Finite Group Databasebeta

← #13382884

Subgroups — #13382884

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic
2C2×#89890C211yesmaximal, characteristic
3#11739671C211yesmaximal, characteristic
4C2×#89745C211yesmaximal, characteristic
5#11729739C211yesmaximal, characteristic
6#11729335C211yesmaximal, characteristic
7#11729312C211yesmaximal, characteristic
8#11728376C212yesmaximal, complemented
9#11728375C212yesmaximal, complemented
10#11728322C212yesmaximal, complemented
11#11728322C212yesmaximal, complemented
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4—21no—
18C22×C4×Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C7×#428—24no—
21C2×C7×#107C2212yes—
22C2×C7×#106C2212yes—
23C2×C7⋊#95—21no—
24#89890—24no—
25C2×Dic7⋊C8C2212yes—
26C2×Dic7⋊C8C2212yes—
27#89745—21no—
28#89745—21no—
29#89745C2211yescharacteristic
30#89745C2211yescharacteristic
31#89745C2211yescharacteristic
32#89745C2211yescharacteristic
33#89745C2212yes—
34#89745C2212yes—
35#89718—21no—
36#89718—21no—
37C4×Dic7⋊C4—21no—
38C4×Dic7⋊C4—21no—
39#89346C2212yes—
40#89346C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#23670—71nomaximal, Sylow
46C23×C4×C7—21no—
47C23×C4×C7C2311yescharacteristic
48C23×Dic7—21no—
49C23×Dic7D411yescharacteristic
50C23×Dic7D411yescharacteristic
51C2×C7×M16—22no—
52C2×C7×M16—22no—
53C2×C7×M16D414yes—
54C22×C7×C8C2312yes—
55C7×#117—22no—
56C7×#117—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C7×C22⋊C4—21no—
64C2×C42×C7—21no—
65C2×C22⋊Dic7—21no—
66#6489—22no—
67#6489Q814yes—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7Q814yes—
71C2×C4⋊Dic7C2312yes—
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2311yescharacteristic
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—44no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C2×#1136—22no—
89C2×#1136—22no—
90C2×#1136—22no—
91C22×C7⋊C8C2312yes—
92C7×#107—24no—
93C7×#106—24no—
94C7⋊#95—21no—
95C7⋊#95—21no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—22no—
99C7⋊#95—22no—
100Dic7⋊C8—24no—
101Dic7⋊C8—24no—
102C2×#428D711yescharacteristic
103C2×#423—71no—
104#2296—71no—
105C2×#389—71no—
106#2088—72no—
107#2088—72no—
108#2087—72no—
109#2086—72no—
110#1736—71no—
111#1736—71no—
112#1736—71no—
113C24×C7C2×D411yescharacteristic
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—42no—
118C22×C4×C7—42no—
119C22×C4×C7—42no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D412yes—
122C22×C4×C7C2×Q812yes—
123C22×C4×C7C2×Q812yes—
124C22×C4×C7#4112yes—
125C22×C4×C7#4112yes—
126C22×C4×C7#4112yes—
127C22×C4×C7C2411yescharacteristic
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—24no—
131C22×Dic7—21no—
132C22×Dic7—21no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—41no—
137C22×Dic7—41no—
138C22×Dic7—42no—
139C22×Dic7—42no—
140C22×Dic7C2×D411yescharacteristic
141C22×Dic7C2×D411yescharacteristic
142C22×Dic7C2×D411yescharacteristic
143C22×Dic7C2×D411yescharacteristic
144C7×M16—28no—
145C2×C7×C8—22no—
146C2×C7×C8—22no—
147C2×C7×C8C2×D414yes—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—42no—
155C7×C22⋊C4—42no—
156C42×C7—42no—
157C22⋊Dic7—24no—
158C4⋊Dic7—24no—
159C4⋊Dic7—22no—
160C4⋊Dic7—21no—
161C4⋊Dic7—21no—
162C4⋊Dic7C2×Q814yes—
163C4⋊Dic7C2×Q812yes—
164C4⋊Dic7C2×Q812yes—
165Dic7⋊C4—42no—
166Dic7⋊C4—42no—
167Dic7⋊C4—44no—
168Dic7⋊C4—42no—
169Dic7⋊C4—44no—
170Dic7⋊C4—42no—
171C4×Dic7—22no—
172C4×Dic7—22no—
173C4×Dic7—24no—
174C4×Dic7—21no—
175C4×Dic7—21no—
176C4×Dic7—21no—
177C4×Dic7—21no—
178#1136—44no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C22×M16—71no—
183C22×M16C2×D711yescharacteristic
184C2×#117—71no—
185C2×#117C2×D711yescharacteristic
186C22×C4⋊C4—71no—
187C22×C4⋊C4—141no—
188C22×C4⋊C4C2×D711yescharacteristic
189C22×C42—71no—
190#428—24no—
191C2×#107C2×D712yes—
192C2×#106C2×D712yes—
193#423—144no—
194C2×C4⋊C8—72no—
195C2×C4⋊C8—72no—
196#389—71no—
197#389—72no—
198#389—71no—
199#389—72no—
200#389—71no—
201#389—71no—
202#389—141no—
203#389—141no—
204#385—141no—
205#385—141no—
206C4×C4⋊C4—141no—
207C4×C4⋊C4—141no—
208C2×#95—141no—
209#340—72no—
210#340—72no—
211#340—72no—
212#340—72no—
213#340—72no—
214#340—72no—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7#12112yes—
220C23×C7C22⋊Q812yes—
221C23×C7C22⋊Q812yes—
222C23×C7C22×D411yescharacteristic
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—84no—
233C2×C4×C7#12112yes—
234C2×C4×C7C22⋊Q812yes—
235C2×C4×C7C22⋊Q812yes—
236C2×C4×C7#12314yes—
237C2×C4×C7#12514yes—
238C2×C4×C7#12514yes—
239C2×C4×C7C4⋊Q814yes—
240C2×C4×C7C22×D411yescharacteristic
241C2×C4×C7C22×Q811yescharacteristic
242C2×C4×C7C22×Q811yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C2×C4×C7C2×#4111yescharacteristic
245C2×C4×C7C2×#4111yescharacteristic
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—24no—
255C2×Dic7—21no—
256C2×Dic7—21no—
257C2×Dic7—21no—
258C2×Dic7—21no—
259C2×Dic7—44no—
260C2×Dic7—44no—
261C2×Dic7—48no—
262C2×Dic7—42no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—42no—
268C2×Dic7AGL(1,ℤ/8)12yes—
269C2×Dic7#13712yes—
270C7×C8—24no—
271C7⋊C8—42no—
272C23×C4—21no—
273C23×C4—71no—
274C23×C4—71no—
275C23×C4—141no—
276C23×C4C22×D711yescharacteristic
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—142no—
280C2×M16—142no—
281C2×M16—142no—
282C2×M16C22⋊D714yes—
283C22×C8—72no—
284C22×C8C22×D712yes—
285#117—22no—
286#117—22no—
287#117—74no—
288#117—142no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—74no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—71no—
296C2×C4⋊C4—142no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4—141no—
302C2×C4⋊C4—141no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—284no—
305C2×C4⋊C4C22×D712yes—
306C2×C4⋊C4C22×D711yescharacteristic
307C2×C4⋊C4C22×D711yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—141no—
310C2×C42—21no—
311C2×C42—71no—
312C2×C42—71no—
313C2×C42—71no—
314C2×C42—71no—
315C2×C42—141no—
316C2×C42—142no—
317C2×C42—142no—
318#107—24no—
319#106—24no—
320C4⋊C8—144no—
321C4⋊C8—144no—
322#95—142no—
323#95—142no—
324#95—141no—
325#95—141no—
326#95—141no—
327#95—141no—
328C22×C7—22no—
329C22×C7—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7—22no—
334C22×C7#39814yes—
335C22×C7C2×#12111yescharacteristic
336C22×C7C2×C22⋊Q811yescharacteristic
337C22×C7C2×C22⋊Q811yescharacteristic
338C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
339C22×C7C2×#13711yescharacteristic
340C22×C7#57511yescharacteristic
341C22×C7#57511yescharacteristic
342C4×C7—81no—
343C4×C7—82no—
344C4×C7—81no—
345C4×C7#39814yes—
346C4×C7C2×#12311yescharacteristic
347C4×C7C2×#12511yescharacteristic
348C4×C7C2×#12511yescharacteristic
349C4×C7C2×C4⋊Q811yescharacteristic
350Dic7—22no—
351Dic7—22no—
352Dic7—44no—
353Dic7—42no—
354Dic7—42no—
355Dic7—41no—
356Dic7—41no—
357C24D4×D711yescharacteristic
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—71no—
365C22×C4—71no—
366C22×C4—71no—
367C22×C4—71no—
368C22×C4—142no—
369C22×C4—142no—
370C22×C4—144no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—141no—
374C22×C4—141no—
375C22×C4—142no—
376C22×C4—281no—
377C22×C4—281no—
378C22×C4—282no—
379C22×C4—282no—
380C22×C4#115712yes—
381C22×C4#115712yes—
382C22×C4D4×D711yescharacteristic
383C22×C4#115912yes—
384C22×C4Q8×D712yes—
385C22×C4Q8×D712yes—
386C22×C4C2×C22⋊D712yes—
387C22×C4C23×D711yescharacteristic
388M16—28no—
389M16—284no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—142no—
393C2×C8—142no—
394C2×C8—142no—
395C2×C8C2×C22⋊D714yes—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—74no—
404C4⋊C4—72no—
405C4⋊C4—72no—
406C4⋊C4—144no—
407C4⋊C4—142no—
408C4⋊C4—141no—
409C4⋊C4—141no—
410C4⋊C4—282no—
411C4⋊C4—282no—
412C4⋊C4—284no—
413C4⋊C4—282no—
414C4⋊C4—282no—
415C4⋊C4—284no—
416C22⋊C4—42no—
417C22⋊C4—144no—
418C42—42no—
419C42—144no—
420C42—142no—
421C42—142no—
422C42—141no—
423C42—141no—
424C42—141no—
425C42—141no—
426C2×C7—24no—
427C2×C7C2×#39811yescharacteristic
428C2×C7#305511yescharacteristic
429C2×C7#305611yescharacteristic
430C2×C7#307711yescharacteristic
431C2×C7#307811yescharacteristic
432C2×C7#307911yescharacteristic
433C2×C7#308011yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23#645912yes—
439C23#645912yes—
440C23#650112yes—
441C23C2×D4×D711yescharacteristic
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—72no—
452C2×C4—72no—
453C2×C4—84no—
454C2×C4—141no—
455C2×C4—141no—
456C2×C4—141no—
457C2×C4—141no—
458C2×C4—142no—
459C2×C4—142no—
460C2×C4—144no—
461C2×C4—141no—
462C2×C4—141no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—284no—
468C2×C4—284no—
469C2×C4—288no—
470C2×C4—282no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—282no—
476C2×C4#645214yes—
477C2×C4#645214yes—
478C2×C4#645912yes—
479C2×C4#645912yes—
480C2×C4#649214yes—
481C2×C4#650112yes—
482C2×C4#650714yes—
483C2×C4C2×#115711yescharacteristic
484C2×C4C2×#115711yescharacteristic
485C2×C4C2×#115911yescharacteristic
486C2×C4C2×Q8×D711yescharacteristic
487C2×C4C2×Q8×D711yescharacteristic
488C2×C4C22×C22⋊D711yescharacteristic
489C8—24no—
490C8—282no—
491C7#2367011yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22#8975214yes—
499C22C2×#645911yescharacteristic
500C22C2×#645911yescharacteristic
501C22D7×AGL(1,ℤ/8)11yescharacteristic
502C22#9046411yescharacteristic
503C22D7×#13711yescharacteristic
504C22#9046811yescharacteristic
505C22C2×#650111yescharacteristic
506C4—82no—
507C4—81no—
508C4—81no—
509C4—142no—
510C4—142no—
511C4—284no—
512C4—282no—
513C4—282no—
514C4—281no—
515C4—281no—
516C4#8975214yes—
517C4C2×#645211yescharacteristic
518C4C2×#645211yescharacteristic
519C4C2×#649211yescharacteristic
520C4C2×#650711yescharacteristic
521C2—24no—
522C2#1173668411yescharacteristic
523C2#1173668511yescharacteristic
524C2#1173668611yescharacteristic
525C2#1173669011yescharacteristic
526C2C2×#8975211yescharacteristic
527C2#1174139511yescharacteristic
528C2#1174143011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#428C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#428D7
hypocenterC7#23670
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
8. #11728376C2semidirect
9. #11728375C2semidirect
10. #11728322C2semidirect
11. #11728322C2semidirect
491. C7#23670semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×#898902
#117396712
C2×#897452
#117297392
#117293352
#117293122
#117283762
#117283752
#117283222
#236707