Finite Group Databasebeta

← #13380901

Subgroups — #13380901

Order 1792.

1938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#137C211yesmaximal, characteristic
2C2×#90516C211yesmaximal, characteristic
3C23×#1144C211yesmaximal, characteristic, complemented
4C2×#90005C211yesmaximal, characteristic
5#11741507C211yesmaximal, characteristic, complemented
6C2×#89892C211yesmaximal, characteristic, complemented
7#11729827C214yesmaximal, complemented
8#11729812C214yesmaximal, complemented
9#11729734C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C23×Dic14C2211yescharacteristic
13C2×C7×#137—24no—
14C2×C7×#137—24no—
15C2×C7×#137—24no—
16C22×C7×Q16C2212yes—
17C22×C7×SD16C2212yes—
18C22×C7×M16C2211yescharacteristic
19#90516—24no—
20#90516—24no—
21C2×#6507C2212yes—
22C22×#1144—24no—
23C22×#1144—24no—
24C22×#1144—24no—
25C22×#1144C2214yes—
26C2×#6499C2212yes—
27C2×#6498—22no—
28C2×#6489C2211yescharacteristic
29C2×#6489C2211yescharacteristic
30C2×#6487—22no—
31C23×C7⋊C8C2211yescharacteristic
32#90005—24no—
33#90002—22no—
34#90002C2214yes—
35C2×#6409C2212yes—
36C2×#6406C2212yes—
37#89983—22no—
38#89892—24no—
39C2×#6390C2214yes—
40#89345C2214yes—
41#23425—71nomaximal, Sylow
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41D414yes—
48C22×C7×Q8—21no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8D414yes—
55C22×C7×Q8C2312yes—
56C22×C7×Q8C2311yescharacteristic
57C22×C7×D4—21no—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C22×Dic14—22no—
64C22×Dic14—22no—
65C22×Dic14—22no—
66C22×Dic14—22no—
67C22×Dic14—21no—
68C22×Dic14—22no—
69C22×Dic14D414yes—
70C22×Dic14C2312yes—
71C7×#137—232no—
72C2×C7×Q16—24no—
73C2×C7×Q16—24no—
74C2×C7×Q16—24no—
75C2×C7×SD16—24no—
76C2×C7×SD16—24no—
77C2×C7×SD16—24no—
78C2×C7×M16—22no—
79C2×C7×M16—22no—
80C2×C7×M16D414yes—
81C22×C7×C8C2312yes—
82C2×C22⋊Dic7—24no—
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7D412yes—
85#6507—24no—
86#6507—24no—
87C2×#1144—232no—
88C2×#1144—28no—
89C2×#1144—28no—
90C2×#1144—28no—
91#6499—24no—
92#6499—24no—
93#6498—216no—
94#6489—22no—
95#6489—22no—
96#6489—22no—
97#6489—22no—
98C2×C4⋊Dic7D412yes—
99C2×C4⋊Dic7C2312yes—
100#6487—216no—
101C2×Dic7⋊C4—24no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7C2312yes—
104C22×C7⋊C8—22no—
105C22×C7⋊C8—22no—
106C22×C7⋊C8D414yes—
107C22×C7⋊C8C2312yes—
108#6409—24no—
109#6406—24no—
110C7⋊#95—28no—
111#6390—28no—
112C22×#137D711yescharacteristic
113C23×Q16—71no—
114C2×#536—71no—
115C2×#419—71no—
116C2×#416—71no—
117#2310—71no—
118#2016—74no—
119#2015—74no—
120#1729—71no—
121C24×C7—21no—
122C24×C7C2×D411yescharacteristic
123C7×#41—28no—
124C7×#41—28no—
125C7×#41—44no—
126C7×#41—44no—
127C2×C7×Q8—24no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—28no—
131C2×C7×Q8—28no—
132C2×C7×Q8—21no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—42no—
141C2×C7×Q8—44no—
142C2×C7×Q8—44no—
143C2×C7×Q8—42no—
144C2×C7×Q8C2×D414yes—
145C2×C7×Q8C2×D412yes—
146C2×C7×D4—24no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—42no—
153C2×C7×D4—42no—
154C2×C7×D4C2×D412yes—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7#4112yes—
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—24no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—42no—
175C22×Dic7—42no—
176C22×Dic7C2×D414yes—
177C2×Dic14—28no—
178C2×Dic14—28no—
179C2×Dic14—22no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14—24no—
184C2×Dic14—22no—
185C2×Dic14—44no—
186C2×Dic14—44no—
187C2×Dic14—42no—
188C2×Dic14—42no—
189C2×Dic14C2×D414yes—
190C7×Q16—216no—
191C7×SD16—216no—
192C7×M16—28no—
193C2×C7×C8—22no—
194C2×C7×C8—22no—
195C2×C7×C8C2×D414yes—
196C22⋊Dic7—216no—
197C22⋊Dic7—24no—
198C22⋊Dic7—42no—
199#1144—232no—
200C4⋊Dic7—24no—
201C4⋊Dic7—22no—
202C4⋊Dic7—22no—
203Dic7⋊C4—216no—
204C4×Dic7—22no—
205C4×Dic7—22no—
206C4×Dic7—42no—
207C2×C7⋊C8—28no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8C2×D414yes—
211C22×#41C2×D711yescharacteristic
212C23×Q8—71no—
213C23×Q8C2×D711yescharacteristic
214C2×#137—24no—
215C2×#137—24no—
216C2×#137—24no—
217C22×Q16—74no—
218C22×Q16—144no—
219C22×Q16—144no—
220C22×Q16—144no—
221C22×Q16C2×D712yes—
222C22×SD16C2×D712yes—
223C22×M16C2×D711yescharacteristic
224C23×C8—71no—
225#536—144no—
226#536—144no—
227C2×C4⋊Q8—72no—
228C2×#124—72no—
229C2×#123—142no—
230C2×C22⋊Q8—142no—
231C2×#117—71no—
232C2×#117—71no—
233#419—144no—
234#416—144no—
235C2×#103—72no—
236C2×#103—74no—
237C2×#102—72no—
238#391—74no—
239#391—142no—
240#381—142no—
241#336—74no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—44no—
247C23×C7#12012yes—
248C23×C7#12112yes—
249C23×C7#12712yes—
250C23×C7C22×D411yescharacteristic
251C7×Q8—24no—
252C7×Q8—24no—
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—41no—
256C7×Q8—42no—
257C7×Q8—42no—
258C7×Q8—41no—
259C7×Q8—48no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—48no—
263C7×D4—41no—
264C7×D4—41no—
265C2×C4×C7—22no—
266C2×C4×C7—24no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—42no—
271C2×C4×C7—48no—
272C2×C4×C7—48no—
273C2×C4×C7#12014yes—
274C2×C4×C7#12012yes—
275C2×C4×C7#12014yes—
276C2×C4×C7#12114yes—
277C2×C4×C7#12114yes—
278C2×C4×C7#12112yes—
279C2×C4×C7#12712yes—
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—28no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—48no—
292C2×Dic7—42no—
293C2×Dic7—44no—
294Dic14—24no—
295Dic14—24no—
296Dic14—42no—
297Dic14—42no—
298Dic14—48no—
299C7×C8—24no—
300C7⋊C8—24no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41C22⋊D714yes—
307C22×Q8—21no—
308C22×Q8—22no—
309C22×Q8—22no—
310C22×Q8—22no—
311C22×Q8—22no—
312C22×Q8—22no—
313C22×Q8—74no—
314C22×Q8—72no—
315C22×Q8—141no—
316C22×Q8—142no—
317C22×Q8—142no—
318C22×Q8—142no—
319C22×Q8—142no—
320C22×Q8—142no—
321C22×Q8C22⋊D714yes—
322C22×Q8C22×D711yescharacteristic
323C22×Q8C22×D712yes—
324C22×D4—21no—
325C22×D4C22×D711yescharacteristic
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—141no—
329C23×C4C22×D711yescharacteristic
330#137—232no—
331C2×Q16—24no—
332C2×Q16—24no—
333C2×Q16—24no—
334C2×Q16—148no—
335C2×Q16—148no—
336C2×Q16—1432no—
337C2×Q16—148no—
338C2×SD16—24no—
339C2×SD16—24no—
340C2×SD16—24no—
341C2×M16—22no—
342C2×M16—22no—
343C2×M16C22⋊D714yes—
344C22×C8—74no—
345C22×C8—72no—
346C22×C8—142no—
347C22×C8—142no—
348C22×C8C22×D712yes—
349C4⋊Q8—144no—
350C4⋊Q8—144no—
351#124—144no—
352#124—144no—
353#123—1416no—
354C22⋊Q8—1416no—
355#117—142no—
356#117—142no—
357#117—142no—
358#117—142no—
359C2×C4⋊C4—72no—
360C2×C4⋊C4—72no—
361C2×C4⋊C4—144no—
362C2×C22⋊C4—72no—
363C2×C22⋊C4—144no—
364C2×C22⋊C4—141no—
365C2×C42—72no—
366C2×C42—141no—
367#103—144no—
368#103—148no—
369#102—144no—
370#95—148no—
371C22×C7—24no—
372C22×C7—24no—
373C22×C7—24no—
374C22×C7—42no—
375C22×C7—44no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7#39214yes—
379C22×C7C2×#12011yescharacteristic
380C22×C7C2×#12111yescharacteristic
381C22×C7C2×#12711yescharacteristic
382C22×C7#57814yes—
383C4×C7—42no—
384C4×C7—44no—
385C4×C7#39214yes—
386C4×C7C2×#12011yescharacteristic
387C4×C7C2×#12011yescharacteristic
388C4×C7C2×#12111yescharacteristic
389C4×C7C2×#12111yescharacteristic
390Dic7—44no—
391Dic7—44no—
392Dic7—82no—
393C24—21no—
394C24D4×D711yescharacteristic
395#41—28no—
396#41—28no—
397#41—44no—
398#41—44no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—74no—
417C2×Q8—144no—
418C2×Q8—142no—
419C2×Q8—148no—
420C2×Q8—148no—
421C2×Q8—142no—
422C2×Q8—142no—
423C2×Q8—142no—
424C2×Q8—142no—
425C2×Q8—282no—
426C2×Q8—284no—
427C2×Q8—282no—
428C2×Q8—284no—
429C2×Q8C2×C22⋊D712yes—
430C2×Q8C2×C22⋊D714yes—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×C22⋊D712yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—74no—
449C22×C4—144no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4D4×D712yes—
455C22×C4D4×D711yescharacteristic
456C22×C4D4×D712yes—
457C22×C4#115912yes—
458C22×C4C2×C22⋊D712yes—
459C22×C4C2×C22⋊D712yes—
460C22×C4C2×C22⋊D712yes—
461C22×C4C23×D711yescharacteristic
462Q16—216no—
463Q16—1432no—
464SD16—216no—
465M16—28no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—74no—
469C2×C8—148no—
470C2×C8—142no—
471C2×C8—142no—
472C2×C8C2×C22⋊D714yes—
473C4⋊C4—1416no—
474C4⋊C4—144no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C22⋊C4—1416no—
478C22⋊C4—144no—
479C22⋊C4—282no—
480C42—142no—
481C42—142no—
482C42—282no—
483C2×C7—24no—
484C2×C7—82no—
485C2×C7C2×#39211yescharacteristic
486C2×C7#301112yes—
487C2×C7#304412yes—
488C2×C7#315612yes—
489C23—24no—
490C23—22no—
491C23—44no—
492C23—41no—
493C23—41no—
494C23#650012yes—
495C23#650212yes—
496C23#650312yes—
497C23C2×D4×D711yescharacteristic
498Q8—24no—
499Q8—22no—
500Q8—24no—
501Q8—22no—
502Q8—41no—
503Q8—42no—
504Q8—41no—
505Q8—42no—
506Q8—48no—
507Q8—144no—
508Q8—144no—
509Q8—282no—
510Q8—288no—
511Q8—282no—
512D4—22no—
513D4—22no—
514D4—48no—
515D4—41no—
516D4—41no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—148no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—282no—
529C2×C4—282no—
530C2×C4—288no—
531C2×C4—284no—
532C2×C4—282no—
533C2×C4#650014yes—
534C2×C4#650012yes—
535C2×C4#650114yes—
536C2×C4#650214yes—
537C2×C4#650212yes—
538C2×C4#650312yes—
539C2×C4#651614yes—
540C2×C4C2×D4×D711yescharacteristic
541C2×C4C2×D4×D711yescharacteristic
542C2×C4C2×#115911yescharacteristic
543C2×C4C22×C22⋊D711yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C2×C4C22×C22⋊D711yescharacteristic
546C8—24no—
547C8—144no—
548C7#2342511yescharacteristic, complemented, Sylow
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—44no—
554C22—82no—
555C22—82no—
556C22#8999514yes—
557C22#9047014yes—
558C22C2×#650011yescharacteristic
559C22C2×#650211yescharacteristic
560C22C2×#650311yescharacteristic
561C4—42no—
562C4—44no—
563C4—284no—
564C4—284no—
565C4—562no—
566C4#8999514yes—
567C4C2×#650011yescharacteristic
568C4C2×#650111yescharacteristic
569C4C2×#650211yescharacteristic
570C4C2×#651611yescharacteristic
571C2—24no—
572C2—82no—
573C2#1174141512yes—
574C2#1174142012yes—
575C2#1174142812yes—
576C2C2×#8999511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C7×#137C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC22×#137D7
hypocenterC7#23425
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C23×#1144C2semidirect
5. #11741507C2semidirect
6. C2×#89892C2semidirect
7. #11729827C2semidirect
8. #11729812C2semidirect
9. #11729734C2semidirect
548. C7#23425semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C2×#905162
C23×#11442
C2×#900052
#117415072
C2×#898922
#117298272
#117298122
#117297342
#234257