Finite Group Databasebeta

← #13395354

Subgroups — #13395354

Order 1792.

946 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#89803C211yesmaximal, characteristic, complemented
3C2×#89800C212yesmaximal
4C2×C4×#1143C211yesmaximal, characteristic
5#11731247C211yesmaximal, characteristic
6#11729386C212yesmaximal, complemented
7#11727388C212yesmaximal, complemented
8#11727382C212yesmaximal
9#11727338C211yesmaximal, characteristic
10#11727333C211yesmaximal, characteristic, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C4×C7×Q8C2212yes—
13C2×C4×C7×Q8C2211yescharacteristic
14C22×#1144C2212yes—
15C22×#1143C2211yescharacteristic
16C2×#6438C2212yes—
17C2×C4×D28C2211yescharacteristic
18C2×#6432C2211yescharacteristic
19C7×#389—22no—
20C7×#389C2212yes—
21C7×#384—21no—
22C7×#384C2211yescharacteristic
23C4×C7×C4⋊C4—21no—
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26#89803—24no—
27#89800—28no—
28C4×#1143—24no—
29#89759—21no—
30#89759—21no—
31C2×#6384C2212yes—
32C2×#6384C2212yes—
33C2×#6384C2212yes—
34C2×#6383C2211yescharacteristic
35C2×#6383C2212yes—
36C2×#6382C2211yescharacteristic
37#89701C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89446—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24729—71nomaximal, Sylow
45C22×C7×Q8C2311yescharacteristic
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C22×Dic14C2312yes—
49C4×C7×Q8—22no—
50C4×C7×Q8—24no—
51C4×C7×Q8—42no—
52C4×C7×Q8D414yes—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2311yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7—22no—
64C2×C42×C7C2312yes—
65C2×C42×C7C2311yescharacteristic
66C2×#1144—24no—
67C2×#1144—24no—
68C2×#1144—24no—
69C2×#1143—22no—
70C2×#1143—22no—
71C2×#1143D414yes—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2311yescharacteristic
78C22×C7⋊C8C2311yescharacteristic
79C22×C7⋊C8C2312yes—
80C22×C7⋊C8C2311yescharacteristic
81#6438—24no—
82#6438—42no—
83C4×D28—22no—
84C4×D28—22no—
85#6432—22no—
86#6432—22no—
87C7×#95—22no—
88C7×#95D412yes—
89#6409—22no—
90C7⋊#95—22no—
91C7⋊#95—22no—
92#6384—24no—
93#6384—24no—
94#6384—42no—
95#6383—24no—
96#6383—22no—
97#6382—22no—
98C4⋊(C7⋊C8)—22no—
99C4⋊(C7⋊C8)—24no—
100C4×C7⋊C8—22no—
101C2×#464—72no—
102C2×#462—71no—
103C2×C4×SD16—71no—
104#2547—71no—
105Q8×C4⋊C4D711yescharacteristic, complemented
106#2066—72no—
107#2042—72no—
108#1956—71no—
109#1939—71no—
110#1878—72no—
111#1802—71no—
112C23×D7—21no—
113C2×C7×Q8—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8—21no—
116C2×C7×Q8C2×D412yes—
117C2×C7×Q8C2×D412yes—
118C2×C7×Q8#4112yes—
119C22×C4×C7—21no—
120C22×C4×C7—21no—
121C22×C4×C7—22no—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C22×Dic7—22no—
129C2×D28—21no—
130C2×D28—21no—
131C2×D28—21no—
132C2×D28—21no—
133C2×D28—21no—
134C2×D28C2×D412yes—
135C2×C4×D7—42no—
136C2×C4×D7—42no—
137C2×C4×D7—42no—
138C2×Dic14—22no—
139C2×Dic14—22no—
140C2×Dic14—22no—
141C2×Dic14—22no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—44no—
151C7×C4⋊C4—42no—
152C7×C4⋊C4—81no—
153C7×C4⋊C4—82no—
154C7×C4⋊C4C2×D412yes—
155C42×C7—24no—
156C42×C7—22no—
157C42×C7—22no—
158C42×C7—44no—
159C42×C7C2×D412yes—
160C42×C7#4112yes—
161#1144—48no—
162#1143—28no—
163(C2×D7)⋊C4—44no—
164(C2×D7)⋊C4—44no—
165(C2×D7)⋊C4—42no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—42no—
169C4⋊Dic7—42no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—21no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—21no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8C2×D412yes—
179C22×Q16—72no—
180C22×SD16—71no—
181C2×C4⋊Q8—71no—
182C2×#124—72no—
183C2×C4×Q8C2×D712yes—
184C2×C4×Q8C2×D711yescharacteristic
185C2×C4×D4—71no—
186#464—148no—
187#462—144no—
188C4×SD16—144no—
189C2×#106—71no—
190C2×C4⋊C8—71no—
191C2×C4⋊C8—72no—
192C2×#103—71no—
193C2×#103—72no—
194C2×#103—72no—
195C2×#103—72no—
196C2×#102—72no—
197C2×#102—71no—
198C2×C4×C8—71no—
199#397—142no—
200#391—72no—
201#389—22no—
202#389C2×D712yes—
203#384—21no—
204#384C2×D711yescharacteristic
205#380—141no—
206#380—141no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4C2×D711yescharacteristic
209#340—72no—
210#336—72no—
211C23×C7C22×D411yescharacteristic
212C22×D7—22no—
213C22×D7—41no—
214C22×D7—41no—
215C22×D7—41no—
216C22×D7—41no—
217C22×D7—41no—
218C22×D7—41no—
219C7×Q8—22no—
220C7×Q8—22no—
221C7×Q8—22no—
222C7×Q8#12114yes—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—41no—
236C2×C4×C7—41no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7#12112yes—
241C2×C4×C7#12112yes—
242C2×C4×C7#13512yes—
243C2×C4×C7#13712yes—
244C2×C4×C7C22×D411yescharacteristic
245C2×C4×C7C22×D411yescharacteristic
246C2×C4×C7C2×#4111yescharacteristic
247C2×C4×C7C2×#4111yescharacteristic
248C2×C4×C7H(5,𝔽2)11yescharacteristic
249C2×C4×C72−1+411yescharacteristic
250C2×Dic7—41no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256D28—22no—
257D28—22no—
258D28—41no—
259D28—41no—
260C4×D7—84no—
261Dic14—42no—
262Dic14—42no—
263Dic14—42no—
264Dic14—42no—
265C7⋊C8—22no—
266C7⋊C8—42no—
267C7⋊C8—41no—
268C22×Q8—72no—
269C22×Q8C22×D711yescharacteristic
270C22×D4—71no—
271C23×C4—141no—
272C2×Q16—144no—
273C2×Q16—144no—
274C2×Q16—144no—
275C2×SD16—74no—
276C2×SD16—142no—
277C2×SD16—142no—
278C22×C8—71no—
279C22×C8—72no—
280C22×C8—71no—
281C4⋊Q8—142no—
282C4⋊Q8—142no—
283#124—144no—
284#124—282no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—42no—
288C4×Q8D2814yes—
289C4×D4—142no—
290C4×D4—142no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—71no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—141no—
299C2×C4⋊C4C22⋊D711yescharacteristic
300C2×C4⋊C4C22⋊D711yescharacteristic
301C2×C4⋊C4C22×D712yes—
302C2×C4⋊C4C22×D711yescharacteristic
303C2×C22⋊C4—141no—
304C2×C22⋊C4—142no—
305C2×C22⋊C4—142no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42C22×D712yes—
309C2×C42C22×D711yescharacteristic
310#106—142no—
311C4⋊C8—144no—
312C4⋊C8—142no—
313#103—144no—
314#103—142no—
315#103—144no—
316#103—282no—
317#102—144no—
318#102—142no—
319C4×C8—142no—
320#95—22no—
321#95—142no—
322#95—142no—
323#95C22⋊D712yes—
324C22×C7C2×#12111yescharacteristic
325C22×C7D4211yescharacteristic
326C22×C7#54811yescharacteristic
327C22×C7C2×#13511yescharacteristic
328C22×C7C2×#13711yescharacteristic
329C22×C7#57711yescharacteristic
330C22×C7#57811yescharacteristic
331C2×D7—42no—
332C2×D7—42no—
333C2×D7—42no—
334C2×D7—81no—
335C2×D7—81no—
336C2×D7—81no—
337C2×D7—81no—
338C4×C7—24no—
339C4×C7—22no—
340C4×C7—42no—
341C4×C7—42no—
342C4×C7—42no—
343C4×C7—82no—
344C4×C7#46414yescomplemented
345C4×C7C2×#12111yescharacteristic
346C4×C7#53711yescharacteristic
347C4×C7#54711yescharacteristic
348C4×C7#55311yescharacteristic
349Dic7—82no—
350Dic7—81no—
351C24—141no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—142no—
356C2×Q8—142no—
357C2×Q8—142no—
358C2×Q8—142no—
359C2×Q8—142no—
360C2×Q8—142no—
361C2×Q8C2×D2812yes—
362C2×Q8D4×D712yes—
363C2×Q8#116112yes—
364C2×D4—72no—
365C2×D4—141no—
366C2×D4—141no—
367C2×D4—141no—
368C2×D4—141no—
369C2×D4—141no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—141no—
374C22×C4—142no—
375C22×C4—282no—
376C22×C4—282no—
377C22×C4—282no—
378C22×C4C2×C22⋊D712yes—
379C22×C4C2×C22⋊D712yes—
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C2×C22⋊D711yescharacteristic
382C22×C4C23×D711yescharacteristic
383Q16—288no—
384SD16—148no—
385C2×C8—72no—
386C2×C8—141no—
387C2×C8—141no—
388C2×C8—141no—
389C2×C8—142no—
390C2×C8—142no—
391C2×C8—141no—
392C2×C8—142no—
393C2×C8—141no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—82no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4—282no—
407C4⋊C4—282no—
408C4⋊C4C2×D2812yes—
409C22⋊C4—284no—
410C22⋊C4—284no—
411C22⋊C4—282no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—44no—
416C42C2×D2812yes—
417C42#115712yes—
418C2×C7#241611yescharacteristic
419C2×C7C2×#46411yescharacteristic
420C2×C7#304611yescharacteristic
421C2×C7#327611yescharacteristic
422C2×C7#328411yescharacteristic
423C2×C7#336411yescharacteristic
424C2×C7#336911yescharacteristic
425D7—82no—
426C23—142no—
427C23—281no—
428C23—281no—
429C23—281no—
430C23—281no—
431C23—281no—
432C23—281no—
433C23C22×C22⋊D711yescharacteristic
434Q8—22no—
435Q8—22no—
436Q8—22no—
437Q8—282no—
438Q8—282no—
439Q8—282no—
440Q8—282no—
441Q8#645814yes—
442D4—142no—
443D4—142no—
444D4—281no—
445D4—281no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—281no—
464C2×C4—282no—
465C2×C4—282no—
466C2×C4—281no—
467C2×C4—282no—
468C2×C4—281no—
469C2×C4—564no—
470C2×C4#649312yes—
471C2×C4#649312yes—
472C2×C4#650512yes—
473C2×C4#651312yes—
474C2×C4C22×D2811yescharacteristic
475C2×C4C2×#115711yescharacteristic
476C2×C4C2×D4×D711yescharacteristic
477C2×C4#654811yescharacteristic
478C2×C4C2×#116111yescharacteristic
479C2×C4#655111yescharacteristic
480C8—142no—
481C8—282no—
482C8—281no—
483C7#2472911yescharacteristic, complemented, Sylow
484C22—282no—
485C22—282no—
486C22—282no—
487C22—561no—
488C22—561no—
489C22—561no—
490C22—561no—
491C22C2×#649311yescharacteristic
492C22D4×C22⋊D711yescharacteristic
493C22C2×#650511yescharacteristic
494C22#9050811yescharacteristic
495C22C2×#651311yescharacteristic
496C22#9052711yescharacteristic
497C22#9052911yescharacteristic
498C4—22no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—82no—
504C4—561no—
505C4—562no—
506C4#8980014yescomplemented
507C4C2×#645811yescharacteristic
508C4#9019811yescharacteristic
509C4#9020111yescharacteristic
510C4#9020511yescharacteristic
511C2—562no—
512C2#1173963811yescharacteristic
513C2C2×#8980011yescharacteristic
514C2#1174001711yescharacteristic
515C2#1174100511yescharacteristic
516C2#1174102411yescharacteristic
517C2#1174113411yescharacteristic
518C2#1174114911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×Q8×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterQ8×C4⋊C4D7
hypocenterC7#24729
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C7×Q8×C4⋊C4C2semidirect
2. C2×#89803C2semidirect
6. #11729386C2semidirect
7. #11727388C2semidirect
10. #11727333C2semidirect
11. #11727238C2semidirect
105. Q8×C4⋊C4D7semidirect
344. C4×C7#464semidirect
483. C7#24729semidirect
506. C4#89800semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C4⋊C42
C2×#898032
C2×#898002
C2×C4×#11432
#117312472
#117293862
#117273882
#117273822
#117273382
#117273332
#117272382
#247297