Finite Group Databasebeta

← #13403923

Subgroups — #13403923

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C8×D4C211yesmaximal, characteristic
2#11739829C211yesmaximal, characteristic
3#11739821C211yesmaximal, characteristic
4C2×#89622C211yesmaximal, characteristic, complemented
5#11729726C212yesmaximal, complemented
6#11729725C212yesmaximal, complemented
7#11728207C211yesmaximal, characteristic, complemented
8#11727693C212yesmaximal, complemented
9#11727691C212yesmaximal, complemented
10#11727531C211yesmaximal, characteristic, complemented
11C4×#6390C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6487C2212yes—
15C2×#6432C2211yescharacteristic
16C2×C4×Dic14C2211yescharacteristic
17C7×C8×D4—24no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C22⋊C8C2212yes—
20C2×C4×C7×C8C2211yescharacteristic
21#89993—21no—
22#89993—21no—
23#89984C2212yes—
24#89984C2212yes—
25#89984C2212yes—
26C2×#6392C2212yes—
27C2×#6392C2212yes—
28C2×#6390C2211yescharacteristic
29C2×#6390C2212yes—
30C2×#6390C2211yescharacteristic
31#89749—21no—
32#89745—21no—
33#89720—22no—
34#89719—22no—
35#89707C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89622—24no—
38#89346C2211yescharacteristic
39#89346C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89346C2211yescharacteristic
43#25340—71nomaximal, Sylow
44C22×C7×D4D411yescharacteristic
45C23×C4×C7C2312yes—
46C22×Dic14—21no—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8—24no—
49C22×C7×C8—24no—
50C22×C7×C8—42no—
51C22×C7×C8C2311yescharacteristic
52C22×C7×C8C2312yes—
53C22×C7×C8C2311yescharacteristic
54C4×C7×D4—22no—
55C4×C7×D4—22no—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C22⋊C4D412yes—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—22no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2312yes—
69C2×C4⋊Dic7C2311yescharacteristic
70C2×C4⋊Dic7C2311yescharacteristic
71#6487—24no—
72#6487—42no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—22no—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—21no—
77#6432—22no—
78#6432—22no—
79C4×Dic14—22no—
80C4×Dic14—22no—
81C7×C4⋊C8D414yes—
82C7×C22⋊C8—24no—
83C4×C7×C8—22no—
84C7⋊#95—21no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88#6392—24no—
89#6392—42no—
90#6390—22no—
91#6390—24no—
92#6390—22no—
93C2×#477—71no—
94C2×C8×D4D711yescharacteristic
95#2563—71no—
96#2538—71no—
97#2076—72no—
98#2036—72no—
99#1990—71no—
100#1947—71no—
101#1892—72no—
102#1808—72no—
103C4×#103—71no—
104C24×C7C2×D412yes—
105C2×C7×D4—22no—
106C2×C7×D4—22no—
107C2×C7×D4Q1614yes—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7C2×D412yes—
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2411yescharacteristic
118C22×Dic7—21no—
119C22×Dic7—21no—
120C22×Dic7—21no—
121C22×Dic7—21no—
122C22×Dic7—22no—
123C22×Dic7—22no—
124C2×Dic14—21no—
125C2×Dic14—21no—
126C2×Dic14—21no—
127C2×Dic14—21no—
128C2×Dic14—21no—
129C2×Dic14—42no—
130C2×Dic14—81no—
131C2×Dic14—81no—
132C2×Dic14#4112yes—
133C2×C7×C8—21no—
134C2×C7×C8—22no—
135C2×C7×C8—21no—
136C2×C7×C8—22no—
137C2×C7×C8—21no—
138C2×C7×C8—21no—
139C2×C7×C8—21no—
140C2×C7×C8—48no—
141C2×C7×C8C2×D414yes—
142C2×C7×C8#4112yes—
143C7×C4⋊C4—22no—
144C7×C22⋊C4—24no—
145C42×C7C2×D412yes—
146C42×C7#4112yes—
147C22⋊Dic7—44no—
148C22⋊Dic7—82no—
149C22⋊Dic7—82no—
150C22⋊Dic7—82no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—21no—
157C4⋊Dic7—21no—
158C4⋊Dic7—22no—
159C4⋊Dic7—21no—
160C4⋊Dic7—21no—
161C4⋊Dic7—42no—
162C4⋊Dic7—42no—
163C4⋊Dic7—81no—
164Dic7⋊C4—44no—
165Dic7⋊C4—42no—
166C4×Dic7—42no—
167C4×Dic7—42no—
168C23×C8C2×D712yes—
169C2×C4⋊Q8—71no—
170C2×C22⋊Q8—72no—
171C2×C4×Q8—71no—
172C2×C4×D4C2×D711yescharacteristic
173#477—144no—
174C8×D4—24no—
175C2×#107—72no—
176C2×#107—72no—
177C2×C4⋊C8C2×D711yescharacteristic
178C2×#103—72no—
179C2×#103—71no—
180C2×#103—71no—
181C2×C22⋊C8C2×D712yes—
182C2×C4×C8C2×D711yescharacteristic
183#396—142no—
184#393—142no—
185#390—141no—
186#390—141no—
187#389—141no—
188#384—71no—
189#382—141no—
190#380—72no—
191#380—72no—
192#380—72no—
193C4×C4⋊C4—71no—
194#340—71no—
195#340—72no—
196#340—71no—
197#340—72no—
198#340—72no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7C2×Q1614yes—
206C23×C7C22×D411yescharacteristic
207C7×D4—28no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7#12112yes—
218C2×C4×C7#12112yes—
219C2×C4×C7C2×Q1612yes—
220C2×C4×C7#13512yes—
221C2×C4×C7C22×D411yescharacteristic
222C2×C4×C7C2×#4111yescharacteristic
223C2×C4×C7C2×#4111yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×C4×C7H(5,𝔽2)11yescharacteristic
226C2×C4×C72−1+411yescharacteristic
227C2×Dic7—22no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—42no—
232C2×Dic7—42no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—42no—
242C2×Dic7—41no—
243C2×Dic7—42no—
244C2×Dic7—41no—
245Dic14—22no—
246Dic14—22no—
247Dic14—41no—
248Dic14—41no—
249Dic14—84no—
250C7×C8—22no—
251C7×C8—24no—
252C7×C8—41no—
253C22×Q8—71no—
254C22×Q8—141no—
255C22×D4D2811yescharacteristic
256C23×C4C22×D712yes—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8—42no—
260C22×C8C22×D711yescharacteristic
261C22×C8C22×D711yescharacteristic
262C22×C8C22×D712yes—
263C4⋊Q8—142no—
264C4⋊Q8—142no—
265C22⋊Q8—144no—
266C22⋊Q8—282no—
267C4×Q8—142no—
268C4×Q8—142no—
269C4×D4—22no—
270C4×D4—22no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—72no—
274C2×C4⋊C4—71no—
275C2×C4⋊C4—71no—
276C2×C4⋊C4—141no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4D2811yescharacteristic
282C2×C22⋊C4—142no—
283C2×C22⋊C4—142no—
284C2×C22⋊C4—142no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4D2812yes—
287C2×C42—141no—
288C2×C42—141no—
289C2×C42C22×D711yescharacteristic
290#107—144no—
291#107—282no—
292C4⋊C8C22⋊D714yes—
293#103—142no—
294#103—142no—
295#103—144no—
296C22⋊C8—24no—
297C4×C8—22no—
298#95—141no—
299#95—142no—
300#95—142no—
301#95—142no—
302C22×C7—24no—
303C22×C7—24no—
304C22×C7—24no—
305C22×C7—42no—
306C22×C7—42no—
307C22×C7—42no—
308C22×C7—42no—
309C22×C7C2×#12111yescharacteristic
310C22×C7#54611yescharacteristic
311C22×C7#54711yescharacteristic
312C22×C7C22×Q1611yescharacteristic
313C22×C7C2×#13511yescharacteristic
314C22×C7#57611yescharacteristic
315C22×C7#57711yescharacteristic
316C4×C7—22no—
317C4×C7—42no—
318C4×C7#46714yes—
319C4×C7#52911yescharacteristic
320C4×C7#54211yescharacteristic
321C4×C7#54611yescharacteristic
322C4×C7#54811yescharacteristic
323Dic7—42no—
324Dic7—81no—
325Dic7—82no—
326Dic7—81no—
327Dic7—82no—
328Dic7—81no—
329C24C2×D2812yes—
330C2×Q8—72no—
331C2×Q8—141no—
332C2×Q8—141no—
333C2×Q8—141no—
334C2×Q8—141no—
335C2×Q8—141no—
336C2×Q8—282no—
337C2×Q8—561no—
338C2×Q8—561no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4Dic2814yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—141no—
349C22×C4—141no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4C2×D2812yes—
355C22×C4C2×D2811yescharacteristic
356C22×C4C2×D2811yescharacteristic
357C22×C4C23×D711yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—48no—
366C2×C8#115912yes—
367C2×C8C2×C22⋊D714yes—
368C4⋊C4—22no—
369C4⋊C4—141no—
370C4⋊C4—142no—
371C4⋊C4—141no—
372C4⋊C4—141no—
373C4⋊C4—142no—
374C4⋊C4—141no—
375C4⋊C4—142no—
376C4⋊C4—141no—
377C4⋊C4—142no—
378C4⋊C4—141no—
379C4⋊C4—284no—
380C4⋊C4—282no—
381C4⋊C4—282no—
382C4⋊C4—282no—
383C4⋊C4—561no—
384C22⋊C4—24no—
385C22⋊C4—284no—
386C22⋊C4—562no—
387C22⋊C4—562no—
388C22⋊C4—562no—
389C42—282no—
390C42—282no—
391C42#115712yes—
392C42C2×C22⋊D712yes—
393C2×C7—44no—
394C2×C7#242711yescharacteristic
395C2×C7C2×#46711yescharacteristic
396C2×C7#306211yescharacteristic
397C2×C7#329911yescharacteristic
398C2×C7#330511yescharacteristic
399C2×C7#333811yescharacteristic
400C2×C7#334011yescharacteristic
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23C2×Dic2814yes—
408C23C22×D2811yescharacteristic
409Q8—142no—
410Q8—142no—
411Q8—281no—
412Q8—281no—
413Q8—564no—
414D4—28no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—142no—
425C2×C4—281no—
426C2×C4—281no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—282no—
430C2×C4—281no—
431C2×C4—282no—
432C2×C4—281no—
433C2×C4—281no—
434C2×C4—281no—
435C2×C4—282no—
436C2×C4—282no—
437C2×C4—281no—
438C2×C4—282no—
439C2×C4—282no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4#646712yes—
443C2×C4C2×Dic2812yes—
444C2×C4#649312yes—
445C2×C4#649312yes—
446C2×C4C2×#115711yescharacteristic
447C2×C4C2×#115911yescharacteristic
448C2×C4#654811yescharacteristic
449C2×C4D7×#4111yescharacteristic
450C2×C4#655411yescharacteristic
451C2×C4C22×C22⋊D711yescharacteristic
452C8—22no—
453C8—24no—
454C8—41no—
455C7#2534011yescharacteristic, complemented, Sylow
456C22—24no—
457C22—24no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22#9024511yescharacteristic
464C22#9024611yescharacteristic
465C22C2×#646711yescharacteristic
466C22C22×Dic2811yescharacteristic
467C22#9044211yescharacteristic
468C22#9044311yescharacteristic
469C22C2×#649311yescharacteristic
470C4—22no—
471C4—42no—
472C4—282no—
473C4—561no—
474C4—562no—
475C4—561no—
476C4—561no—
477C4—562no—
478C4#8988814yes—
479C4#9024811yescharacteristic
480C4#9025211yescharacteristic
481C4#9049711yescharacteristic
482C4#9052211yescharacteristic
483C2—44no—
484C2#1173686011yescharacteristic
485C2#1173686211yescharacteristic
486C2#1173689311yescharacteristic
487C2#1173689711yescharacteristic
488C2#1173985311yescharacteristic
489C2C2×#8988811yescharacteristic
490C2#1174124711yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×C8×D4C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenterC2×C8×D4D7
hypocenterC7#25340
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
4. C2×#89622C2semidirect
5. #11729726C2semidirect
6. #11729725C2semidirect
7. #11728207C2semidirect
8. #11727693C2semidirect
9. #11727691C2semidirect
10. #11727531C2semidirect
11. C4×#6390C2semidirect
455. C7#25340semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C8×D42
#117398292
#117398212
C2×#896222
#117297262
#117297252
#117282072
#117276932
#117276912
#117275312
C4×#63902
#253407