Finite Group Databasebeta

← #13363310

Subgroups — #13363310

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×#125C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic
3#11731198C211yesmaximal, characteristic
4C7×#1989C211yesmaximal, characteristic, complemented
5#11729492C211yesmaximal, characteristic, complemented
6#11728215C211yesmaximal, characteristic, complemented
7#11727245C211yesmaximal, characteristic, complemented
8#11726626C212yesmaximal
9#11726625C212yesmaximal
10#11726559C212yesmaximal, complemented
11#11726559C212yesmaximal, complemented
12C2×C7×#125C2211yescharacteristic
13D7×#125—24no—
14D7×#125C418yescomplemented
15C2×D7×C4⋊C4—21no—
16C2×D7×C4⋊C4—21no—
17C2×D7×C4⋊C4C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×#6453C2211yescharacteristic
20C2×#6453C2211yescharacteristic
21C2×#6452—21no—
22C2×#6452—21no—
23C2×#6435—21no—
24C2×C42×D7C2211yescharacteristic
25C2×#6433C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C8C2212yes—
29C2×#6389C2212yes—
30#89759C2212yes—
31#89748C2212yes—
32C4×C4⋊Dic7C2211yescharacteristic
33C4×Dic7⋊C4—21no—
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35#89607—24no—
36#89439—21no—
37#89422—21no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21838—71nomaximal, Sylow
47C22×C4×D7—21no—
48C22×C4×D7—21no—
49C22×C4×D7D411yescharacteristic
50C22×C4×D7D411yescharacteristic
51C22×C4×D7C2311yescharacteristic
52C22×C7×C8C2312yes—
53C7×#125—22no—
54C7×#125C2×C414yes—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C2×(C2×D7)⋊C4—22no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7C2312yes—
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—21no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—21no—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C22×C7⋊C8C2312yes—
81D7×C4⋊C4—28no—
82D7×C4⋊C4—28no—
83D7×C4⋊C4—22no—
84D7×C4⋊C4—22no—
85D7×C4⋊C4C2×C414yes—
86D7×C4⋊C4C2×C414yes—
87#6453—22no—
88#6453—22no—
89#6453C2×C414yes—
90#6453C2×C414yes—
91#6452—28no—
92#6452—28no—
93#6435—44no—
94C42×D7C2×C414yes—
95C42×D7D414yes—
96#6433—22no—
97#6433C2×C414yes—
98C7×C4⋊C8—22no—
99C7×#95—21no—
100C7⋊#95—21no—
101C7⋊#95—22no—
102(C2×D7)⋊C8—24no—
103#6389—24no—
104C4⋊(C7⋊C8)—22no—
105C22×#125—71no—
106C2×#432—71no—
107#2285—71no—
108#1992—72no—
109#1992—72no—
110#1989—71no—
111#1989—71no—
112#1989—71no—
113#1989D711yescharacteristic, complemented
114#1824—72no—
115#1823—72no—
116C23×D7C2×D411yescharacteristic
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2411yescharacteristic
123C22×Dic7—21no—
124C22×Dic7—21no—
125C22×Dic7—22no—
126C22×Dic7C2×D411yescharacteristic
127C22×Dic7C2×D411yescharacteristic
128C22×Dic7C2×D411yescharacteristic
129C2×C4×D7—22no—
130C2×C4×D7—22no—
131C2×C4×D7—22no—
132C2×C4×D7—22no—
133C2×C4×D7—24no—
134C2×C4×D7—24no—
135C2×C4×D7—42no—
136C2×C4×D7—44no—
137C2×C4×D7—42no—
138C2×C4×D7—42no—
139C2×C4×D7C22⋊C414yes—
140C2×C4×D7C22⋊C414yes—
141C2×C4×D7C22×C412yes—
142C2×C4×D7C2×D412yes—
143C2×C7×C8—22no—
144C2×C7×C8—24no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—21no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4C22×C412yes—
152C7×C4⋊C4C22×C412yes—
153C42×C7—42no—
154C42×C7C22×C412yes—
155C42×C7C2×D412yes—
156(C2×D7)⋊C4—44no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—24no—
160C4⋊Dic7—24no—
161C4⋊Dic7—22no—
162C4⋊Dic7—22no—
163C4⋊Dic7C22×C412yes—
164C4⋊Dic7C22×C412yes—
165Dic7⋊C4—24no—
166Dic7⋊C4—24no—
167Dic7⋊C4—24no—
168Dic7⋊C4—24no—
169Dic7⋊C4—24no—
170Dic7⋊C4—24no—
171Dic7⋊C4—44no—
172Dic7⋊C4—44no—
173C4×Dic7—42no—
174C4×Dic7C22⋊C414yes—
175C4×Dic7C22⋊C414yes—
176C4×Dic7C22×C412yes—
177C4×Dic7C2×D412yes—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—24no—
180C2×#125—78no—
181C2×#125—71no—
182C2×#125—71no—
183C2×#125—71no—
184C2×#125—141no—
185C2×#125—141no—
186C2×#125—144no—
187C2×#125C2×D711yescharacteristic
188C2×#117—141no—
189C22×C4⋊C4—71no—
190C22×C4⋊C4—71no—
191C22×C4⋊C4—141no—
192C22×C4⋊C4—141no—
193C22×C42—71no—
194#432—144no—
195C2×C4⋊C8—71no—
196C2×C4⋊C8C2×D711yescharacteristic
197C2×C22⋊C8—72no—
198C2×#97—72no—
199#384—72no—
200#382—141no—
201#380—72no—
202#380—141no—
203C4×C4⋊C4—71no—
204C4×C4⋊C4—141no—
205C4×C4⋊C4C2×D711yescharacteristic
206#340—72no—
207#340—72no—
208#340—72no—
209#340—72no—
210#340—72no—
211#340—72no—
212#340C2×D712yes—
213#340C2×D712yes—
214C23×C7C22×D411yescharacteristic
215C22×D7—22no—
216C22×D7—22no—
217C22×D7C2×C22⋊C412yes—
218C22×D7AGL(1,ℤ/8)12yes—
219C22×D7#13712yes—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7C2×C22⋊C412yes—
232C2×C4×C7C2×C22⋊C412yes—
233C2×C4×C7C23×C411yescharacteristic
234C2×C4×C7C22×D411yescharacteristic
235C2×C4×C72−1+412yes—
236C2×C4×C72−1+412yes—
237C2×Dic7—22no—
238C2×Dic7—21no—
239C2×Dic7—21no—
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7C2×C22⋊C412yes—
250C2×Dic7C2×C22⋊C412yes—
251C2×Dic7C2×C22⋊C412yes—
252C4×D7—28no—
253C4×D7—24no—
254C4×D7—22no—
255C4×D7—22no—
256C4×D7—44no—
257C4×D7—48no—
258C4×D7—48no—
259C7×C8—42no—
260C7⋊C8—42no—
261C23×C4—71no—
262C23×C4—71no—
263C23×C4—71no—
264C23×C4—141no—
265C23×C4—141no—
266C22×C8—72no—
267C22×C8C22×D712yes—
268#125—22no—
269#125—74no—
270#125—74no—
271#125—74no—
272#125—148no—
273#125—148no—
274#125—142no—
275#125—142no—
276#125—142no—
277#125C4×D714yes—
278#117—284no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—71no—
282C2×C4⋊C4—71no—
283C2×C4⋊C4—74no—
284C2×C4⋊C4—74no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—142no—
287C2×C4⋊C4—142no—
288C2×C4⋊C4—148no—
289C2×C4⋊C4—148no—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—141no—
292C2×C4⋊C4—141no—
293C2×C4⋊C4—141no—
294C2×C4⋊C4—141no—
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C4⋊C4—141no—
298C2×C4⋊C4—142no—
299C2×C4⋊C4—142no—
300C2×C4⋊C4C22×D712yes—
301C2×C4⋊C4C22×D711yescharacteristic
302C2×C4⋊C4C22×D711yescharacteristic
303C2×C22⋊C4—142no—
304C2×C42—21no—
305C2×C42—71no—
306C2×C42—71no—
307C2×C42—71no—
308C2×C42—74no—
309C2×C42—74no—
310C2×C42—141no—
311C2×C42C22×D711yescharacteristic
312C4⋊C8—22no—
313C4⋊C8—142no—
314C22⋊C8—144no—
315#97—144no—
316#95—21no—
317#95—142no—
318#95—141no—
319C22×C7C22×C22⋊C411yescharacteristic
320C22×C7#53612yes—
321C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
322C22×C7C2×#13711yescharacteristic
323C22×C7#57512yes—
324C2×D7—22no—
325C2×D7—22no—
326C2×D7—24no—
327C2×D7—24no—
328C2×D7#41714yes—
329C4×C7—22no—
330C4×C7—22no—
331C4×C7—42no—
332C4×C7—42no—
333C4×C7—82no—
334C4×C7#51912yes—
335C4×C7#53611yescharacteristic
336C4×C7#53611yescharacteristic
337Dic7—22no—
338Dic7—22no—
339Dic7—22no—
340Dic7—42no—
341Dic7—42no—
342Dic7—82no—
343C24—71no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—71no—
350C22×C4—74no—
351C22×C4—72no—
352C22×C4—72no—
353C22×C4—74no—
354C22×C4—142no—
355C22×C4—142no—
356C22×C4—144no—
357C22×C4—144no—
358C22×C4—142no—
359C22×C4—142no—
360C22×C4—141no—
361C22×C4—141no—
362C22×C4—142no—
363C22×C4—282no—
364C22×C4—282no—
365C22×C4—282no—
366C22×C4—284no—
367C22×C4D4×D711yescharacteristic
368C22×C4D4×D711yescharacteristic
369C22×C4C23×D711yescharacteristic
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—142no—
373C2×C8—144no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—72no—
381C4⋊C4—72no—
382C4⋊C4—144no—
383C4⋊C4—144no—
384C4⋊C4—144no—
385C4⋊C4—144no—
386C4⋊C4—144no—
387C4⋊C4—144no—
388C4⋊C4—144no—
389C4⋊C4—144no—
390C4⋊C4—141no—
391C4⋊C4—141no—
392C4⋊C4—142no—
393C4⋊C4—142no—
394C4⋊C4—284no—
395C4⋊C4—284no—
396C4⋊C4C2×C4×D712yes—
397C4⋊C4C2×C4×D712yes—
398C22⋊C4—284no—
399C42—42no—
400C42—74no—
401C42—74no—
402C42—72no—
403C42—72no—
404C42—282no—
405C42C2×C4×D712yes—
406C42D4×D712yes—
407C2×C7#231011yescharacteristic
408C2×C7C2×#41711yescharacteristic
409C2×C7#289611yescharacteristic
410C2×C7#323412yes—
411C2×C7#323612yes—
412D7—24no—
413C23—72no—
414C23—72no—
415C23—72no—
416C23—142no—
417C23—142no—
418C23C2×D4×D711yescharacteristic
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—72no—
431C2×C4—72no—
432C2×C4—72no—
433C2×C4—148no—
434C2×C4—142no—
435C2×C4—141no—
436C2×C4—141no—
437C2×C4—142no—
438C2×C4—142no—
439C2×C4—144no—
440C2×C4—142no—
441C2×C4—142no—
442C2×C4—142no—
443C2×C4—284no—
444C2×C4—288no—
445C2×C4—288no—
446C2×C4—282no—
447C2×C4—281no—
448C2×C4—281no—
449C2×C4—282no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4D7×C22⋊C412yes—
453C2×C4D7×C22⋊C412yes—
454C2×C4C22×C4×D711yescharacteristic
455C2×C4C2×D4×D711yescharacteristic
456C2×C4#655112yes—
457C2×C4#655412yes—
458C8—42no—
459C8—282no—
460C7#2183811yescharacteristic, complemented, Sylow
461C22—74no—
462C22—142no—
463C22—144no—
464C22—142no—
465C22—144no—
466C22C2×D7×C22⋊C411yescharacteristic
467C22#9031912yes—
468C22#9045912yes—
469C22D7×AGL(1,ℤ/8)11yescharacteristic
470C22D7×#13711yescharacteristic
471C4—22no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—142no—
477C4—142no—
478C4—142no—
479C4—282no—
480C4—282no—
481C4—562no—
482C4#9026312yes—
483C4#9036611yescharacteristic
484C4#9041511yescharacteristic
485C2—144no—
486C2#1173405511yescharacteristic
487C2#1173627511yescharacteristic
488C2D7×#41711yescharacteristic
489C2#1173861912yes—
490C2#1173868312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1989C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1989D7
hypocenterC7#21838
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. C7×#1989C2semidirect
5. #11729492C2semidirect
6. #11728215C2semidirect
7. #11727245C2semidirect
10. #11726559C2semidirect
11. #11726559C2semidirect
14. D7×#125C4semidirect
113. #1989D7semidirect
460. C7#21838semidirect

Maximal subgroups

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

subgroupindex
C2×D7×#1252
C2×#896072
#117311982
C7×#19892
#117294922
#117282152
#117272452
#117266262
#117266252
#117265592
#218387