Finite Group Databasebeta

← #13372184

Subgroups — #13372184

Order 1792.

1134 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Dic7C211yesmaximal, characteristic, complemented
2C2×C4×#1141C211yesmaximal, characteristic
3C2×#89528C212yesmaximal
4#11731805C211yesmaximal, characteristic, complemented
5C7×#1802C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic, complemented
7#11728219C211yesmaximal, characteristic, complemented
8#11727921C212yesmaximal, complemented
9#11727331C211yesmaximal, characteristic, complemented
10#11726648C212yesmaximal, complemented
11#11726613C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13C2×D4×Dic7C2212yes—
14C22×#1141C2211yescharacteristic
15C22×C4⋊Dic7—21no—
16C2×#6456C2212yes—
17C2×D7×C4⋊C4—21no—
18C2×C4×D28C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#102C2212yes—
21C7×#384C2211yescharacteristic
22C2×#6406C2211yescharacteristic
23#89984—21no—
24C2×#6395C2212yes—
25C2×C8×Dic7C2212yes—
26C4×#1141C418yescomplemented
27#89761—22no—
28#89759—21no—
29#89757C2212yes—
30#89745C2212yes—
31C2×#6383C2211yescharacteristic
32C2×#6381C2212yes—
33C2×#6381C2212yes—
34C2×#6381C2211yescharacteristic
35#89722—22no—
36#89707C2211yescharacteristic
37C4×C4⋊Dic7C2211yescharacteristic
38C2×C4×C7⋊C8C2211yescharacteristic
39#89528—28no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89345C2212yes—
43C4⋊C4×D8—71nomaximal, Sylow
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7—21no—
46C23×Dic7—22no—
47C22×D28C2311yescharacteristic
48C22×C4×D7—21no—
49C22×C4×D7—22no—
50C22×C7×C8C2312yes—
51C4×C7×D4—22no—
52C4×C7×D4C2×C414yescomplemented
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C22⋊C4—21no—
58C2×C42×C7C2311yescharacteristic
59C2×C22⋊Dic7—22no—
60D4×Dic7—24no—
61D4×Dic7—42no—
62C2×#1141C2×C414yes—
63C2×#1141D414yes—
64C2×#1141Q814yes—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—22no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—44no—
71C2×C4⋊Dic7D411yescharacteristic
72C2×C4⋊Dic7D411yescharacteristic
73C2×C4⋊Dic7C2312yes—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7C2312yes—
77C22×C7⋊C8C2311yescharacteristic
78C22×C7⋊C8C2311yescharacteristic
79#6456—24no—
80#6456—42no—
81D7×C4⋊C4—44no—
82C4×D28—22no—
83C4×D28C2×C414yescomplemented
84C7×C4⋊C8—22no—
85C7×#102—24no—
86#6406C2×C414yescomplemented
87C7⋊#95—22no—
88C7⋊#95—22no—
89C7⋊#95D412yes—
90#6395—24no—
91C8×Dic7—24no—
92#6383C2×C414yescomplemented
93#6381—24no—
94#6381—42no—
95#6381C2×C414yescomplemented
96C4×C7⋊C8C2×C414yescomplemented
97C2×C4×D8—72no—
98C2×C4×D8—71no—
99D4×C4⋊C4—71no—
100D4×C4⋊C4—71no—
101#1942—72no—
102#1862—72no—
103#1862—72no—
104#1845—71no—
105#1802—71no—
106#1802D711yescharacteristic, complemented
107C8×C4⋊C4—71no—
108C24×C7—21no—
109C23×D7—21no—
110C2×C7×D4—21no—
111C2×C7×D4—21no—
112C2×C7×D4—21no—
113C2×C7×D4C22×C412yes—
114C2×C7×D4C2×D412yes—
115C2×C7×D4C2×Q812yes—
116C22×C4×C7—21no—
117C22×C4×C7—24no—
118C22×C4×C7—22no—
119C22×C4×C7—42no—
120C22×C4×C7—42no—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2411yescharacteristic
124C22×Dic7—21no—
125C22×Dic7—22no—
126C22×Dic7—44no—
127C22×Dic7—44no—
128C22×Dic7—82no—
129C22×Dic7C2×D412yes—
130C22×Dic7C2×D412yes—
131C2×D28—21no—
132C2×D28—21no—
133C2×D28—21no—
134C2×D28C22×C412yes—
135C2×D28C2×D412yes—
136C2×D28C2×Q812yes—
137C2×C4×D7—24no—
138C2×C4×D7—42no—
139C2×C4×D7—42no—
140C2×C4×D7—44no—
141C2×C4×D7—44no—
142C2×C4×D7—82no—
143C2×C7×C8—22no—
144C2×C7×C8—22no—
145C2×C7×C8—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—42no—
150C7×C4⋊C4—81no—
151C7×C4⋊C4C22×C412yes—
152C7×C22⋊C4—24no—
153C42×C7C22×C412yes—
154C42×C7#4112yes—
155C22⋊Dic7—44no—
156#1141C4⋊C4116yescomplemented
157(C2×D7)⋊C4—24no—
158(C2×D7)⋊C4—44no—
159C4⋊Dic7—22no—
160C4⋊Dic7—21no—
161C4⋊Dic7—22no—
162C4⋊Dic7—22no—
163C4⋊Dic7—42no—
164C4⋊Dic7—44no—
165C4⋊Dic7—81no—
166C4⋊Dic7D814yescomplemented
167C4⋊Dic7C22×C412yes—
168Dic7⋊C4—44no—
169C4×Dic7—22no—
170C4×Dic7—22no—
171C2×C7⋊C8—21no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8C22×C412yes—
174C2×C7⋊C8C22×C412yes—
175C2×C7⋊C8C2×D412yes—
176C2×C7⋊C8C2×Q812yes—
177C22×D8—71no—
178C2×C4×D4—72no—
179C2×C4×D4—72no—
180C2×C4×D4—71no—
181C2×C4×D4C2×D711yescharacteristic
182C22×C4⋊C4—141no—
183C22×C4⋊C4—141no—
184C4×D8—78no—
185C4×D8—148no—
186C2×#107—71no—
187C2×#107—72no—
188C2×#107—72no—
189C2×C4⋊C8C2×D711yescharacteristic
190C2×#102—71no—
191C2×#102—71no—
192C2×#102—72no—
193C2×#102C2×D712yes—
194C2×C4×C8—71no—
195C2×C4×C8—72no—
196#389—72no—
197#389—72no—
198#385—142no—
199#385—142no—
200#384—71no—
201#384C2×D711yescharacteristic
202#380—141no—
203#380—141no—
204C4×C4⋊C4—71no—
205#340—72no—
206#340C2×D712yes—
207#336—72no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7C22×D411yescharacteristic
216C22×D7—22no—
217C22×D7—22no—
218C22×D7—22no—
219C22×D7—41no—
220C22×D7—41no—
221C22×D7—41no—
222C22×D7—41no—
223C7×D4—22no—
224C7×D4—22no—
225C7×D4—22no—
226C7×D4C2×C4⋊C414yescomplemented
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—48no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7C4×D412yes—
237C2×C4×C7C4×D412yes—
238C2×C4×C7C2×D812yes—
239C2×C4×C7#13512yes—
240C2×C4×C7C23×C411yescharacteristic
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×Q811yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C2×C4×C7H(5,𝔽2)11yescharacteristic
245C2×C4×C72−1+411yescharacteristic
246C2×Dic7—22no—
247C2×Dic7—22no—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—22no—
251C2×Dic7—22no—
252C2×Dic7—41no—
253C2×Dic7—41no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—88no—
258C2×Dic7C2×D814yes—
259D28—22no—
260D28—22no—
261D28—22no—
262D28C2×C4⋊C414yescomplemented
263C4×D7—48no—
264C4×D7—88no—
265C7×C8—42no—
266C7⋊C8—22no—
267C7⋊C8C2×C4⋊C414yescomplemented
268C22×D4—71no—
269C22×D4C22×D711yescharacteristic
270C23×C4—21no—
271C23×C4—142no—
272C23×C4—142no—
273C23×C4—141no—
274C2×D8—74no—
275C2×D8—74no—
276C2×D8—74no—
277C22×C8—71no—
278C22×C8—71no—
279C22×C8C22×D712yes—
280C4×D4—22no—
281C4×D4—74no—
282C4×D4—144no—
283C4×D4—144no—
284C4×D4—142no—
285C4×D4—282no—
286C4×D4—282no—
287C4×D4C4×D714yescomplemented
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—71no—
291C2×C4⋊C4—71no—
292C2×C4⋊C4—71no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—142no—
295C2×C4⋊C4—142no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—284no—
299C2×C4⋊C4—284no—
300C2×C4⋊C4C22×D712yes—
301C2×C4⋊C4C22×D711yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—141no—
304C2×C22⋊C4—142no—
305C2×C22⋊C4—142no—
306C2×C42—72no—
307C2×C42C22×D711yescharacteristic
308#107—74no—
309#107—144no—
310#107—282no—
311C4⋊C8—22no—
312#102—24no—
313#102—74no—
314#102—74no—
315#102—144no—
316C4×C8—74no—
317C4×C8—144no—
318#95—72no—
319#95—142no—
320#95—142no—
321C22×C7—24no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7—42no—
325C22×C7—42no—
326C22×C7—42no—
327C22×C7—42no—
328C22×C7C2×C4×D411yescharacteristic
329C22×C7D4211yescharacteristic
330C22×C7D4×Q811yescharacteristic
331C22×C7C22×D811yescharacteristic
332C22×C7C2×#13511yescharacteristic
333C22×C7#57611yescharacteristic
334C22×C7#57811yescharacteristic
335C2×D7—24no—
336C2×D7—42no—
337C2×D7—42no—
338C2×D7—42no—
339C2×D7—42no—
340C2×D7—42no—
341C2×D7—42no—
342C4×C7—22no—
343C4×C7—42no—
344C4×C7—82no—
345C4×C7C4×D814yescomplemented
346C4×C7C22×C4⋊C411yescharacteristic
347C4×C7#52011yescharacteristic
348C4×C7#54711yescharacteristic
349C4×C7#55411yescharacteristic
350Dic7—24no—
351Dic7—42no—
352Dic7—42no—
353Dic7—82no—
354C24—21no—
355C24—141no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—72no—
360C2×D4—72no—
361C2×D4—72no—
362C2×D4—141no—
363C2×D4—141no—
364C2×D4—141no—
365C2×D4C2×C4×D712yes—
366C2×D4D4×D712yes—
367C2×D4Q8×D712yes—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—72no—
374C22×C4—72no—
375C22×C4—141no—
376C22×C4—144no—
377C22×C4—142no—
378C22×C4—284no—
379C22×C4—284no—
380C22×C4—284no—
381C22×C4—284no—
382C22×C4—282no—
383C22×C4—282no—
384C22×C4—562no—
385C22×C4—562no—
386C22×C4D4×D711yescharacteristic
387C22×C4D4×D711yescharacteristic
388C22×C4C23×D711yescharacteristic
389D8—716no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—72no—
394C2×C8—72no—
395C2×C8—72no—
396C2×C8—72no—
397C2×C8—141no—
398C2×C8—141no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—74no—
404C4⋊C4—72no—
405C4⋊C4—81no—
406C4⋊C4—142no—
407C4⋊C4—141no—
408C4⋊C4—142no—
409C4⋊C4—142no—
410C4⋊C4—284no—
411C4⋊C4—284no—
412C4⋊C4—282no—
413C4⋊C4—561no—
414C4⋊C4C2×C4×D712yes—
415C22⋊C4—24no—
416C22⋊C4—144no—
417C22⋊C4—284no—
418C22⋊C4—284no—
419C42—142no—
420C42—142no—
421C42C2×C4×D712yes—
422C42#115912yes—
423C2×C7—44no—
424C2×C7D4×C4⋊C411yescharacteristic
425C2×C7C2×C4×D811yescharacteristic
426C2×C7#295111yescharacteristic
427C2×C7D4×D811yescharacteristic
428C2×C7#328311yescharacteristic
429C2×C7Q8×D811yescharacteristic
430C2×C7#338011yescharacteristic
431D7—44no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—142no—
440C23—142no—
441C23—142no—
442C23—281no—
443C23—281no—
444C23—281no—
445C23—281no—
446C23C2×D4×D711yescharacteristic
447D4—22no—
448D4—22no—
449D4—22no—
450D4—74no—
451D4—142no—
452D4—142no—
453D4—142no—
454D4D7×C4⋊C414yescomplemented
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—48no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—74no—
465C2×C4—142no—
466C2×C4—142no—
467C2×C4—142no—
468C2×C4—142no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4—288no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—568no—
478C2×C4—568no—
479C2×C4(C22⋊D7)⋊C412yes—
480C2×C4(C22⋊D7)⋊C412yes—
481C2×C4D7×D812yes—
482C2×C4#647912yes—
483C2×C4C22×C4×D711yescharacteristic
484C2×C4C2×D4×D711yescharacteristic
485C2×C4C2×#115911yescharacteristic
486C2×C4C2×Q8×D711yescharacteristic
487C2×C4#655311yescharacteristic
488C2×C4#655411yescharacteristic
489C8—42no—
490C8—74no—
491C8—142no—
492C7C4⋊C4×D811yescharacteristic, complemented, Sylow
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—144no—
501C22—282no—
502C22—282no—
503C22—282no—
504C22—282no—
505C22—282no—
506C22—282no—
507C22C2×(C22⋊D7)⋊C411yescharacteristic
508C22#9030011yescharacteristic
509C22#9032311yescharacteristic
510C22C2×D7×D811yescharacteristic
511C22C2×#647911yescharacteristic
512C22#9046511yescharacteristic
513C22#9047011yescharacteristic
514C4—22no—
515C4—42no—
516C4—82no—
517C4—144no—
518C4—282no—
519C4—282no—
520C4—562no—
521C4#8952814yescomplemented
522C4C2×D7×C4⋊C411yescharacteristic
523C4#9021411yescharacteristic
524C4#9030211yescharacteristic
525C4#9032611yescharacteristic
526C2—44no—
527C2—284no—
528C2#1173145511yescharacteristic
529C2C2×#8952811yescharacteristic
530C2#1173631311yescharacteristic
531C2#1173779511yescharacteristic
532C2#1173789411yescharacteristic
533C2#1173840211yescharacteristic
534C2#1173849911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
cosocleC22×C4×C7C24
hypercenter#1802D7
hypocenterC7C4⋊C4×D8
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. D4×C4⋊Dic7C2semidirect
4. #11731805C2semidirect
5. C7×#1802C2semidirect
6. #11728233C2semidirect
7. #11728219C2semidirect
8. #11727921C2semidirect
9. #11727331C2semidirect
10. #11726648C2semidirect
11. #11726613C2semidirect
26. C4×#1141C4semidirect
52. C4×C7×D4C2×C4semidirect
83. C4×D28C2×C4semidirect
86. #6406C2×C4semidirect
92. #6383C2×C4semidirect
95. #6381C2×C4semidirect
96. C4×C7⋊C8C2×C4semidirect
106. #1802D7semidirect
156. #1141C4⋊C4semidirect
166. C4⋊Dic7D8semidirect
226. C7×D4C2×C4⋊C4semidirect
262. D28C2×C4⋊C4semidirect
267. C7⋊C8C2×C4⋊C4semidirect
287. C4×D4C4×D7semidirect
345. C4×C7C4×D8semidirect
454. D4D7×C4⋊C4semidirect
492. C7C4⋊C4×D8semidirect
521. C4#89528semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Dic72
C2×C4×#11412
C2×#895282
#117318052
C7×#18022
#117282332
#117282192
#117279212
#117273312
#117266482
#117266132
C4⋊C4×D87