Finite Group Databasebeta

← #13375605

Subgroups — #13375605

Order 1792.

898 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89614C211yesmaximal, characteristic
2#11731805C211yesmaximal, characteristic
3#11731101C211yesmaximal, characteristic, complemented
4C7×#1948C211yesmaximal, characteristic, complemented
5#11728338C212yesmaximal
6#11728313C212yesmaximal, complemented
7#11728219C211yesmaximal, characteristic, complemented
8#11727336C211yesmaximal, characteristic, complemented
9C4×#6383C211yesmaximal, characteristic
10#11726570C212yesmaximal, complemented
11#11726534C212yesmaximal
12C2×#6456C2212yes—
13C2×D7×C4⋊C4—21no—
14C2×C4×D28C2211yescharacteristic
15C2×#6432C2211yescharacteristic
16C2×C7×#106C2212yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#384C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20C2×#6395C2212yes—
21C2×#6391C2212yes—
22C2×C8×Dic7C2212yes—
23#89761—22no—
24#89759—21no—
25C4⋊C4×Dic7C2212yes—
26#89745C2212yes—
27#89745C2212yes—
28C2×#6383C2211yescharacteristic
29C2×#6383C2211yescharacteristic
30C4×C4⋊Dic7—21no—
31C4×C4⋊Dic7C2211yescharacteristic
32#89700—21no—
33C2×C4×C7⋊C8C2211yescharacteristic
34#89614C418yescomplemented
35#89418—22no—
36C7×#340C2212yes—
37#89345C2212yes—
38#89324C2211yescharacteristic
39#89324C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#22963—71nomaximal, Sylow
43C22×D28C2311yescharacteristic
44C22×C4×D7—21no—
45C22×C4×D7—22no—
46C22×Dic14—21no—
47C22×C7×C8C2312yes—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4C2312yes—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2311yescharacteristic
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C2×(C2×D7)⋊C4—21no—
56C2×(C2×D7)⋊C4—22no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7D411yescharacteristic
59C2×C4⋊Dic7D411yescharacteristic
60C2×C4⋊Dic7C2311yescharacteristic
61C2×C4⋊Dic7C2312yes—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×C4×Dic7—22no—
65C2×C4×Dic7C2312yes—
66C22×C7⋊C8C2311yescharacteristic
67C22×C7⋊C8C2311yescharacteristic
68#6456—24no—
69#6456—42no—
70D7×C4⋊C4—44no—
71C4×D28—22no—
72C4×D28C2×C414yes—
73#6432—22no—
74#6432C2×C414yescomplemented
75C7×#106—42no—
76C7×C4⋊C8C2×C414yescomplemented
77C7×#95—21no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95D412yes—
82#6395C2×C418yes—
83#6391C2×C418yescomplemented
84C8×Dic7—42no—
85#6383—22no—
86#6383Q814yescomplemented
87C4×C7⋊C8—22no—
88C2×#476—71no—
89#2369—71no—
90D4×C4⋊C4—71no—
91#1990—71no—
92#1948D711yescharacteristic, complemented
93#1941—72no—
94#1919—72no—
95#1864—72no—
96C8×C4⋊C4—71no—
97C4×#102—71no—
98C4×#102—72no—
99C23×D7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—22no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2411yescharacteristic
106C22×Dic7—22no—
107C22×Dic7—21no—
108C22×Dic7C2×D412yes—
109C22×Dic7C2×D412yes—
110C2×D28—21no—
111C2×D28—21no—
112C2×D28—21no—
113C2×D28C22×C412yes—
114C2×D28C2×Q812yes—
115C2×D28#4112yes—
116C2×C4×D7—24no—
117C2×C4×D7—42no—
118C2×C4×D7—42no—
119C2×C4×D7—44no—
120C2×C4×D7—44no—
121C2×C4×D7—82no—
122C2×Dic14—24no—
123C2×Dic14—81no—
124C2×Dic14—81no—
125C2×C7×C8—22no—
126C2×C7×C8—22no—
127C2×C7×C8C22×C414yes—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—22no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—42no—
134C7×C4⋊C4—81no—
135C7×C4⋊C4C2×Q812yes—
136C42×C7—42no—
137C42×C7C22×C412yes—
138C42×C7#4112yes—
139(C2×D7)⋊C4—24no—
140(C2×D7)⋊C4—44no—
141C4⋊Dic7—24no—
142C4⋊Dic7—22no—
143C4⋊Dic7—22no—
144C4⋊Dic7—21no—
145C4⋊Dic7SD1614yes—
146C4⋊Dic7C22×C414yes—
147C4⋊Dic7C22×C412yes—
148Dic7⋊C4—44no—
149Dic7⋊C4—82no—
150C4×Dic7—22no—
151C4×Dic7—22no—
152C4×Dic7—82no—
153C2×C7⋊C8—21no—
154C2×C7⋊C8—21no—
155C2×C7⋊C8—21no—
156C2×C7⋊C8—21no—
157C2×C7⋊C8—21no—
158C2×C7⋊C8C2×Q812yes—
159C2×C4⋊Q8—71no—
160C2×C4×D4—72no—
161C2×C4×D4—71no—
162C22×C4⋊C4—141no—
163#476—78no—
164C2×#106—72no—
165C2×#106C2×D712yes—
166C2×C4⋊C8C2×D711yescharacteristic
167C2×#102—72no—
168C2×#102—71no—
169C2×#102—71no—
170C2×C4×C8—71no—
171C2×C4×C8—72no—
172#391—142no—
173#389—72no—
174#389—72no—
175#389—141no—
176#385—142no—
177#384C2×D711yescharacteristic
178#380—141no—
179C4×C4⋊C4—71no—
180C4×C4⋊C4—72no—
181C4×C4⋊C4—141no—
182C4×C4⋊C4C2×D711yescharacteristic
183#340—72no—
184#340—71no—
185#340—71no—
186#340—72no—
187#340C2×D712yes—
188#336—72no—
189C23×C7C22×D411yescharacteristic
190C22×D7—22no—
191C22×D7—22no—
192C22×D7—22no—
193C22×D7—41no—
194C22×D7—41no—
195C22×D7—41no—
196C22×D7—41no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—22no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—42no—
205C2×C4×C7—41no—
206C2×C4×C7—42no—
207C2×C4×C7—41no—
208C2×C4×C7C4×D412yes—
209C2×C4×C7C4×D412yes—
210C2×C4×C7C2×SD1612yes—
211C2×C4×C7#13512yes—
212C2×C4×C7C23×C411yescharacteristic
213C2×C4×C7C22×Q811yescharacteristic
214C2×C4×C7C2×#4111yescharacteristic
215C2×C4×C7C2×#4111yescharacteristic
216C2×C4×C7H(5,𝔽2)11yescharacteristic
217C2×C4×C72−1+411yescharacteristic
218C2×Dic7—22no—
219C2×Dic7—22no—
220C2×Dic7—24no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—41no—
226C2×Dic7—41no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7C2×SD1614yes—
230D28—22no—
231D28—22no—
232D28—22no—
233D28C4×Q814yes—
234C4×D7—48no—
235C4×D7—88no—
236Dic14—84no—
237C7×C8—24no—
238C7⋊C8—22no—
239C7⋊C8—41no—
240C22×Q8—141no—
241C22×D4—71no—
242C23×C4—142no—
243C23×C4—141no—
244C22×C8—71no—
245C22×C8—71no—
246C22×C8C22×D712yes—
247C4⋊Q8—74no—
248C4⋊Q8—142no—
249C4×D4—74no—
250C4×D4—144no—
251C4×D4—142no—
252C4×D4—282no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—71no—
256C2×C4⋊C4—71no—
257C2×C4⋊C4—72no—
258C2×C4⋊C4—71no—
259C2×C4⋊C4—141no—
260C2×C4⋊C4—142no—
261C2×C4⋊C4—142no—
262C2×C4⋊C4—284no—
263C2×C4⋊C4C22×D712yes—
264C2×C4⋊C4C22×D711yescharacteristic
265C2×C4⋊C4C22×D711yescharacteristic
266C2×C22⋊C4—141no—
267C2×C22⋊C4—142no—
268C2×C42—21no—
269C2×C42—72no—
270C2×C42—142no—
271C2×C42C22×D711yescharacteristic
272#106—42no—
273#106—78no—
274C4⋊C8C4×D714yescomplemented
275#102—78no—
276#102—74no—
277#102—142no—
278C4×C8—142no—
279C4×C8—282no—
280#95—21no—
281#95—72no—
282#95—142no—
283#95—142no—
284#95—142no—
285C22×C7C2×C4×D411yescharacteristic
286C22×C7#54611yescharacteristic
287C22×C7D4×Q811yescharacteristic
288C22×C7C22×SD1611yescharacteristic
289C22×C7C2×#13511yescharacteristic
290C22×C7#57611yescharacteristic
291C22×C7#57711yescharacteristic
292C2×D7—24no—
293C2×D7—42no—
294C2×D7—42no—
295C2×D7—42no—
296C2×D7—42no—
297C2×D7—42no—
298C2×D7—42no—
299C4×C7—22no—
300C4×C7—42no—
301C4×C7—82no—
302C4×C7—81no—
303C4×C7C4×SD1614yes—
304C4×C7C2×C4×Q811yescharacteristic
305C4×C7#52011yescharacteristic
306C4×C7#55211yescharacteristic
307C4×C7#55411yescharacteristic
308Dic7—24no—
309Dic7—42no—
310Dic7—44no—
311Dic7—42no—
312C24—141no—
313C2×Q8—144no—
314C2×Q8—561no—
315C2×Q8—561no—
316C2×D4—72no—
317C2×D4—72no—
318C2×D4—72no—
319C2×D4—141no—
320C2×D4—141no—
321C2×D4—141no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—72no—
326C22×C4—72no—
327C22×C4—142no—
328C22×C4—141no—
329C22×C4—144no—
330C22×C4—284no—
331C22×C4—284no—
332C22×C4—282no—
333C22×C4—282no—
334C22×C4—562no—
335C22×C4D4×D711yescharacteristic
336C22×C4D4×D711yescharacteristic
337C22×C4C23×D711yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—72no—
341C2×C8—141no—
342C2×C8—141no—
343C2×C8—141no—
344C2×C8—141no—
345C2×C8—141no—
346C2×C8C2×C4×D714yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—74no—
354C4⋊C4—74no—
355C4⋊C4—72no—
356C4⋊C4—81no—
357C4⋊C4—144no—
358C4⋊C4—142no—
359C4⋊C4—141no—
360C4⋊C4—142no—
361C4⋊C4—284no—
362C4⋊C4—562no—
363C4⋊C4Q8×D712yes—
364C22⋊C4—144no—
365C22⋊C4—284no—
366C42—42no—
367C42—142no—
368C42—142no—
369C42—562no—
370C42C2×C4×D712yes—
371C42#116112yes—
372C2×C7#236011yescharacteristic
373C2×C7C2×C4×SD1611yescharacteristic
374C2×C7#294611yescharacteristic
375C2×C7#329511yescharacteristic
376C2×C7#330311yescharacteristic
377C2×C7Q8×SD1611yescharacteristic
378C2×C7#338011yescharacteristic
379D7—44no—
380C23—142no—
381C23—142no—
382C23—142no—
383C23—281no—
384C23—281no—
385C23—281no—
386C23—281no—
387C23C2×D4×D711yescharacteristic
388Q8—564no—
389D4—74no—
390D4—142no—
391D4—142no—
392D4—142no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—74no—
405C2×C4—142no—
406C2×C4—142no—
407C2×C4—142no—
408C2×C4—144no—
409C2×C4—142no—
410C2×C4—142no—
411C2×C4—142no—
412C2×C4—281no—
413C2×C4—281no—
414C2×C4—282no—
415C2×C4—282no—
416C2×C4—288no—
417C2×C4—568no—
418C2×C4#645612yes—
419C2×C4#645612yes—
420C2×C4D7×SD1612yes—
421C2×C4#647912yes—
422C2×C4C22×C4×D711yescharacteristic
423C2×C4#654811yescharacteristic
424C2×C4C2×Q8×D711yescharacteristic
425C2×C4C2×#116111yescharacteristic
426C2×C4D7×#4111yescharacteristic
427C2×C4#655411yescharacteristic
428C8—24no—
429C8—142no—
430C8—281no—
431C7#2296311yescharacteristic, complemented, Sylow
432C22—144no—
433C22—282no—
434C22—282no—
435C22—282no—
436C22—282no—
437C22—282no—
438C22—282no—
439C22C2×#645611yescharacteristic
440C22#9030311yescharacteristic
441C22#9032311yescharacteristic
442C22C2×D7×SD1611yescharacteristic
443C22C2×#647911yescharacteristic
444C22#9046511yescharacteristic
445C22#9046911yescharacteristic
446C4—22no—
447C4—42no—
448C4—81no—
449C4—82no—
450C4—144no—
451C4—282no—
452C4—284no—
453C4—282no—
454C4#8962414yes—
455C4#9023711yescharacteristic
456C4C4×Q8×D711yescharacteristic
457C4#9038511yescharacteristic
458C4#9042111yescharacteristic
459C2—284no—
460C2#1173365011yescharacteristic
461C2C2×#8962411yescharacteristic
462C2#1173664611yescharacteristic
463C2#1173784011yescharacteristic
464C2#1173789611yescharacteristic
465C2#1173848411yescharacteristic
466C2#1173849911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1948D7
hypocenterC7#22963
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
3. #11731101C2semidirect
4. C7×#1948C2semidirect
6. #11728313C2semidirect
7. #11728219C2semidirect
8. #11727336C2semidirect
10. #11726570C2semidirect
34. #89614C4semidirect
74. #6432C2×C4semidirect
76. C7×C4⋊C8C2×C4semidirect
83. #6391C2×C4semidirect
86. #6383Q8semidirect
92. #1948D7semidirect
274. C4⋊C8C4×D7semidirect
431. C7#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#896142
#117318052
#117311012
C7×#19482
#117283382
#117283132
#117282192
#117273362
C4×#63832
#117265702
#117265342
#229637