Finite Group Databasebeta

← #13355973

Subgroups — #13355973

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739672C211yesmaximal, characteristic
2C2×#89767C211yesmaximal, characteristic
3C2×C4⋊C4×Dic7C211yesmaximal, characteristic
4C7×#1736C211yesmaximal, characteristic
5#11729739C211yesmaximal, characteristic
6#11729733C211yesmaximal, characteristic
7#11729312C211yesmaximal, characteristic
8#11726575C212yesmaximal, complemented
9#11726573C212yesmaximal, complemented
10#11726571C212yesmaximal, complemented
11#11726569C212yesmaximal, complemented
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7—21no—
18C22×C4×Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C2×C7⋊#95—21no—
21#89767C418yescomplemented
22#89757C2211yescharacteristic
23#89757C2211yescharacteristic
24#89757C2211yescharacteristic
25#89757C2211yescharacteristic
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7C2212yes—
29C4⋊C4×Dic7C2212yes—
30C2×#6382C2212yes—
31C2×#6381C2212yes—
32#89722—21no—
33#89722—21no—
34C4×C4⋊Dic7—21no—
35C4×C4⋊Dic7—21no—
36C7×#340C2212yes—
37C7×#340C2212yes—
38#89346C2212yes—
39#89346C2212yes—
40#89345C2212yes—
41#89345C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21033—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C2×C7×M16—22no—
51C2×C7×M16—22no—
52C2×C7×M16—22no—
53C22×C7×C8C2312yes—
54C7×#117—22no—
55C7×#117C2×C414yes—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4Q814yes—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C22⋊C4—21no—
63C2×C42×C7—21no—
64C2×C22⋊Dic7—21no—
65#6489—22no—
66#6489—22no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7C2×C414yes—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2312yes—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—21no—
80C2×C4×Dic7—22no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—44no—
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7C2311yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136C2×C414yes—
90C22×C7⋊C8C2312yes—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—21no—
94C7⋊#95—21no—
95C7⋊#95—22no—
96C7⋊#95—22no—
97#6382C2×C418yescomplemented
98#6381C2×C418yescomplemented
99C2×#428—71no—
100#2296—71no—
101C2×C4×C4⋊C4—71no—
102#1921—72no—
103#1920—72no—
104#1919—72no—
105#1918—72no—
106#1736—71no—
107#1736—71no—
108#1736D711yescharacteristic
109#1728—71no—
110C24×C7C2×D411yescharacteristic
111C22×C4×C7—24no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—42no—
116C22×C4×C7—42no—
117C22×C4×C7C22×C412yes—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×Q812yes—
120C22×C4×C7#4112yes—
121C22×C4×C7#4112yes—
122C22×C4×C7#4112yes—
123C22×C4×C7#4112yes—
124C22×C4×C7C2411yescharacteristic
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—24no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—21no—
132C22×Dic7—21no—
133C22×Dic7—41no—
134C22×Dic7—41no—
135C22×Dic7—42no—
136C22×Dic7—42no—
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C22×Dic7C2×D411yescharacteristic
141C7×M16—44no—
142C2×C7×C8—22no—
143C2×C7×C8—22no—
144C2×C7×C8—22no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4C22×C412yes—
150C7×C4⋊C4C22×C412yes—
151C7×C4⋊C4C2×Q814yes—
152C7×C22⋊C4—24no—
153C42×C7—24no—
154C22⋊Dic7—42no—
155C4⋊Dic7—24no—
156C4⋊Dic7—22no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—21no—
161C4⋊Dic7—42no—
162C4⋊Dic7—42no—
163C4⋊Dic7—42no—
164C4⋊Dic7—42no—
165C4⋊Dic7C22×C414yes—
166C4×Dic7—22no—
167C4×Dic7—22no—
168C4×Dic7—21no—
169C4×Dic7—21no—
170C4×Dic7—21no—
171C4×Dic7—21no—
172C4×Dic7—44no—
173C4×Dic7—42no—
174C4×Dic7—44no—
175#1136—28no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8C22×C414yes—
179C22×M16—71no—
180C22×M16C2×D711yescharacteristic
181C2×#117—71no—
182C2×#117C2×D711yescharacteristic
183C22×C4⋊C4—71no—
184C22×C4⋊C4C2×D711yescharacteristic
185C22×C42—71no—
186C22×C42—141no—
187#428—78no—
188C2×#107—72no—
189C2×#106—72no—
190#389—71no—
191#389—71no—
192#389—71no—
193#389—71no—
194#385—141no—
195#385—141no—
196C4×C4⋊C4—72no—
197C4×C4⋊C4—72no—
198C4×C4⋊C4—141no—
199C4×C4⋊C4—141no—
200C4×C4⋊C4—141no—
201C4×C4⋊C4—141no—
202C2×#95—141no—
203#340—72no—
204#340—72no—
205#340—72no—
206#340—72no—
207#340C2×D712yes—
208#340C2×D712yes—
209#336—72no—
210#336—72no—
211C23×C7—21no—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7C4×D412yes—
216C23×C7C22⋊Q812yes—
217C23×C7#12312yes—
218C23×C7C22×D411yescharacteristic
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—48no—
223C2×C4×C7—42no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—42no—
229C2×C4×C7#11714yes—
230C2×C4×C7C4×D412yes—
231C2×C4×C7C4×Q814yes—
232C2×C4×C7C22⋊Q812yes—
233C2×C4×C7#12312yes—
234C2×C4×C7#12514yes—
235C2×C4×C7#12614yes—
236C2×C4×C7C23×C411yescharacteristic
237C2×C4×C7C22×Q811yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C2×C4×C7C2×#4111yescharacteristic
241C2×C4×C7C2×#4111yescharacteristic
242C2×Dic7—28no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—21no—
247C2×Dic7—24no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—21no—
253C2×Dic7—21no—
254C2×Dic7—44no—
255C2×Dic7—48no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—41no—
264C2×Dic7AGL(1,ℤ/8)12yes—
265C2×Dic7#13712yes—
266C7×C8—42no—
267C7⋊C8—24no—
268C23×C4—21no—
269C23×C4—71no—
270C23×C4—71no—
271C23×C4—141no—
272C23×C4C22×D711yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—74no—
277C2×M16—142no—
278C2×M16—142no—
279C22×C8—72no—
280C22×C8C22×D712yes—
281#117—22no—
282#117—142no—
283#117—142no—
284#117C4×D714yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—74no—
288C2×C4⋊C4—72no—
289C2×C4⋊C4—71no—
290C2×C4⋊C4—71no—
291C2×C4⋊C4—141no—
292C2×C4⋊C4—142no—
293C2×C4⋊C4—141no—
294C2×C4⋊C4—141no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4Dic1414yes—
298C2×C4⋊C4C22×D712yes—
299C2×C4⋊C4C22×D711yescharacteristic
300C2×C4⋊C4C22×D711yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—141no—
303C2×C42—21no—
304C2×C42—71no—
305C2×C42—71no—
306C2×C42—71no—
307C2×C42—71no—
308C2×C42—142no—
309C2×C42—142no—
310C2×C42—142no—
311C2×C42—142no—
312C2×C42—141no—
313C2×C42—284no—
314#107—78no—
315#106—78no—
316#95—142no—
317#95—142no—
318#95—141no—
319#95—141no—
320#95—141no—
321#95—141no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—22no—
326C22×C7—22no—
327C22×C7—22no—
328C22×C7#38714yes—
329C22×C7C2×C4×D411yescharacteristic
330C22×C7C2×C22⋊Q811yescharacteristic
331C22×C7C2×#12311yescharacteristic
332C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
333C22×C7C2×#13711yescharacteristic
334C22×C7#57511yescharacteristic
335C22×C7#57511yescharacteristic
336C4×C7—44no—
337C4×C7—42no—
338C4×C7—42no—
339C4×C7#38714yes—
340C4×C7C2×#11711yescharacteristic
341C4×C7C2×C4×Q811yescharacteristic
342C4×C7C2×#12511yescharacteristic
343C4×C7C2×#12611yescharacteristic
344Dic7—22no—
345Dic7—22no—
346Dic7—41no—
347Dic7—41no—
348Dic7—44no—
349Dic7—81no—
350Dic7—81no—
351C24D4×D711yescharacteristic
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—71no—
359C22×C4—71no—
360C22×C4—71no—
361C22×C4—71no—
362C22×C4—142no—
363C22×C4—142no—
364C22×C4—142no—
365C22×C4—144no—
366C22×C4—142no—
367C22×C4—142no—
368C22×C4—141no—
369C22×C4—141no—
370C22×C4—281no—
371C22×C4—281no—
372C22×C4—282no—
373C22×C4—282no—
374C22×C4C2×Dic1412yes—
375C22×C4C2×C4×D712yes—
376C22×C4#115712yes—
377C22×C4D4×D711yescharacteristic
378C22×C4#115912yes—
379C22×C4#115912yes—
380C22×C4#116112yes—
381C22×C4C23×D711yescharacteristic
382M16—44no—
383M16—148no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—74no—
388C2×C8—142no—
389C2×C8—142no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—74no—
395C4⋊C4—144no—
396C4⋊C4—142no—
397C4⋊C4—141no—
398C4⋊C4—141no—
399C4⋊C4—141no—
400C4⋊C4—141no—
401C4⋊C4—282no—
402C4⋊C4—282no—
403C4⋊C4—282no—
404C4⋊C4—282no—
405C4⋊C4C2×Dic1414yes—
406C4⋊C4C2×C4×D712yes—
407C4⋊C4C2×C4×D712yes—
408C22⋊C4—24no—
409C22⋊C4—282no—
410C42—24no—
411C42—142no—
412C42—142no—
413C42—141no—
414C42—141no—
415C42—141no—
416C42—141no—
417C42—284no—
418C42—284no—
419C42—282no—
420C2×C7—24no—
421C2×C7C2×#38711yescharacteristic
422C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
423C2×C7C4×#13711yescharacteristic
424C2×C7#307711yescharacteristic
425C2×C7#307811yescharacteristic
426C2×C7#309211yescharacteristic
427C2×C7#309311yescharacteristic
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23#644112yes—
433C23(C22⋊D7)⋊C412yes—
434C23#645712yes—
435C23C2×D4×D711yescharacteristic
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—72no—
447C2×C4—72no—
448C2×C4—148no—
449C2×C4—141no—
450C2×C4—141no—
451C2×C4—141no—
452C2×C4—141no—
453C2×C4—144no—
454C2×C4—141no—
455C2×C4—141no—
456C2×C4—141no—
457C2×C4—141no—
458C2×C4—141no—
459C2×C4—141no—
460C2×C4—284no—
461C2×C4—288no—
462C2×C4—282no—
463C2×C4—282no—
464C2×C4—281no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—281no—
468C2×C4—281no—
469C2×C4—281no—
470C2×C4C4×Dic1414yes—
471C2×C4#644112yes—
472C2×C4#644214yes—
473C2×C4(C22⋊D7)⋊C412yes—
474C2×C4#645314yes—
475C2×C4#645514yes—
476C2×C4#645712yes—
477C2×C4C22×Dic1411yescharacteristic
478C2×C4C22×C4×D711yescharacteristic
479C2×C4C2×#115711yescharacteristic
480C2×C4C2×#115911yescharacteristic
481C2×C4C2×#115911yescharacteristic
482C2×C4C2×#116111yescharacteristic
483C8—42no—
484C8—144no—
485C7#2103311yescharacteristic, complemented, Sylow
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22#8942714yes—
493C22C2×#644111yescharacteristic
494C22C2×(C22⋊D7)⋊C411yescharacteristic
495C22C2×#645711yescharacteristic
496C22D7×AGL(1,ℤ/8)11yescharacteristic
497C22#9046411yescharacteristic
498C22D7×#13711yescharacteristic
499C22#9046811yescharacteristic
500C4—44no—
501C4—42no—
502C4—42no—
503C4—142no—
504C4—142no—
505C4—281no—
506C4—281no—
507C4—284no—
508C4—561no—
509C4—561no—
510C4#8942714yes—
511C4C2×C4×Dic1411yescharacteristic
512C4C2×#644211yescharacteristic
513C4C2×#645311yescharacteristic
514C4C2×#645511yescharacteristic
515C2—24no—
516C2C2×#8942711yescharacteristic
517C2#1173631111yescharacteristic
518C2#1173631211yescharacteristic
519C2#1173636211yescharacteristic
520C2#1173639411yescharacteristic
521C2#1173666011yescharacteristic
522C2#1173666111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1736C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1736D7
hypocenterC7#21033
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
8. #11726575C2semidirect
9. #11726573C2semidirect
10. #11726571C2semidirect
11. #11726569C2semidirect
21. #89767C4semidirect
97. #6382C2×C4semidirect
98. #6381C2×C4semidirect
485. C7#21033semidirect

Maximal subgroups

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

subgroupindex
#117396722
C2×#897672
C2×C4⋊C4×Dic72
C7×#17362
#117297392
#117297332
#117293122
#117265752
#117265732
#117265712
#117265692
#210337