Finite Group Databasebeta

← #13358641

Subgroups — #13358641

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89605C211yesmaximal, characteristic
2#11731763C211yesmaximal, characteristic
3C4×D7×C4⋊C4C211yesmaximal, characteristic
4C7×#1948C211yesmaximal, characteristic, complemented
5#11728333C212yesmaximal
6Dic7×#106C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727322C211yesmaximal, characteristic, complemented
9#11727221C211yesmaximal, characteristic, complemented
10#11726629C212yesmaximal
11#11726538C212yesmaximal, complemented
12C2×#6455—21no—
13C2×D7×C4⋊C4C2211yescharacteristic
14C2×D7×C4⋊C4C2211yescharacteristic
15C2×C42×D7—21no—
16C2×C42×D7C2211yescharacteristic
17C2×C7×#106C2212yes—
18C2×C7×C4⋊C8C2211yescharacteristic
19C7×#384C2211yescharacteristic
20C4×C7×C4⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C8C2212yes—
22C2×#6391C2212yes—
23C2×C8×Dic7C2212yes—
24#89759—21no—
25#89759C2212yes—
26#89749—21no—
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7C2212yes—
29C4⋊C4×Dic7C2211yescharacteristic
30C4⋊C4×Dic7C2211yescharacteristic
31C2×#6382C2212yes—
32C2×#6382C2212yes—
33#89707C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35C4×Dic7⋊C4—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89605C418yescomplemented
38D7×#95—21no—
39#89421—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21359—71nomaximal, Sylow
45C22×C4×D7—21no—
46C22×C4×D7—21no—
47C22×C4×D7D411yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7C2311yescharacteristic
50C22×C7×C8C2312yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C42×C7—21no—
57C2×C42×C7C2311yescharacteristic
58C2×(C2×D7)⋊C4—22no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4⋊Dic7C2311yescharacteristic
64C2×Dic7⋊C4—21no—
65C2×Dic7⋊C4—21no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7—21no—
69C2×C4×Dic7—22no—
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74#6455—44no—
75D7×C4⋊C4—22no—
76D7×C4⋊C4—22no—
77D7×C4⋊C4—22no—
78D7×C4⋊C4—22no—
79C42×D7—44no—
80C42×D7C2×C414yes—
81C42×D7Q814yes—
82C7×#106—24no—
83C7×C4⋊C8C2×C414yescomplemented
84C7×#95—21no—
85C7⋊#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90(C2×D7)⋊C8C2×C418yes—
91#6391—24no—
92C8×Dic7C2×C418yescomplemented
93#6382—24no—
94#6382—24no—
95C4⋊(C7⋊C8)C2×C414yescomplemented
96C2×#431—71no—
97#2285—71no—
98C2×C4×C4⋊C4—71no—
99#1948—71no—
100#1948—71no—
101#1948—71no—
102#1948D711yescharacteristic, complemented
103#1853—72no—
104#1853—72no—
105C4×#106—72no—
106C4×#106—72no—
107C23×D7C2×D411yescharacteristic
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—22no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2411yescharacteristic
114C22×Dic7—21no—
115C22×Dic7—21no—
116C22×Dic7—22no—
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C22×Dic7C2×D411yescharacteristic
120C2×C4×D7—21no—
121C2×C4×D7—21no—
122C2×C4×D7—21no—
123C2×C4×D7—21no—
124C2×C4×D7—24no—
125C2×C4×D7—24no—
126C2×C4×D7—41no—
127C2×C4×D7—41no—
128C2×C4×D7—42no—
129C2×C4×D7—42no—
130C2×C4×D7—42no—
131C2×C4×D7—42no—
132C2×C4×D7C4⋊C414yes—
133C2×C4×D7C4⋊C414yes—
134C2×C4×D7SD1614yes—
135C2×C4×D7SD1614yes—
136C2×C4×D7C22×C412yes—
137C2×C4×D7C2×Q812yes—
138C2×C7×C8—22no—
139C2×C7×C8—22no—
140C2×C7×C8C22×C414yes—
141C7×C4⋊C4—21no—
142C7×C4⋊C4—21no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—42no—
146C7×C4⋊C4—42no—
147C7×C4⋊C4#4112yes—
148C7×C4⋊C4#4112yes—
149C42×C7—42no—
150C42×C7C22×C412yes—
151C42×C7C2×Q812yes—
152(C2×D7)⋊C4—44no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7—42no—
158C4⋊Dic7—42no—
159C4⋊Dic7#4112yes—
160C4⋊Dic7#4112yes—
161Dic7⋊C4—24no—
162Dic7⋊C4—24no—
163C4×Dic7—42no—
164C4×Dic7—42no—
165C4×Dic7—42no—
166C4×Dic7—44no—
167C4×Dic7C4⋊C414yes—
168C4×Dic7C4⋊C414yes—
169C4×Dic7C22×C412yes—
170C4×Dic7C2×Q812yes—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8C22×C414yes—
174C2×#117—141no—
175C22×C4⋊C4—71no—
176C22×C4⋊C4—71no—
177C22×C42—71no—
178C22×C42—141no—
179#431—78no—
180C2×#106—72no—
181C2×#106—72no—
182C2×#106—72no—
183C2×#106C2×D712yes—
184C2×C4⋊C8—71no—
185C2×C4⋊C8C2×D711yescharacteristic
186C2×C22⋊C8—72no—
187C2×C4×C8—72no—
188#384—71no—
189#384C2×D711yescharacteristic
190#382—141no—
191#380—72no—
192#380—141no—
193C4×C4⋊C4—71no—
194C4×C4⋊C4—71no—
195C4×C4⋊C4—72no—
196C4×C4⋊C4—71no—
197C4×C4⋊C4—141no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4—141no—
200C4×C4⋊C4C2×D711yescharacteristic
201C2×#95—141no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340C2×D712yes—
206C23×C7C22×D411yescharacteristic
207C22×D7—21no—
208C22×D7—21no—
209C22×D7—21no—
210C22×D7—21no—
211C22×D7C2×C4⋊C412yes—
212C22×D7C2×SD1612yes—
213C22×D7C2×SD1612yes—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7C2×C4⋊C412yes—
226C2×C4×C7C2×C4⋊C412yes—
227C2×C4×C7C2×SD1612yes—
228C2×C4×C7C2×SD1612yes—
229C2×C4×C7C23×C411yescharacteristic
230C2×C4×C7C22×Q811yescharacteristic
231C2×C4×C7C2×#4111yescharacteristic
232C2×C4×C7C2×#4111yescharacteristic
233C2×C4×C7C2×#4111yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×Dic7—21no—
236C2×Dic7—21no—
237C2×Dic7—21no—
238C2×Dic7—21no—
239C2×Dic7—22no—
240C2×Dic7—22no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7C2×C4⋊C412yes—
249C2×Dic7C2×C4⋊C412yes—
250C2×Dic7C2×C4⋊C412yes—
251C2×Dic7C2×SD1612yes—
252C2×Dic7C2×SD1612yes—
253C4×D7—22no—
254C4×D7—22no—
255C4×D7—22no—
256C4×D7—22no—
257C4×D7—44no—
258C4×D7—48no—
259C4×D7—48no—
260C4×D7#106116yescomplemented
261C7×C8—24no—
262C7⋊C8—24no—
263C23×C4—71no—
264C23×C4—71no—
265C23×C4—71no—
266C23×C4—141no—
267C23×C4—141no—
268C22×C8—72no—
269C22×C8C22×D712yes—
270#117—284no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—71no—
274C2×C4⋊C4—71no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—142no—
281C2×C4⋊C4—142no—
282C2×C4⋊C4—141no—
283C2×C4⋊C4—141no—
284C2×C4⋊C4C22×D712yes—
285C2×C4⋊C4C22×D711yescharacteristic
286C2×C4⋊C4C22×D711yescharacteristic
287C2×C22⋊C4—142no—
288C2×C42—21no—
289C2×C42—71no—
290C2×C42—71no—
291C2×C42—71no—
292C2×C42—74no—
293C2×C42—74no—
294C2×C42—142no—
295C2×C42—141no—
296C2×C42—141no—
297C2×C42—141no—
298C2×C42—284no—
299C2×C42C22×D711yescharacteristic
300#106—24no—
301#106—144no—
302#106—144no—
303#106—144no—
304C4⋊C8—74no—
305C4⋊C8C4×D714yescomplemented
306C22⋊C8—78no—
307C4×C8—78no—
308#95—21no—
309#95—141no—
310#95—141no—
311#95—142no—
312#95—142no—
313#95—141no—
314C22×C7C22×C4⋊C411yescharacteristic
315C22×C7#52511yescharacteristic
316C22×C7#52511yescharacteristic
317C22×C7C22×SD1611yescharacteristic
318C22×C7C22×SD1611yescharacteristic
319C22×C7#57511yescharacteristic
320C22×C7#57511yescharacteristic
321C2×D7—22no—
322C2×D7—22no—
323C2×D7—22no—
324C2×D7—22no—
325C2×D7—22no—
326C2×D7—22no—
327C2×D7C2×#10614yes—
328C4×C7—22no—
329C4×C7—42no—
330C4×C7—42no—
331C4×C7—82no—
332C4×C7C2×#10614yes—
333C4×C7C4×#4111yescharacteristic
334C4×C7C4×#4111yescharacteristic
335C4×C7#53311yescharacteristic
336C4×C7#53311yescharacteristic
337Dic7—22no—
338Dic7—22no—
339Dic7—42no—
340Dic7—42no—
341Dic7—82no—
342Dic7C2×#10614yes—
343C24—71no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—71no—
348C22×C4—74no—
349C22×C4—74no—
350C22×C4—74no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4—72no—
354C22×C4—72no—
355C22×C4—74no—
356C22×C4—144no—
357C22×C4—144no—
358C22×C4—141no—
359C22×C4—141no—
360C22×C4—141no—
361C22×C4—141no—
362C22×C4—141no—
363C22×C4—141no—
364C22×C4—142no—
365C22×C4—281no—
366C22×C4—281no—
367C22×C4—282no—
368C22×C4—282no—
369C22×C4—282no—
370C22×C4—282no—
371C22×C4D4×D711yescharacteristic
372C22×C4D4×D711yescharacteristic
373C22×C4C23×D711yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—74no—
377C2×C8—142no—
378C2×C8—142no—
379C2×C8C2×C4×D714yes—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—72no—
387C4⋊C4—72no—
388C4⋊C4—144no—
389C4⋊C4—144no—
390C4⋊C4—141no—
391C4⋊C4—141no—
392C4⋊C4—142no—
393C4⋊C4—142no—
394C4⋊C4—282no—
395C4⋊C4—282no—
396C4⋊C4#115912yes—
397C4⋊C4#116112yes—
398C22⋊C4—284no—
399C42—42no—
400C42—74no—
401C42—74no—
402C42—72no—
403C42—72no—
404C42—282no—
405C42—282no—
406C42—284no—
407C42—282no—
408C42C2×C4×D712yes—
409C42Q8×D712yes—
410C2×C7#229611yescharacteristic
411C2×C7C22×#10611yescharacteristic
412C2×C7#291111yescharacteristic
413C2×C7#310111yescharacteristic
414C2×C7#310111yescharacteristic
415C2×C7#310311yescharacteristic
416C2×C7#310311yescharacteristic
417D7—24no—
418C23—72no—
419C23—72no—
420C23—72no—
421C23—141no—
422C23—141no—
423C23—141no—
424C23—141no—
425C23C2×D4×D711yescharacteristic
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—72no—
438C2×C4—72no—
439C2×C4—72no—
440C2×C4—716no—
441C2×C4—72no—
442C2×C4—72no—
443C2×C4—141no—
444C2×C4—141no—
445C2×C4—141no—
446C2×C4—141no—
447C2×C4—142no—
448C2×C4—142no—
449C2×C4—142no—
450C2×C4—142no—
451C2×C4—142no—
452C2×C4—142no—
453C2×C4—284no—
454C2×C4—288no—
455C2×C4—288no—
456C2×C4—281no—
457C2×C4—281no—
458C2×C4—281no—
459C2×C4—281no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4D7×C4⋊C412yes—
464C2×C4D7×C4⋊C412yes—
465C2×C4D7×SD1612yes—
466C2×C4D7×SD1612yes—
467C2×C4C22×C4×D711yescharacteristic
468C2×C4C2×#115911yescharacteristic
469C2×C4C2×Q8×D711yescharacteristic
470C2×C4C2×#116111yescharacteristic
471C2×C4D7×#4111yescharacteristic
472C2×C4D7×#4111yescharacteristic
473C8—24no—
474C8—144no—
475C7#2135911yescharacteristic, complemented, Sylow
476C22—74no—
477C22—142no—
478C22—142no—
479C22—142no—
480C22—142no—
481C22—142no—
482C22—142no—
483C22C2×D7×C4⋊C411yescharacteristic
484C22#9029711yescharacteristic
485C22#9031811yescharacteristic
486C22C2×D7×SD1611yescharacteristic
487C22C2×D7×SD1611yescharacteristic
488C22#9046411yescharacteristic
489C22#9046811yescharacteristic
490C4—22no—
491C4—42no—
492C4—42no—
493C4—74no—
494C4—82no—
495C4—142no—
496C4—142no—
497C4—282no—
498C4—282no—
499C4—562no—
500C4D7×#10614yes—
501C4C4×#115911yescharacteristic
502C4C4×#116111yescharacteristic
503C4#9037911yescharacteristic
504C4#9041911yescharacteristic
505C2—144no—
506C2#1173340811yescharacteristic
507C2C2×D7×#10611yescharacteristic
508C2#1173665311yescharacteristic
509C2#1173780511yescharacteristic
510C2#1173785311yescharacteristic
511C2#1173844111yescharacteristic
512C2#1173848911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1948D7
hypocenterC7#21359
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
4. C7×#1948C2semidirect
6. Dic7×#106C2semidirect
7. #11728208C2semidirect
8. #11727322C2semidirect
9. #11727221C2semidirect
11. #11726538C2semidirect
37. #89605C4semidirect
83. C7×C4⋊C8C2×C4semidirect
92. C8×Dic7C2×C4semidirect
95. C4⋊(C7⋊C8)C2×C4semidirect
102. #1948D7semidirect
260. C4×D7#106semidirect
305. C4⋊C8C4×D7semidirect
475. C7#21359semidirect

Maximal subgroups

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

subgroupindex
C2×#896052
#117317632
C4×D7×C4⋊C42
C7×#19482
#117283332
Dic7×#1062
#117282082
#117273222
#117272212
#117266292
#117265382
#213597