Finite Group Databasebeta

← #13388152

Subgroups — #13388152

Order 1792.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic, complemented
3C2×#89862C211yesmaximal, characteristic, complemented
4C4×#6487C211yesmaximal, characteristic
5#11729533C211yesmaximal, characteristic, complemented
6#11729532C211yesmaximal, characteristic, complemented
7#11729454C212yesmaximal
8#11729449C212yesmaximal
9#11729386C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11#11729284C211yesmaximal, characteristic
12C7×#533—24no—
13C7×#533—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2212yes—
21C2×C7×#117—21no—
22C22×C42×C7C2211yescharacteristic
23C22×#1144C2212yes—
24C2×#6487C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29#89984C2211yescharacteristic
30#89982—21no—
31C4×C22⋊Dic7—21no—
32C2×C22⋊(C7⋊C8)C2212yes—
33#89863—24no—
34#89862—24no—
35C2×#6384C2212yes—
36C2×#6381C2212yes—
37C4×C4⋊Dic7C2211yescharacteristic
38#89705—21no—
39C4×Dic7⋊C4—21no—
40#89701C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24155—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8C2312yes—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×Dic14C2311yescharacteristic
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8D414yes—
57C7×C4⋊Q8D414yes—
58C7×#125—28no—
59C7×C22⋊Q8—28no—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8—24no—
62C4×C7×Q8—28no—
63C4×C7×Q8—24no—
64C4×C7×Q8—24no—
65C7×#117—28no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C22⋊C4—22no—
73C2×C42×C7—22no—
74C2×C42×C7—22no—
75C2×C42×C7—22no—
76C2×C42×C7D411yescharacteristic
77C2×C42×C7D411yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×#1144—24no—
83C2×#1144—24no—
84C2×#1144—42no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88#6487—22no—
89#6487—22no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2312yes—
97C4×Dic14—22no—
98C4×Dic14—22no—
99#6409—24no—
100#6409—24no—
101#6409—42no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C22⋊(C7⋊C8)—42no—
108#6384—24no—
109#6381—24no—
110C4⋊(C7⋊C8)—22no—
111C4⋊(C7⋊C8)—22no—
112C2×#533D711yescharacteristic
113C2×#477—71no—
114C2×#462—71no—
115C4×C22⋊Q8—71no—
116#2036—72no—
117#2001—72no—
118#1987—71no—
119#1981—71no—
120#1878—72no—
121#1863—72no—
122#1797—71no—
123C24×C7C2×D411yescharacteristic
124C2×C7×Q8—24no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—42no—
131C2×C7×Q8—42no—
132C2×C7×Q8C2×D414yes—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7Q1614yescomplemented
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C2×Dic14—21no—
158C2×Dic14—21no—
159C2×Dic14—21no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14#4112yes—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—28no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4C2×D414yes—
170C7×C22⋊C4—28no—
171C42×C7—28no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7C2×D412yes—
177C42×C7C2×D412yes—
178C22⋊Dic7—24no—
179C22⋊Dic7—42no—
180#1144—48no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7#4112yes—
187Dic7⋊C4—24no—
188Dic7⋊C4—42no—
189Dic7⋊C4—42no—
190C4×Dic7—42no—
191C4×Dic7—42no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C22×Q16—72no—
199#533—24no—
200#533—24no—
201C2×C4⋊Q8C2×D711yescharacteristic
202C2×C4⋊Q8C2×D711yescharacteristic
203C2×#125—21no—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8—71no—
206C2×C22⋊Q8C2×D712yes—
207C2×C4×Q8—21no—
208C2×C4×Q8—71no—
209C2×C4×Q8C2×D712yes—
210C2×#117—21no—
211C22×C42C2×D711yescharacteristic
212#477—144no—
213#462—144no—
214C2×#107—72no—
215C2×C4⋊C8—71no—
216C2×C4⋊C8—71no—
217C2×#103—72no—
218C2×#103—72no—
219C2×#103—72no—
220C2×#103—72no—
221C2×C22⋊C8—72no—
222#391—71no—
223#389—71no—
224#387—141no—
225#385—141no—
226#380—71no—
227C4×C4⋊C4—71no—
228C4×C4⋊C4—141no—
229C4×C22⋊C4—141no—
230#340—72no—
231#340—72no—
232#336—72no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7C2×Q1612yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—24no—
242C7×Q8—24no—
243C7×Q8—48no—
244C7×Q8—42no—
245C7×Q8—42no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—28no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—44no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7C2×Q1612yes—
271C2×C4×C7C2×Q1612yes—
272C2×C4×C7#13712yes—
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×C4×C72−1+411yescharacteristic
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291Dic14—22no—
292Dic14—22no—
293Dic14—41no—
294Dic14—41no—
295C7⋊C8—42no—
296C7⋊C8—42no—
297C22×Q8—21no—
298C22×Q8—71no—
299C22×Q8C22×D712yes—
300C23×C4C22⋊D711yescharacteristic
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22×D711yescharacteristic
303C2×Q16—144no—
304C2×Q16—144no—
305C2×Q16—282no—
306C22×C8—72no—
307C22×C8—72no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8C22⋊D714yes—
311C4⋊Q8C22⋊D714yes—
312#125—28no—
313C22⋊Q8—28no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—142no—
317C22⋊Q8—142no—
318C4×Q8—28no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—142no—
322C4×Q8—142no—
323#117—28no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4C22×D712yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—141no—
338C2×C22⋊C4—141no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—141no—
343C2×C42—141no—
344C2×C42C22⋊D711yescharacteristic
345C2×C42C22⋊D711yescharacteristic
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348#107—144no—
349C4⋊C8—142no—
350C4⋊C8—142no—
351#103—144no—
352#103—144no—
353#103—144no—
354#103—282no—
355C22⋊C8—282no—
356#95—141no—
357#95—141no—
358#95—141no—
359#95—141no—
360#95—141no—
361C22×C7—22no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#52511yescharacteristic
370C22×C7#53711yescharacteristic
371C22×C7C22×Q1611yescharacteristic
372C22×C7C2×#13711yescharacteristic
373C22×C7#57511yescharacteristic
374C22×C7#57511yescharacteristic
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—41no—
378C4×C7—44no—
379C4×C7—41no—
380C4×C7—44no—
381C4×C7#45114yes—
382C4×C7#54611yescharacteristic
383C4×C7#54711yescharacteristic
384C4×C7#54811yescharacteristic
385C4×C7#55011yescharacteristic
386Dic7—42no—
387Dic7—42no—
388Dic7—81no—
389Dic7—81no—
390C24C2×C22⋊D711yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—72no—
400C2×Q8—141no—
401C2×Q8—141no—
402C2×Q8—141no—
403C2×Q8—141no—
404C2×Q8—141no—
405C2×Q8C2×C22⋊D714yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4#114414yescomplemented
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C23×D711yescharacteristic
430Q16—288no—
431C2×C8—142no—
432C2×C8—142no—
433C2×C8—142no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—72no—
444C4⋊C4—144no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—282no—
451C4⋊C4—282no—
452C4⋊C4C2×C22⋊D714yes—
453C22⋊C4—28no—
454C22⋊C4—144no—
455C22⋊C4—282no—
456C42—28no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42—282no—
462C42—282no—
463C42C2×C22⋊D712yes—
464C42C2×C22⋊D712yes—
465C2×C7—42no—
466C2×C7#240711yescharacteristic
467C2×C7C2×#45111yescharacteristic
468C2×C7#300511yescharacteristic
469C2×C7#310411yescharacteristic
470C2×C7#311611yescharacteristic
471C2×C7#324911yescharacteristic
472C2×C7#325211yescharacteristic
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23C2×#114412yes—
480C23C22×C22⋊D711yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—48no—
484Q8—42no—
485Q8—42no—
486Q8—142no—
487Q8—142no—
488Q8—281no—
489Q8—281no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—28no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—281no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4#650512yes—
525C2×C4C2×#114412yes—
526C2×C4C2×#114412yes—
527C2×C4#651612yes—
528C2×C4#651612yes—
529C2×C4D7×#4111yescharacteristic
530C2×C4D7×#4111yescharacteristic
531C2×C4#655311yescharacteristic
532C2×C4#655411yescharacteristic
533C2×C4C22×C22⋊D711yescharacteristic
534C2×C4C22×C22⋊D711yescharacteristic
535C8—282no—
536C8—282no—
537C7#2415511yescharacteristic, complemented, Sylow
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#650511yescharacteristic
546C22C22×#114411yescharacteristic
547C22#9051311yescharacteristic
548C22#9051311yescharacteristic
549C22#9051911yescharacteristic
550C22#9052411yescharacteristic
551C22C2×#651611yescharacteristic
552C4—22no—
553C4—22no—
554C4—41no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—282no—
559C4—282no—
560C4—561no—
561C4—561no—
562C4#9000014yes—
563C4#9051511yescharacteristic
564C4#9052011yescharacteristic
565C4#9052211yescharacteristic
566C4#9052511yescharacteristic
567C2—42no—
568C2#1174079911yescharacteristic
569C2#1174081011yescharacteristic
570C2#1174081311yescharacteristic
571C2#1174081811yescharacteristic
572C2C2×#9000011yescharacteristic
573C2#1174152611yescharacteristic
574C2#1174154211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#533C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#533D7
hypocenterC7#24155
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. C2×#89863C2semidirect
3. C2×#89862C2semidirect
5. #11729533C2semidirect
6. #11729532C2semidirect
9. #11729386C2semidirect
10. #11729382C2semidirect
146. C22×C4×C7Q16semidirect
423. C22×C4#1144semidirect
537. C7#24155semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×#898632
C2×#898622
C4×#64872
#117295332
#117295322
#117294542
#117294492
#117293862
#117293822
#117292842
#241557