Finite Group Databasebeta

← #13382882

Subgroups — #13382882

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×#423C211yesmaximal, characteristic
2C2×#89893C211yesmaximal, characteristic
3#11739514C211yesmaximal, characteristic
4C2×#89700C211yesmaximal, characteristic
5#11729739C211yesmaximal, characteristic
6#11729739C211yesmaximal, characteristic
7#11729739C211yesmaximal, characteristic
8#11728218C212yesmaximal, complemented
9#11728218C212yesmaximal, complemented
10#11728217C212yesmaximal, complemented
11#11728216C212yesmaximal, complemented
12C22×C7×M16C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C2×#6489C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4—21no—
20C7×#423—24no—
21C2×C7×C4⋊C8C2212yes—
22C2×C7×C4⋊C8C2212yes—
23#89982—21no—
24#89982—21no—
25C2×C7⋊#95—21no—
26#89893—24no—
27C2×#6392C2212yes—
28C2×#6391C2212yes—
29C4×Dic7⋊C4—21no—
30C4×Dic7⋊C4—21no—
31#89700—21no—
32#89700—21no—
33#89700C2211yescharacteristic
34#89700C2212yes—
35#89700C2211yescharacteristic
36#89700C2212yes—
37#89700C2211yescharacteristic
38#89700C2211yescharacteristic
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#89346C2212yes—
45#23670—71nomaximal, Sylow
46C23×C4×C7D411yescharacteristic
47C23×C4×C7D411yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7—21no—
50C23×Dic7—21no—
51C2×C7×M16—22no—
52C2×C7×M16—22no—
53C2×C7×M16—22no—
54C2×C7×M16—22no—
55C2×C7×M16—22no—
56C2×C7×M16D414yes—
57C22×C7×C8C2312yes—
58C22×C7×C8C2312yes—
59C2×C42×C7—22no—
60C2×C42×C7—22no—
61C2×C42×C7D411yescharacteristic
62C2×C42×C7D411yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×C42×C7C2311yescharacteristic
65C2×C22⋊Dic7—21no—
66C2×C22⋊Dic7—21no—
67#6489—22no—
68#6489—22no—
69#6489—22no—
70#6489Q814yes—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7Q814yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2312yes—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2312yes—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—44no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C7×C4⋊C8—24no—
93C7×C4⋊C8—24no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100#6392—24no—
101#6391—24no—
102C2×#428—71no—
103C2×#423D711yescharacteristic
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—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—41no—
119C22×C4×C7—41no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×Q812yes—
127C22×C4×C7C2×Q812yes—
128C22×C4×C7#4112yes—
129C22×C4×C7#4112yes—
130C22×C4×C7#4112yes—
131C22×C4×C7C2411yescharacteristic
132C22×Dic7—24no—
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7—22no—
136C22×Dic7—21no—
137C22×Dic7—21no—
138C22×Dic7—22no—
139C22×Dic7—42no—
140C22×Dic7—42no—
141C22×Dic7—42no—
142C22×Dic7—42no—
143C22×Dic7—42no—
144C7×M16—28no—
145C7×M16—44no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C2×C7×C8—22no—
151C2×C7×C8C2×D414yes—
152C42×C7—22no—
153C42×C7—22no—
154C42×C7—21no—
155C42×C7—21no—
156C42×C7—21no—
157C42×C7—21no—
158C22⋊Dic7—24no—
159C22⋊Dic7—42no—
160C4⋊Dic7—24no—
161C4⋊Dic7—22no—
162C4⋊Dic7—21no—
163C4⋊Dic7—21no—
164C4⋊Dic7—22no—
165C4⋊Dic7—21no—
166C4⋊Dic7—21no—
167C4⋊Dic7—22no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—42no—
171C4⋊Dic7C2×Q814yes—
172C4⋊Dic7C2×Q812yes—
173C4⋊Dic7C2×Q812yes—
174Dic7⋊C4—42no—
175Dic7⋊C4—42no—
176Dic7⋊C4—44no—
177Dic7⋊C4—42no—
178Dic7⋊C4—44no—
179Dic7⋊C4—42no—
180C4×Dic7—24no—
181C4×Dic7—42no—
182C22×M16C2×D711yescharacteristic
183C22×M16C2×D711yescharacteristic
184C2×#117—71no—
185C2×#117—71no—
186C22×C4⋊C4—71no—
187C22×C4⋊C4—71no—
188C22×C4⋊C4—141no—
189C22×C42C2×D711yescharacteristic
190#428—144no—
191C2×#107—72no—
192C2×#106—72no—
193#423—24no—
194C2×C4⋊C8C2×D712yes—
195C2×C4⋊C8C2×D712yes—
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—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
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—44no—
234C2×C4×C7—44no—
235C2×C4×C7#12112yes—
236C2×C4×C7C22⋊Q812yes—
237C2×C4×C7C22⋊Q812yes—
238C2×C4×C7#12314yes—
239C2×C4×C7#12514yes—
240C2×C4×C7#12514yes—
241C2×C4×C7C4⋊Q814yes—
242C2×C4×C7AGL(1,ℤ/8)12yes—
243C2×C4×C7#13712yes—
244C2×C4×C7C22×D411yescharacteristic
245C2×C4×C7C22×Q811yescharacteristic
246C2×C4×C7C22×Q811yescharacteristic
247C2×C4×C7C2×#4111yescharacteristic
248C2×C4×C7C2×#4111yescharacteristic
249C2×C4×C7C2×#4111yescharacteristic
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—24no—
253C2×Dic7—48no—
254C2×Dic7—42no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—42no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—42no—
269C2×Dic7—84no—
270C7×C8—24no—
271C7×C8—42no—
272C23×C4—141no—
273C23×C4—141no—
274C23×C4D2811yescharacteristic
275C23×C4D2811yescharacteristic
276C23×C4C22×D711yescharacteristic
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16C22⋊D714yes—
283C22×C8C22×D712yes—
284C22×C8C22×D712yes—
285#117—74no—
286#117—142no—
287#117—142no—
288#117—142no—
289C2×C4⋊C4—74no—
290C2×C4⋊C4—72no—
291C2×C4⋊C4—71no—
292C2×C4⋊C4—71no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—71no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—142no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—284no—
308C2×C22⋊C4—141no—
309C2×C22⋊C4—141no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—141no—
313C2×C42—141no—
314C2×C42D2811yescharacteristic
315C2×C42D2811yescharacteristic
316C2×C42C22×D711yescharacteristic
317C2×C42C22×D711yescharacteristic
318#107—144no—
319#106—144no—
320C4⋊C8—24no—
321C4⋊C8—24no—
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—22no—
343C4×C7—22no—
344C4×C7—41no—
345C4×C7—41no—
346C4×C7#39814yes—
347C4×C7C2×#12311yescharacteristic
348C4×C7C2×#12511yescharacteristic
349C4×C7C2×#12511yescharacteristic
350C4×C7C2×C4⋊Q811yescharacteristic
351Dic7—44no—
352Dic7—42no—
353Dic7—42no—
354Dic7—81no—
355Dic7—82no—
356Dic7—81no—
357C24C2×D2811yescharacteristic
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—144no—
365C22×C4—141no—
366C22×C4—141no—
367C22×C4—142no—
368C22×C4—141no—
369C22×C4—141no—
370C22×C4—142no—
371C22×C4—282no—
372C22×C4—282no—
373C22×C4—282no—
374C22×C4—282no—
375C22×C4—282no—
376C22×C4C2×D2811yescharacteristic
377C22×C4C2×D2811yescharacteristic
378C22×C4C2×D2811yescharacteristic
379C22×C4C2×D2811yescharacteristic
380C22×C4C2×D2811yescharacteristic
381C22×C4#115712yes—
382C22×C4#115912yes—
383C22×C4#115912yes—
384C22×C4Q8×D712yes—
385C22×C4Q8×D712yes—
386C22×C4C2×C22⋊D712yes—
387C22×C4C23×D711yescharacteristic
388M16—28no—
389M16—44no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8C2×C22⋊D714yes—
396C4⋊C4—74no—
397C4⋊C4—72no—
398C4⋊C4—72no—
399C4⋊C4—144no—
400C4⋊C4—142no—
401C4⋊C4—141no—
402C4⋊C4—141no—
403C4⋊C4—142no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4—141no—
407C4⋊C4—142no—
408C4⋊C4—141no—
409C4⋊C4—282no—
410C4⋊C4—282no—
411C4⋊C4—284no—
412C4⋊C4—282no—
413C4⋊C4—284no—
414C4⋊C4—282no—
415C4⋊C4—282no—
416C22⋊C4—144no—
417C22⋊C4—282no—
418C42—22no—
419C42—22no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—144no—
425C42—282no—
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#646012yes—
439C23#646012yes—
440C23#649312yes—
441C23C22×D2811yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—142no—
455C2×C4—142no—
456C2×C4—144no—
457C2×C4—288no—
458C2×C4—282no—
459C2×C4—281no—
460C2×C4—281no—
461C2×C4—281no—
462C2×C4—281no—
463C2×C4—282no—
464C2×C4—282no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—281no—
468C2×C4—281no—
469C2×C4—282no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—282no—
473C2×C4—564no—
474C2×C4#645214yes—
475C2×C4#645214yes—
476C2×C4#646012yes—
477C2×C4#646012yes—
478C2×C4#647112yes—
479C2×C4#647212yes—
480C2×C4#649312yes—
481C2×C4#649814yes—
482C2×C4#650714yes—
483C2×C4C2×#115711yescharacteristic
484C2×C4C2×#115911yescharacteristic
485C2×C4C2×#115911yescharacteristic
486C2×C4C2×Q8×D711yescharacteristic
487C2×C4C2×Q8×D711yescharacteristic
488C2×C4C22×C22⋊D711yescharacteristic
489C8—24no—
490C8—42no—
491C7#2367011yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22#8975514yes—
499C22C2×#646011yescharacteristic
500C22C2×#646011yescharacteristic
501C22C2×#647111yescharacteristic
502C22C2×#647211yescharacteristic
503C22#9043911yescharacteristic
504C22#9043911yescharacteristic
505C22C2×#649311yescharacteristic
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—284no—
511C4—282no—
512C4—282no—
513C4—562no—
514C4—561no—
515C4—561no—
516C4#8975514yes—
517C4C2×#645211yescharacteristic
518C4C2×#645211yescharacteristic
519C4C2×#649811yescharacteristic
520C4C2×#650711yescharacteristic
521C2—24no—
522C2#1173650411yescharacteristic
523C2#1173650511yescharacteristic
524C2#1173650611yescharacteristic
525C2#1173651311yescharacteristic
526C2C2×#8975511yescharacteristic
527C2#1174121711yescharacteristic
528C2#1174122111yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#423C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#423D7
hypocenterC7#23670
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
8. #11728218C2semidirect
9. #11728218C2semidirect
10. #11728217C2semidirect
11. #11728216C2semidirect
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×#4232
C2×#898932
#117395142
C2×#897002
#117297392
#117282182
#117282172
#117282162
#236707