Finite Group Databasebeta

← #13363300

Subgroups — #13363300

Order 1792.

1030 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90379C211yesmaximal, characteristic
2C2×#89457C211yesmaximal, characteristic
3#11731592C211yesmaximal, characteristic
4C7×#1991C211yesmaximal, characteristic, complemented
5#11729494C211yesmaximal, characteristic, complemented
6#11727532C211yesmaximal, characteristic, complemented
7#11727308C211yesmaximal, characteristic, complemented
8#11726627C212yesmaximal
9#11726567C212yesmaximal
10#11726554C212yesmaximal, complemented
11#11726553C212yesmaximal, complemented
12C2×C7×#125C2211yescharacteristic
13#90379—24no—
14#90379C418yescomplemented
15C2×#6459—21no—
16C2×#6459—21no—
17C2×#6455C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×#6451C2211yescharacteristic
21C2×#6450—21no—
22C2×#6450—21no—
23C2×C42×D7C2211yescharacteristic
24C2×#6433C2211yescharacteristic
25C2×C4×C7×C8C2211yescharacteristic
26C7×#382C2211yescharacteristic
27C2×(C2×D7)⋊C8C2212yes—
28C2×Dic7⋊C8C2212yes—
29#89759C2212yes—
30C4⋊C4×Dic7C2212yes—
31#89708C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33#89457—24no—
34#89438—21no—
35#89436—21no—
36#89423—21no—
37#89420—21no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21837—71nomaximal, Sylow
47C22×C4×D7D411yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7C2311yescharacteristic
50C22×Dic14—21no—
51C22×Dic14—21no—
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×C7C2311yescharacteristic
61C2×(C2×D7)⋊C4—21no—
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—22no—
64C2×C4⋊Dic7C2312yes—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×Dic7⋊C4—21no—
68C2×Dic7⋊C4—21no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—21no—
71C2×C4×Dic7—21no—
72C2×C4×Dic7—21no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7D411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7C2311yescharacteristic
77C22×C7⋊C8C2312yes—
78#6459—28no—
79#6459—28no—
80#6455—22no—
81#6455—22no—
82#6455C2×C414yes—
83#6455C2×C414yes—
84#6451—22no—
85#6451—22no—
86#6451C2×C414yes—
87#6451C2×C414yes—
88#6450—28no—
89#6450—28no—
90C42×D7C2×C414yes—
91C42×D7D414yes—
92#6433—22no—
93#6433C2×C414yes—
94C4×C7×C8—22no—
95C7×#95—21no—
96C7×#95—21no—
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101(C2×D7)⋊C8—24no—
102Dic7⋊C8—24no—
103C4×C7⋊C8—22no—
104C2×#533—71no—
105C2×#431—71no—
106#2283—71no—
107#1991—71no—
108#1991—71no—
109#1991—71no—
110#1991D711yescharacteristic, complemented
111#1988—72no—
112#1987—72no—
113#1825—72no—
114#1825—72no—
115C23×D7C2×D411yescharacteristic
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2411yescharacteristic
122C22×Dic7—21no—
123C22×Dic7—21no—
124C22×Dic7—22no—
125C22×Dic7C2×D411yescharacteristic
126C22×Dic7C2×D411yescharacteristic
127C22×Dic7C2×D411yescharacteristic
128C2×C4×D7—22no—
129C2×C4×D7—22no—
130C2×C4×D7—24no—
131C2×C4×D7—42no—
132C2×C4×D7C22⋊C414yes—
133C2×C4×D7C22⋊C414yes—
134C2×C4×D7C22×C412yes—
135C2×C4×D7C2×D412yes—
136C2×Dic14—24no—
137C2×Dic14—24no—
138C2×Dic14—44no—
139C2×Dic14—44no—
140C2×C7×C8—22no—
141C2×C7×C8—22no—
142C2×C7×C8—22no—
143C7×C4⋊C4—24no—
144C7×C4⋊C4—24no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4C22×C412yes—
149C7×C4⋊C4C22×C412yes—
150C42×C7C22×C412yes—
151C42×C7C2×D412yes—
152(C2×D7)⋊C4—24no—
153(C2×D7)⋊C4—24no—
154(C2×D7)⋊C4—44no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—24no—
158C4⋊Dic7C22×C412yes—
159C4⋊Dic7C22×C412yes—
160Dic7⋊C4—24no—
161Dic7⋊C4—24no—
162Dic7⋊C4—24no—
163Dic7⋊C4—24no—
164C4×Dic7—24no—
165C4×Dic7—24no—
166C4×Dic7—44no—
167C4×Dic7C22⋊C414yes—
168C4×Dic7C22⋊C414yes—
169C4×Dic7C22×C412yes—
170C4×Dic7C2×D412yes—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—22no—
174#533—78no—
175#533—144no—
176C2×C4⋊Q8—71no—
177C2×C4⋊Q8—71no—
178C2×#125—71no—
179C2×#125C2×D711yescharacteristic
180C2×C22⋊Q8—141no—
181C2×C22⋊Q8—141no—
182C2×C4×Q8—141no—
183C2×C4×Q8—141no—
184C2×#117—71no—
185C2×#117—71no—
186C22×C42—71no—
187#431—144no—
188C2×C4⋊C8—72no—
189C2×C22⋊C8—72no—
190C2×C4×C8—71no—
191C2×C4×C8C2×D711yescharacteristic
192#383—141no—
193#382—71no—
194#382C2×D711yescharacteristic
195#381—141no—
196#380—72no—
197C4×C4⋊C4—72no—
198C4×C22⋊C4—141no—
199#376—141no—
200#340—72no—
201#340—72no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340—72no—
206#340C2×D712yes—
207#340C2×D712yes—
208C23×C7C22×D411yescharacteristic
209C22×D7—21no—
210C22×D7—21no—
211C22×D7—22no—
212C22×D7—22no—
213C22×D7C2×C22⋊C412yes—
214C2×C4×C7—22no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7C2×C22⋊C412yes—
224C2×C4×C7C2×C22⋊C412yes—
225C2×C4×C7C23×C411yescharacteristic
226C2×C4×C7C22×D411yescharacteristic
227C2×C4×C72−1+412yes—
228C2×C4×C72−1+412yes—
229C2×Dic7—21no—
230C2×Dic7—21no—
231C2×Dic7—22no—
232C2×Dic7—22no—
233C2×Dic7—22no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236C2×Dic7—42no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239C2×Dic7C2×C22⋊C412yes—
240C2×Dic7C2×C22⋊C412yes—
241C2×Dic7C2×C22⋊C412yes—
242C2×Dic7#13514yes—
243C4×D7—24no—
244C4×D7—24no—
245C4×D7—28no—
246C4×D7—44no—
247Dic14—48no—
248Dic14—48no—
249C7×C8—42no—
250C7⋊C8—42no—
251C22×Q8—141no—
252C22×Q8—141no—
253C23×C4—71no—
254C23×C4—71no—
255C23×C4—71no—
256C22×C8—72no—
257C22×C8C22×D712yes—
258C4⋊Q8—74no—
259C4⋊Q8—74no—
260C4⋊Q8—142no—
261C4⋊Q8—142no—
262#125—22no—
263#125—74no—
264#125—142no—
265#125C4×D714yes—
266C22⋊Q8—148no—
267C22⋊Q8—148no—
268C4×Q8—148no—
269C4×Q8—148no—
270#117—74no—
271#117—74no—
272#117—142no—
273#117—142no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—71no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—72no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—141no—
283C2×C4⋊C4C22×D712yes—
284C2×C4⋊C4C22×D711yescharacteristic
285C2×C4⋊C4C22×D711yescharacteristic
286C2×C22⋊C4—141no—
287C2×C22⋊C4—141no—
288C2×C22⋊C4—142no—
289C2×C42—71no—
290C2×C42—71no—
291C2×C42—71no—
292C2×C42—74no—
293C2×C42—74no—
294C2×C42—141no—
295C2×C42—141no—
296C2×C42—142no—
297C2×C42C22×D711yescharacteristic
298C4⋊C8—144no—
299C22⋊C8—144no—
300C4×C8—22no—
301C4×C8—142no—
302#95—21no—
303#95—21no—
304#95—142no—
305#95—142no—
306#95—141no—
307#95—141no—
308C22×C7C22×C22⋊C411yescharacteristic
309C22×C7#53611yescharacteristic
310C22×C7#53611yescharacteristic
311C22×C7C2×#13512yes—
312C22×C7#57512yes—
313C2×D7—22no—
314C2×D7—22no—
315C2×D7—22no—
316C2×D7—42no—
317C2×D7—42no—
318C4×C7—22no—
319C4×C7—22no—
320C4×C7—42no—
321C4×C7—42no—
322C4×C7—82no—
323C4×C7#51912yes—
324C4×C7#53612yes—
325Dic7—22no—
326Dic7—22no—
327Dic7—42no—
328Dic7—42no—
329Dic7—82no—
330Dic7#41614yes—
331C24—71no—
332C2×Q8—144no—
333C2×Q8—144no—
334C2×Q8—284no—
335C2×Q8—284no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—71no—
340C22×C4—71no—
341C22×C4—71no—
342C22×C4—74no—
343C22×C4—72no—
344C22×C4—72no—
345C22×C4—74no—
346C22×C4—144no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—141no—
350C22×C4—141no—
351C22×C4—142no—
352C22×C4—282no—
353C22×C4D4×D711yescharacteristic
354C22×C4D4×D711yescharacteristic
355C22×C4C23×D711yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—142no—
360C2×C8—142no—
361C2×C8—142no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—24no—
367C4⋊C4—72no—
368C4⋊C4—72no—
369C4⋊C4—144no—
370C4⋊C4—144no—
371C4⋊C4—144no—
372C4⋊C4—144no—
373C4⋊C4—141no—
374C4⋊C4—141no—
375C4⋊C4—144no—
376C4⋊C4C2×C4×D712yes—
377C4⋊C4C2×C4×D712yes—
378C22⋊C4—144no—
379C22⋊C4—144no—
380C22⋊C4—284no—
381C42—74no—
382C42—74no—
383C42—72no—
384C42—72no—
385C42—144no—
386C42—144no—
387C42—284no—
388C42C2×C4×D712yes—
389C42D4×D712yes—
390C2×C7#231011yescharacteristic
391C2×C7C2×#41611yescharacteristic
392C2×C7#289611yescharacteristic
393C2×C7#323312yes—
394C2×C7#323312yes—
395D7—42no—
396C23—72no—
397C23—141no—
398C23—142no—
399C23—142no—
400C23—141no—
401C23C2×D4×D711yescharacteristic
402Q8—288no—
403Q8—288no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—72no—
414C2×C4—74no—
415C2×C4—72no—
416C2×C4—72no—
417C2×C4—148no—
418C2×C4—141no—
419C2×C4—142no—
420C2×C4—141no—
421C2×C4—144no—
422C2×C4—144no—
423C2×C4—142no—
424C2×C4—142no—
425C2×C4—284no—
426C2×C4—282no—
427C2×C4—282no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4D7×C22⋊C412yes—
432C2×C4D7×C22⋊C412yes—
433C2×C4C22×C4×D711yescharacteristic
434C2×C4C2×D4×D711yescharacteristic
435C2×C4#655112yes—
436C2×C4#655412yes—
437C8—42no—
438C8—282no—
439C7#2183711yescharacteristic, complemented, Sylow
440C22—142no—
441C22—142no—
442C22—142no—
443C22—282no—
444C22—282no—
445C22C2×D7×C22⋊C411yescharacteristic
446C22#9036611yescharacteristic
447C22#9041511yescharacteristic
448C22D7×#13512yes—
449C22#9045912yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—74no—
455C4—82no—
456C4—142no—
457C4—142no—
458C4—282no—
459C4—282no—
460C4—562no—
461C4#9026312yes—
462C4#9031912yes—
463C2—282no—
464C2#1173405511yescharacteristic
465C2D7×#41611yescharacteristic
466C2#1173627511yescharacteristic
467C2#1173909412yes—
468C2#1173936612yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1991C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1991D7
hypocenterC7#21837
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. C7×#1991C2semidirect
5. #11729494C2semidirect
6. #11727532C2semidirect
7. #11727308C2semidirect
10. #11726554C2semidirect
11. #11726553C2semidirect
14. #90379C4semidirect
110. #1991D7semidirect
439. C7#21837semidirect

Maximal subgroups

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

subgroupindex
C2×#903792
C2×#894572
#117315922
C7×#19912
#117294942
#117275322
#117273082
#117266272
#117265672
#117265542
#117265532
#218377