Finite Group Databasebeta

← #13404533

Subgroups — #13404533

Order 1792.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C8×D4C211yesmaximal, characteristic
2C2×#89888C212yesmaximal, complemented
3#11739824C211yesmaximal, characteristic
4#11739821C211yesmaximal, characteristic
5C2×#89621C211yesmaximal, characteristic, complemented
6#11729726C212yesmaximal, complemented
7#11728207C211yesmaximal, characteristic, complemented
8#11727718C212yesmaximal, complemented
9#11727691C212yesmaximal, complemented
10#11727525C211yesmaximal, characteristic, complemented
11C4×#6390C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6487C2212yes—
15C2×#6487C2212yes—
16C2×#6487C2212yes—
17C22×Dic28C2212yes—
18C2×#6432C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C7×C8×D4—24no—
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×C22⋊C8C2212yes—
23C2×C4×C7×C8C2211yescharacteristic
24#89993—21no—
25#89993—21no—
26#89984C2212yes—
27#89888—28no—
28C2×#6392C2212yes—
29C2×#6390C2212yes—
30C2×#6390C2212yes—
31C2×#6390C2211yescharacteristic
32C2×#6390C2212yes—
33C2×#6390C2211yescharacteristic
34#89748—21no—
35#89745—21no—
36#89721—22no—
37#89720—22no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40#89621—24no—
41#89346C2211yescharacteristic
42#89346C2212yes—
43#89346C2211yescharacteristic
44#25386—71nomaximal, Sylow
45C22×C7×D4D411yescharacteristic
46C23×C4×C7C2312yes—
47C22×Dic14C2312yes—
48C22×Dic14C2311yescharacteristic
49C22×C7×C8—24no—
50C22×C7×C8—24no—
51C22×C7×C8—42no—
52C22×C7×C8C2311yescharacteristic
53C22×C7×C8C2311yescharacteristic
54C22×C7×C8C2312yes—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4D412yes—
59C2×C42×C7C2311yescharacteristic
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—22no—
62C2×C22⋊Dic7—22no—
63C2×C22⋊Dic7—22no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2311yescharacteristic
72#6487—24no—
73#6487—24no—
74#6487—24no—
75#6487—42no—
76#6487—42no—
77#6487—42no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—21no—
84C2×Dic28—24no—
85C2×Dic28—24no—
86C2×Dic28—42no—
87#6432—22no—
88#6432—22no—
89C4×Dic14—22no—
90C4×Dic14—22no—
91C7×C4⋊C8D414yes—
92C7×C22⋊C8—24no—
93C4×C7×C8—22no—
94C7⋊#95—21no—
95C7⋊#95—22no—
96#6392—24no—
97#6390—24no—
98#6390—24no—
99#6390—22no—
100#6390—42no—
101#6390D414yescomplemented
102C2×#467—72no—
103C2×#462—71no—
104C2×C8×D4D711yescharacteristic
105#2542—71no—
106#2538—71no—
107#2036—72no—
108#2001—72no—
109#1956—71no—
110#1947—71no—
111#1892—72no—
112C4×#103—71no—
113C24×C7C2×D412yes—
114C2×C7×D4—22no—
115C2×C7×D4—22no—
116C2×C7×D4Q1614yescomplemented
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—21no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C2×Dic14—21no—
134C2×Dic14—21no—
135C2×Dic14—21no—
136C2×Dic14—21no—
137C2×Dic14—21no—
138C2×Dic14—22no—
139C2×Dic14—22no—
140C2×Dic14—22no—
141C2×Dic14—22no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C2×Dic14C2×D412yes—
145C2×C7×C8—21no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8—21no—
149C2×C7×C8—21no—
150C2×C7×C8—21no—
151C2×C7×C8—21no—
152C2×C7×C8—48no—
153C2×C7×C8C2×D412yes—
154C2×C7×C8C2×D414yes—
155C7×C4⋊C4—22no—
156C7×C22⋊C4—24no—
157C42×C7C2×D412yes—
158C42×C7#4112yes—
159C22⋊Dic7—44no—
160C22⋊Dic7—44no—
161C22⋊Dic7—44no—
162C22⋊Dic7—82no—
163C4⋊Dic7—21no—
164C4⋊Dic7—21no—
165C4⋊Dic7—21no—
166C4⋊Dic7—22no—
167C4⋊Dic7—21no—
168C4⋊Dic7—22no—
169C4⋊Dic7—42no—
170C4⋊Dic7—42no—
171C4⋊Dic7—42no—
172C4⋊Dic7—81no—
173C4⋊Dic7C2×D412yes—
174C4⋊Dic7#4112yes—
175Dic7⋊C4—44no—
176Dic7⋊C4—44no—
177Dic7⋊C4—44no—
178Dic7⋊C4—42no—
179C4×Dic7—42no—
180C4×Dic7—42no—
181Dic28—48no—
182C22×Q16—72no—
183C23×C8C2×D712yes—
184C2×C4⋊Q8—71no—
185C2×C22⋊Q8—72no—
186C2×C22⋊Q8—72no—
187C2×C22⋊Q8—72no—
188C2×C4×Q8—71no—
189C2×C4×D4C2×D711yescharacteristic
190#467—148no—
191#462—144no—
192C8×D4—24no—
193C2×#107—72no—
194C2×C4⋊C8C2×D711yescharacteristic
195C2×#103—71no—
196C2×#103—72no—
197C2×#103—72no—
198C2×#103—71no—
199C2×#103—72no—
200C2×C22⋊C8C2×D712yes—
201C2×C4×C8C2×D711yescharacteristic
202#399—142no—
203#396—142no—
204#390—141no—
205#390—141no—
206#389—141no—
207#384—71no—
208#384—141no—
209#380—72no—
210C4×C4⋊C4—71no—
211#340—71no—
212#340—72no—
213#340—71no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7C2×Q1614yes—
221C23×C7C22×D411yescharacteristic
222C7×D4—28no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7#12112yes—
233C2×C4×C7#12112yes—
234C2×C4×C7C2×Q1612yes—
235C2×C4×C7#13512yes—
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C2×C4×C7H(5,𝔽2)11yescharacteristic
241C2×C4×C7H(5,𝔽2)11yescharacteristic
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—22no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—41no—
251C2×Dic7—41no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—41no—
260Dic14—22no—
261Dic14—22no—
262Dic14—41no—
263Dic14—41no—
264Dic14—42no—
265Dic14—42no—
266Dic14—42no—
267Dic14—42no—
268C7×C8—24no—
269C7×C8—22no—
270C7×C8—41no—
271C22×Q8—71no—
272C22×Q8—72no—
273C22×D4D2811yescharacteristic
274C23×C4C22×D712yes—
275C2×Q16—144no—
276C2×Q16—144no—
277C2×Q16—282no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—42no—
281C22×C8C22×D711yescharacteristic
282C22×C8C22×D711yescharacteristic
283C22×C8C22×D712yes—
284C4⋊Q8—142no—
285C4⋊Q8—142no—
286C22⋊Q8—144no—
287C22⋊Q8—144no—
288C22⋊Q8—144no—
289C22⋊Q8—282no—
290C22⋊Q8—282no—
291C22⋊Q8—282no—
292C4×Q8—142no—
293C4×Q8—142no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—71no—
297C2×C4⋊C4—71no—
298C2×C4⋊C4—71no—
299C2×C4⋊C4—72no—
300C2×C4⋊C4—141no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4—142no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—141no—
308C2×C4⋊C4D2811yescharacteristic
309C2×C22⋊C4—142no—
310C2×C22⋊C4—142no—
311C2×C22⋊C4—142no—
312C2×C22⋊C4—142no—
313C2×C22⋊C4D2812yes—
314C2×C42—141no—
315C2×C42—141no—
316C2×C42C22×D711yescharacteristic
317#107—144no—
318C4⋊C8C22⋊D714yes—
319#103—74no—
320#103—142no—
321#103—144no—
322#103—144no—
323#103—282no—
324C22⋊C8—24no—
325C4×C8—22no—
326#95—141no—
327#95—142no—
328C22×C7—24no—
329C22×C7—24no—
330C22×C7—24no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7C2×#12111yescharacteristic
336C22×C7D4211yescharacteristic
337C22×C7#54611yescharacteristic
338C22×C7C22×Q1611yescharacteristic
339C22×C7C2×#13511yescharacteristic
340C22×C7#57711yescharacteristic
341C22×C7#57811yescharacteristic
342C4×C7—22no—
343C4×C7—42no—
344C4×C7#46714yescomplemented
345C4×C7#53211yescharacteristic
346C4×C7#54011yescharacteristic
347C4×C7D4211yescharacteristic
348C4×C7#55011yescharacteristic
349Dic7—42no—
350Dic7—42no—
351Dic7—42no—
352Dic7—82no—
353Dic7—82no—
354Dic7—81no—
355C24C2×D2812yes—
356C2×Q8—72no—
357C2×Q8—141no—
358C2×Q8—141no—
359C2×Q8—142no—
360C2×Q8—142no—
361C2×Q8—141no—
362C2×Q8—142no—
363C2×Q8—141no—
364C2×Q8—142no—
365C2×Q8—142no—
366C2×Q8—141no—
367C2×Q8—142no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4Dic2814yescomplemented
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—141no—
378C22×C4—141no—
379C22×C4—141no—
380C22×C4—141no—
381C22×C4—142no—
382C22×C4—142no—
383C22×C4C2×D2812yes—
384C22×C4C2×D2811yescharacteristic
385C22×C4C2×D2811yescharacteristic
386C22×C4C23×D711yescharacteristic
387Q16—288no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—48no—
396C2×C8D4×D712yes—
397C2×C8C2×C22⋊D714yes—
398C4⋊C4—22no—
399C4⋊C4—72no—
400C4⋊C4—72no—
401C4⋊C4—141no—
402C4⋊C4—141no—
403C4⋊C4—141no—
404C4⋊C4—142no—
405C4⋊C4—142no—
406C4⋊C4—141no—
407C4⋊C4—284no—
408C4⋊C4—284no—
409C4⋊C4—284no—
410C4⋊C4—282no—
411C4⋊C4—282no—
412C4⋊C4—282no—
413C4⋊C4—282no—
414C4⋊C4—561no—
415C22⋊C4—24no—
416C22⋊C4—284no—
417C22⋊C4—284no—
418C22⋊C4—284no—
419C22⋊C4—562no—
420C42—282no—
421C42—282no—
422C42#115712yes—
423C42C2×C22⋊D712yes—
424C2×C7—44no—
425C2×C7#242811yescharacteristic
426C2×C7C2×#46711yescharacteristic
427C2×C7#306311yescharacteristic
428C2×C7#328411yescharacteristic
429C2×C7D4×Q1611yescharacteristic
430C2×C7#329911yescharacteristic
431C2×C7#330211yescharacteristic
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23C2×Dic2814yes—
439C23C22×D2811yescharacteristic
440Q8—142no—
441Q8—142no—
442Q8—281no—
443Q8—281no—
444Q8—282no—
445Q8—282no—
446Q8—282no—
447Q8—282no—
448D4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—142no—
459C2×C4—142no—
460C2×C4—142no—
461C2×C4—281no—
462C2×C4—281no—
463C2×C4—281no—
464C2×C4—281no—
465C2×C4—281no—
466C2×C4—281no—
467C2×C4—282no—
468C2×C4—281no—
469C2×C4—282no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4#646712yes—
477C2×C4C2×Dic2812yes—
478C2×C4#649312yes—
479C2×C4#649312yes—
480C2×C4C2×#115711yescharacteristic
481C2×C4C2×D4×D711yescharacteristic
482C2×C4#654811yescharacteristic
483C2×C4D7×#4111yescharacteristic
484C2×C4#655311yescharacteristic
485C2×C4C22×C22⋊D711yescharacteristic
486C8—22no—
487C8—24no—
488C8—41no—
489C7#2538611yescharacteristic, complemented, Sylow
490C22—24no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22D4×D2811yescharacteristic
498C22#9024511yescharacteristic
499C22C2×#646711yescharacteristic
500C22C22×Dic2811yescharacteristic
501C22#9044211yescharacteristic
502C22#9044411yescharacteristic
503C22C2×#649311yescharacteristic
504C4—22no—
505C4—42no—
506C4—282no—
507C4—282no—
508C4—282no—
509C4—562no—
510C4—561no—
511C4—562no—
512C4#8988814yescomplemented
513C4#9023911yescharacteristic
514C4#9025311yescharacteristic
515C4D4×C22⋊D711yescharacteristic
516C4#9052511yescharacteristic
517C2—44no—
518C2#1173688211yescharacteristic
519C2#1173688711yescharacteristic
520C2D4×Dic2811yescharacteristic
521C2#1173689311yescharacteristic
522C2#1173985411yescharacteristic
523C2C2×#8988811yescharacteristic
524C2#1174125311yescharacteristic
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#25386
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
2. C2×#89888C2semidirect
5. C2×#89621C2semidirect
6. #11729726C2semidirect
7. #11728207C2semidirect
8. #11727718C2semidirect
9. #11727691C2semidirect
10. #11727525C2semidirect
11. C4×#6390C2semidirect
101. #6390D4semidirect
116. C2×C7×D4Q16semidirect
344. C4×C7#467semidirect
370. C2×D4Dic28semidirect
489. C7#25386semidirect
512. C4#89888semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C8×D42
C2×#898882
#117398242
#117398212
C2×#896212
#117297262
#117282072
#117277182
#117276912
#117275252
C4×#63902
#253867