Finite Group Databasebeta

← #13364978

Subgroups — #13364978

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#451C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic
3Dic7×C22⋊Q8C211yesmaximal, characteristic
4#11729886C211yesmaximal, characteristic
5#11729820C212yesmaximal, complemented
6#11729819C212yesmaximal, complemented
7#11729446C211yesmaximal, characteristic
8#11728007C212yesmaximal, complemented
9#11727978C212yesmaximal, complemented
10#11727681C211yesmaximal, characteristic
11Dic7×C22⋊C8C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×C7×Q16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×Q8×Dic7C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C7×#451C418yescomplemented
19C2×C7×#103C2212yes—
20C2×C7×C22⋊C8C2211yescharacteristic
21#90002—21no—
22#90002—21no—
23#90002C2211yescharacteristic
24#90002C2212yes—
25#90002C2212yes—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29#89984C2211yescharacteristic
30C2×C7⋊#95—21no—
31C2×C7⋊#95—21no—
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6392C2212yes—
34C2×C8×Dic7C2212yes—
35#89757C2211yescharacteristic
36#89753—21no—
37C4⋊C4×Dic7—21no—
38C4⋊C4×Dic7C2211yescharacteristic
39#89722—21no—
40C22⋊C4×Dic7—21no—
41#89346C2212yes—
42#89345C2212yes—
43#89324C2212yes—
44#22044—71nomaximal, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yes—
50C22×C7×Q8C2312yes—
51C22×C7×Q8C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C23×Dic7D411yescharacteristic
57C2×C7×Q16—42no—
58C2×C7×Q16—42no—
59C2×C7×Q16C2×C418yescomplemented
60C22×C7×C8C2312yes—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8C2×C414yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C22⋊Dic7—21no—
67Q8×Dic7—42no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—42no—
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2312yes—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×C4×Dic7—21no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7—42no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C22×C7⋊C8C2312yes—
83C7×#103C2×C418yescomplemented
84C7×C22⋊C8C2×C414yes—
85#6409—42no—
86#6409—42no—
87#6409—42no—
88C7⋊#95—21no—
89C7⋊#95—21no—
90C7⋊#95—21no—
91C7⋊#95—21no—
92C7⋊#95—21no—
93C7⋊#95—22no—
94C7⋊#95—22no—
95C7⋊#95—22no—
96C7⋊#95—22no—
97C22⋊(C7⋊C8)—22no—
98#6392—42no—
99C8×Dic7—42no—
100C2×#451D711yescharacteristic
101C4×C22⋊Q8—71no—
102C2×#391—71no—
103#1969—72no—
104#1933—72no—
105#1928—72no—
106#1878—72no—
107#1872—71no—
108#1852—71no—
109#1787—71no—
110C4×C22⋊C8—71no—
111C24×C7C2×D411yescharacteristic
112C2×C7×Q8—24no—
113C2×C7×Q8—21no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—48no—
122C2×C7×Q8—42no—
123C2×C7×Q8—42no—
124C2×C7×Q8—48no—
125C2×C7×Q8C22×C414yes—
126C2×C7×Q8C22×C412yes—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7#4112yes—
138C22×C4×C7C2411yescharacteristic
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—21no—
144C22×Dic7—41no—
145C22×Dic7—41no—
146C22×Dic7—41no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C22×Dic7Q1614yescomplemented
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C7×Q16—48no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8C22×C414yes—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4C22×C412yes—
162C7×C22⋊C4—24no—
163C22⋊Dic7—42no—
164C4⋊Dic7—22no—
165C4⋊Dic7—22no—
166C4⋊Dic7—22no—
167C4⋊Dic7—22no—
168C4⋊Dic7—42no—
169C4⋊Dic7—42no—
170C4⋊Dic7—42no—
171C4×Dic7—22no—
172C4×Dic7—22no—
173C4×Dic7—41no—
174C4×Dic7—41no—
175C4×Dic7—42no—
176C4×Dic7—42no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C23×Q8C2×D711yescharacteristic
181C22×Q16C2×D712yes—
182C2×C22⋊Q8C2×D711yescharacteristic
183C2×C4×Q8—71no—
184C22×C4⋊C4—71no—
185C22×C42—71no—
186#451Dic718yescomplemented
187C2×#107—72no—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#103C2×D712yes—
192C2×C22⋊C8—71no—
193C2×C22⋊C8C2×D711yescharacteristic
194C2×C4×C8—72no—
195#391—72no—
196#391—72no—
197#391—71no—
198#391—141no—
199#391—141no—
200#389—71no—
201#387—141no—
202#385—141no—
203#380—71no—
204C4×C4⋊C4—71no—
205C4×C4⋊C4—141no—
206C4×C22⋊C4—141no—
207C2×#95—141no—
208C2×#95—141no—
209#340—72no—
210#340—72no—
211#336—72no—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7C4×D412yes—
217C23×C7C2×Q1612yes—
218C23×C7#13512yes—
219C23×C7C22×D411yescharacteristic
220C7×Q8—24no—
221C7×Q8—24no—
222C7×Q8—24no—
223C7×Q8—48no—
224C7×Q8—42no—
225C7×Q8—42no—
226C7×Q8—42no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—24no—
234C2×C4×C7—48no—
235C2×C4×C7—41no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7C4×D412yes—
242C2×C4×C7C23×C411yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C2×C4×C7H(5,𝔽2)12yes—
245C2×C4×C7H(5,𝔽2)12yes—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—21no—
253C2×Dic7—21no—
254C2×Dic7—21no—
255C2×Dic7—44no—
256C2×Dic7—44no—
257C2×Dic7—41no—
258C2×Dic7—42no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—42no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—84no—
267C2×Dic7C2×Q1612yes—
268C2×Dic7C2×Q1612yes—
269C2×Dic7#13712yes—
270C7×C8—24no—
271C7⋊C8—42no—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8—42no—
275C22×Q8—42no—
276C22×Q8C2×Dic714yes—
277C22×Q8C22×D711yescharacteristic
278C22×Q8C22×D712yes—
279C23×C4—21no—
280C23×C4—71no—
281C23×C4—71no—
282C23×C4—141no—
283C23×C4C22×D711yescharacteristic
284C2×Q16—42no—
285C2×Q16—42no—
286C2×Q16C2×Dic718yescomplemented
287C22×C8—72no—
288C22×C8C22×D712yes—
289C22⋊Q8—22no—
290C22⋊Q8C2×Dic714yes—
291C4×Q8—282no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—72no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—71no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—141no—
299C2×C4⋊C4—282no—
300C2×C4⋊C4C22×D711yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—141no—
303C2×C42—71no—
304C2×C42—71no—
305C2×C42—71no—
306C2×C42—71no—
307C2×C42—141no—
308C2×C42—141no—
309C2×C42—282no—
310#107—282no—
311#103—282no—
312#103—282no—
313#103—282no—
314#103C2×Dic718yescomplemented
315C22⋊C8—142no—
316C22⋊C8C2×Dic714yes—
317C4×C8—282no—
318#95—142no—
319#95—142no—
320#95—142no—
321#95—142no—
322#95—141no—
323#95—141no—
324#95—141no—
325#95—141no—
326#95—141no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—22no—
333C22×C7C4×Q1614yes—
334C22×C7C2×C4×D411yescharacteristic
335C22×C7#53411yescharacteristic
336C22×C7#53511yescharacteristic
337C22×C7C22×Q1611yescharacteristic
338C22×C7C2×#13511yescharacteristic
339C22×C7C2×#13711yescharacteristic
340C22×C7#57511yescharacteristic
341C4×C7—22no—
342C4×C7—44no—
343C4×C7—42no—
344C4×C7—42no—
345C4×C7#51812yes—
346C4×C7#53812yes—
347Dic7—22no—
348Dic7—22no—
349Dic7—22no—
350Dic7—41no—
351Dic7—81no—
352Dic7—81no—
353Dic7—82no—
354C24D4×D711yescharacteristic
355C2×Q8—24no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—48no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—48no—
368C2×Q8C22×Dic714yes—
369C2×Q8C22×Dic712yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—71no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—74no—
382C22×C4—71no—
383C22×C4—142no—
384C22×C4—142no—
385C22×C4—141no—
386C22×C4—142no—
387C22×C4—141no—
388C22×C4—281no—
389C22×C4—281no—
390C22×C4—281no—
391C22×C4—282no—
392C22×C4—282no—
393C22×C4—282no—
394C22×C4D4×D711yescharacteristic
395C22×C4#115912yes—
396C22×C4C22×Dic712yes—
397C22×C4C23×D711yescharacteristic
398Q16—48no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—142no—
402C2×C8—142no—
403C2×C8—142no—
404C2×C8C22×Dic714yes—
405C4⋊C4—24no—
406C4⋊C4—21no—
407C4⋊C4—142no—
408C4⋊C4—142no—
409C4⋊C4—142no—
410C4⋊C4—142no—
411C4⋊C4—282no—
412C4⋊C4—282no—
413C4⋊C4—282no—
414C4⋊C4C22×Dic712yes—
415C22⋊C4—24no—
416C22⋊C4—282no—
417C42—142no—
418C42—142no—
419C42—281no—
420C42—281no—
421C42—282no—
422C42—282no—
423C2×C7—24no—
424C2×C7#232111yescharacteristic
425C2×C7C2×C4×Q1611yescharacteristic
426C2×C7C4×#13711yescharacteristic
427C2×C7#318911yescharacteristic
428C2×C7#319011yescharacteristic
429C2×C7#320911yescharacteristic
430C2×C7#321111yescharacteristic
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23D7×Q1612yes—
436C23#648212yes—
437C23D4×Dic712yes—
438C23C2×D4×D711yescharacteristic
439Q8—24no—
440Q8—24no—
441Q8—24no—
442Q8—48no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—72no—
461C2×C4—72no—
462C2×C4—72no—
463C2×C4—141no—
464C2×C4—141no—
465C2×C4—141no—
466C2×C4—141no—
467C2×C4—141no—
468C2×C4—141no—
469C2×C4—141no—
470C2×C4—141no—
471C2×C4—141no—
472C2×C4—284no—
473C2×C4—284no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—282no—
477C2×C4—281no—
478C2×C4—281no—
479C2×C4—282no—
480C2×C4—281no—
481C2×C4—282no—
482C2×C4—281no—
483C2×C4—564no—
484C2×C4D4×Dic712yes—
485C2×C4C2×#115911yescharacteristic
486C2×C4#654812yes—
487C2×C4#654812yes—
488C2×C4C23×Dic711yescharacteristic
489C8—24no—
490C8—282no—
491C7#2204411yescharacteristic, complemented, Sylow
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22Q16×Dic714yes—
499C22#9028011yescharacteristic
500C22#9028311yescharacteristic
501C22C2×D7×Q1611yescharacteristic
502C22C2×#648211yescharacteristic
503C22D7×#13711yescharacteristic
504C22#9046811yescharacteristic
505C22C2×D4×Dic711yescharacteristic
506C4—22no—
507C4—44no—
508C4—42no—
509C4—42no—
510C4—142no—
511C4—142no—
512C4—142no—
513C4—281no—
514C4—561no—
515C4—562no—
516C4—561no—
517C4#9028412yes—
518C4#9048912yes—
519C2—24no—
520C2#1173762311yescharacteristic
521C2#1173763211yescharacteristic
522C2#1173763311yescharacteristic
523C2#1173764611yescharacteristic
524C2#1174002311yescharacteristic
525C2C2×Q16×Dic711yescharacteristic
526C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#451C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×#451D7
hypocenterC7#22044
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
5. #11729820C2semidirect
6. #11729819C2semidirect
8. #11728007C2semidirect
9. #11727978C2semidirect
18. C7×#451C4semidirect
59. C2×C7×Q16C2×C4semidirect
83. C7×#103C2×C4semidirect
150. C22×Dic7Q16semidirect
186. #451Dic7semidirect
286. C2×Q16C2×Dic7semidirect
314. #103C2×Dic7semidirect
491. C7#22044semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4512
C2×#900022
Dic7×C22⋊Q82
#117298862
#117298202
#117298192
#117294462
#117280072
#117279782
#117276812
Dic7×C22⋊C82
#220447