Finite Group Databasebeta

← #13361444

Subgroups — #13361444

Order 1792.

988 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#476C211yesmaximal, characteristic
2Dic7×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×D4×Dic7C211yesmaximal, characteristic
4#11729533C211yesmaximal, characteristic, complemented
5#11729358C211yesmaximal, characteristic
6#11728311C212yesmaximal, complemented
7#11728307C212yesmaximal, complemented
8#11728207C211yesmaximal, characteristic, complemented
9#11728195C211yesmaximal, characteristic, complemented
10#11727922C212yesmaximal
11#11727906C212yesmaximal
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C2×Q8×Dic7—21no—
15C2×D4×Dic7C2211yescharacteristic
16C2×D4×Dic7C2211yescharacteristic
17C22×C4×Dic7—21no—
18C7×#476C418yescomplemented
19C2×C7×#106C2212yes—
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#102C2212yes—
22#90002—21no—
23C2×#6406C2212yes—
24C4×C22⋊Dic7—21no—
25C2×#6392C2212yes—
26C2×#6389C2212yes—
27#89757—21no—
28#89757C2212yes—
29C4⋊C4×Dic7—21no—
30C4⋊C4×Dic7C2212yes—
31C4⋊C4×Dic7C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C22⋊C4×Dic7—21no—
34C22⋊C4×Dic7—21no—
35#89707C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37C42×Dic7C2211yescharacteristic
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21614—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×Dic7—21no—
49C23×Dic7—21no—
50C22×C7×C8C2312yes—
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8C2×C414yescomplemented
53C4×C7×D4—22no—
54C4×C7×D4C2×C414yes—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C2×C22⋊Dic7—21no—
61C2×C22⋊Dic7—21no—
62Q8×Dic7—44no—
63D4×Dic7—22no—
64D4×Dic7—22no—
65D4×Dic7—22no—
66D4×Dic7Q814yescomplemented
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7C2312yes—
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4×Dic7—21no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7—21no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—44no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85C7×#106C2×C418yescomplemented
86C7×C4⋊C8C2×C414yescomplemented
87C7×#102C2×C418yes—
88#6406—24no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94#6392—24no—
95#6389—24no—
96C4⋊(C7⋊C8)—22no—
97C2×#476D711yescharacteristic
98C4×C4⋊Q8—71no—
99C42×D4—71no—
100#1987—71no—
101#1947—71no—
102#1919—72no—
103#1862—72no—
104#1802—71no—
105#1776—72no—
106#1767—71no—
107#1762—72no—
108C24×C7—21no—
109C2×C7×Q8—24no—
110C2×C7×Q8—42no—
111C2×C7×Q8—42no—
112C2×C7×D4—21no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C2×C7×D4C22×C412yes—
116C2×C7×D4C2×Q812yes—
117C2×C7×D4#4112yes—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7—24no—
121C22×C4×C7—42no—
122C22×C4×C7—42no—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2411yescharacteristic
126C22×Dic7—24no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—24no—
130C22×Dic7—22no—
131C22×Dic7—42no—
132C22×Dic7—42no—
133C22×Dic7—42no—
134C22×Dic7—42no—
135C22×Dic7C2×D411yescharacteristic
136C22×Dic7C2×D411yescharacteristic
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C2×C7×C8—22no—
140C2×C7×C8—22no—
141C2×C7×C8C22×C414yes—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4C22×C414yes—
146C7×C4⋊C4C22×C412yes—
147C7×C22⋊C4—24no—
148C42×C7C22×C412yes—
149C42×C7#4112yes—
150C22⋊Dic7—24no—
151C22⋊Dic7—24no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—22no—
156C4⋊Dic7—42no—
157C4⋊Dic7—42no—
158C4⋊Dic7—44no—
159C4⋊Dic7C2×Q812yes—
160C4⋊Dic7#4112yes—
161C4×Dic7—22no—
162C4×Dic7—22no—
163C4×Dic7—21no—
164C4×Dic7—21no—
165C4×Dic7—22no—
166C4×Dic7—41no—
167C4×Dic7—42no—
168C4×Dic7—42no—
169C4×Dic7—44no—
170C4×Dic7—42no—
171C4×Dic7C2×Q812yes—
172C4×Dic7#4112yes—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C4⋊Q8C2×D711yescharacteristic
177C2×C4×Q8—141no—
178C2×C4×D4—71no—
179C2×C4×D4—71no—
180C2×C4×D4C2×D711yescharacteristic
181C22×C42—141no—
182#476Dic718yescomplemented
183C2×#107—72no—
184C2×#106C2×D712yes—
185C2×C4⋊C8—71no—
186C2×C4⋊C8C2×D711yescharacteristic
187C2×#102—72no—
188C2×#102C2×D712yes—
189C2×#97—72no—
190#391—141no—
191#389—72no—
192#389—141no—
193#384—71no—
194C4×C4⋊C4—72no—
195C4×C4⋊C4—71no—
196C4×C4⋊C4—71no—
197C4×C4⋊C4—71no—
198C4×C4⋊C4—141no—
199C4×C22⋊C4—141no—
200C4×C22⋊C4—141no—
201C4×C22⋊C4—141no—
202C43—71no—
203#340—72no—
204#340—72no—
205#340—72no—
206#340—72no—
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
216C7×Q8—48no—
217C7×D4—22no—
218C7×D4—22no—
219C7×D4—22no—
220C7×D4C4×Q814yescomplemented
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—22no—
224C2×C4×C7—24no—
225C2×C4×C7—41no—
226C2×C4×C7—42no—
227C2×C4×C7—41no—
228C2×C4×C7—42no—
229C2×C4×C7—48no—
230C2×C4×C7C4×D412yes—
231C2×C4×C7C4×D412yes—
232C2×C4×C7AGL(1,ℤ/8)12yes—
233C2×C4×C7#13712yes—
234C2×C4×C7C23×C411yescharacteristic
235C2×C4×C7C22×Q811yescharacteristic
236C2×C4×C7C2×#4111yescharacteristic
237C2×C4×C7C2×#4111yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C2×C4×C7C2×#4111yescharacteristic
240C2×Dic7—21no—
241C2×Dic7—21no—
242C2×Dic7—21no—
243C2×Dic7—21no—
244C2×Dic7—21no—
245C2×Dic7—21no—
246C2×Dic7—22no—
247C2×Dic7—21no—
248C2×Dic7—22no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—48no—
252C2×Dic7—48no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7C22⋊Q812yes—
261C2×Dic7C22⋊Q812yes—
262C2×Dic7AGL(1,ℤ/8)12yes—
263C7×C8—24no—
264C7⋊C8—42no—
265C22×Q8—21no—
266C22×D4C22×D711yescharacteristic
267C23×C4—21no—
268C23×C4—141no—
269C23×C4—141no—
270C22×C8—72no—
271C22×C8C22×D712yes—
272C4⋊Q8—22no—
273C4⋊Q8C2×Dic714yescomplemented
274C4×Q8—284no—
275C4×D4—22no—
276C4×D4—74no—
277C4×D4—142no—
278C4×D4—142no—
279C4×D4—142no—
280C4×D4C2×Dic714yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—71no—
283C2×C4⋊C4—71no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—142no—
286C2×C4⋊C4—141no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4C22×D711yescharacteristic
289C2×C4⋊C4C22×D712yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—141no—
292C2×C22⋊C4—141no—
293C2×C42—71no—
294C2×C42—71no—
295C2×C42—71no—
296C2×C42—71no—
297C2×C42—71no—
298C2×C42—71no—
299C2×C42—142no—
300C2×C42—141no—
301C2×C42—141no—
302C2×C42—141no—
303C2×C42—284no—
304C2×C42C22×D711yescharacteristic
305#107—144no—
306#106C2×Dic718yescomplemented
307C4⋊C8—142no—
308C4⋊C8C2×Dic714yescomplemented
309#102—144no—
310#102C2×Dic718yes—
311#97—144no—
312#95—141no—
313#95—141no—
314#95—142no—
315#95—142no—
316#95—141no—
317C22×C7—24no—
318C22×C7—42no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—42no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7C2×C4×D411yescharacteristic
325C22×C7C2×C22⋊Q811yescharacteristic
326C22×C7#52511yescharacteristic
327C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C7C2×#13711yescharacteristic
330C22×C7#57511yescharacteristic
331C4×C7—22no—
332C4×C7—42no—
333C4×C7—44no—
334C4×C7#44014yes—
335C4×C7C2×C4×Q811yescharacteristic
336C4×C7C4×#4111yescharacteristic
337C4×C7#52911yescharacteristic
338C4×C7#53311yescharacteristic
339Dic7—22no—
340Dic7—22no—
341Dic7—22no—
342Dic7—42no—
343Dic7—42no—
344Dic7—41no—
345Dic7—82no—
346C24—21no—
347C2×Q8—24no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4Q8×D712yes—
354C2×D4#116112yes—
355C2×D4C22×Dic712yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—71no—
362C22×C4—71no—
363C22×C4—71no—
364C22×C4—71no—
365C22×C4—144no—
366C22×C4—144no—
367C22×C4—141no—
368C22×C4—141no—
369C22×C4—142no—
370C22×C4—282no—
371C22×C4—282no—
372C22×C4—282no—
373C22×C4—282no—
374C22×C4D4×D711yescharacteristic
375C22×C4D4×D711yescharacteristic
376C22×C4C23×D711yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—142no—
380C2×C8—142no—
381C2×C8—142no—
382C2×C8C22×Dic714yes—
383C4⋊C4—24no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—72no—
387C4⋊C4—72no—
388C4⋊C4—141no—
389C4⋊C4—141no—
390C4⋊C4—142no—
391C4⋊C4—142no—
392C4⋊C4—282no—
393C4⋊C4—284no—
394C4⋊C4—282no—
395C4⋊C4C22×Dic714yes—
396C4⋊C4C22×Dic712yes—
397C22⋊C4—24no—
398C22⋊C4—144no—
399C22⋊C4—144no—
400C42—72no—
401C42—72no—
402C42—142no—
403C42—142no—
404C42—142no—
405C42—141no—
406C42—141no—
407C42—281no—
408C42—282no—
409C42—282no—
410C42—284no—
411C42—282no—
412C42#115912yes—
413C42C22×Dic712yes—
414C2×C7—44no—
415C2×C7C4×C22⋊Q811yescharacteristic
416C2×C7C2×#44011yescharacteristic
417C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
418C2×C7#307711yescharacteristic
419C2×C7#307911yescharacteristic
420C2×C7#310911yescharacteristic
421C2×C7#311611yescharacteristic
422C23—22no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23C2×D4×D711yescharacteristic
430Q8—48no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4Q8×Dic714yescomplemented
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—72no—
445C2×C4—72no—
446C2×C4—72no—
447C2×C4—141no—
448C2×C4—141no—
449C2×C4—141no—
450C2×C4—141no—
451C2×C4—141no—
452C2×C4—141no—
453C2×C4—141no—
454C2×C4—141no—
455C2×C4—142no—
456C2×C4—141no—
457C2×C4—142no—
458C2×C4—288no—
459C2×C4—288no—
460C2×C4—281no—
461C2×C4—281no—
462C2×C4—281no—
463C2×C4—281no—
464C2×C4—282no—
465C2×C4—282no—
466C2×C4—282no—
467C2×C4#647712yes—
468C2×C4#647812yes—
469C2×C4D4×Dic712yes—
470C2×C4D4×Dic712yes—
471C2×C4C2×#115911yescharacteristic
472C2×C4C2×Q8×D711yescharacteristic
473C2×C4C2×#116111yescharacteristic
474C2×C4D7×#4111yescharacteristic
475C2×C4D7×#4111yescharacteristic
476C2×C4C23×Dic711yescharacteristic
477C8—24no—
478C8—282no—
479C7#2161411yescharacteristic, complemented, Sylow
480C22—24no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22D7×C22⋊Q811yescharacteristic
488C22#9031811yescharacteristic
489C22C2×#647711yescharacteristic
490C22C2×#647811yescharacteristic
491C22D7×AGL(1,ℤ/8)11yescharacteristic
492C22#9046411yescharacteristic
493C22C2×D4×Dic711yescharacteristic
494C4—22no—
495C4—42no—
496C4—44no—
497C4—142no—
498C4—142no—
499C4—142no—
500C4—282no—
501C4—281no—
502C4—282no—
503C4—562no—
504C4#8993614yes—
505C4#9031311yescharacteristic
506C4#9031511yescharacteristic
507C4C2×Q8×Dic711yescharacteristic
508C4#41×Dic711yescharacteristic
509C2—44no—
510C2#1173849111yescharacteristic
511C2#1173849211yescharacteristic
512C2#1173850511yescharacteristic
513C2#1173850611yescharacteristic
514C2Dic7×C22⋊Q811yescharacteristic
515C2C2×#8993611yescharacteristic
516C2Dic7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#476C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#476D7
hypocenterC7#21614
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. Dic7×C4⋊Q8C2semidirect
4. #11729533C2semidirect
6. #11728311C2semidirect
7. #11728307C2semidirect
8. #11728207C2semidirect
9. #11728195C2semidirect
18. C7×#476C4semidirect
52. C7×C4⋊Q8C2×C4semidirect
66. D4×Dic7Q8semidirect
85. C7×#106C2×C4semidirect
86. C7×C4⋊C8C2×C4semidirect
182. #476Dic7semidirect
220. C7×D4C4×Q8semidirect
273. C4⋊Q8C2×Dic7semidirect
306. #106C2×Dic7semidirect
308. C4⋊C8C2×Dic7semidirect
434. D4Q8×Dic7semidirect
479. C7#21614semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4762
Dic7×C4⋊Q82
C4×D4×Dic72
#117295332
#117293582
#117283112
#117283072
#117282072
#117281952
#117279222
#117279062
#216147