Finite Group Databasebeta

← #13320090

Subgroups — #13320090

Order 1792.

1022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90256C211yesmaximal, characteristic
2#11736781C212yesmaximal, complemented
3C7×#1466C211yesmaximal, characteristic, complemented
4#11729355C211yesmaximal, characteristic, complemented
5#11728585C212yesmaximal, complemented
6#11725876C214yesmaximal, complemented
7#11725875C212yesmaximal, complemented
8#11725873C212yesmaximal, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×#6502C412yes—
11C2×#6493C411yescharacteristic
12C2×#6493C411yescharacteristic
13C2×#6489C411yescharacteristic
14C22×C4⋊Dic7C411yescharacteristic
15#90256—22no—
16#90256—22no—
17#90256—24no—
18C2×#6461C412yes—
19C2×#6446C2212yes—
20C2×(C22⋊D7)⋊C4C2212yes—
21C2×C4×D28C2211yescharacteristic
22C2×#6433C2211yescharacteristic
23C7×#432C2212yes—
24#89697C2212yes—
25#89610—22no—
26#89606—22no—
27#89498C2214yes—
28D7×C22⋊C8C2214yes—
29#89478—24no—
30C7×#323C2212yes—
31C7×#323C2212yes—
32#89320C2211yescharacteristic
33#89320C2211yescharacteristic
34#89320C2212yes—
35#89264C2214yes—
36#89264C2214yes—
37#17424—71nomaximal, Sylow
38C22×C7×D4C2×C411yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2×C411yescharacteristic
41C22×C22⋊D7C2×C412yes—
42C22×C22⋊D7C2×C412yes—
43C23×Dic7C2×C412yes—
44C22×D28C2×C411yescharacteristic
45C22×C4×D7C2312yes—
46C4×C7×D4—22no—
47C4×C7×D4—22no—
48C2×C7×C4⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4C2311yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C2×C22⋊Dic7C2×C412yes—
53#6502—24no—
54#6502—24no—
55#6493—22no—
56#6493—22no—
57#6493—22no—
58#6493—22no—
59C2×(C2×D7)⋊C4C2×C412yes—
60C2×(C2×D7)⋊C4C2×C412yes—
61C2×(C2×D7)⋊C4C2×C412yes—
62#6489—22no—
63#6489—21no—
64#6489—21no—
65C2×C4⋊Dic7—22no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7C2×C411yescharacteristic
69C2×C4⋊Dic7C2×C411yescharacteristic
70C2×C4⋊Dic7C2×C412yes—
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×Dic7⋊C4C2×C412yes—
73C2×Dic7⋊C4C2312yes—
74C2×C4×Dic7C2×C412yes—
75C2×#1131—24no—
76C2×C8×D7—24no—
77#6461—24no—
78#6461—24no—
79#6446—24no—
80#6446—24no—
81(C22⋊D7)⋊C4—24no—
82(C22⋊D7)⋊C4—24no—
83C4×D28D412yes—
84C4×D28D412yes—
85C4×D28D414yes—
86#6433—22no—
87#6433—22no—
88C7×C4⋊C8—22no—
89C7×C22⋊C8C2314yes—
90C7×#97—22no—
91C22⋊(C7⋊C8)C2312yes—
92C22⋊(C7⋊C8)C2312yes—
93(C2×D7)⋊C8—24no—
94(C2×D7)⋊C8—24no—
95C4⋊(C7⋊C8)—22no—
96C4×C7⋊C8—22no—
97C2×#553—71no—
98#2974—72no—
99#1470—72no—
100#1468—74no—
101#1466D711yescharacteristic, complemented
102#1465—72no—
103#1460—71no—
104#1459—72no—
105C24×C7C22×C411yescharacteristic
106C24×C7C22×C411yescharacteristic
107C23×D7C22×C412yes—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—41no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—42no—
116C22×C4×C7—42no—
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C22×C411yescharacteristic
119C22×C4×C7C22×C411yescharacteristic
120C22×C4×C7C22×C411yescharacteristic
121C22×C4×C7C2411yescharacteristic
122C2×C22⋊D7—24no—
123C2×C22⋊D7—24no—
124C2×C22⋊D7—24no—
125C2×C22⋊D7—42no—
126C2×C22⋊D7—82no—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—82no—
130C22×Dic7C22×C412yes—
131C22×Dic7C22×C412yes—
132C22×Dic7C22×C412yes—
133C2×D28—42no—
134C2×D28C22⋊C412yes—
135C2×D28C22⋊C412yes—
136C2×C4×D7—82no—
137C2×C4×D7C2×D414yes—
138C2×C4×D7C2×D414yes—
139C2×C7×C8—24no—
140C7×C4⋊C4—22no—
141C7×C22⋊C4—22no—
142C7×C22⋊C4—22no—
143C42×C7C2×D412yes—
144C42×C7C2×D412yes—
145C22⋊Dic7—22no—
146C22⋊Dic7—22no—
147(C2×D7)⋊C4—24no—
148(C2×D7)⋊C4—24no—
149(C2×D7)⋊C4C22⋊C414yes—
150(C2×D7)⋊C4C22⋊C414yes—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7C22⋊C412yes—
158C4⋊Dic7C22⋊C411yescharacteristic
159C4⋊Dic7C22⋊C411yescharacteristic
160Dic7⋊C4—24no—
161Dic7⋊C4—24no—
162C4×Dic7—22no—
163C4×Dic7—22no—
164C2×C7⋊C8—22no—
165C2×C7⋊C8—22no—
166#1131—84no—
167C8×D7—84no—
168#553—142no—
169#553—144no—
170#553—142no—
171C2×#126—72no—
172C2×#125—71no—
173C2×#123—72no—
174C2×#121—72no—
175C2×#121—71no—
176C2×#121—71no—
177C2×C4×D4—72no—
178C2×C4×D4—71no—
179C2×C4×D4C2×D711yescharacteristic
180C2×#117—71no—
181C22×C4⋊C4—71no—
182#436—142no—
183#435—144no—
184C8×D4—142no—
185#432C2×D712yes—
186#431—72no—
187#407—74no—
188C2×C22⋊C8—74no—
189#324—74no—
190#324—74no—
191#323—71no—
192#323—71no—
193#323—72no—
194#323C2×D712yes—
195#323C2×D712yes—
196C23×C7—21no—
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—41no—
202C23×C7—41no—
203C23×C7—41no—
204C23×C7—41no—
205C23×C7#13112yes—
206C23×C7C23×C411yescharacteristic
207C22×D7—22no—
208C22×D7—22no—
209C22×D7—42no—
210C22×D7—42no—
211C22×D7C2×C22⋊C414yes—
212C7×D4—44no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—22no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—41no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7C2×C22⋊C412yes—
226C2×C4×C7C2×C22⋊C411yescharacteristic
227C2×C4×C7C2×C22⋊C411yescharacteristic
228C2×C4×C7C22×D412yes—
229C22⋊D7—48no—
230C22⋊D7—84no—
231C2×Dic7—22no—
232C2×Dic7—22no—
233C2×Dic7—22no—
234C2×Dic7—22no—
235C2×Dic7—42no—
236C2×Dic7—42no—
237C2×Dic7—82no—
238C2×Dic7—82no—
239C2×Dic7C2×C22⋊C412yes—
240C2×Dic7C2×C22⋊C412yes—
241D28—24no—
242D28—42no—
243C4×D7—44no—
244C4×D7—44no—
245C7×C8—44no—
246C7⋊C8—42no—
247C7⋊C8—42no—
248C22×D4—72no—
249C22×D4—72no—
250C22×D4—71no—
251C22×D4C4×D711yescharacteristic
252C23×C4—72no—
253C23×C4—72no—
254C23×C4C4×D711yescharacteristic
255C23×C4C4×D711yescharacteristic
256C2×M16—144no—
257C22×C8—144no—
258#126—144no—
259#126—144no—
260#125—142no—
261#125—142no—
262#123—144no—
263#123—144no—
264#121—144no—
265#121—144no—
266#121—142no—
267#121—142no—
268#121—142no—
269#121—142no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—72no—
273C4×D4—74no—
274C4×D4—72no—
275C4×D4—144no—
276C4×D4—144no—
277#117—142no—
278#117—141no—
279#117—141no—
280C2×C4⋊C4—71no—
281C2×C4⋊C4—71no—
282C2×C4⋊C4—72no—
283C2×C4⋊C4—71no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—142no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4C4×D711yescharacteristic
290C2×C22⋊C4—72no—
291C2×C22⋊C4—72no—
292C2×C22⋊C4—72no—
293C2×C22⋊C4—72no—
294C2×C22⋊C4C22×D711yescharacteristic
295C2×C22⋊C4C22×D711yescharacteristic
296C2×C42—72no—
297C2×C42C22×D711yescharacteristic
298C4⋊C8—22no—
299C4⋊C8—142no—
300C22⋊C8—72no—
301C22⋊C8—72no—
302C22⋊C8—144no—
303C22⋊C8—144no—
304C22⋊C8C22×D714yes—
305#97—22no—
306C4×C8—142no—
307C22×C7—21no—
308C22×C7—21no—
309C22×C7—22no—
310C22×C7—42no—
311C22×C7—42no—
312C22×C7—41no—
313C22×C7—41no—
314C22×C7—41no—
315C22×C7—41no—
316C22×C7—41no—
317C22×C7—81no—
318C22×C7—81no—
319C22×C7C22×C22⋊C411yescharacteristic
320C22×C7C2×#13111yescharacteristic
321C22×C7#56811yescharacteristic
322C2×D7—24no—
323C2×D7—24no—
324C2×D7—24no—
325C2×D7—82no—
326C2×D7—82no—
327C4×C7—22no—
328C4×C7—22no—
329C4×C7—21no—
330C4×C7—41no—
331C4×C7—41no—
332C4×C7#41312yes—
333Dic7—22no—
334Dic7—22no—
335Dic7—42no—
336Dic7—42no—
337C24—72no—
338C24C2×C4×D711yescharacteristic
339C24C2×C4×D711yescharacteristic
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—72no—
344C2×D4—72no—
345C2×D4—144no—
346C2×D4—144no—
347C2×D4—144no—
348C2×D4—282no—
349C2×D4—282no—
350C2×D4—562no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—72no—
358C22×C4—74no—
359C22×C4—74no—
360C22×C4—72no—
361C22×C4—72no—
362C22×C4—142no—
363C22×C4—142no—
364C22×C4—562no—
365C22×C4—562no—
366C22×C4C2×C4×D711yescharacteristic
367C22×C4C2×C4×D711yescharacteristic
368C22×C4C2×C4×D711yescharacteristic
369C22×C4C2×C4×D711yescharacteristic
370C22×C4C23×D711yescharacteristic
371M16—564no—
372C2×C8—24no—
373C2×C8—142no—
374C2×C8—142no—
375C2×C8—564no—
376C4⋊C4—22no—
377C4⋊C4—72no—
378C4⋊C4—71no—
379C4⋊C4—71no—
380C4⋊C4—142no—
381C4⋊C4—142no—
382C4⋊C4—144no—
383C4⋊C4—141no—
384C4⋊C4—141no—
385C4⋊C4—141no—
386C4⋊C4—141no—
387C4⋊C4—144no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—74no—
391C22⋊C4—74no—
392C22⋊C4—144no—
393C22⋊C4—144no—
394C22⋊C4—142no—
395C22⋊C4—142no—
396C42—142no—
397C42—142no—
398C42D4×D712yes—
399C42D4×D712yes—
400C2×C7—21no—
401C2×C7—21no—
402C2×C7—42no—
403C2×C7—81no—
404C2×C7#288011yescharacteristic
405C2×C7#288311yescharacteristic
406C2×C7C2×#41311yescharacteristic
407D7—44no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—74no—
418C23—142no—
419C23—142no—
420C23—282no—
421C23—282no—
422C23#647012yes—
423C23C22×C4×D711yescharacteristic
424D4—44no—
425D4—144no—
426D4—282no—
427D4—288no—
428D4—564no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—72no—
442C2×C4—72no—
443C2×C4—142no—
444C2×C4—142no—
445C2×C4—142no—
446C2×C4—142no—
447C2×C4—284no—
448C2×C4—284no—
449C2×C4—282no—
450C2×C4—282no—
451C2×C4—562no—
452C2×C4—562no—
453C2×C4D7×C22⋊C412yes—
454C2×C4D7×C22⋊C411yescharacteristic
455C2×C4D7×C22⋊C411yescharacteristic
456C2×C4C2×D4×D712yes—
457C8—44no—
458C8—282no—
459C8—282no—
460C7#1742411yescharacteristic, complemented, Sylow
461C22—21no—
462C22—21no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—81no—
472C22—81no—
473C22—144no—
474C22—144no—
475C22—144no—
476C22—562no—
477C22—562no—
478C22C2×D7×C22⋊C411yescharacteristic
479C22C2×#647011yescharacteristic
480C22#9043611yescharacteristic
481C4—22no—
482C4—22no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—142no—
487C4—142no—
488C4—282no—
489C4—282no—
490C4#8951712yes—
491C2—21no—
492C2—21no—
493C2—42no—
494C2—81no—
495C2—284no—
496C2#1173607611yescharacteristic
497C2#1173613911yescharacteristic
498C2C2×#8951711yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1466C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#90256C2
cosocleC22×C4×C7C24
hypercenter#1466D7
hypocenterC7#17424
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #11736781C2semidirect
3. C7×#1466C2semidirect
4. #11729355C2semidirect
5. #11728585C2semidirect
6. #11725876C2semidirect
7. #11725875C2semidirect
8. #11725873C2semidirect
101. #1466D7semidirect
460. C7#17424semidirect

Maximal subgroups

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

subgroupindex
C2×#902562
#117367812
C7×#14662
#117293552
#117285852
#117258762
#117258752
#117258732
#174247