Finite Group Databasebeta

← #13410073

Subgroups — #13410073

Order 1792.

1230 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×#89854C211yesmaximal, characteristic, complemented
3C2×#89828C211yesmaximal, characteristic, complemented
4#11739710C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal
6#11729886C212yesmaximal, complemented
7#11729539C211yesmaximal, characteristic, complemented
8#11729535C212yesmaximal, complemented
9#11729480C211yesmaximal, characteristic, complemented
10#11729476C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#536—24no—
15C7×#536—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C7×#124C2211yescharacteristic
20C2×C7×#123—22no—
21C2×C7×C22⋊Q8—22no—
22C2×C7×#117C2211yescharacteristic
23C2×#6433C2211yescharacteristic
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32#89984C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34#89854—24no—
35#89828—24no—
36#89756—22no—
37#89720—22no—
38#89707C2211yescharacteristic
39#89700C2212yes—
40C2×#6377C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89319C2212yes—
43#25868—71nomaximal, Sylow
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—24no—
47C2×C7×#41—42no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—21no—
51C22×C7×Q8—24no—
52C22×C7×Q8—42no—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×Q8C2311yescharacteristic
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D414yes—
65C7×#124—22no—
66C7×#124—22no—
67C7×#124—22no—
68C7×#124D414yes—
69C7×#123—216no—
70C7×C22⋊Q8—216no—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4D412yes—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4D412yes—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C22⋊Dic7—22no—
82C2×C4⋊Dic7—22no—
83C2×C4⋊Dic7C2312yes—
84C2×C4⋊Dic7C2312yes—
85C2×Dic7⋊C4—22no—
86C2×Dic7⋊C4—22no—
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2312yes—
89#6433—22no—
90#6433—22no—
91#6409—24no—
92#6409—24no—
93#6409—24no—
94#6409—42no—
95#6409—42no—
96#6409—42no—
97#6406—24no—
98#6406—42no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101C22⋊(C7⋊C8)—24no—
102#6377—22no—
103C4×C7⋊C8—22no—
104C2×#536D711yescharacteristic
105C2×#491—71no—
106C2×#488—71no—
107#2629—71no—
108#2043—72no—
109#2042—72no—
110#1959—71no—
111#1956—71no—
112#1788—72no—
113#1787—72no—
114#1750—71no—
115C24×C7—21no—
116C24×C7C2×D411yescharacteristic
117C7×#41—48no—
118C7×#41—48no—
119C2×C7×Q8—24no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—48no—
130C2×C7×Q8—48no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×D412yes—
135C2×C7×Q8C2×D412yes—
136C2×C7×D4—24no—
137C2×C7×D4—21no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4C2×D412yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7#4114yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—22no—
161C22×Dic7—22no—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—24no—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—24no—
168C42×C7—21no—
169C42×C7—21no—
170C42×C7C2×D412yes—
171C42×C7C2×D412yes—
172C22⋊Dic7—82no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—82no—
178Dic7⋊C4—44no—
179Dic7⋊C4—82no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C22×#41C2×D711yescharacteristic
187C23×Q8C2×D711yescharacteristic
188#536—24no—
189#536—24no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×C4⋊Q8C2×D711yescharacteristic
192C2×#125—71no—
193C2×#124C2×D711yescharacteristic
194C2×#124C2×D711yescharacteristic
195C2×#123—22no—
196C2×C22⋊Q8—22no—
197C2×#117C2×D711yescharacteristic
198#491—144no—
199#488—144no—
200C2×#103—72no—
201C2×#103—72no—
202C2×#103—72no—
203C2×#103—72no—
204C2×#103—72no—
205C2×#103—72no—
206C2×#102—72no—
207C2×#102—72no—
208C2×C22⋊C8—72no—
209C2×#97—71no—
210C2×C4×C8—71no—
211#398—142no—
212#396—142no—
213#389—72no—
214#384—71no—
215#380—72no—
216#336—72no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—22no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7#12412yes—
227C23×C7C22×D411yescharacteristic
228C7×Q8—24no—
229C7×Q8—24no—
230C7×Q8—24no—
231C7×Q8—48no—
232C7×Q8—42no—
233C7×Q8—42no—
234C7×Q8—42no—
235C7×D4—24no—
236C7×D4—48no—
237C7×D4—42no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—24no—
243C2×C4×C7—24no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—48no—
251C2×C4×C7—48no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7#12014yes—
259C2×C4×C7#12412yes—
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×C4×C7C2×#4112yes—
263C2×C4×C7H(5,𝔽2)12yes—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C7⋊C8—42no—
271C7⋊C8—42no—
272C2×#41—22no—
273C2×#41—22no—
274C2×#41—24no—
275C2×#41—42no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8—21no—
279C22×Q8—24no—
280C22×Q8—42no—
281C22×Q8C22×D711yescharacteristic
282C22×Q8C22×D711yescharacteristic
283C22×Q8C22×D711yescharacteristic
284C22×D4—21no—
285C22×D4C22×D711yescharacteristic
286C23×C4—21no—
287C23×C4—21no—
288C23×C4C22×D711yescharacteristic
289C22×C8—72no—
290C22×C8—72no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8C22⋊D714yes—
295#125—142no—
296#125—142no—
297#124—22no—
298#124—22no—
299#124—22no—
300#124C22⋊D714yes—
301#123—216no—
302C22⋊Q8—216no—
303#117—22no—
304#117—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—72no—
308C2×C4⋊C4—72no—
309C2×C4⋊C4—142no—
310C2×C4⋊C4—142no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4C22⋊D712yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—142no—
316C2×C22⋊C4C22⋊D712yes—
317C2×C42C22×D711yescharacteristic
318C2×C42C22×D711yescharacteristic
319#103—144no—
320#103—144no—
321#103—144no—
322#103—282no—
323#103—282no—
324#103—282no—
325#102—144no—
326#102—282no—
327C22⋊C8—144no—
328#97—142no—
329C4×C8—142no—
330#95—142no—
331#95—142no—
332C22×C7—24no—
333C22×C7—22no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7—42no—
337C22×C7—42no—
338C22×C7—44no—
339C22×C7—82no—
340C22×C7—81no—
341C22×C7—81no—
342C22×C7C2×#12011yescharacteristic
343C22×C7C2×#12411yescharacteristic
344C22×C7#53511yescharacteristic
345C22×C7#57712yes—
346C22×C7#57812yes—
347C4×C7—22no—
348C4×C7—24no—
349C4×C7—42no—
350C4×C7—42no—
351C4×C7—42no—
352C4×C7—42no—
353C4×C7#54612yes—
354C4×C7#54612yes—
355Dic7—82no—
356Dic7—82no—
357C24—21no—
358C24C2×C22⋊D711yescharacteristic
359#41—48no—
360#41—48no—
361C2×Q8—24no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8C2×C22⋊D712yes—
376C2×Q8C2×C22⋊D712yes—
377C2×Q8C2×C22⋊D712yes—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4C2×C22⋊D712yes—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4#115914yes—
400C22×C4C2×C22⋊D712yes—
401C22×C4C2×C22⋊D711yescharacteristic
402C22×C4C2×C22⋊D712yes—
403C22×C4C23×D711yescharacteristic
404C2×C8—142no—
405C2×C8—142no—
406C2×C8—142no—
407C2×C8—142no—
408C2×C8—142no—
409C2×C8—142no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—142no—
414C4⋊C4—142no—
415C4⋊C4—142no—
416C4⋊C4—142no—
417C4⋊C4—284no—
418C4⋊C4—562no—
419C4⋊C4—562no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—24no—
423C22⋊C4—562no—
424C42—21no—
425C42—21no—
426C42C2×C22⋊D712yes—
427C42C2×C22⋊D712yes—
428C2×C7—42no—
429C2×C7—82no—
430C2×C7#243511yescharacteristic
431C2×C7#301312yes—
432C2×C7#313712yes—
433C2×C7#322111yescharacteristic
434C2×C7#322311yescharacteristic
435C23—21no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23#649912yes—
445C23C22×C22⋊D711yescharacteristic
446Q8—24no—
447Q8—24no—
448Q8—24no—
449Q8—48no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453D4—24no—
454D4—48no—
455D4—42no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—282no—
482C2×C4#649912yes—
483C2×C4#651614yes—
484C2×C4C2×#115912yes—
485C2×C4#654812yes—
486C2×C4C22×C22⋊D711yescharacteristic
487C2×C4C22×C22⋊D711yescharacteristic
488C8—282no—
489C8—282no—
490C7#2586811yescharacteristic, complemented, Sylow
491C22—22no—
492C22—24no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—42no—
497C22—44no—
498C22—82no—
499C22—81no—
500C22—81no—
501C22C2×#649911yescharacteristic
502C22#9049611yescharacteristic
503C22#9052812yes—
504C22#9052912yes—
505C22C2×#651611yescharacteristic
506C4—22no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—562no—
513C4—562no—
514C4#9049712yes—
515C4#9049712yes—
516C2—42no—
517C2—82no—
518C2#1174084612yes—
519C2#1174085411yescharacteristic
520C2#1174085911yescharacteristic
521C2#1174149511yescharacteristic
522C2#1174155912yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×#536D7
hypocenterC7#25868
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C2×#89854C2semidirect
3. C2×#89828C2semidirect
4. #11739710C2semidirect
6. #11729886C2semidirect
7. #11729539C2semidirect
8. #11729535C2semidirect
9. #11729480C2semidirect
10. #11729476C2semidirect
11. #11729336C2semidirect
490. C7#25868semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×#898542
C2×#898282
#117397102
#117298892
#117298862
#117295392
#117295352
#117294802
#117294762
#117293362
#258687