Finite Group Databasebeta

← #13364310

Subgroups — #13364310

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89605C211yesmaximal, characteristic
2D7×#384C211yesmaximal, characteristic
3#11731198C211yesmaximal, characteristic
4C7×#1948C211yesmaximal, characteristic, complemented
5#11728333C212yesmaximal
6#11728308C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727322C211yesmaximal, characteristic, complemented
9#11727221C211yesmaximal, characteristic, complemented
10#11726629C212yesmaximal
11#11726601C212yesmaximal, complemented
12C2×D7×C4⋊C4—21no—
13C2×D7×C4⋊C4—21no—
14C2×D7×C4⋊C4C2212yes—
15C2×#6435—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#89759C2211yescharacteristic
25#89759C2211yescharacteristic
26#89748—21no—
27#89748—21no—
28#89748C2211yescharacteristic
29#89748C2212yes—
30#89748C2211yescharacteristic
31C2×#6382C2212yes—
32C2×#6382C2212yes—
33#89707C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35C4×Dic7⋊C4—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89605C418yescomplemented
38#89439—21no—
39#89422—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21981—71nomaximal, Sylow
45C22×C4×D7—22no—
46C22×C4×D7D411yescharacteristic
47C22×C4×D7D411yescharacteristic
48C22×C4×D7C2311yescharacteristic
49C22×C7×C8C2312yes—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2311yescharacteristic
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2311yescharacteristic
64C2×Dic7⋊C4—21no—
65C2×Dic7⋊C4—21no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—21no—
71C2×C4×Dic7—21no—
72C2×C4×Dic7D411yescharacteristic
73C2×C4×Dic7D411yescharacteristic
74C2×C4×Dic7C2311yescharacteristic
75C22×C7⋊C8C2312yes—
76D7×C4⋊C4—42no—
77D7×C4⋊C4—42no—
78D7×C4⋊C4—44no—
79D7×C4⋊C4—44no—
80#6435—44no—
81C42×D7C2×C414yes—
82C42×D7Q814yes—
83C7×#106—42no—
84C7×C4⋊C8C2×C414yescomplemented
85C7×#95—21no—
86C7⋊#95—21no—
87C7⋊#95—21no—
88C7⋊#95—21no—
89(C2×D7)⋊C8C2×C418yes—
90#6391—42no—
91C8×Dic7C2×C418yescomplemented
92#6382—42no—
93#6382—42no—
94C4⋊(C7⋊C8)C2×C414yescomplemented
95C2×#431—71no—
96#2285—71no—
97C2×#384—71no—
98#1948—71no—
99#1948—71no—
100#1948—71no—
101#1948D711yescharacteristic, complemented
102#1853—72no—
103#1853—72no—
104#1844—72no—
105#1834—72no—
106C23×D7C2×D411yescharacteristic
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—21no—
114C22×Dic7—21no—
115C22×Dic7—22no—
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C2×C4×D7—21no—
120C2×C4×D7—21no—
121C2×C4×D7—21no—
122C2×C4×D7—21no—
123C2×C4×D7—22no—
124C2×C4×D7—22no—
125C2×C4×D7—44no—
126C2×C4×D7—44no—
127C2×C4×D7—44no—
128C2×C4×D7C4⋊C414yes—
129C2×C4×D7C4⋊C414yes—
130C2×C4×D7SD1614yes—
131C2×C4×D7SD1614yes—
132C2×C4×D7C22×C412yes—
133C2×C4×D7C2×Q812yes—
134C2×C7×C8—22no—
135C2×C7×C8—22no—
136C2×C7×C8C22×C414yes—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—21no—
142C7×C4⋊C4—21no—
143C7×C4⋊C4—42no—
144C7×C4⋊C4—42no—
145C42×C7—42no—
146C42×C7C22×C412yes—
147C42×C7C2×Q812yes—
148(C2×D7)⋊C4—42no—
149(C2×D7)⋊C4—42no—
150C4⋊Dic7—22no—
151C4⋊Dic7—22no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C4⋊Dic7—42no—
157C4⋊Dic7—42no—
158Dic7⋊C4—44no—
159Dic7⋊C4—42no—
160Dic7⋊C4—42no—
161Dic7⋊C4—44no—
162Dic7⋊C4—44no—
163Dic7⋊C4—42no—
164Dic7⋊C4—42no—
165C4×Dic7—42no—
166C4×Dic7C4⋊C414yes—
167C4×Dic7C4⋊C414yes—
168C4×Dic7C22×C412yes—
169C4×Dic7C2×Q812yes—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8C22×C414yes—
173C2×#117—141no—
174C22×C4⋊C4—72no—
175C22×C4⋊C4—141no—
176C22×C4⋊C4—141no—
177C22×C42—71no—
178#431—78no—
179C2×#106—72no—
180C2×#106—72no—
181C2×#106—72no—
182C2×#106C2×D712yes—
183C2×C4⋊C8—71no—
184C2×C4⋊C8C2×D711yescharacteristic
185C2×C22⋊C8—72no—
186C2×C4×C8—72no—
187#384—71no—
188#384—72no—
189#384—71no—
190#384—71no—
191#384—141no—
192#384—141no—
193#384C2×D711yescharacteristic
194#382—141no—
195#380—71no—
196#380—71no—
197#380—141no—
198C4×C4⋊C4—71no—
199C4×C4⋊C4—141no—
200C4×C4⋊C4C2×D711yescharacteristic
201#340—72no—
202#340—72no—
203#340—72no—
204#340C2×D712yes—
205C23×C7C22×D411yescharacteristic
206C22×D7—21no—
207C22×D7—21no—
208C22×D7—21no—
209C22×D7—21no—
210C22×D7C2×C4⋊C412yes—
211C22×D7C2×SD1612yes—
212C22×D7C2×SD1612yes—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—42no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7C2×C4⋊C412yes—
225C2×C4×C7C2×C4⋊C412yes—
226C2×C4×C7C2×SD1612yes—
227C2×C4×C7C2×SD1612yes—
228C2×C4×C7C23×C411yescharacteristic
229C2×C4×C7C22×Q811yescharacteristic
230C2×C4×C7H(5,𝔽2)11yescharacteristic
231C2×C4×C7H(5,𝔽2)11yescharacteristic
232C2×C4×C72−1+411yescharacteristic
233C2×C4×C72−1+411yescharacteristic
234C2×Dic7—21no—
235C2×Dic7—21no—
236C2×Dic7—42no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7C2×C4⋊C412yes—
246C2×Dic7C2×C4⋊C412yes—
247C2×Dic7C2×C4⋊C412yes—
248C2×Dic7C2×SD1612yes—
249C2×Dic7C2×SD1612yes—
250C2×Dic7AGL(1,ℤ/8)12yes—
251C2×Dic7#13712yes—
252C4×D7—22no—
253C4×D7—22no—
254C4×D7—22no—
255C4×D7—22no—
256C4×D7—28no—
257C4×D7—88no—
258C4×D7#106116yescomplemented
259C7×C8—24no—
260C7⋊C8—24no—
261C23×C4—71no—
262C23×C4—71no—
263C23×C4—71no—
264C23×C4—142no—
265C22×C8—72no—
266C22×C8C22×D712yes—
267#117—284no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—72no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—71no—
273C2×C4⋊C4—142no—
274C2×C4⋊C4—142no—
275C2×C4⋊C4—142no—
276C2×C4⋊C4—141no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—282no—
283C2×C4⋊C4—282no—
284C2×C4⋊C4—284no—
285C2×C4⋊C4—284no—
286C2×C4⋊C4C22×D712yes—
287C2×C4⋊C4C22×D711yescharacteristic
288C2×C4⋊C4C22×D711yescharacteristic
289C2×C22⋊C4—141no—
290C2×C22⋊C4—141no—
291C2×C42—21no—
292C2×C42—71no—
293C2×C42—71no—
294C2×C42—71no—
295C2×C42—74no—
296C2×C42—74no—
297C2×C42—141no—
298C2×C42C22×D711yescharacteristic
299#106—42no—
300#106—282no—
301#106—282no—
302#106—282no—
303C4⋊C8—74no—
304C4⋊C8C4×D714yescomplemented
305C22⋊C8—78no—
306C4×C8—78no—
307#95—21no—
308#95—141no—
309#95—141no—
310#95—141no—
311C22×C7C22×C4⋊C411yescharacteristic
312C22×C7#53511yescharacteristic
313C22×C7#53611yescharacteristic
314C22×C7C22×SD1611yescharacteristic
315C22×C7C22×SD1611yescharacteristic
316C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C7C2×#13711yescharacteristic
318C2×D7—22no—
319C2×D7—22no—
320C2×D7—22no—
321C2×D7—22no—
322C2×D7—22no—
323C2×D7—22no—
324C2×D7C2×#10614yes—
325C4×C7—22no—
326C4×C7—81no—
327C4×C7—82no—
328C4×C7—81no—
329C4×C7C2×#10614yes—
330C4×C7#52011yescharacteristic
331C4×C7#52011yescharacteristic
332C4×C7#54411yescharacteristic
333C4×C7#54411yescharacteristic
334Dic7—22no—
335Dic7—81no—
336Dic7—82no—
337Dic7—81no—
338Dic7C2×#10614yes—
339Dic7#42814yes—
340C24—71no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—71no—
345C22×C4—71no—
346C22×C4—74no—
347C22×C4—74no—
348C22×C4—74no—
349C22×C4—71no—
350C22×C4—74no—
351C22×C4—72no—
352C22×C4—72no—
353C22×C4—142no—
354C22×C4—142no—
355C22×C4—141no—
356C22×C4—141no—
357C22×C4—141no—
358C22×C4—141no—
359C22×C4—141no—
360C22×C4—141no—
361C22×C4—142no—
362C22×C4—284no—
363C22×C4—284no—
364C22×C4—284no—
365C22×C4D4×D711yescharacteristic
366C22×C4D4×D711yescharacteristic
367C22×C4C23×D711yescharacteristic
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—74no—
371C2×C8—142no—
372C2×C8—142no—
373C2×C8C2×C4×D714yes—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—141no—
383C4⋊C4—142no—
384C4⋊C4—141no—
385C4⋊C4—141no—
386C4⋊C4—142no—
387C4⋊C4—141no—
388C4⋊C4—282no—
389C4⋊C4—282no—
390C4⋊C4—282no—
391C4⋊C4—284no—
392C4⋊C4—284no—
393C4⋊C4—282no—
394C4⋊C4—282no—
395C4⋊C4—284no—
396C4⋊C4—282no—
397C22⋊C4—282no—
398C22⋊C4—282no—
399C42—42no—
400C42—74no—
401C42—74no—
402C42—72no—
403C42—72no—
404C42—282no—
405C42C2×C4×D712yes—
406C42Q8×D712yes—
407C2×C7#231711yescharacteristic
408C2×C7C22×#10611yescharacteristic
409C2×C7C2×#42811yescharacteristic
410C2×C7#320611yescharacteristic
411C2×C7#320811yescharacteristic
412C2×C7#322711yescharacteristic
413C2×C7#322911yescharacteristic
414D7—24no—
415C23—72no—
416C23—72no—
417C23—72no—
418C23—141no—
419C23—141no—
420C23—141no—
421C23—141no—
422C23C2×D4×D711yescharacteristic
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—72no—
435C2×C4—72no—
436C2×C4—72no—
437C2×C4—72no—
438C2×C4—72no—
439C2×C4—72no—
440C2×C4—716no—
441C2×C4—72no—
442C2×C4—148no—
443C2×C4—141no—
444C2×C4—141no—
445C2×C4—142no—
446C2×C4—142no—
447C2×C4—142no—
448C2×C4—142no—
449C2×C4—281no—
450C2×C4—281no—
451C2×C4—282no—
452C2×C4—282no—
453C2×C4—281no—
454C2×C4—281no—
455C2×C4—282no—
456C2×C4—281no—
457C2×C4—281no—
458C2×C4—568no—
459C2×C4D7×C4⋊C412yes—
460C2×C4D7×C4⋊C412yes—
461C2×C4D7×SD1612yes—
462C2×C4D7×SD1612yes—
463C2×C4C22×C4×D711yescharacteristic
464C2×C4#654811yescharacteristic
465C2×C4C2×Q8×D711yescharacteristic
466C2×C4#655111yescharacteristic
467C2×C4#655311yescharacteristic
468C2×C4#655411yescharacteristic
469C8—24no—
470C8—144no—
471C7#2198111yescharacteristic, complemented, Sylow
472C22—74no—
473C22—142no—
474C22—142no—
475C22—142no—
476C22—142no—
477C22—142no—
478C22—142no—
479C22C2×D7×C4⋊C411yescharacteristic
480C22#9029811yescharacteristic
481C22#9031911yescharacteristic
482C22C2×D7×SD1611yescharacteristic
483C22C2×D7×SD1611yescharacteristic
484C22D7×AGL(1,ℤ/8)11yescharacteristic
485C22D7×#13711yescharacteristic
486C4—22no—
487C4—74no—
488C4—74no—
489C4—81no—
490C4—81no—
491C4—82no—
492C4—142no—
493C4—561no—
494C4—561no—
495C4—562no—
496C4D7×#10614yes—
497C4#9023711yescharacteristic
498C4#9026511yescharacteristic
499C4#9038011yescharacteristic
500C4#9042011yescharacteristic
501C2—144no—
502C2#1173341311yescharacteristic
503C2C2×D7×#10611yescharacteristic
504C2D7×#42811yescharacteristic
505C2#1173783211yescharacteristic
506C2#1173787711yescharacteristic
507C2#1173846011yescharacteristic
508C2#1173850911yescharacteristic
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#21981
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. #11728308C2semidirect
7. #11728208C2semidirect
8. #11727322C2semidirect
9. #11727221C2semidirect
11. #11726601C2semidirect
37. #89605C4semidirect
84. C7×C4⋊C8C2×C4semidirect
91. C8×Dic7C2×C4semidirect
94. C4⋊(C7⋊C8)C2×C4semidirect
101. #1948D7semidirect
258. C4×D7#106semidirect
304. C4⋊C8C4×D7semidirect
471. C7#21981semidirect

Maximal subgroups

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

subgroupindex
C2×#896052
D7×#3842
#117311982
C7×#19482
#117283332
#117283082
#117282082
#117273222
#117272212
#117266292
#117266012
#219817