Finite Group Databasebeta

← #13403302

Subgroups — #13403302

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
2#11739945C211yesmaximal, characteristic
3#11739924C211yesmaximal, characteristic, complemented
4C2×#89613C211yesmaximal, characteristic
5#11728242C212yesmaximal, complemented
6#11728236C212yesmaximal
7#11728208C211yesmaximal, characteristic, complemented
8#11727528C212yesmaximal
9#11727525C211yesmaximal, characteristic, complemented
10#11727521C212yesmaximal, complemented
11C4×#6395C211yesmaximal, characteristic
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6509—21no—
14C22×#1132C2212yes—
15C2×C4×D28C2211yescharacteristic
16C2×#6432C2211yescharacteristic
17C7×C8×Q8—24no—
18C2×C7×C4⋊C8C2212yes—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C4×C7×C8C2211yescharacteristic
21C2×C4×C7×C8C2212yes—
22#90002—21no—
23C2×#6395C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6391C2212yes—
26C2×#6390C2212yes—
27C2×#6390C2212yes—
28C2×#6390C2212yes—
29#89766—22no—
30#89759—21no—
31#89758—22no—
32#89749—21no—
33#89711C2212yes—
34#89707C2211yescharacteristic
35C4×C4⋊Dic7C2211yescharacteristic
36#89701C2212yes—
37#89701C2212yes—
38#89700C2212yes—
39#89613—24no—
40#89346C2212yes—
41#89346C2211yescharacteristic
42#89346C2211yescharacteristic
43#25293—71nomaximal, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×D28C2311yescharacteristic
46C22×C4×D7—21no—
47C22×Dic14C2312yes—
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2312yes—
50C22×C7×C8C2311yescharacteristic
51C4×C7×Q8—22no—
52C4×C7×Q8—22no—
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C4⋊C4D412yes—
55C2×C42×C7C2312yes—
56C2×C42×C7C2311yescharacteristic
57#6509—44no—
58C2×(C2×D7)⋊C4—21no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×Dic7⋊C4—22no—
70C2×C4×Dic7—21no—
71C2×#1132—24no—
72C2×#1132—24no—
73C2×#1132—42no—
74C4×D28—22no—
75C4×D28—22no—
76#6432—22no—
77#6432—22no—
78C7×C4⋊C8—24no—
79C7×C4⋊C8D414yes—
80C4×C7×C8—24no—
81C4×C7×C8—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C7⋊#95—21no—
88#6395—22no—
89#6395—22no—
90#6391—42no—
91#6390—24no—
92#6390—24no—
93#6390—42no—
94C2×#461—71no—
95C2×C8×Q8D711yescharacteristic, complemented
96#2568—71no—
97#2538—71no—
98#2066—72no—
99#2055—72no—
100#1968—72no—
101#1956—71no—
102#1948—71no—
103#1933—72no—
104C4×#102—71no—
105C23×D7—21no—
106C2×C7×Q8—22no—
107C2×C7×Q8—22no—
108C2×C7×Q8SD1614yescomplemented
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2411yescharacteristic
114C22×Dic7—21no—
115C22×Dic7—21no—
116C22×Dic7—22no—
117C22×Dic7—21no—
118C22×Dic7—22no—
119C2×D28—21no—
120C2×D28—21no—
121C2×D28—21no—
122C2×D28—21no—
123C2×D28—21no—
124C2×D28#4112yes—
125C2×C4×D7—42no—
126C2×C4×D7—42no—
127C2×C4×D7—42no—
128C2×Dic14—22no—
129C2×Dic14—22no—
130C2×Dic14—22no—
131C2×Dic14—22no—
132C2×Dic14—22no—
133C2×Dic14—22no—
134C2×C7×C8—21no—
135C2×C7×C8—22no—
136C2×C7×C8—22no—
137C2×C7×C8—21no—
138C2×C7×C8—21no—
139C2×C7×C8—21no—
140C2×C7×C8—21no—
141C2×C7×C8C2×D414yes—
142C2×C7×C8#4112yes—
143C7×C4⋊C4—24no—
144C7×C4⋊C4—22no—
145C42×C7—22no—
146C42×C7—22no—
147C42×C7C2×D412yes—
148C42×C7#4112yes—
149(C2×D7)⋊C4—42no—
150(C2×D7)⋊C4—44no—
151(C2×D7)⋊C4—82no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—22no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—42no—
161C4⋊Dic7—42no—
162C4⋊Dic7—42no—
163C4⋊Dic7—81no—
164Dic7⋊C4—44no—
165C4×Dic7—42no—
166#1132—48no—
167C22×SD16—72no—
168C2×C4⋊Q8—71no—
169C2×C22⋊Q8—141no—
170C2×C4×Q8C2×D711yescharacteristic
171C2×C4×D4—71no—
172#461—144no—
173C8×Q8—24no—
174C2×#106—72no—
175C2×C4⋊C8C2×D712yes—
176C2×C4⋊C8C2×D711yescharacteristic
177C2×#103—72no—
178C2×#103—72no—
179C2×#103—72no—
180C2×#102—71no—
181C2×#102—71no—
182C2×C4×C8C2×D712yes—
183C2×C4×C8C2×D711yescharacteristic
184#400—142no—
185#396—142no—
186#391—72no—
187#391—72no—
188#391—141no—
189#390—72no—
190#389—72no—
191#384—71no—
192#382—141no—
193#380—141no—
194C4×C4⋊C4—71no—
195#340—72no—
196#340—71no—
197#340—71no—
198C23×C7C22×D411yescharacteristic
199C22×D7—22no—
200C22×D7—41no—
201C22×D7—41no—
202C22×D7—41no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C7×Q8—28no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7#12112yes—
215C2×C4×C7#12112yes—
216C2×C4×C7C2×SD1614yes—
217C2×C4×C7C2×SD1612yes—
218C2×C4×C7#13512yes—
219C2×C4×C7C22×D411yescharacteristic
220C2×C4×C7C2×#4111yescharacteristic
221C2×C4×C7C2×#4111yescharacteristic
222C2×C4×C7C2×#4111yescharacteristic
223C2×C4×C7H(5,𝔽2)11yescharacteristic
224C2×C4×C72−1+411yescharacteristic
225C2×Dic7—41no—
226C2×Dic7—42no—
227C2×Dic7—41no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—41no—
231C2×Dic7—42no—
232C2×Dic7—41no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—42no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240D28—22no—
241D28—22no—
242D28—41no—
243D28—41no—
244C4×D7—84no—
245Dic14—42no—
246Dic14—42no—
247Dic14—42no—
248Dic14—42no—
249C7×C8—24no—
250C7×C8—22no—
251C7×C8—41no—
252C22×Q8—72no—
253C22×Q8D2811yescharacteristic
254C22×D4—71no—
255C23×C4—141no—
256C2×SD16—144no—
257C2×SD16—144no—
258C2×SD16—282no—
259C22×C8C22×D711yescharacteristic
260C22×C8C22×D711yescharacteristic
261C22×C8C22×D712yes—
262C4⋊Q8—142no—
263C4⋊Q8—142no—
264C22⋊Q8—284no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×D4—142no—
268C4×D4—142no—
269C2×C4⋊C4—71no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—142no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—141no—
277C2×C4⋊C4—141no—
278C2×C4⋊C4D2811yescharacteristic
279C2×C4⋊C4D2812yes—
280C2×C22⋊C4—141no—
281C2×C22⋊C4—142no—
282C2×C22⋊C4—142no—
283C2×C42—141no—
284C2×C42C22×D712yes—
285C2×C42C22×D711yescharacteristic
286#106—282no—
287C4⋊C8—24no—
288C4⋊C8C22⋊D714yes—
289#103—144no—
290#103—144no—
291#103—282no—
292#102—142no—
293#102—142no—
294C4×C8—24no—
295C4×C8—22no—
296#95—141no—
297#95—142no—
298#95—142no—
299#95—142no—
300#95—142no—
301#95—142no—
302C22×C7C2×#12111yescharacteristic
303C22×C7#54811yescharacteristic
304C22×C7#55011yescharacteristic
305C22×C7C22×SD1611yescharacteristic
306C22×C7C2×#13511yescharacteristic
307C22×C7#57711yescharacteristic
308C22×C7#57811yescharacteristic
309C2×D7—42no—
310C2×D7—42no—
311C2×D7—42no—
312C2×D7—81no—
313C2×D7—81no—
314C2×D7—81no—
315C2×D7—81no—
316C4×C7—24no—
317C4×C7—22no—
318C4×C7—42no—
319C4×C7#46514yes—
320C4×C7#52911yescharacteristic
321C4×C7#54211yescharacteristic
322C4×C7#54611yescharacteristic
323C4×C7#54811yescharacteristic
324Dic7—81no—
325Dic7—82no—
326Dic7—81no—
327Dic7—82no—
328Dic7—81no—
329C24—141no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—142no—
333C2×Q8—142no—
334C2×Q8—142no—
335C2×Q8—142no—
336C2×Q8—142no—
337C2×Q8—142no—
338C2×Q8#113214yescomplemented
339C2×D4—72no—
340C2×D4—141no—
341C2×D4—141no—
342C2×D4—141no—
343C2×D4—141no—
344C2×D4—141no—
345C22×C4—141no—
346C22×C4—141no—
347C22×C4—142no—
348C22×C4—141no—
349C22×C4—142no—
350C22×C4—282no—
351C22×C4—282no—
352C22×C4—282no—
353C22×C4C2×D2812yes—
354C22×C4C2×D2812yes—
355C22×C4C2×D2811yescharacteristic
356C22×C4C2×D2811yescharacteristic
357C22×C4C23×D711yescharacteristic
358SD16—288no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8#116112yes—
367C2×C8C2×C22⋊D714yes—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—141no—
371C4⋊C4—141no—
372C4⋊C4—142no—
373C4⋊C4—141no—
374C4⋊C4—141no—
375C4⋊C4—142no—
376C4⋊C4—141no—
377C4⋊C4—141no—
378C4⋊C4—284no—
379C4⋊C4—282no—
380C4⋊C4—282no—
381C4⋊C4—282no—
382C4⋊C4—561no—
383C22⋊C4—284no—
384C22⋊C4—282no—
385C22⋊C4—562no—
386C42—22no—
387C42—22no—
388C42—282no—
389C42#115712yes—
390C42C2×C22⋊D712yes—
391C2×C7#242611yescharacteristic
392C2×C7C2×#46511yescharacteristic
393C2×C7#306311yescharacteristic
394C2×C7#335911yescharacteristic
395C2×C7#336011yescharacteristic
396C2×C7#339211yescharacteristic
397C2×C7#339311yescharacteristic
398D7—82no—
399C23—142no—
400C23—281no—
401C23—281no—
402C23—281no—
403C23—281no—
404C23—281no—
405C23—281no—
406C23C22×D2811yescharacteristic
407Q8—28no—
408Q8—282no—
409Q8—282no—
410Q8—282no—
411Q8—282no—
412D4—142no—
413D4—142no—
414D4—281no—
415D4—281no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—282no—
424C2×C4—281no—
425C2×C4—281no—
426C2×C4—281no—
427C2×C4—281no—
428C2×C4—282no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4—281no—
432C2×C4—282no—
433C2×C4—281no—
434C2×C4—282no—
435C2×C4—282no—
436C2×C4—281no—
437C2×C4—282no—
438C2×C4—564no—
439C2×C4C2×#113214yes—
440C2×C4C2×#113212yes—
441C2×C4#646712yes—
442C2×C4#649312yes—
443C2×C4#649312yes—
444C2×C4C2×#115711yescharacteristic
445C2×C4C2×#116111yescharacteristic
446C2×C4#655111yescharacteristic
447C2×C4D7×#4111yescharacteristic
448C2×C4#655311yescharacteristic
449C2×C4C22×C22⋊D711yescharacteristic
450C8—22no—
451C8—24no—
452C8—41no—
453C7#2529311yescharacteristic, complemented, Sylow
454C22—282no—
455C22—282no—
456C22—282no—
457C22—561no—
458C22—561no—
459C22—561no—
460C22—561no—
461C22#9026711yescharacteristic
462C22#9026811yescharacteristic
463C22C22×#113211yescharacteristic
464C22C2×#646711yescharacteristic
465C22#9044211yescharacteristic
466C22#9044411yescharacteristic
467C22C2×#649311yescharacteristic
468C4—24no—
469C4—22no—
470C4—42no—
471C4—561no—
472C4—562no—
473C4—561no—
474C4—561no—
475C4—562no—
476C4#8988614yes—
477C4#9027111yescharacteristic
478C4#9027311yescharacteristic
479C4#9050811yescharacteristic
480C4#9052011yescharacteristic
481C2—562no—
482C2#1173735611yescharacteristic
483C2#1173735711yescharacteristic
484C2#1173737011yescharacteristic
485C2#1173737111yescharacteristic
486C2#1173994011yescharacteristic
487C2C2×#8988611yescharacteristic
488C2#1174125311yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×C8×Q8C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterC2×C8×Q8D7
hypocenterC7#25293
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C8×Q8C2semidirect
3. #11739924C2semidirect
5. #11728242C2semidirect
7. #11728208C2semidirect
9. #11727525C2semidirect
10. #11727521C2semidirect
95. C2×C8×Q8D7semidirect
108. C2×C7×Q8SD16semidirect
338. C2×Q8#1132semidirect
453. C7#25293semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C8×Q82
#117399452
#117399242
C2×#896132
#117282422
#117282362
#117282082
#117275282
#117275252
#117275212
C4×#63952
#252937