C4×Dic7—21no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7C2×C414yes—
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D414yes—
86C2×C4×Dic7C2311yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8—22no—
89C22×C7⋊C8—42no—
90C22×C7⋊C8C2312yes—
91#6461—28no—
92#6456—22no—
93#6456C2×C414yes—
94#6455—28no—
95#6453—22no—
96#6453C2×C414yes—
97#6449—28no—
98(C22⋊D7)⋊C4—28no—
99C7×#103—42no—
100C7×#95—21no—
101#6409—42no—
102C7⋊#95—21no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106#6395—42no—
107Dic7⋊C8—24no—
108Dic7⋊C8—24no—
109#6383—42no—
110C2×#529—71no—
111C22×C4⋊C8—71no—
112#2283—71no—
113#1989—72no—
114#1983—72no—
115#1893—71no—
116#1893—71no—
117#1892—71no—
118#1892D711yescharacteristic, complemented
119#1805—72no—
120#1804—72no—
121C24×C7C2×D411yescharacteristic
122C23×D7—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8C22×C412yes—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2411yescharacteristic
136C2×C22⋊D7—24no—
137C2×C22⋊D7—42no—
138C2×C22⋊D7—42no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—42no—
144C22×Dic7C22⋊C414yes—
145C22×Dic7C22⋊C414yes—
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—21no—
153C2×D28C22×C412yes—
154C2×C4×D7—24no—
155C2×C4×D7—44no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—44no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4C22×C412yes—
165C7×C22⋊C4—24no—
166C7×C22⋊C4—42no—
167C22⋊Dic7—42no—
168(C2×D7)⋊C4—24no—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—24no—
171(C2×D7)⋊C4—24no—
172C4⋊Dic7—21no—
173C4⋊Dic7—24no—
174C4⋊Dic7—24no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7C22×C412yes—
178Dic7⋊C4—24no—
179C4×Dic7—24no—
180C4×Dic7—44no—
181C4×Dic7C22⋊C414yes—
182C4×Dic7C22⋊C414yes—
183C4×Dic7C22×C412yes—
184C4×Dic7C22×C412yes—
185C4×Dic7C2×D412yes—
186C4×Dic7C2×D412yes—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—44no—
191C23×C8—71no—
192C23×C8C2×D711yescharacteristic
193#529—78no—
194#529—144no—
195C2×#126—141no—
196C2×#125—71no—
197C2×#124—71no—
198C2×#123—141no—
199C2×C22⋊Q8C2×D711yescharacteristic
200C2×#121—71no—
201C2×C4×Q8—71no—
202C2×C4×D4—71no—
203C2×C4×D4—141no—
204C2×#117—141no—
205C22×C42—71no—
206C2×C4⋊C8—72no—
207C2×C4⋊C8—72no—
208C2×C4⋊C8—144no—
209C2×#103—72no—
210C2×#103C2×D712yes—
211C2×#102—72no—
212C2×#102—72no—
213#383—141no—
214#382—72no—
215#381—141no—
216#380—71no—
217#380C2×D711yescharacteristic
218C4×C4⋊C4—72no—
219C4×C22⋊C4—141no—
220#376—141no—
221#340—72no—
222#340—72no—
223#340—72no—
224#340C2×D712yes—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7C2×C22⋊C412yes—
228C23×C7#13512yes—
229C23×C7#13512yes—
230C23×C7C22×D411yescharacteristic
231C22×D7—22no—
232C22×D7—42no—
233C22×D7—41no—
234C22×D7—41no—
235C22×D7—42no—
236C7×Q8—24no—
237C7×Q8—41no—
238C7×Q8—41no—
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7C2×C22⋊C412yes—
251C2×C4×C7C23×C411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7H(5,𝔽2)12yes—
254C2×C4×C72−1+412yes—
255C22⋊D7—48no—
256C2×Dic7—21no—
257C2×Dic7—21no—
258C2×Dic7—21no—
259C2×Dic7—21no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—28no—
263C2×Dic7—22no—
264C2×Dic7—44no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7C2×C22⋊C412yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7C2×C22⋊C412yes—
273C2×Dic7C2×C22⋊C412yes—
274D28—24no—
275D28—41no—
276D28—41no—
277C4×D7—48no—
278C7×C8—42no—
279C7⋊C8—42no—
280C22×Q8C22×D711yescharacteristic
281C22×D4—71no—
282C22×D4—141no—
283C23×C4—71no—
284C23×C4—71no—
285C23×C4—141no—
286C23×C4C22×D711yescharacteristic
287C22×C8—22no—
288C22×C8—42no—
289C22×C8—72no—
290C22×C8—142no—
291C22×C8—282no—
292C22×C8C22×D712yes—
293#126—148no—
294#125—74no—
295#125—142no—
296#124—74no—
297#124—142no—
298#123—148no—
299C22⋊Q8—22no—
300C22⋊Q8C4×D714yes—
301#121—74no—
302#121—142no—
303C4×Q8—74no—
304C4×Q8—142no—
305C4×D4—74no—
306C4×D4—142no—
307C4×D4—148no—
308#117—148no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4C22×D712yes—
316C2×C4⋊C4C22×D711yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—141no—
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—141no—
323C2×C22⋊C4—141no—
324C2×C42—74no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—74no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—141no—
331C2×C42—142no—
332C4⋊C8—144no—
333C4⋊C8—144no—
334#103—42no—
335#103—282no—
336#102—282no—
337#102—282no—
338#95—21no—
339#95—142no—
340#95—142no—
341#95—142no—
342#95—141no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—24no—
346C22×C7—24no—
347C22×C7#41614yes—
348C22×C7C22×C22⋊C411yescharacteristic
349C22×C7#53712yes—
350C22×C7C2×#13511yescharacteristic
351C22×C7C2×#13511yescharacteristic
352C22×C7#57512yes—
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—42no—
356C2×D7—82no—
357C2×D7—82no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7—42no—
361C4×C7—82no—
362C4×C7#52012yes—
363C4×C7#53511yescharacteristic
364C4×C7#53611yescharacteristic
365Dic7—22no—
366Dic7—22no—
367Dic7—22no—
368Dic7—22no—
369Dic7—42no—
370Dic7—82no—
371C24—141no—
372C24D4×D711yescharacteristic
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C2×C4×D712yes—
377C2×D4—72no—
378C2×D4—144no—
379C2×D4—141no—
380C2×D4—142no—
381C2×D4—142no—
382C2×D4—282no—
383C2×D4—282no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—74no—
390C22×C4—71no—
391C22×C4—71no—
392C22×C4—71no—
393C22×C4—71no—
394C22×C4—74no—
395C22×C4—144no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—141no—
399C22×C4—142no—
400C22×C4—284no—
401C22×C4—282no—
402C22×C4C2×C4×D712yes—
403C22×C4D4×D711yescharacteristic
404C22×C4D4×D712yes—
405C22×C4C23×D711yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—44no—
410C2×C8—142no—
411C2×C8—142no—
412C2×C8—142no—
413C2×C8—284no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—72no—
419C4⋊C4—144no—
420C4⋊C4—144no—
421C4⋊C4—144no—
422C4⋊C4—141no—
423C4⋊C4—142no—
424C4⋊C4—142no—
425C4⋊C4C2×C4×D712yes—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—144no—
429C22⋊C4—144no—
430C22⋊C4—144no—
431C22⋊C4—144no—
432C22⋊C4—282no—
433C42—74no—
434C42—74no—
435C42—72no—
436C42—72no—
437C42—72no—
438C42—72no—
439C42—144no—
440C42—284no—
441C2×C7—24no—
442C2×C7#231111yescharacteristic
443C2×C7C2×#41611yescharacteristic
444C2×C7#289611yescharacteristic
445C2×C7#325012yes—
446C2×C7#325012yes—
447D7—82no—
448C23—22no—
449C23—22no—
450C23—142no—
451C23—281no—
452C23—281no—
453C23—282no—
454C23—282no—
455C23D7×C22⋊C412yes—
456C23#647912yes—
457C23#648212yes—
458C23C2×D4×D711yescharacteristic
459Q8—24no—
460Q8—41no—
461Q8—41no—
462D4—144no—
463D4—288no—
464D4—281no—
465D4—281no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—72no—
478C2×C4—72no—
479C2×C4—72no—
480C2×C4—72no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—141no—
484C2×C4—141no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—148no—
488C2×C4—142no—
489C2×C4—288no—
490C2×C4—284no—
491C2×C4—281no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4D7×C22⋊C412yes—
497C2×C4C22×C4×D711yescharacteristic
498C2×C4C2×D4×D711yescharacteristic
499C2×C4#654812yes—
500C2×C4#655412yes—
501C8—42no—
502C8—282no—
503C7#2188611yescharacteristic, complemented, Sylow
504C22—24no—
505C22—22no—
506C22—22no—
507C22—24no—
508C22—282no—
509C22—282no—
510C22—282no—
511C22—562no—
512C22—562no—
513C22#8957614yes—
514C22C2×D7×C22⋊C411yescharacteristic
515C22#9028812yes—
516C22C2×#647911yescharacteristic
517C22C2×#648211yescharacteristic
518C22#9045912yes—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—142no—
524C4—142no—
525C4—142no—
526C4—142no—
527C4—282no—
528C4—562no—
529C4#9023712yes—
530C4#9028311yescharacteristic
531C4#9033911yescharacteristic
532C2—24no—
533C2—562no—
534C2#1173210711yescharacteristic
535C2C2×#8957611yescharacteristic
536C2#1173627511yescharacteristic
537C2#1173808812yes—
538C2#1173819312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1892C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
cosocleC22×C4×C7C24
hypercenter#1892D7
hypocenterC7#21886
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C8C2semidirect
3. #11731308C2semidirect
4. C7×#1892C2semidirect
6. #11729450C2semidirect
7. #11727984C2semidirect
8. #11727976C2semidirect
10. #11726647C2semidirect
11. #11726555C2semidirect
20. #90315C4semidirect
118. #1892D7semidirect
503. C7#21886semidirect

Maximal subgroups

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

subgroupindex
C2×#903152
C22×Dic7⋊C82
#117313082
C7×#18922
#117297272
#117294502
#117279842
#117279762
#117272692
#117266472
#117265552
#218867