Finite Group Databasebeta

← C2×#11728071 #13382330

Subgroups — C2×#11728071 #13382330

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#418C211yesmaximal, characteristic
2C22×#6407C211yesmaximal, characteristic
3C22×#6406C211yesmaximal, characteristic
4C22×#6396C211yesmaximal, characteristic, complemented
5C2×#89767C211yesmaximal, characteristic, complemented
6C2×#89771C211yesmaximal, characteristic, complemented
7C2×#89349C211yesmaximal, characteristic, complemented
8#11728071C218yesmaximal, direct product factor
9C23×C7×D4C2211yescharacteristic
10C22×C7×M16C2211yescharacteristic
11C2×C7×#117C2211yescharacteristic
12C22×C4⋊Dic7C2211yescharacteristic
13C22×#1136C2211yescharacteristic
14C22×#1136C2211yescharacteristic
15C23×C7⋊C8C2211yescharacteristic
16C7×#418—22no—
17C7×#418C2214yes—
18C2×C7×#102—21no—
19C2×C7×#102—21no—
20C2×#6407—24no—
21C2×#6407C2214yes—
22C2×#6406—22no—
23C2×#6406—22no—
24C2×#6406C2214yes—
25C2×#6396—24no—
26C2×#6396C2214yescomplemented
27#89771—22no—
28#89771C2214yescomplemented
29#89767—22no—
30#89767C2214yescomplemented
31C2×#6382—21no—
32C2×#6381—21no—
33C2×C4⋊(C7⋊C8)—22no—
34#89349C2214yescomplemented
35C2×#2063—71nomaximal, Sylow
36C25×C7—21no—
37C22×C7×D4—21no—
38C22×C7×D4—21no—
39C22×C7×D4—21no—
40C22×C7×D4—21no—
41C22×C7×D4—22no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4—21no—
45C22×C7×D4Q814yes—
46C22×C7×D4C2312yes—
47C23×C4×C7C2311yescharacteristic
48C23×Dic7—21no—
49C2×C7×M16—21no—
50C2×C7×M16—22no—
51C2×C7×M16—22no—
52C2×C7×M16C2312yes—
53C22×C7×C8—21no—
54C7×#117—21no—
55C7×#117—21no—
56C7×#117D412yes—
57C7×#117C2312yes—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C42×C7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7C2312yes—
68C2×#1136—21no—
69C2×#1136—22no—
70C2×#1136—21no—
71C2×#1136—22no—
72C2×#1136—22no—
73C2×#1136D414yes—
74C2×#1136C2312yes—
75C2×#1136C2312yes—
76C22×C7⋊C8—22no—
77C22×C7⋊C8—21no—
78C22×C7⋊C8—22no—
79C22×C7⋊C8—21no—
80C22×C7⋊C8D412yes—
81C22×C7⋊C8D412yes—
82C22×C7⋊C8C2312yes—
83C7×#102—24no—
84C7×#102—24no—
85#6407—28no—
86#6406—28no—
87#6396—28no—
88#6382—24no—
89#6381—24no—
90C4⋊(C7⋊C8)—28no—
91C22×#108—71no—
92C2×#428—71no—
93C2×#424—71no—
94C2×#418D711yescharacteristic
95C22×#102—71no—
96C22×#100—71no—
97#2063—78no—
98C2×#343—71no—
99C24×C7—24no—
100C24×C7—22no—
101C24×C7—41no—
102C24×C7—41no—
103C24×C7—42no—
104C24×C7—82no—
105C24×C7—82no—
106C24×C7#4111yescharacteristic
107C2×C7×D4—24no—
108C2×C7×D4—24no—
109C2×C7×D4—24no—
110C2×C7×D4—24no—
111C2×C7×D4—24no—
112C2×C7×D4—22no—
113C2×C7×D4—22no—
114C2×C7×D4—22no—
115C2×C7×D4—22no—
116C2×C7×D4—22no—
117C2×C7×D4—42no—
118C2×C7×D4—42no—
119C2×C7×D4—42no—
120C2×C7×D4—42no—
121C2×C7×D4—41no—
122C2×C7×D4—41no—
123C2×C7×D4—42no—
124C2×C7×D4C2×Q814yes—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×Q812yes—
133C22×C4×C7#4111yescharacteristic
134C22×C4×C7#4112yes—
135C22×C4×C7#4112yes—
136C22×C4×C7C2411yescharacteristic
137C22×Dic7—22no—
138C22×Dic7—41no—
139C22×Dic7—41no—
140C22×Dic7—82no—
141C7×M16—22no—
142C7×M16—22no—
143C7×M16—42no—
144C2×C7×C8—22no—
145C2×C7×C8—42no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4C2×D412yes—
149C7×C4⋊C4C2×D412yes—
150C7×C22⋊C4—22no—
151C7×C22⋊C4—22no—
152C42×C7—22no—
153C42×C7—22no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—41no—
157C4⋊Dic7—41no—
158#1136—24no—
159#1136—22no—
160#1136—22no—
161#1136—22no—
162#1136—42no—
163#1136C2×D414yes—
164C2×C7⋊C8—24no—
165C2×C7⋊C8—24no—
166C2×C7⋊C8—24no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—42no—
172C2×C7⋊C8C2×D414yes—
173C23×D4C2×D711yescharacteristic
174C22×M16—71no—
175C22×M16—71no—
176C22×M16C2×D711yescharacteristic
177C23×C8—71no—
178C2×#117C2×D711yescharacteristic
179C22×C4⋊C4—71no—
180C2×#108—74no—
181C2×#108—144no—
182#428—74no—
183#428—142no—
184C2×#107—141no—
185C2×#106—141no—
186#424—74no—
187#424—142no—
188C2×C4⋊C8—142no—
189#418—22no—
190#418C2×D714yes—
191C2×#102—21no—
192C2×#102—21no—
193C2×#102—74no—
194C2×#102—142no—
195C2×#102—142no—
196C2×#100—74no—
197C2×#100—144no—
198#343—74no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7—24no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—44no—
205C23×C7—44no—
206C23×C7—88no—
207C23×C7—82no—
208C23×C7—82no—
209C23×C7—88no—
210C23×C7—82no—
211C23×C7—82no—
212C23×C7—82no—
213C23×C7—82no—
214C23×C7#12111yescharacteristic
215C23×C7#12111yescharacteristic
216C23×C7C22⋊Q812yes—
217C23×C7#12612yes—
218C23×C7C2×#4111yescharacteristic
219C7×D4—24no—
220C7×D4—24no—
221C7×D4—42no—
222C7×D4—42no—
223C7×D4—44no—
224C7×D4—44no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7#12112yes—
230C2×C4×C7#12112yes—
231C2×C4×C7#12112yes—
232C2×C4×C7#12112yes—
233C2×C4×C7#12111yescharacteristic
234C2×C4×C7#12111yescharacteristic
235C2×C4×C7C22⋊Q812yes—
236C2×C4×C7C22⋊Q812yes—
237C2×C4×C7C22⋊Q812yes—
238C2×C4×C7C22⋊Q812yes—
239C2×C4×C7C22⋊Q812yes—
240C2×C4×C7#12612yes—
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C22×D411yescharacteristic
244C2×C4×C7C22×Q811yescharacteristic
245C2×C4×C7C2×#4111yescharacteristic
246C2×C4×C7C2×#4111yescharacteristic
247C2×Dic7—42no—
248C2×Dic7—82no—
249C2×Dic7—82no—
250C2×Dic7—82no—
251C7×C8—42no—
252C7⋊C8—24no—
253C7⋊C8—24no—
254C7⋊C8—42no—
255C25—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4Dic1414yes—
265C22×D4C22×D712yes—
266C23×C4—141no—
267C23×C4C22×D711yescharacteristic
268C2×M16—21no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—74no—
272C2×M16—72no—
273C2×M16—72no—
274C2×M16—142no—
275C2×M16—141no—
276C2×M16—141no—
277C2×M16—142no—
278C2×M16—142no—
279C2×M16C22×D712yes—
280C22×C8—21no—
281C22×C8—72no—
282C22×C8—72no—
283C22×C8—72no—
284C22×C8—142no—
285C22×C8—142no—
286C22×C8—141no—
287C22×C8—141no—
288#117—21no—
289#117—21no—
290#117C22⋊D712yes—
291#117C22×D712yes—
292C2×C4⋊C4—72no—
293C2×C4⋊C4—141no—
294C2×C4⋊C4—141no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4C22⋊D711yescharacteristic
299C2×C4⋊C4C22⋊D711yescharacteristic
300C2×C22⋊C4—21no—
301C2×C42—21no—
302#108—148no—
303#107—144no—
304#106—144no—
305C4⋊C8—148no—
306#102—24no—
307#102—24no—
308#102—148no—
309#100—148no—
310C22×C7—24no—
311C22×C7—24no—
312C22×C7—21no—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—44no—
317C22×C7—84no—
318C22×C7—84no—
319C22×C7—88no—
320C22×C7—88no—
321C22×C7—82no—
322C22×C7—82no—
323C22×C7#39612yes—
324C22×C7#39612yes—
325C22×C7#47112yes—
326C22×C7#47212yes—
327C22×C7C2×#12111yescharacteristic
328C22×C7C2×C22⋊Q811yescharacteristic
329C22×C7C2×#12611yescharacteristic
330C4×C7—42no—
331C4×C7—42no—
332C4×C7#39612yes—
333C4×C7#39612yes—
334C4×C7C2×#12111yescharacteristic
335C4×C7C2×#12111yescharacteristic
336C4×C7C2×C22⋊Q811yescharacteristic
337C4×C7C2×C22⋊Q811yescharacteristic
338Dic7—82no—
339C24—24no—
340C24—22no—
341C24—42no—
342C24—41no—
343C24—41no—
344C24—82no—
345C24—82no—
346C24#115911yescharacteristic
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4C2×Dic1414yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—142no—
368C22×C4—281no—
369C22×C4—281no—
370C22×C4—562no—
371C22×C4C2×Dic1412yes—
372C22×C4#115712yes—
373C22×C4D4×D711yescharacteristic
374C22×C4D4×D711yescharacteristic
375C22×C4D4×D712yes—
376C22×C4#115911yescharacteristic
377C22×C4#115912yes—
378C22×C4C2×C22⋊D711yescharacteristic
379C22×C4C2×C22⋊D711yescharacteristic
380C22×C4C23×D711yescharacteristic
381M16—22no—
382M16—22no—
383M16—42no—
384M16—74no—
385M16—144no—
386M16—142no—
387M16—142no—
388M16—142no—
389M16—282no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—74no—
393C2×C8—144no—
394C2×C8—144no—
395C2×C8—144no—
396C2×C8—142no—
397C2×C8—142no—
398C2×C8—142no—
399C2×C8—142no—
400C2×C8—282no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—142no—
404C4⋊C4—142no—
405C4⋊C4—281no—
406C4⋊C4—281no—
407C4⋊C4C2×C22⋊D712yes—
408C4⋊C4C2×C22⋊D712yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C42—22no—
412C42—22no—
413C2×C7—21no—
414C2×C7—21no—
415C2×C7—22no—
416C2×C7—84no—
417C2×C7#206314yescomplemented
418C2×C7C2×#39611yescharacteristic
419C2×C7C2×#47111yescharacteristic
420C2×C7C2×#47211yescharacteristic
421C23—22no—
422C23—22no—
423C23—24no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—88no—
429C23—82no—
430C23—82no—
431C23—88no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23#644112yes—
437C23#644212yes—
438C23#650211yescharacteristic
439C23#650211yescharacteristic
440C23C2×#115911yescharacteristic
441D4—24no—
442D4—24no—
443D4—42no—
444D4—44no—
445D4—44no—
446D4—42no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—282no—
452C2×C4—562no—
453C2×C4—562no—
454C2×C4—562no—
455C2×C4#644112yes—
456C2×C4#644112yes—
457C2×C4#644112yes—
458C2×C4#644212yes—
459C2×C4#644712yes—
460C2×C4#644712yes—
461C2×C4#648712yes—
462C2×C4#648712yes—
463C2×C4#650112yes—
464C2×C4#650112yes—
465C2×C4#650211yescharacteristic
466C2×C4#650211yescharacteristic
467C2×C4C22×Dic1411yescharacteristic
468C2×C4C2×#115711yescharacteristic
469C2×C4C2×D4×D711yescharacteristic
470C2×C4C2×D4×D711yescharacteristic
471C2×C4C2×#115911yescharacteristic
472C2×C4C22×C22⋊D711yescharacteristic
473C8—42no—
474C8—144no—
475C8—144no—
476C8—282no—
477C7C2×#206311yescharacteristic, complemented, Sylow
478C22—24no—
479C22—24no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—44no—
485C22—84no—
486C22—84no—
487C22—88no—
488C22—88no—
489C22—82no—
490C22—82no—
491C22#8972012yes—
492C22#8972012yes—
493C22#8997112yes—
494C22#8997212yes—
495C22C2×#644111yescharacteristic
496C22C2×#644211yescharacteristic
497C22C2×#650211yescharacteristic
498C4—42no—
499C4—42no—
500C4—562no—
501C4#8972012yes—
502C4#8972012yes—
503C4C2×#644111yescharacteristic
504C4C2×#644711yescharacteristic
505C4C2×#648711yescharacteristic
506C4C2×#650111yescharacteristic
507C2—21no—
508C2—21no—
509C2—22no—
510C2—84no—
511C2#1172807114yesdirect product factor
512C2C2×#8972011yescharacteristic
513C2C2×#8997211yescharacteristic
514C2C2×#8997111yescharacteristic
subgroupG/H
centerC23#6502
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#418C2
socleC23×C7#121
prime radicalC23×C7×D4C22
hypercenterC2×#418D7
hypocenterC7C2×#2063
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
4. C22×#6396C2semidirect
5. C2×#89767C2semidirect
6. C2×#89771C2semidirect
7. C2×#89349C2semidirect
8. #11728071C2direct (N × H)
26. C2×#6396C22semidirect
28. #89771C22semidirect
30. #89767C22semidirect
34. #89349C22semidirect
417. C2×C7#2063semidirect
477. C7C2×#2063semidirect
511. C2#11728071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4182
C22×#64072
C22×#64062
C22×#63962
C2×#897672
C2×#897712
C2×#893492
#117280712
C2×#20637