Finite Group Databasebeta

← #13514916

Subgroups — #13514916

Order 1792.

2390 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90265C211yesmaximal, characteristic
2C7×#2276C211yesmaximal, characteristic, complemented
3#11739906C211yesmaximal, characteristic, complemented
4Q8×Dic7⋊C4C211yesmaximal, characteristic, complemented
5#11739902C211yesmaximal, characteristic
6#11739902C211yesmaximal, characteristic
7C4×Q8×Dic7C211yesmaximal, characteristic, complemented
8#11731213C213yesmaximal, complemented
9#11731212C213yesmaximal, complemented
10#11731205C213yesmaximal, complemented
11#11731091C213yesmaximal, complemented
12#11731084C213yesmaximal, complemented
13#11731062C213yesmaximal, complemented
14#11730860C213yesmaximal, complemented
15#11730830C213yesmaximal, complemented
16C22×#1161C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23#90265C4116yescomplemented
24C2×#6456C2213yes—
25C2×D7×C4⋊C4C2213yes—
26C2×C4×D28C2213yes—
27C2×#6435C2216yes—
28C2×#6435C2213yes—
29C4×C7×C4⋊C4C2213yes—
30C4×C7×C4⋊C4C2213yes—
31C4×C7×C4⋊C4C2213yes—
32C7×#376C2213yes—
33#89760C2213yes—
34#89760C2213yes—
35#89759C2213yes—
36#89759C2213yes—
37#89749C2213yes—
38#89748C2213yes—
39C4⋊C4×Dic7C2213yes—
40C4⋊C4×Dic7C2213yes—
41#89745C2213yes—
42#89745C2213yes—
43#89712C2216yes—
44#89711C2213yes—
45#89711C2213yes—
46C4×(C2×D7)⋊C4C2213yes—
47C4×(C2×D7)⋊C4C2216yes—
48C4×(C2×D7)⋊C4C2213yes—
49C4×C4⋊Dic7C2213yes—
50#89705C2216yes—
51C4×Dic7⋊C4C2213yes—
52C4×Dic7⋊C4C2213yes—
53C42×Dic7C2213yes—
54#89700C2213yes—
55#89700C2213yes—
56#89441C2216yes—
57#89441C2213yes—
58#89441C2213yes—
59#89441C2213yes—
60#89441C2213yes—
61#89439C2216yes—
62#89436C2216yes—
63#89424C2213yes—
64#89424C2213yes—
65#89424C2213yes—
66#89424C2211yescharacteristic
67#34263—71nomaximal, Sylow
68C22×C7×Q8C2311yescharacteristic
69C2×#1161—24no—
70C2×#1161—24no—
71C2×#1161C2×C418yes—
72C22×D28C2313yes—
73C22×C4×D7C2313yes—
74C22×C4×D7C2316yes—
75C4×C7×Q8—24no—
76C4×C7×Q8—24no—
77C4×C7×Q8C2×C418yes—
78C2×C7×C4⋊C4C2313yes—
79C2×C7×C4⋊C4C2313yes—
80C2×C7×C4⋊C4C2313yes—
81C2×C42×C7C2313yes—
82C2×C42×C7C2313yes—
83C2×C42×C7C2316yes—
84C2×C42×C7C2313yes—
85Q8×Dic7—24no—
86Q8×Dic7—24no—
87Q8×Dic7C2×C418yes—
88C2×(C2×D7)⋊C4C2313yes—
89C2×(C2×D7)⋊C4C2313yes—
90C2×(C2×D7)⋊C4C2316yes—
91C2×(C2×D7)⋊C4C2313yes—
92C2×(C2×D7)⋊C4C2313yes—
93C2×(C2×D7)⋊C4C2316yes—
94C2×(C2×D7)⋊C4C2313yes—
95C2×(C2×D7)⋊C4C2313yes—
96C2×C4⋊Dic7C2313yes—
97C2×C4⋊Dic7C2313yes—
98C2×C4⋊Dic7C2313yes—
99C2×Dic7⋊C4C2313yes—
100C2×Dic7⋊C4C2311yescharacteristic
101C2×Dic7⋊C4C2316yes—
102C2×Dic7⋊C4C2313yes—
103C2×Dic7⋊C4C2313yes—
104C2×Dic7⋊C4C2311yescharacteristic
105C2×Dic7⋊C4C2313yes—
106C2×C4×Dic7C2313yes—
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2313yes—
109C2×C4×Dic7C2313yes—
110C2×C4×Dic7C2313yes—
111C2×C4×Dic7C2313yes—
112C2×C4×Dic7C2313yes—
113#6456C2×C4124yes—
114D7×C4⋊C4C2×C4124yes—
115C4×D28C2×C4124yes—
116#6435—412no—
117#6435C2×C4124yes—
118C7×#95C2313yes—
119C7×#95C2313yes—
120C7×#95C2311yescharacteristic
121C7×#95C2313yes—
122C7⋊#95C2311yescharacteristic
123C7⋊#95C2311yescharacteristic
124C7⋊#95C2313yes—
125C7⋊#95C2313yes—
126C7⋊#95C2313yes—
127C7⋊#95C2311yescharacteristic
128C7⋊#95C2313yes—
129C7⋊#95C2313yes—
130C7⋊#95C2316yes—
131C7⋊#95C2316yes—
132C2×#520—71no—
133#2357—73no—
134#2354—73no—
135#2351—71no—
136Q8×C4⋊C4—71no—
137#2345—73no—
138#2344—73no—
139#2341—71no—
140#2341—71no—
141#2319—73no—
142#2314—73no—
143#2298—73no—
144#2293—73no—
145#2276D711yescharacteristic, complemented
146C42×Q8—71no—
147C23×D7C2413yes—
148C2×C7×Q8C22×C414yes—
149C2×C7×Q8#4114yes—
150C2×C7×Q8#4114yes—
151C22×C4×C7C2411yescharacteristic
152C22×C4×C7C2413yes—
153C22×C4×C7C2411yescharacteristic
154C22×C4×C7C2413yes—
155C22×C4×C7C2413yes—
156C22×C4×C7C2411yescharacteristic
157C22×C4×C7C2413yes—
158C22×Dic7C2413yes—
159C22×Dic7C2413yes—
160C22×Dic7C2411yescharacteristic
161C22×Dic7C2411yescharacteristic
162C22×Dic7C2413yes—
163C22×Dic7C2411yescharacteristic
164C22×Dic7C2411yescharacteristic
165#1161—232no—
166C2×D28—26no—
167C2×D28—26no—
168C2×D28C22×C4112yes—
169C2×C4×D7—26no—
170C2×C4×D7—26no—
171C2×C4×D7—46no—
172C2×C4×D7—46no—
173C2×C4×D7C22×C4112yes—
174C2×C4×D7C22×C4124yes—
175C7×C4⋊C4—26no—
176C7×C4⋊C4—26no—
177C7×C4⋊C4C22×C4112yes—
178C42×C7—26no—
179C42×C7—26no—
180C42×C7—46no—
181C42×C7C22×C4112yes—
182(C2×D7)⋊C4—26no—
183(C2×D7)⋊C4—26no—
184(C2×D7)⋊C4—46no—
185(C2×D7)⋊C4—43no—
186(C2×D7)⋊C4—43no—
187(C2×D7)⋊C4—46no—
188(C2×D7)⋊C4C22×C4112yes—
189(C2×D7)⋊C4C22×C4112yes—
190C4⋊Dic7—26no—
191C4⋊Dic7—26no—
192C4⋊Dic7C22×C4112yes—
193Dic7⋊C4—26no—
194Dic7⋊C4—22no—
195Dic7⋊C4—46no—
196Dic7⋊C4—43no—
197Dic7⋊C4—43no—
198Dic7⋊C4C22×C414yes—
199Dic7⋊C4C22×C4112yes—
200C4×Dic7—26no—
201C4×Dic7—26no—
202C4×Dic7—26no—
203C4×Dic7—22no—
204C4×Dic7—43no—
205C4×Dic7—43no—
206C4×Dic7C22×C4112yes—
207C22×#41—71no—
208#520—716no—
209C2×C4×Q8—71no—
210C2×C4×Q8—71no—
211C2×C4×Q8—71no—
212C2×C4×Q8C2×D711yescharacteristic
213C2×C4×Q8C2×D711yescharacteristic
214C2×C4×Q8C2×D711yescharacteristic
215C2×C4×D4—73no—
216C2×C4×D4—73no—
217C2×#117—73no—
218C2×#117—76no—
219C22×C4⋊C4—73no—
220#390—73no—
221#390—73no—
222#390—73no—
223#390—73no—
224#389—73no—
225#389—73no—
226#389—73no—
227#389—73no—
228#388—76no—
229#388—76no—
230#388—73no—
231#388—73no—
232#388—73no—
233#388—73no—
234#387—76no—
235#387—73no—
236#387—73no—
237#387—73no—
238#387—71no—
239#384—73no—
240#382—73no—
241#381—76no—
242#380—73no—
243#380—73no—
244#380—76no—
245C4×C4⋊C4—73no—
246C4×C4⋊C4—73no—
247C4×C4⋊C4—73no—
248C4×C4⋊C4—73no—
249C4×C4⋊C4—73no—
250C4×C4⋊C4C2×D713yes—
251C4×C4⋊C4C2×D713yes—
252C4×C4⋊C4C2×D713yes—
253C4×C22⋊C4—73no—
254C4×C22⋊C4—73no—
255C4×C22⋊C4—76no—
256#376C2×D713yes—
257C43—73no—
258C23×C7C2511yescharacteristic
259C22×D7—23no—
260C22×D7—23no—
261C22×D7—23no—
262C22×D7—23no—
263C22×D7—23no—
264C22×D7—23no—
265C22×D7C23×C416yes—
266C7×Q8#117116yescomplemented
267C2×C4×C7—23no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—23no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—23no—
274C2×C4×C7—23no—
275C2×C4×C7—23no—
276C2×C4×C7—23no—
277C2×C4×C7—21no—
278C2×C4×C7—23no—
279C2×C4×C7—21no—
280C2×C4×C7—23no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7C23×C412yes—
284C2×C4×C7C23×C416yes—
285C2×C4×C7C23×C416yes—
286C2×C4×C7C2×#4116yes—
287C2×C4×C7C2×#4116yes—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—23no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—23no—
298C2×Dic7—23no—
299C2×Dic7—23no—
300C2×Dic7—23no—
301C2×Dic7—23no—
302C2×Dic7—23no—
303C2×Dic7—23no—
304C2×Dic7C23×C412yes—
305C2×Dic7C23×C416yes—
306C2×Dic7C23×C412yes—
307C2×Dic7H(5,𝔽2)12yes—
308C2×Dic72−1+412yes—
309D28—224no—
310C4×D7—224no—
311C4×D7—424no—
312C2×#41—78no—
313C2×#41—144no—
314C2×#41—144no—
315C22×Q8C22×D711yescharacteristic
316C22×D4—73no—
317C23×C4—73no—
318C23×C4—76no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—78no—
322C4×Q8—144no—
323C4×Q8—144no—
324C4×Q8C4×D718yes—
325C4×D4—724no—
326C4×D4—724no—
327#117—724no—
328#117—2812no—
329C2×C4⋊C4—73no—
330C2×C4⋊C4—73no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—724no—
333C2×C4⋊C4—73no—
334C2×C4⋊C4—73no—
335C2×C4⋊C4—73no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—76no—
338C2×C4⋊C4—73no—
339C2×C4⋊C4—73no—
340C2×C4⋊C4C22×D713yes—
341C2×C4⋊C4C22×D713yes—
342C2×C4⋊C4C22×D713yes—
343C2×C22⋊C4—73no—
344C2×C22⋊C4—73no—
345C2×C22⋊C4—73no—
346C2×C22⋊C4—73no—
347C2×C22⋊C4—76no—
348C2×C22⋊C4—76no—
349C2×C22⋊C4—73no—
350C2×C22⋊C4—73no—
351C2×C42—73no—
352C2×C42—73no—
353C2×C42—73no—
354C2×C42—73no—
355C2×C42—71no—
356C2×C42—73no—
357C2×C42—73no—
358C2×C42C22×D713yes—
359C2×C42C22×D713yes—
360C2×C42C22×D716yes—
361C2×C42C22×D713yes—
362#95—71no—
363#95—71no—
364#95—73no—
365#95—73no—
366#95—73no—
367#95—76no—
368#95—76no—
369#95—71no—
370#95—73no—
371#95—73no—
372#95C22×D711yescharacteristic
373#95C22×D713yes—
374#95C22×D713yes—
375#95C22×D713yes—
376C22×C7C24×C411yescharacteristic
377C22×C7C22×#4111yescharacteristic
378C22×C7C22×#4111yescharacteristic
379C22×C7C2×H(5,𝔽2)11yescharacteristic
380C22×C7C2×2−1+411yescharacteristic
381C22×C7#58511yescharacteristic
382C22×C7#58511yescharacteristic
383C2×D7—26no—
384C2×D7—26no—
385C2×D7—26no—
386C2×D7—43no—
387C2×D7—43no—
388C2×D7—43no—
389C2×D7—43no—
390C4×C7—26no—
391C4×C7—22no—
392C4×C7—41no—
393C4×C7—43no—
394C4×C7—43no—
395C4×C7—41no—
396C4×C7C2×#117112yes—
397Dic7—22no—
398Dic7—26no—
399Dic7—22no—
400Dic7—22no—
401Dic7—22no—
402Dic7—43no—
403Dic7—43no—
404C24—73no—
405#41—1432no—
406C2×Q8C2×C4×D714yes—
407C2×Q8#115714yes—
408C2×Q8#115714yes—
409C2×D4—712no—
410C2×D4—146no—
411C2×D4—146no—
412C22×C4—71no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—73no—
416C22×C4—71no—
417C22×C4—712no—
418C22×C4—724no—
419C22×C4—73no—
420C22×C4—73no—
421C22×C4—146no—
422C22×C4—146no—
423C22×C4—286no—
424C22×C4—286no—
425C22×C4C23×D711yescharacteristic
426C22×C4C23×D713yes—
427C22×C4C23×D711yescharacteristic
428C22×C4C23×D713yes—
429C22×C4C23×D713yes—
430C22×C4C23×D711yescharacteristic
431C22×C4C23×D713yes—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—712no—
435C4⋊C4—74no—
436C4⋊C4—712no—
437C4⋊C4—142no—
438C4⋊C4—146no—
439C4⋊C4—146no—
440C4⋊C4—146no—
441C4⋊C4—283no—
442C4⋊C4—283no—
443C4⋊C4—286no—
444C4⋊C4C2×C4×D7112yes—
445C22⋊C4—712no—
446C22⋊C4—712no—
447C22⋊C4—146no—
448C22⋊C4—146no—
449C22⋊C4—283no—
450C22⋊C4—283no—
451C22⋊C4—286no—
452C22⋊C4—286no—
453C42—26no—
454C42—26no—
455C42—46no—
456C42—712no—
457C42—142no—
458C42—146no—
459C42—146no—
460C42—146no—
461C42—283no—
462C42—283no—
463C42C2×C4×D7112yes—
464C2×C7C22×#11711yescharacteristic
465C2×C7C4×H(5,𝔽2)11yescharacteristic
466C2×C7C4×2−1+411yescharacteristic
467C2×C7#351611yescharacteristic
468C2×C7#351711yescharacteristic
469C2×C7#352211yescharacteristic
470C2×C7#352411yescharacteristic
471D7—46no—
472C23—76no—
473C23—143no—
474C23—143no—
475C23—143no—
476C23—143no—
477C23—143no—
478C23—143no—
479C23C24×D711yescharacteristic
480Q8#6435116yescomplemented
481D4—1424no—
482C2×C4—23no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—23no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—23no—
489C2×C4—23no—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—23no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—23no—
497C2×C4—21no—
498C2×C4—72no—
499C2×C4—76no—
500C2×C4—72no—
501C2×C4—72no—
502C2×C4—72no—
503C2×C4—141no—
504C2×C4—141no—
505C2×C4—141no—
506C2×C4—141no—
507C2×C4—143no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—143no—
513C2×C4—1424no—
514C2×C4—143no—
515C2×C4—143no—
516C2×C4—143no—
517C2×C4—143no—
518C2×C4—143no—
519C2×C4—143no—
520C2×C4—2824no—
521C2×C4C22×C4×D712yes—
522C2×C4C22×C4×D716yes—
523C2×C4C22×C4×D716yes—
524C2×C4C2×#115716yes—
525C2×C4C2×#115716yes—
526C7#3426311yescharacteristic, complemented, Sylow
527C22—146no—
528C22—146no—
529C22—146no—
530C22—283no—
531C22—283no—
532C22—283no—
533C22—283no—
534C22C23×C4×D711yescharacteristic
535C22C22×#115711yescharacteristic
536C22C22×#115711yescharacteristic
537C22D7×H(5,𝔽2)11yescharacteristic
538C22D7×2−1+411yescharacteristic
539C22#9062011yescharacteristic
540C22#9062011yescharacteristic
541C4—22no—
542C4—26no—
543C4—43no—
544C4—41no—
545C4—41no—
546C4—43no—
547C4—142no—
548C4—142no—
549C4—142no—
550C4—142no—
551C4—146no—
552C4—283no—
553C4—283no—
554C4C2×#6435112yes—
555C2—286no—
556C2C22×#643511yescharacteristic
557C2#1174294111yescharacteristic
558C2#1174294711yescharacteristic
559C2#1174295111yescharacteristic
560C2#1174295411yescharacteristic
561C2#1174296411yescharacteristic
562C2#1174297611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2276C2
socle, cosocleC23×C7C25
prime radicalC22×#1161C22
hypercenter#2276D7
hypocenterC7#34263
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C7×#2276C2semidirect
3. #11739906C2semidirect
4. Q8×Dic7⋊C4C2semidirect
7. C4×Q8×Dic7C2semidirect
8. #11731213C2semidirect
9. #11731212C2semidirect
10. #11731205C2semidirect
11. #11731091C2semidirect
12. #11731084C2semidirect
13. #11731062C2semidirect
14. #11730860C2semidirect
15. #11730830C2semidirect
23. #90265C4semidirect
145. #2276D7semidirect
266. C7×Q8#117semidirect
480. Q8#6435semidirect
526. C7#34263semidirect

Maximal subgroups

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

subgroupindex
C2×#902652
C7×#22762
#117399062
Q8×Dic7⋊C42
#117399022
C4×Q8×Dic72
#117312132
#117312122
#117312052
#117310912
#117310842
#117310622
#117308602
#117308302
#342637