Finite Group Databasebeta

← #13320009

Subgroups — #13320009

Order 1792.

1008 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#11736029C212yesmaximal, complemented
3C7×#1459C211yesmaximal, characteristic, complemented
4#11729356C211yesmaximal, characteristic, complemented
5#11727675C212yesmaximal, complemented
6#11725883C214yesmaximal, complemented
7#11725857C212yesmaximal, complemented
8#11725855C212yesmaximal, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×#6502C412yes—
11C2×#6493C411yescharacteristic
12C2×#6493C2211yescharacteristic
13C2×#6489C2211yescharacteristic
14C22×C4⋊Dic7C411yescharacteristic
15#90256—22no—
16#90256—22no—
17#90256—24no—
18C2×#6461C2212yes—
19C2×#6446C412yes—
20C2×(C22⋊D7)⋊C4C2212yes—
21C2×C4×D28C411yescharacteristic
22C2×#6433C411yescharacteristic
23C2×C7×C22⋊C8C2212yes—
24#89979C2212yes—
25#89874—22no—
26#89873—22no—
27#89492—24no—
28#89491C2214yes—
29#89490C2214yes—
30C7×#324C2212yes—
31C7×#323C2211yescharacteristic
32C7×#323C2211yescharacteristic
33#89323C2212yes—
34#89320C2212yes—
35#89263C2214yes—
36#89263C2214yes—
37#17415—71nomaximal, Sylow
38C22×C7×D4C2×C411yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C22×C22⋊D7C2×C412yes—
42C22×C22⋊D7C2×C412yes—
43C23×Dic7C2×C412yes—
44C22×D28C2×C411yescharacteristic
45C22×C4×D7C2×C412yes—
46C22×C7×C8—22no—
47C4×C7×D4—22no—
48C4×C7×D4—22no—
49C2×C7×C4⋊C4C2311yescharacteristic
50C2×C7×C22⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4C2311yescharacteristic
52C2×C42×C7C2×C411yescharacteristic
53C2×C22⋊Dic7C2×C412yes—
54#6502—24no—
55#6502—24no—
56#6493—22no—
57#6493—22no—
58#6493—22no—
59#6493—22no—
60C2×(C2×D7)⋊C4C2×C412yes—
61C2×(C2×D7)⋊C4C2×C412yes—
62C2×(C2×D7)⋊C4C2312yes—
63#6489D412yes—
64#6489D412yes—
65#6489D414yes—
66C2×C4⋊Dic7—22no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7C2×C411yescharacteristic
70C2×C4⋊Dic7C2×C412yes—
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×C4⋊Dic7C2×C411yescharacteristic
73C2×Dic7⋊C4C2×C412yes—
74C2×Dic7⋊C4C2×C412yes—
75C2×C4×Dic7C2312yes—
76C2×#1136—22no—
77#6461—24no—
78#6461—24no—
79#6446—24no—
80#6446—24no—
81(C22⋊D7)⋊C4—24no—
82(C22⋊D7)⋊C4—24no—
83C4×D28—22no—
84C4×D28—21no—
85C4×D28—21no—
86#6433—22no—
87#6433—22no—
88C7×C22⋊C8—22no—
89C7×C22⋊C8C2312yes—
90C7×C22⋊C8C2312yes—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2314yes—
93#6389—24no—
94Dic7⋊C8—24no—
95Dic7⋊C8—24no—
96C8×Dic7—24no—
97C2×#553—71no—
98#2924—72no—
99#1473—74no—
100#1471—72no—
101#1466—72no—
102#1465—71no—
103#1460—72no—
104#1459D711yescharacteristic, complemented
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—42no—
114C22×C4×C7—42no—
115C22×C4×C7C22×C411yescharacteristic
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C22×C411yescharacteristic
119C22×C4×C7C2×D412yes—
120C22×C4×C7C2×D412yes—
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—21no—
134C2×D28—21no—
135C2×D28—42no—
136C2×C4×D7—22no—
137C2×C4×D7—22no—
138C2×C4×D7—82no—
139C2×C7×C8—22no—
140C2×C7×C8—22no—
141C2×C7×C8—44no—
142C7×C4⋊C4—22no—
143C7×C22⋊C4—22no—
144C7×C22⋊C4—22no—
145C42×C7—21no—
146C42×C7—21no—
147C22⋊Dic7C22⋊C414yes—
148C22⋊Dic7C22⋊C414yes—
149(C2×D7)⋊C4—24no—
150(C2×D7)⋊C4—22no—
151(C2×D7)⋊C4—22no—
152(C2×D7)⋊C4—24no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—22no—
156C4⋊Dic7—22no—
157C4⋊Dic7C22⋊C412yes—
158C4⋊Dic7C22⋊C412yes—
159C4⋊Dic7C22⋊C412yes—
160C4⋊Dic7C22⋊C411yescharacteristic
161C4⋊Dic7C22⋊C411yescharacteristic
162Dic7⋊C4—24no—
163Dic7⋊C4—24no—
164C4×Dic7C2×D414yes—
165C4×Dic7C2×D414yes—
166#1136—44no—
167C2×C7⋊C8—24no—
168#553—142no—
169#553—142no—
170#553—144no—
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#433—144no—
183#432—74no—
184#431—74no—
185#424—142no—
186#407—72no—
187C2×C22⋊C8C2×D712yes—
188#405—142no—
189#324—72no—
190#324C2×D712yes—
191#323—72no—
192#323—74no—
193#323—74no—
194#323C2×D711yescharacteristic
195#323C2×D711yescharacteristic
196C23×C7—21no—
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—41no—
201C23×C7—41no—
202C23×C7—41no—
203C23×C7—41no—
204C23×C7C2×C22⋊C412yes—
205C23×C7#13112yes—
206C23×C7C23×C411yescharacteristic
207C22×D7—22no—
208C22×D7—22no—
209C22×D7—22no—
210C22×D7—42no—
211C22×D7—42no—
212C7×D4—44no—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—21no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
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—42no—
235C2×Dic7—42no—
236C2×Dic7—82no—
237C2×Dic7—82no—
238C2×Dic7C2×C22⋊C414yes—
239C2×Dic7C2×C22⋊C412yes—
240C2×Dic7C2×C22⋊C412yes—
241D28—42no—
242D28—42no—
243C4×D7—82no—
244C4×D7—82no—
245C7×C8—42no—
246C7×C8—42no—
247C7⋊C8—44no—
248C22×D4—72no—
249C22×D4—72no—
250C22×D4—71no—
251C22×D4C4×D711yescharacteristic
252C23×C4—72no—
253C23×C4—72no—
254C23×C4C4×D711yescharacteristic
255C23×C4C22×D711yescharacteristic
256C2×M16—142no—
257C22×C8—22no—
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—142no—
273C4×D4—141no—
274C4×D4—141no—
275C4×D4—144no—
276C4×D4—144no—
277#117—72no—
278#117—72no—
279#117—74no—
280C2×C4⋊C4—71no—
281C2×C4⋊C4—71no—
282C2×C4⋊C4—71no—
283C2×C4⋊C4—72no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—142no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4C22×D711yescharacteristic
290C2×C22⋊C4—72no—
291C2×C22⋊C4—72no—
292C2×C22⋊C4—72no—
293C2×C22⋊C4—72no—
294C2×C22⋊C4C4×D711yescharacteristic
295C2×C22⋊C4C22×D711yescharacteristic
296C2×C42—72no—
297C2×C42C4×D711yescharacteristic
298C4⋊C8—144no—
299C4⋊C8—144no—
300C22⋊C8—22no—
301C22⋊C8—74no—
302C22⋊C8—142no—
303C22⋊C8C22×D712yes—
304C22⋊C8C22×D712yes—
305#97—144no—
306C4×C8—144no—
307C22×C7—21no—
308C22×C7—21no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7—22no—
312C22×C7—22no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7—41no—
316C22×C7—41no—
317C22×C7—81no—
318C22×C7—81no—
319C22×C7C22×C22⋊C411yescharacteristic
320C22×C7C2×#13111yescharacteristic
321C22×C7#56811yescharacteristic
322C2×D7—42no—
323C2×D7—42no—
324C2×D7—42no—
325C2×D7—82no—
326C2×D7—82no—
327C4×C7—22no—
328C4×C7—21no—
329C4×C7—41no—
330C4×C7—41no—
331C4×C7—41no—
332C4×C7#41012yes—
333Dic7—24no—
334Dic7—22no—
335Dic7—22no—
336Dic7—42no—
337C24—72no—
338C24C2×C4×D711yescharacteristic
339C24C2×C4×D711yescharacteristic
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—144no—
344C2×D4—144no—
345C2×D4—144no—
346C2×D4—141no—
347C2×D4—141no—
348C2×D4—282no—
349C2×D4—282no—
350C2×D4—562no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—72no—
356C22×C4—72no—
357C22×C4—72no—
358C22×C4—142no—
359C22×C4—142no—
360C22×C4—142no—
361C22×C4—142no—
362C22×C4—562no—
363C22×C4—562no—
364C22×C4C2×C4×D711yescharacteristic
365C22×C4C2×C4×D711yescharacteristic
366C22×C4C2×C4×D711yescharacteristic
367C22×C4C2×C4×D711yescharacteristic
368C22×C4D4×D712yes—
369C22×C4D4×D712yes—
370C22×C4C23×D711yescharacteristic
371M16—284no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—44no—
375C2×C8—144no—
376C4⋊C4—22no—
377C4⋊C4—72no—
378C4⋊C4—71no—
379C4⋊C4—71no—
380C4⋊C4—72no—
381C4⋊C4—72no—
382C4⋊C4—144no—
383C4⋊C4—144no—
384C4⋊C4—142no—
385C4⋊C4—142no—
386C4⋊C4—141no—
387C4⋊C4—141no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—74no—
391C22⋊C4—74no—
392C22⋊C4—142no—
393C22⋊C4—142no—
394C22⋊C4—144no—
395C22⋊C4—144no—
396C42—21no—
397C42—21no—
398C42—74no—
399C42—74no—
400C2×C7—21no—
401C2×C7—21no—
402C2×C7—42no—
403C2×C7—42no—
404C2×C7#288011yescharacteristic
405C2×C7C2×#41011yescharacteristic
406C2×C7#288811yescharacteristic
407D7—82no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—142no—
417C23—142no—
418C23—142no—
419C23—282no—
420C23—282no—
421C23D7×C22⋊C412yes—
422C23#646412yes—
423C23C22×C4×D711yescharacteristic
424D4—44no—
425D4—282no—
426D4—288no—
427D4—282no—
428D4—564no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—74no—
443C2×C4—72no—
444C2×C4—72no—
445C2×C4—142no—
446C2×C4—142no—
447C2×C4—142no—
448C2×C4—282no—
449C2×C4—282no—
450C2×C4—562no—
451C2×C4—562no—
452C2×C4—562no—
453C2×C4—562no—
454C2×C4D7×C22⋊C411yescharacteristic
455C2×C4D7×C22⋊C411yescharacteristic
456C2×C4C2×D4×D712yes—
457C8—42no—
458C8—42no—
459C8—284no—
460C7#1741511yescharacteristic, complemented, Sylow
461C22—21no—
462C22—21no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—41no—
470C22—41no—
471C22—81no—
472C22—81no—
473C22—282no—
474C22—282no—
475C22—282no—
476C22—562no—
477C22—562no—
478C22C2×D7×C22⋊C411yescharacteristic
479C22C2×#646411yescharacteristic
480C22#9043611yescharacteristic
481C4—22no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—142no—
487C4—142no—
488C4—144no—
489C4—282no—
490C4#8951212yes—
491C2—21no—
492C2—21no—
493C2—42no—
494C2—42no—
495C2—562no—
496C2#1173604511yescharacteristic
497C2C2×#8951211yescharacteristic
498C2#1173615911yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1459C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#90256C2
cosocleC22×C4×C7C24
hypercenter#1459D7
hypocenterC7#17415
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #11736029C2semidirect
3. C7×#1459C2semidirect
4. #11729356C2semidirect
5. #11727675C2semidirect
6. #11725883C2semidirect
7. #11725857C2semidirect
8. #11725855C2semidirect
104. #1459D7semidirect
460. C7#17415semidirect

Maximal subgroups

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

subgroupindex
C2×#902562
#117360292
C7×#14592
#117293562
#117276752
#117258832
#117258572
#117258552
#174157