Finite Group Databasebeta

← #13409937

Subgroups — #13409937

Order 1792.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90270C211yesmaximal, characteristic
2C7×#2546C211yesmaximal, characteristic, complemented
3C2×#89856C211yesmaximal, characteristic
4C2×#89852C211yesmaximal, characteristic, complemented
5#11729389C211yesmaximal, characteristic, complemented
6#11727380C212yesmaximal
7#11727379C212yesmaximal
8#11727360C212yesmaximal, complemented
9#11727359C212yesmaximal, complemented
10#11727245C211yesmaximal, characteristic, complemented
11#11727228C211yesmaximal, characteristic
12C2×C7×C4⋊Q8—21no—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×#6509—21no—
16#90270—24no—
17#90270—24no—
18C2×#6459—21no—
19C2×#6459—21no—
20C2×D7×C4⋊C4—21no—
21C2×C4×D28C2211yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×C4×D28C2211yescharacteristic
24C2×#6433C2211yescharacteristic
25C2×#6433C2211yescharacteristic
26C2×#6432C2211yescharacteristic
27C7×#398—21no—
28C7×#398—21no—
29C7×#391—21no—
30C7×#389C2212yes—
31C7×#389C2212yes—
32C4×C7×C4⋊C4C2211yescharacteristic
33#89856—24no—
34#89852—24no—
35C2×#6383C2212yes—
36C2×#6383C2212yes—
37C2×#6382C2212yes—
38C2×#6382C2212yes—
39C2×#6381C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89319C2212yes—
46#89319C2212yes—
47#25851—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C22×C4×D7—21no—
53C22×C4×D7—21no—
54C22×Dic14—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8—44no—
58C4×C7×Q8—22no—
59C4×C7×Q8—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4D411yescharacteristic
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C4⋊C4C2312yes—
67C2×C7×C4⋊C4C2312yes—
68C2×C42×C7—21no—
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72#6509—28no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C22×C7⋊C8C2312yes—
85C22×C7⋊C8C2312yes—
86#6459—28no—
87#6459—28no—
88D7×C4⋊C4—28no—
89C4×D28—22no—
90C4×D28—22no—
91C4×D28—22no—
92C4×D28—22no—
93C4×D28—22no—
94C4×D28D414yes—
95#6433—22no—
96#6433—22no—
97#6433—22no—
98#6433D414yes—
99#6432—22no—
100#6432—22no—
101C7×#95—21no—
102C7×#95—22no—
103C7×#95—22no—
104#6383—24no—
105#6383Q818yescomplemented
106#6382—24no—
107#6382—24no—
108#6381—24no—
109C4⋊(C7⋊C8)—22no—
110#6377—22no—
111C2×#554—71no—
112C2×#501—71no—
113C2×#476—71no—
114#2546D711yescharacteristic, complemented
115#2059—72no—
116#2058—72no—
117#1989—71no—
118#1985—71no—
119#1864—72no—
120#1862—72no—
121#1762—71no—
122C23×D7—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—41no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C2×D28—21no—
142C2×D28—22no—
143C2×D28C2×D412yes—
144C2×D28C2×Q814yes—
145C2×C4×D7—24no—
146C2×C4×D7—24no—
147C2×C4×D7—24no—
148C2×C4×D7—42no—
149C2×C4×D7—42no—
150C2×C4×D7—82no—
151C2×C4×D7—82no—
152C2×Dic14—24no—
153C2×Dic14—82no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—44no—
161C7×C4⋊C4—42no—
162C7×C4⋊C4—44no—
163C7×C4⋊C4C2×Q814yes—
164C42×C7—21no—
165C42×C7—21no—
166C42×C7—21no—
167C42×C7—21no—
168C42×C7—42no—
169C42×C7C2×D412yes—
170C42×C7C2×D412yes—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—24no—
173(C2×D7)⋊C4—24no—
174C4⋊Dic7—24no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7C2×D412yes—
179C4⋊Dic7C2×D412yes—
180C4⋊Dic7C2×D412yes—
181Dic7⋊C4—24no—
182Dic7⋊C4—24no—
183Dic7⋊C4—24no—
184Dic7⋊C4—24no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—24no—
189C2×C7⋊C8C2×Q814yes—
190#554—144no—
191#554—144no—
192C2×C4⋊Q8—21no—
193C2×C4⋊Q8—71no—
194C2×C4⋊Q8C2×D711yescharacteristic
195C2×#125—71no—
196C2×#125—71no—
197C2×C22⋊Q8—141no—
198C2×C22⋊Q8—141no—
199C2×C22⋊Q8—141no—
200C2×C4×Q8C2×D711yescharacteristic
201C2×C4×D4—71no—
202C2×C4×D4—71no—
203C2×C4×D4—71no—
204C22×C4⋊C4—141no—
205#501—144no—
206#476—144no—
207C2×#107—72no—
208C2×#106—72no—
209C2×#106—72no—
210C2×C4⋊C8—71no—
211C2×#102—72no—
212C2×#102—72no—
213C2×#97—71no—
214#398—21no—
215#398—21no—
216#391—21no—
217#389C2×D712yes—
218#389C2×D712yes—
219C4×C4⋊C4C2×D711yescharacteristic
220#340—72no—
221#340—72no—
222#340—72no—
223#336—72no—
224#336—72no—
225C23×C7C22×D411yescharacteristic
226C22×D7—22no—
227C22×D7—24no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—42no—
231C7×Q8—44no—
232C7×Q8—84no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—22no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—24no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7#12012yes—
252C2×C4×C7#12012yes—
253C2×C4×C7AGL(1,ℤ/8)12yes—
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×Q812yes—
257C2×C4×C72−1+412yes—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265D28—22no—
266D28—24no—
267D28C4⋊Q814yescomplemented
268C4×D7—48no—
269C4×D7—84no—
270C4×D7—84no—
271Dic14—84no—
272C7⋊C8—24no—
273C7⋊C8—42no—
274C22×Q8—21no—
275C22×Q8—141no—
276C22×Q8C22⋊D711yescharacteristic
277C22×D4—71no—
278C23×C4—141no—
279C23×C4—141no—
280C23×C4—141no—
281C22×C8—72no—
282C22×C8—72no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—44no—
286C4⋊Q8—142no—
287C4⋊Q8—142no—
288#125—74no—
289#125—142no—
290#125—142no—
291#125—142no—
292C22⋊Q8—148no—
293C22⋊Q8—148no—
294C22⋊Q8—148no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×D4—74no—
298C4×D4—142no—
299C4×D4—142no—
300C4×D4—142no—
301C4×D4—142no—
302C4×D4—142no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—148no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4C22⋊D711yescharacteristic
316C2×C4⋊C4C22⋊D711yescharacteristic
317C2×C4⋊C4C22⋊D711yescharacteristic
318C2×C4⋊C4C22×D712yes—
319C2×C4⋊C4C22×D712yes—
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—141no—
323C2×C42—21no—
324C2×C42C22×D711yescharacteristic
325C2×C42C22×D711yescharacteristic
326C2×C42C22×D711yescharacteristic
327#107—144no—
328#106—144no—
329#106—144no—
330C4⋊C8—142no—
331#102—78no—
332#102—144no—
333#97—142no—
334#95—21no—
335#95—22no—
336#95—22no—
337C22×C7C2×#12011yescharacteristic
338C22×C7D4×Q812yes—
339C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
340C22×C7#57511yescharacteristic
341C22×C7#57712yes—
342C2×D7—24no—
343C2×D7—44no—
344C2×D7—44no—
345C2×D7—42no—
346C2×D7—42no—
347C4×C7—22no—
348C4×C7—22no—
349C4×C7—22no—
350C4×C7—44no—
351C4×C7—41no—
352C4×C7—41no—
353C4×C7—41no—
354C4×C7—82no—
355C4×C7C2×C4⋊Q811yescharacteristic
356C4×C7#53611yescharacteristic
357C4×C7D4×Q812yes—
358Dic7—42no—
359Dic7—42no—
360Dic7—42no—
361C24—141no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—144no—
369C2×Q8—562no—
370C2×D4—72no—
371C2×D4—74no—
372C2×D4—141no—
373C2×D4—142no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—144no—
377C22×C4—144no—
378C22×C4—141no—
379C22×C4—141no—
380C22×C4—141no—
381C22×C4—144no—
382C22×C4—282no—
383C22×C4—282no—
384C22×C4—562no—
385C22×C4—562no—
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388C22×C4C2×C22⋊D711yescharacteristic
389C22×C4C2×C22⋊D711yescharacteristic
390C22×C4C2×C22⋊D711yescharacteristic
391C22×C4C2×C22⋊D711yescharacteristic
392C22×C4C23×D711yescharacteristic
393C2×C8—74no—
394C2×C8—142no—
395C2×C8—142no—
396C2×C8—142no—
397C2×C8—144no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—72no—
408C4⋊C4—72no—
409C4⋊C4—72no—
410C4⋊C4—144no—
411C4⋊C4—144no—
412C4⋊C4—144no—
413C4⋊C4—144no—
414C4⋊C4—144no—
415C4⋊C4—141no—
416C4⋊C4—141no—
417C4⋊C4—141no—
418C4⋊C4Q8×D714yes—
419C22⋊C4—144no—
420C22⋊C4—144no—
421C22⋊C4—144no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—42no—
427C42D4×D712yes—
428C42D4×D712yes—
429C2×C7#243411yescharacteristic
430C2×C7#300611yescharacteristic
431C2×C7#300911yescharacteristic
432C2×C7#338412yes—
433C2×C7#338812yes—
434D7—44no—
435C23—142no—
436C23—144no—
437C23—281no—
438C23—281no—
439C23—282no—
440C23C22×C22⋊D711yescharacteristic
441Q8—44no—
442Q8—84no—
443Q8—564no—
444D4—74no—
445D4—142no—
446D4—144no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—142no—
466C2×C4—142no—
467C2×C4—142no—
468C2×C4—288no—
469C2×C4—281no—
470C2×C4—281no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—564no—
474C2×C4—564no—
475C2×C4#649512yes—
476C2×C4#650012yes—
477C2×C4#650012yes—
478C2×C4C2×D4×D711yescharacteristic
479C2×C4C2×D4×D711yescharacteristic
480C2×C4C2×Q8×D712yes—
481C2×C4#655112yes—
482C8—144no—
483C8—282no—
484C7#2585111yescharacteristic, complemented, Sylow
485C22—144no—
486C22—284no—
487C22—282no—
488C22—282no—
489C22—284no—
490C22C2×#649511yescharacteristic
491C22C2×#650011yescharacteristic
492C22Q8×C22⋊D712yes—
493C22#9051311yescharacteristic
494C22#9052812yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—44no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—82no—
503C4—282no—
504C4—282no—
505C4—282no—
506C4D7×C4⋊Q811yescharacteristic
507C4#9041511yescharacteristic
508C4#9041812yes—
509C2—284no—
510C2#1174006911yescharacteristic
511C2#1174067711yescharacteristic
512C2#1174084911yescharacteristic
513C2#1174100912yes—
514C2#1174114012yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2546C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2546D7
hypocenterC7#25851
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C7×#2546C2semidirect
4. C2×#89852C2semidirect
5. #11729389C2semidirect
8. #11727360C2semidirect
9. #11727359C2semidirect
10. #11727245C2semidirect
105. #6383Q8semidirect
114. #2546D7semidirect
267. D28C4⋊Q8semidirect
484. C7#25851semidirect

Maximal subgroups

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

subgroupindex
C2×#902702
C7×#25462
C2×#898562
C2×#898522
#117293892
#117273802
#117273792
#117273602
#117273592
#117272452
#117272282
#258517