Finite Group Databasebeta

← #13363712

Subgroups — #13363712

Order 1792.

1118 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2C2×D7×C4⋊C8C211yesmaximal, characteristic
3#11731592C211yesmaximal, characteristic
4C7×#1805C211yesmaximal, characteristic, complemented
5#11729381C211yesmaximal, characteristic, complemented
6#11728240C211yesmaximal, characteristic, complemented
7#11728024C212yesmaximal
8#11727983C212yesmaximal, complemented
9#11727313C211yesmaximal, characteristic, complemented
10#11726636C212yesmaximal
11#11726554C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×#6507C2211yescharacteristic
15C22×C8×D7C2212yes—
16#90269—24no—
17#90269C418yescomplemented
18C2×#6460C2211yescharacteristic
19C2×#6459—21no—
20C2×#6452—21no—
21C2×#6451C2211yescharacteristic
22C2×#6435—21no—
23C2×C42×D7C2211yescharacteristic
24C2×C4×Dic14—21no—
25C2×C4×Dic14C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C2×C7×#103C2212yes—
28C7×#382C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6390C2212yes—
31C2×Dic7⋊C8C2212yes—
32#89759C2212yes—
33C4⋊C4×Dic7C2212yes—
34C2×#6384C2212yes—
35#89708C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37D7×C4⋊C8—24no—
38#89438—21no—
39#89436—21no—
40#89423—21no—
41#89420—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21889—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7D411yescharacteristic
50C22×C4×D7C2311yescharacteristic
51C22×Dic14—21no—
52C22×Dic14C2311yescharacteristic
53C22×C7×C8C2312yes—
54C4×C7×Q8—22no—
55C4×C7×Q8C2×C414yes—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C42×C7—21no—
60C2×C42×C7C2311yescharacteristic
61#6509—22no—
62#6509C2×C414yes—
63#6507—22no—
64#6507C2×C414yes—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—22no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—21no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C22×C7⋊C8C2312yes—
82C2×C8×D7—24no—
83C2×C8×D7—42no—
84C2×C8×D7—42no—
85#6460—22no—
86#6460C2×C414yes—
87#6459—28no—
88#6452—28no—
89#6451—22no—
90#6451C2×C414yes—
91#6435—28no—
92C42×D7C2×C414yes—
93C42×D7D414yes—
94C4×Dic14—22no—
95C4×Dic14—28no—
96C4×Dic14C2×C414yes—
97C7×C4⋊C8—22no—
98C7×#103—42no—
99C7×#95—21no—
100C7×#95—21no—
101#6409—42no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106#6390—42no—
107Dic7⋊C8—24no—
108#6384—42no—
109C4⋊(C7⋊C8)—22no—
110C2×#533—71no—
111C22×C4⋊C8—71no—
112#2283—71no—
113#1988—72no—
114#1981—72no—
115#1892—72no—
116#1892—72no—
117#1805—71no—
118#1805—71no—
119#1805—71no—
120#1805D711yescharacteristic, complemented
121C23×D7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8C22×C412yes—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—21no—
136C22×Dic7—22no—
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C2×C4×D7—24no—
141C2×C4×D7—21no—
142C2×C4×D7—21no—
143C2×C4×D7—21no—
144C2×C4×D7—21no—
145C2×C4×D7—24no—
146C2×C4×D7C22⋊C414yes—
147C2×C4×D7C22⋊C414yes—
148C2×C4×D7C22×C412yes—
149C2×C4×D7C2×D412yes—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—24no—
156C2×Dic14—44no—
157C2×Dic14C22×C412yes—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4C22×C412yes—
166C42×C7—24no—
167C42×C7C22×C412yes—
168C42×C7C2×D412yes—
169(C2×D7)⋊C4—24no—
170(C2×D7)⋊C4—24no—
171(C2×D7)⋊C4—44no—
172C4⋊Dic7—21no—
173C4⋊Dic7—24no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7C22×C412yes—
177Dic7⋊C4—24no—
178Dic7⋊C4—24no—
179Dic7⋊C4—24no—
180Dic7⋊C4—24no—
181Dic7⋊C4—24no—
182C4×Dic7—24no—
183C4×Dic7—44no—
184C4×Dic7C22⋊C414yes—
185C4×Dic7C22⋊C414yes—
186C4×Dic7C22×C412yes—
187C4×Dic7C2×D412yes—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C8×D7—48no—
192C23×C8—72no—
193#533—78no—
194#533—144no—
195C2×C4⋊Q8—71no—
196C2×C4⋊Q8—71no—
197C2×#125—141no—
198C2×C22⋊Q8—71no—
199C2×C22⋊Q8—71no—
200C2×C22⋊Q8—141no—
201C2×C4×Q8—71no—
202C2×C4×Q8—141no—
203C2×C4×Q8C2×D711yescharacteristic
204C2×#117—141no—
205C22×C42—71no—
206C2×C4⋊C8—72no—
207C2×C4⋊C8—71no—
208C2×C4⋊C8—144no—
209C2×C4⋊C8C2×D711yescharacteristic
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103C2×D712yes—
214#383—141no—
215#382—71no—
216#382C2×D711yescharacteristic
217#381—141no—
218#380—72no—
219C4×C4⋊C4—72no—
220C4×C22⋊C4—141no—
221#376—141no—
222#340—72no—
223#340—72no—
224#340—72no—
225#340C2×D712yes—
226C23×C7C22×D411yescharacteristic
227C22×D7—21no—
228C22×D7—21no—
229C22×D7—22no—
230C22×D7C2×C22⋊C412yes—
231C22×D7#13714yes—
232C7×Q8—22no—
233C7×Q8—22no—
234C7×Q8—41no—
235C7×Q8—41no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7C2×C22⋊C412yes—
247C2×C4×C7C2×C22⋊C412yes—
248C2×C4×C7C23×C411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7H(5,𝔽2)11yescharacteristic
251C2×C4×C7H(5,𝔽2)11yescharacteristic
252C2×C4×C72−1+411yescharacteristic
253C2×C4×C72−1+411yescharacteristic
254C2×Dic7—21no—
255C2×Dic7—21no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7C2×C22⋊C412yes—
266C2×Dic7C2×C22⋊C412yes—
267C2×Dic7C2×C22⋊C412yes—
268C2×Dic7#13514yes—
269C4×D7—28no—
270C4×D7—22no—
271C4×D7—22no—
272C4×D7—22no—
273C4×D7—22no—
274C4×D7—28no—
275Dic14—22no—
276Dic14—22no—
277Dic14—41no—
278Dic14—41no—
279Dic14—48no—
280C7×C8—42no—
281C7⋊C8—42no—
282C22×Q8—71no—
283C22×Q8—141no—
284C22×Q8C22×D711yescharacteristic
285C23×C4—71no—
286C23×C4—71no—
287C23×C4—71no—
288C22×C8—72no—
289C22×C8—144no—
290C22×C8—282no—
291C22×C8—282no—
292C22×C8C22×D712yes—
293C4⋊Q8—74no—
294C4⋊Q8—74no—
295C4⋊Q8—142no—
296C4⋊Q8—142no—
297#125—148no—
298C22⋊Q8—74no—
299C22⋊Q8—74no—
300C22⋊Q8—142no—
301C22⋊Q8—148no—
302C22⋊Q8—142no—
303C4×Q8—22no—
304C4×Q8—74no—
305C4×Q8—148no—
306C4×Q8—142no—
307C4×Q8C4×D714yes—
308#117—148no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4C22×D712yes—
319C2×C4⋊C4C22×D711yescharacteristic
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—142no—
323C2×C42—21no—
324C2×C42—71no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—74no—
328C2×C42—74no—
329C2×C42—141no—
330C2×C42—142no—
331C2×C42C22×D711yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—144no—
334C4⋊C8—142no—
335#103—42no—
336#103—282no—
337#103—282no—
338#103—282no—
339#95—21no—
340#95—21no—
341#95—142no—
342#95—142no—
343#95—141no—
344#95—141no—
345C22×C7C22×C22⋊C411yescharacteristic
346C22×C7#53711yescharacteristic
347C22×C7#53711yescharacteristic
348C22×C7C2×#13512yes—
349C22×C7C2×#13712yes—
350C2×D7—22no—
351C2×D7—22no—
352C2×D7—24no—
353C2×D7—24no—
354C2×D7#41914yes—
355C4×C7—22no—
356C4×C7—22no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7—82no—
360C4×C7#52011yescharacteristic
361C4×C7#52011yescharacteristic
362C4×C7#53511yescharacteristic
363C4×C7#53611yescharacteristic
364Dic7—22no—
365Dic7—22no—
366Dic7—42no—
367Dic7—42no—
368Dic7—82no—
369Dic7#41614yes—
370C24—71no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—72no—
377C2×Q8—144no—
378C2×Q8—141no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8—284no—
384C2×Q8C2×C4×D712yes—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—71no—
389C22×C4—71no—
390C22×C4—71no—
391C22×C4—74no—
392C22×C4—74no—
393C22×C4—72no—
394C22×C4—72no—
395C22×C4—144no—
396C22×C4—144no—
397C22×C4—141no—
398C22×C4—141no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—142no—
404C22×C4D4×D711yescharacteristic
405C22×C4D4×D711yescharacteristic
406C22×C4C23×D711yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—142no—
411C2×C8—142no—
412C2×C8—142no—
413C2×C8—288no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—72no—
419C4⋊C4—144no—
420C4⋊C4—144no—
421C4⋊C4—144no—
422C4⋊C4—144no—
423C4⋊C4—144no—
424C4⋊C4—144no—
425C4⋊C4—141no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4C2×C4×D712yes—
429C22⋊C4—144no—
430C22⋊C4—144no—
431C22⋊C4—284no—
432C42—24no—
433C42—74no—
434C42—74no—
435C42—72no—
436C42—72no—
437C42—144no—
438C42—284no—
439C42C2×C4×D712yes—
440C42D4×D712yes—
441C2×C7#231111yescharacteristic
442C2×C7C2×#41611yescharacteristic
443C2×C7C2×#41911yescharacteristic
444C2×C7#324812yes—
445C2×C7#324812yes—
446D7—24no—
447C23—72no—
448C23—74no—
449C23—142no—
450C23—141no—
451C23—141no—
452C23C2×D4×D711yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457Q8—142no—
458Q8—142no—
459Q8—288no—
460Q8—281no—
461Q8—281no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—72no—
473C2×C4—74no—
474C2×C4—72no—
475C2×C4—72no—
476C2×C4—148no—
477C2×C4—148no—
478C2×C4—142no—
479C2×C4—141no—
480C2×C4—141no—
481C2×C4—142no—
482C2×C4—142no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—282no—
488C2×C4—281no—
489C2×C4—281no—
490C2×C4—282no—
491C2×C4—282no—
492C2×C4—282no—
493C2×C4D7×C22⋊C412yes—
494C2×C4D7×C22⋊C412yes—
495C2×C4C22×C4×D711yescharacteristic
496C2×C4C2×D4×D711yescharacteristic
497C2×C4#654811yescharacteristic
498C2×C4#655111yescharacteristic
499C2×C4#655311yescharacteristic
500C2×C4#655411yescharacteristic
501C8—42no—
502C8—282no—
503C7#2188911yescharacteristic, complemented, Sylow
504C22—74no—
505C22—144no—
506C22—144no—
507C22—142no—
508C22—142no—
509C22C2×D7×C22⋊C411yescharacteristic
510C22#9029911yescharacteristic
511C22#9032011yescharacteristic
512C22D7×#13512yes—
513C22D7×#13712yes—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—74no—
519C4—82no—
520C4—142no—
521C4—142no—
522C4—282no—
523C4—282no—
524C4—562no—
525C4#9019511yescharacteristic
526C4#9021411yescharacteristic
527C4#9029811yescharacteristic
528C4#9031911yescharacteristic
529C2—144no—
530C2#1173144811yescharacteristic
531C2D7×#41611yescharacteristic
532C2D7×#41911yescharacteristic
533C2#1173802112yes—
534C2#1173865712yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1805C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1805D7
hypocenterC7#21889
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. C7×#1805C2semidirect
5. #11729381C2semidirect
6. #11728240C2semidirect
8. #11727983C2semidirect
9. #11727313C2semidirect
11. #11726554C2semidirect
17. #90269C4semidirect
120. #1805D7semidirect
503. C7#21889semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
C2×D7×C4⋊C82
#117315922
C7×#18052
#117293812
#117282402
#117280242
#117279832
#117273132
#117266362
#117265542
#218897