Finite Group Databasebeta

← #13367608

Subgroups — #13367608

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#89858C211yesmaximal, characteristic
3C4×#6438C211yesmaximal, characteristic
4#11729540C211yesmaximal, characteristic, complemented
5#11729387C212yesmaximal
6#11729386C212yesmaximal, complemented
7#11727444C212yesmaximal, complemented
8#11727441C212yesmaximal
9#11727335C211yesmaximal, characteristic
10#11727329C211yesmaximal, characteristic, complemented
11#11727143C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2212yes—
14C22×#1144C2212yes—
15C22×#1143C2212yes—
16C2×#6438C2211yescharacteristic
17C2×C4×D28C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C7×#391C2212yes—
20C7×#389—21no—
21C7×#389—21no—
22C7×#384C2211yescharacteristic
23C4×C7×C4⋊C4—21no—
24C4×C7×C4⋊C4—21no—
25C43×C7C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28#89858C418yescomplemented
29C2×#6384C2212yes—
30C2×#6383C2212yes—
31#89712—21no—
32#89712—21no—
33#89711C2211yescharacteristic
34C4×(C2×D7)⋊C4—21no—
35C4×(C2×D7)⋊C4—21no—
36#89708C2211yescharacteristic
37#89701C2211yescharacteristic
38C2×#6377C2211yescharacteristic
39C2×#6377C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#89319C2212yes—
44#22273—71nomaximal, Sylow
45C22×C7×Q8C2312yes—
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C22×Dic14C2311yescharacteristic
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8C2×C414yescomplemented
51C4×C7×Q8—42no—
52C4×C7×Q8—42no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4C2312yes—
59C2×C42×C7—22no—
60C2×C42×C7D411yescharacteristic
61C2×C42×C7D411yescharacteristic
62C2×C42×C7D411yescharacteristic
63C2×C42×C7D411yescharacteristic
64C2×C42×C7C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2311yescharacteristic
67C2×#1144—42no—
68C2×#1144—42no—
69C2×#1144C2×C418yescomplemented
70C2×#1143—42no—
71C2×#1143—42no—
72C2×#1143C2×C418yes—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×C4×Dic7—21no—
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2312yes—
84#6438—22no—
85#6438C2×C414yes—
86C4×D28—42no—
87C4×Dic14—42no—
88C7×#95—22no—
89C7×#95—22no—
90#6409—42no—
91#6409—42no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96#6384—42no—
97#6383—42no—
98#6377—22no—
99#6377C2×C414yescomplemented
100C2×#497—71no—
101C4×C4⋊Q8D711yescharacteristic, complemented
102C4×#124—71no—
103#1960—71no—
104#1959—71no—
105#1880—72no—
106#1878—72no—
107#1878—72no—
108#1877—72no—
109#1836—71no—
110#1745—71no—
111C23×D7—21no—
112C2×C7×Q8—22no—
113C2×C7×Q8—22no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8C22×C414yes—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C2×D28—22no—
131C2×D28—22no—
132C2×D28—21no—
133C2×D28C22×C412yes—
134C2×C4×D7—42no—
135C2×C4×D7—42no—
136C2×C4×D7—42no—
137C2×Dic14—22no—
138C2×Dic14—22no—
139C2×Dic14—21no—
140C2×Dic14C22×C412yes—
141C7×C4⋊C4—24no—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—42no—
146C7×C4⋊C4—42no—
147C7×C4⋊C4—44no—
148C42×C7—22no—
149C42×C7—22no—
150C42×C7—22no—
151C42×C7—22no—
152C42×C7—41no—
153C42×C7—41no—
154C42×C7—44no—
155C42×C7C22×C412yes—
156C42×C7#4112yes—
157#1144—48no—
158#1143—48no—
159(C2×D7)⋊C4—24no—
160(C2×D7)⋊C4—24no—
161(C2×D7)⋊C4—42no—
162(C2×D7)⋊C4—42no—
163(C2×D7)⋊C4—42no—
164C4⋊Dic7—22no—
165C4⋊Dic7—22no—
166Dic7⋊C4—42no—
167C4×Dic7—42no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8C22×C414yes—
174C22×Q16—72no—
175C22×SD16—72no—
176C2×C4⋊Q8C2×D711yescharacteristic
177C2×#124—71no—
178C2×C4×Q8—71no—
179C2×C4×Q8C2×D712yes—
180C2×C4×D4—71no—
181#497—78no—
182C2×#103—72no—
183C2×#103—72no—
184C2×#103—72no—
185C2×#102—72no—
186C2×#97—71no—
187C2×#97—71no—
188#391—71no—
189#391C2×D712yes—
190#390—71no—
191#389—21no—
192#389—21no—
193#388—141no—
194#388—141no—
195#384C2×D711yescharacteristic
196#382—71no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C22⋊C4—141no—
200C4×C22⋊C4—141no—
201C43C2×D711yescharacteristic
202#340—72no—
203#340—72no—
204#336—72no—
205#336—72no—
206C23×C7C22×D411yescharacteristic
207C22×D7—22no—
208C22×D7—41no—
209C22×D7—41no—
210C22×D7—41no—
211C22×D7—41no—
212C22×D7—41no—
213C22×D7—41no—
214C7×Q8—24no—
215C7×Q8—24no—
216C7×Q8—42no—
217C7×Q8—42no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—24no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7C4×D412yes—
239C2×C4×C7C4×D412yes—
240C2×C4×C7#13712yes—
241C2×C4×C7#13712yes—
242C2×C4×C7C23×C411yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C2×C4×C7H(5,𝔽2)12yes—
245C2×C4×C7H(5,𝔽2)12yes—
246C2×Dic7—22no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—41no—
251C2×Dic7—41no—
252C2×Dic7—41no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255D28—24no—
256D28—42no—
257C4×D7—84no—
258Dic14—24no—
259Dic14—42no—
260C7⋊C8—24no—
261C7⋊C8—42no—
262C22×Q8—71no—
263C22×Q8C22×D712yes—
264C22×D4—71no—
265C23×C4—141no—
266C2×Q16—78no—
267C2×Q16—282no—
268C2×Q16—282no—
269C2×SD16—78no—
270C2×SD16—282no—
271C2×SD16—282no—
272C22×C8—72no—
273C22×C8—72no—
274C4⋊Q8—22no—
275C4⋊Q8C4×D714yescomplemented
276#124—74no—
277#124—142no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—282no—
281C4×D4—282no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—71no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4C22×D712yes—
291C2×C22⋊C4—141no—
292C2×C22⋊C4—141no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—141no—
296C2×C42—22no—
297C2×C42—141no—
298C2×C42C22⋊D711yescharacteristic
299C2×C42C22⋊D711yescharacteristic
300C2×C42C22⋊D711yescharacteristic
301C2×C42C22⋊D711yescharacteristic
302C2×C42C22×D711yescharacteristic
303C2×C42C22×D711yescharacteristic
304C2×C42C22×D711yescharacteristic
305#103—282no—
306#103—282no—
307#103—282no—
308#102—282no—
309#97—74no—
310#97—142no—
311#95—22no—
312#95—22no—
313#95—141no—
314#95—141no—
315#95—141no—
316#95—141no—
317C22×C7C2×C4×D411yescharacteristic
318C22×C7#53511yescharacteristic
319C22×C7#53511yescharacteristic
320C22×C7C2×#13711yescharacteristic
321C22×C7C2×#13711yescharacteristic
322C22×C7#57511yescharacteristic
323C22×C7#57511yescharacteristic
324C2×D7—42no—
325C2×D7—42no—
326C2×D7—42no—
327C2×D7—81no—
328C2×D7—81no—
329C2×D7—81no—
330C2×D7—81no—
331C4×C7—22no—
332C4×C7—22no—
333C4×C7—22no—
334C4×C7—22no—
335C4×C7—44no—
336C4×C7—41no—
337C4×C7—41no—
338C4×C7—82no—
339C4×C7#51812yes—
340C4×C7#54012yes—
341Dic7—42no—
342Dic7—81no—
343Dic7—81no—
344C24—141no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—72no—
351C2×Q8—141no—
352C2×Q8—142no—
353C2×Q8—142no—
354C2×Q8C2×C4×D714yes—
355C2×D4—72no—
356C2×D4—141no—
357C2×D4—142no—
358C2×D4—142no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—141no—
362C22×C4—141no—
363C22×C4—141no—
364C22×C4—282no—
365C22×C4—282no—
366C22×C4—282no—
367C22×C4C2×C22⋊D711yescharacteristic
368C22×C4C2×C22⋊D711yescharacteristic
369C22×C4C2×C22⋊D711yescharacteristic
370C22×C4C2×C22⋊D711yescharacteristic
371C22×C4C2×C22⋊D711yescharacteristic
372C22×C4C2×C22⋊D711yescharacteristic
373C22×C4C23×D711yescharacteristic
374Q16—288no—
375SD16—288no—
376C2×C8—74no—
377C2×C8—142no—
378C2×C8—142no—
379C2×C8—142no—
380C2×C8—142no—
381C2×C8—142no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—142no—
390C4⋊C4—142no—
391C4⋊C4—282no—
392C22⋊C4—144no—
393C22⋊C4—144no—
394C22⋊C4—282no—
395C22⋊C4—282no—
396C22⋊C4—282no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—41no—
402C42—41no—
403C42—44no—
404C42—282no—
405C42C2×C4×D712yes—
406C42#115712yes—
407C2×C7#232811yescharacteristic
408C2×C7C4×#13711yescharacteristic
409C2×C7C4×#13711yescharacteristic
410C2×C7#321511yescharacteristic
411C2×C7#321611yescharacteristic
412C2×C7#321811yescharacteristic
413C2×C7#321911yescharacteristic
414D7—82no—
415C23—142no—
416C23—281no—
417C23—281no—
418C23—281no—
419C23—281no—
420C23—281no—
421C23—281no—
422C23C22×C22⋊D711yescharacteristic
423Q8—24no—
424Q8—24no—
425Q8—42no—
426Q8—42no—
427Q8—144no—
428Q8—282no—
429D4—144no—
430D4—282no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—142no—
452C2×C4—281no—
453C2×C4—281no—
454C2×C4—281no—
455C2×C4—281no—
456C2×C4—281no—
457C2×C4—281no—
458C2×C4—281no—
459C2×C4—281no—
460C2×C4—564no—
461C2×C4C4×C22⋊D712yes—
462C2×C4C4×C22⋊D712yes—
463C2×C4#650512yes—
464C2×C4#650512yes—
465C2×C4C22×C4×D711yescharacteristic
466C2×C4C2×#115711yescharacteristic
467C2×C4#654812yes—
468C2×C4#655312yes—
469C8—144no—
470C8—282no—
471C7#2227311yescharacteristic, complemented, Sylow
472C22—282no—
473C22—282no—
474C22—282no—
475C22—561no—
476C22—561no—
477C22—561no—
478C22—561no—
479C22C2×C4×C22⋊D711yescharacteristic
480C22#9049611yescharacteristic
481C22C2×#650511yescharacteristic
482C22C2×#650511yescharacteristic
483C22#9051311yescharacteristic
484C22#9051311yescharacteristic
485C22#9052111yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—44no—
492C4—41no—
493C4—82no—
494C4—282no—
495C4—561no—
496C4—561no—
497C4#9023612yes—
498C4#9025312yes—
499C2—562no—
500C2#1173977411yescharacteristic
501C2C4×#650511yescharacteristic
502C2C4×#650511yescharacteristic
503C2#1174065911yescharacteristic
504C2#1174075411yescharacteristic
505C2#1174080211yescharacteristic
506C2#1174080311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×C4⋊Q8C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenterC4×C4⋊Q8D7
hypocenterC7#22273
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×C7×C4⋊Q8C2semidirect
4. #11729540C2semidirect
6. #11729386C2semidirect
7. #11727444C2semidirect
10. #11727329C2semidirect
11. #11727143C2semidirect
28. #89858C4semidirect
50. C7×C4⋊Q8C2×C4semidirect
69. C2×#1144C2×C4semidirect
99. #6377C2×C4semidirect
101. C4×C4⋊Q8D7semidirect
275. C4⋊Q8C4×D7semidirect
471. C7#22273semidirect

Maximal subgroups

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

subgroupindex
C4×C7×C4⋊Q82
C2×#898582
C4×#64382
#117295402
#117293872
#117293862
#117274442
#117274412
#117273352
#117273292
#117271432
#222737