Finite Group Databasebeta

← #13060077

Subgroups — #13060077

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4246C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3#10621420C211yesmaximal, characteristic, complemented
4C7×D4×Dic3C211yesmaximal, characteristic, complemented
5C3×#6500C211yesmaximal, characteristic, complemented
6#10621137C211yesmaximal, characteristic, complemented
7#10621131C211yesmaximal, characteristic, complemented
8#10621123C211yesmaximal, characteristic, complemented
9#10621112C211yesmaximal, characteristic, complemented
10D7×#217C211yesmaximal, characteristic, complemented
11#10621095C211yesmaximal, characteristic, complemented
12#10620975C211yesmaximal, characteristic, complemented
13#10620957C211yesmaximal, characteristic, complemented
14#10620918C211yesmaximal, characteristic, complemented
15#10620915C211yesmaximal, characteristic, complemented
16#90241—31nomaximal, Hall
17C23×D21C2211yescharacteristic
18C23×C3×D7C2211yescharacteristic
19C2×C3×C7×D4C2211yescharacteristic
20C2×#4268C2211yescharacteristic, complemented
21C2×#4268C2211yescharacteristic, complemented
22C22×C7×Dic3C2211yescharacteristic
23C22×C7×Dic3C2211yescharacteristic, complemented
24C2×C3×C22⋊D7C2211yescharacteristic, complemented
25C2×C3×C22⋊D7C2211yescharacteristic, complemented
26C2×#4248C2211yescharacteristic, complemented
27C2×#4246—24no—
28C2×#4246C2211yescharacteristic
29C2×#4246C2211yescharacteristic
30C2×#4246C2211yescharacteristic
31C2×#4246C2211yescharacteristic
32C2×D21⋊C4C2211yescharacteristic, complemented
33C2×D21⋊C4C2211yescharacteristic, complemented
34#66683—24no—
35C2×Dic3×D7C2211yescharacteristic, complemented
36C2×Dic3×D7C2211yescharacteristic, complemented
37#66636C2211yescharacteristic
38#66631C2211yescharacteristic, complemented
39#66631C2211yescharacteristic, complemented
40C7×#217C2211yescharacteristic
41C7×#217C2211yescharacteristic, complemented
42C7×C4⋊Dic3C2211yescharacteristic, complemented
43C4×C7×Dic3C2211yescharacteristic, complemented
44C3×C22⋊Dic7C2211yescharacteristic
45C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
47Dic3⋊Dic7C2211yescharacteristic, complemented
48#66577C2211yescharacteristic, complemented
49Dic7⋊Dic3C2211yescharacteristic, complemented
50#66575C2211yescharacteristic, complemented
51#66575C2211yescharacteristic, complemented
52(C2×D7)⋊Dic3C2211yescharacteristic, complemented
53(C2×D7)⋊Dic3C2211yescharacteristic, complemented
54C2×#1157—31no—
55C22×D28—31no—
56C4×C7×D4—31no—
57#6500—31no—
58#6500D311yescharacteristic, complemented
59#6493—31no—
60#6487—31no—
61#6457—31no—
62#6457—31no—
63#6447—31no—
64#6447—31no—
65D7×C22⋊C4—31no—
66D7×C22⋊C4—31no—
67#6438—31no—
68C4×D28—31no—
69#5591—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C22×D21—22no—
73C22×D21—22no—
74C22×D21—41no—
75C22×D21C2311yescharacteristic
76C22×D21C2311yescharacteristic
77C22×C3×D7—22no—
78C22×C3×D7—22no—
79C22×C3×D7—41no—
80C22×C3×D7C2311yescharacteristic
81C22×C3×D7C2311yescharacteristic
82C3×C7×D4—22no—
83C2×C3×C4×C7C2311yescharacteristic
84#4268—22no—
85#4268—41no—
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88C2×C7×Dic3—22no—
89C2×C7×Dic3—22no—
90C2×C7×Dic3C2311yescharacteristic
91C2×C7×Dic3C2311yescharacteristic
92C2×C7×Dic3C2311yescharacteristic
93C2×C7×Dic3C2311yescharacteristic
94C3×C22⋊D7—22no—
95C3×C22⋊D7—41no—
96C2×C3×Dic7C2311yescharacteristic
97C2×C3×Dic7C2311yescharacteristic
98#4248—22no—
99#4246—22no—
100#4246—22no—
101#4246—22no—
102#4246D414yescomplemented
103D21⋊C4—22no—
104D21⋊C4—41no—
105Dic3×D7—22no—
106Dic3×D7—41no—
107C23×D7—31no—
108C23×D7C2×D311yescharacteristic
109C2×C7×D4C2×D311yescharacteristic
110C22×C4×C7—31no—
111C22×C4×C7—31no—
112C2×C22⋊D7—31no—
113C2×C22⋊D7—31no—
114C2×C22⋊D7C2×D311yescharacteristic, complemented
115C2×C22⋊D7C2×D311yescharacteristic, complemented
116#1157—64no—
117C2×D28—31no—
118C2×D28—31no—
119C2×D28—31no—
120C2×D28—31no—
121C2×D28—64no—
122C2×C4×D7—31no—
123C2×C4×D7—31no—
124C2×C4×D7—31no—
125C2×C4×D7—31no—
126C2×Dic14—31no—
127C7×C4⋊C4—31no—
128C7×C22⋊C4—31no—
129C7×C22⋊C4—31no—
130C42×C7—31no—
131C22⋊Dic7—31no—
132C22⋊Dic7C2×D311yescharacteristic
133(C2×D7)⋊C4—31no—
134(C2×D7)⋊C4—31no—
135(C2×D7)⋊C4—31no—
136(C2×D7)⋊C4—31no—
137(C2×D7)⋊C4—31no—
138(C2×D7)⋊C4—31no—
139(C2×D7)⋊C4C2×D311yescharacteristic, complemented
140(C2×D7)⋊C4C2×D311yescharacteristic, complemented
141C4⋊Dic7—31no—
142Dic7⋊C4—31no—
143Dic7⋊C4—31no—
144C22×#67—71no—
145C2×#237—71no—
146C3×#120—71no—
147C2×#217—71no—
148#939—71no—
149#939—71no—
150#938—71no—
151#937—71no—
152D4×Dic3D711yescharacteristic, complemented
153#886—71no—
154#885—71no—
155#883—71no—
156#67⋊C4—71no—
157D3×C22⋊C4—71no—
158#878—71no—
159C22×C3×C7—21no—
160C22×C3×C7—21no—
161C22×C3×C7—21no—
162C22×C3×C7—21no—
163C22×C3×C7—21no—
164C22×C3×C7#4112yescomplemented
165C22×C3×C7C2411yescharacteristic
166C2×D21—21no—
167C2×D21—21no—
168C2×D21—21no—
169C2×D21—21no—
170C2×D21—21no—
171C2×D21—44no—
172C2×D21C2×D412yes—
173C2×C3×D7—21no—
174C2×C3×D7—21no—
175C2×C3×D7—21no—
176C2×C3×D7—21no—
177C2×C3×D7—21no—
178C2×C3×D7—44no—
179C2×C3×D7C2×D412yes—
180C3×C4×C7—21no—
181Dic21—21no—
182Dic21—21no—
183C3×Dic7—21no—
184C3×Dic7—21no—
185C7×Dic3—21no—
186C7×Dic3—21no—
187C7×Dic3—21no—
188C7×Dic3C2×D412yes—
189#546—211noSylow
190C23×C7C22×D311yescharacteristic
191C23×C7C22×D311yescharacteristic
192C22×D7—22no—
193C22×D7—22no—
194C22×D7—31no—
195C22×D7—31no—
196C22×D7—41no—
197C22×D7—62no—
198C22×D7—62no—
199C22×D7—121no—
200C22×D7C22×D311yescharacteristic
201C22×D7C22×D311yescharacteristic
202C7×D4—22no—
203C2×C4×C7—31no—
204C2×C4×C7—31no—
205C2×C4×C7—31no—
206C2×C4×C7—31no—
207C2×C4×C7—62no—
208C2×C4×C7—62no—
209C2×C4×C7C22×D311yescharacteristic
210C22⋊D7—22no—
211C22⋊D7—41no—
212C22⋊D7—62no—
213C22⋊D7—121no—
214C2×Dic7—31no—
215C2×Dic7—31no—
216C2×Dic7C22×D311yescharacteristic
217C2×Dic7C22×D311yescharacteristic
218D28—34no—
219D28—62no—
220D28—62no—
221D28—62no—
222C4×D7—62no—
223C4×D7—62no—
224C4×D7—121no—
225C4×D7—121no—
226Dic14—62no—
227C24×C3—71no—
228C23×D3—71no—
229C2×C3×D4—71no—
230C2×C3×D4—71no—
231C2×C3×D4C2×D711yescharacteristic
232C2×#67—71no—
233C2×#67—71no—
234C2×#67—71no—
235C2×#67—71no—
236C2×#67—71no—
237C2×#67—71no—
238C2×#67—144no—
239C22×Dic3—71no—
240C22×Dic3—71no—
241C22×Dic3C2×D711yescharacteristic
242C22×Dic3C2×D711yescharacteristic, complemented
243#237—144no—
244C2×C4×D3—71no—
245C2×C4×D3—71no—
246C2×Dic6—71no—
247C3×C22⋊C4—71no—
248C3×C22⋊C4—71no—
249C3×C22⋊C4—71no—
250#217—71no—
251#217—71no—
252#217—71no—
253#217C2×D711yescharacteristic
254#217C2×D711yescharacteristic, complemented
255(C2×D3)⋊C4—71no—
256(C2×D3)⋊C4—71no—
257(C2×D3)⋊C4—71no—
258(C2×D3)⋊C4—71no—
259C4⋊Dic3—71no—
260C4⋊Dic3C2×D711yescharacteristic, complemented
261Dic3⋊C4—71no—
262Dic3⋊C4—71no—
263C4×Dic3C2×D711yescharacteristic, complemented
264C2×C3×C7—22no—
265C2×C3×C7—41no—
266C2×C3×C7C22×D411yescharacteristic
267C2×C3×C7C2×#4111yescharacteristic
268C2×C3×C7H(5,𝔽2)11yescharacteristic
269D21—22no—
270D21—41no—
271C3×D7—22no—
272C3×D7—41no—
273C2×#41—211no—
274C22×D4—211no—
275#124—211no—
276#123—211no—
277#123—211no—
278C22⋊Q8—211no—
279#121—211no—
280#121—211no—
281#121—211no—
282#120—71no—
283#120—211no—
284C4×D4—31no—
285C4×D4—211no—
286C2×C22⋊C4—211no—
287C2×C22⋊C4—211no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—21no—
291C22×C7—21no—
292C22×C7—21no—
293C22×C7#23712yescomplemented
294C22×C7C23×D311yescharacteristic
295C2×D7—21no—
296C2×D7—21no—
297C2×D7—21no—
298C2×D7—21no—
299C2×D7—21no—
300C2×D7—32no—
301C2×D7—44no—
302C2×D7—61no—
303C2×D7—61no—
304C2×D7—61no—
305C2×D7—61no—
306C2×D7—61no—
307C2×D7—124no—
308C2×D7D3×D412yes—
309C4×C7—21no—
310C4×C7—32no—
311C4×C7—61no—
312C4×C7—61no—
313C4×C7—61no—
314Dic7—21no—
315Dic7—21no—
316Dic7—61no—
317Dic7—61no—
318C23×C3—71no—
319C23×C3—71no—
320C23×C3—142no—
321C23×C3—142no—
322C23×C3—281no—
323C23×C3C22×D711yescharacteristic
324C23×C3C22×D711yescharacteristic
325C22×D3—71no—
326C22×D3—71no—
327C22×D3—142no—
328C22×D3—142no—
329C22×D3—281no—
330C3×D4—22no—
331C3×D4—142no—
332C3×D4—281no—
333C2×C3×C4—71no—
334C2×C3×C4—71no—
335C2×C3×C4C22×D711yescharacteristic
336#67—74no—
337#67—142no—
338#67—142no—
339#67—142no—
340#67—142no—
341#67—281no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—71no—
345C2×Dic3—71no—
346C2×Dic3—142no—
347C2×Dic3—281no—
348C2×Dic3C22×D711yescharacteristic
349C2×Dic3C22×D711yescharacteristic
350C2×Dic3C22×D711yescharacteristic
351C2×Dic3C22×D711yescharacteristic
352C4×D3—142no—
353C4×D3—281no—
354Dic6—142no—
355C3×C7#54611yescharacteristic, complemented, Hall
356C24—71no—
357C24—211no—
358#41—424no—
359C2×Q8—211no—
360C2×D4—71no—
361C2×D4—71no—
362C2×D4—211no—
363C2×D4—211no—
364C2×D4—211no—
365C2×D4—211no—
366C2×D4—211no—
367C2×D4—211no—
368C2×D4—424no—
369C2×D4D3×D711yescharacteristic
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—211no—
373C22×C4—211no—
374C22×C4—211no—
375C22×C4—211no—
376C4⋊C4—31no—
377C4⋊C4—211no—
378C4⋊C4—211no—
379C4⋊C4—211no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—71no—
383C22⋊C4—71no—
384C22⋊C4—71no—
385C22⋊C4—211no—
386C22⋊C4—211no—
387C22⋊C4—211no—
388C22⋊C4—211no—
389C22⋊C4—211no—
390C22⋊C4—211no—
391C22⋊C4—211no—
392C42—31no—
393C2×C7—22no—
394C2×C7—41no—
395C2×C7C2×D3×D411yescharacteristic
396C2×C7C2×#23711yescharacteristic
397C2×C7#100411yescharacteristic
398D7—22no—
399D7—41no—
400D7—62no—
401D7—121no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—72no—
408C22×C3—141no—
409C22×C3—141no—
410C22×C3—141no—
411C22×C3—141no—
412C22×C3—141no—
413C22×C3—284no—
414C22×C3#115712yescomplemented
415C22×C3C23×D711yescharacteristic
416C2×D3—72no—
417C2×D3—141no—
418C2×D3—141no—
419C2×D3—141no—
420C2×D3—141no—
421C2×D3—141no—
422C2×D3—284no—
423C3×C4—21no—
424C3×C4—141no—
425C3×C4—141no—
426Dic3—21no—
427Dic3—21no—
428Dic3—21no—
429Dic3—141no—
430Dic3—141no—
431Dic3D4×D712yes—
432C23—71no—
433C23—71no—
434C23—142no—
435C23—142no—
436C23—211no—
437C23—211no—
438C23—281no—
439C23—422no—
440C23—422no—
441C23—841no—
442C23C2×D3×D711yescharacteristic
443C23C2×D3×D711yescharacteristic
444Q8—422no—
445D4—22no—
446D4—142no—
447D4—214no—
448D4—281no—
449D4—422no—
450D4—422no—
451D4—422no—
452D4—422no—
453D4—841no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—71no—
461C2×C4—71no—
462C2×C4—211no—
463C2×C4—211no—
464C2×C4—422no—
465C2×C4—422no—
466C2×C4—841no—
467C2×C4—841no—
468C2×C4C2×D3×D711yescharacteristic
469C7#559111yescharacteristic, complemented, Sylow
470C2×C3—22no—
471C2×C3—41no—
472C2×C3—142no—
473C2×C3—281no—
474C2×C3C2×#115711yescharacteristic
475C2×C3C2×D4×D711yescharacteristic
476C2×C3#655311yescharacteristic
477D3—142no—
478D3—281no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—72no—
485C22—141no—
486C22—141no—
487C22—141no—
488C22—141no—
489C22—141no—
490C22—212no—
491C22—284no—
492C22—421no—
493C22—421no—
494C22—421no—
495C22—421no—
496C22—421no—
497C22—844no—
498C22#6668312yescomplemented
499C22C22×D3×D711yescharacteristic
500C4—21no—
501C4—32no—
502C4—61no—
503C4—61no—
504C4—61no—
505C4—141no—
506C4—141no—
507C4—421no—
508C4—421no—
509C3#9024111yescharacteristic, complemented, Sylow
510C2—22no—
511C2—41no—
512C2—142no—
513C2—281no—
514C2—422no—
515C2—841no—
516C2D3×D4×D711yescharacteristic
517C2#1062157311yescharacteristic
518C2C2×#6668311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#546
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#4246C2semidirect
2. C2×#66683C2semidirect
3. #10621420C2semidirect
4. C7×D4×Dic3C2semidirect
5. C3×#6500C2semidirect
6. #10621137C2semidirect
7. #10621131C2semidirect
8. #10621123C2semidirect
9. #10621112C2semidirect
10. D7×#217C2semidirect
11. #10621095C2semidirect
12. #10620975C2semidirect
13. #10620957C2semidirect
14. #10620918C2semidirect
15. #10620915C2semidirect
20. C2×#4268C22semidirect
21. C2×#4268C22semidirect
23. C22×C7×Dic3C22semidirect
24. C2×C3×C22⋊D7C22semidirect
25. C2×C3×C22⋊D7C22semidirect
26. C2×#4248C22semidirect
32. C2×D21⋊C4C22semidirect
33. C2×D21⋊C4C22semidirect
35. C2×Dic3×D7C22semidirect
36. C2×Dic3×D7C22semidirect
38. #66631C22semidirect
39. #66631C22semidirect
41. C7×#217C22semidirect
42. C7×C4⋊Dic3C22semidirect
43. C4×C7×Dic3C22semidirect
45. C3×(C2×D7)⋊C4C22semidirect
46. C3×(C2×D7)⋊C4C22semidirect
47. Dic3⋊Dic7C22semidirect
48. #66577C22semidirect
49. Dic7⋊Dic3C22semidirect
50. #66575C22semidirect
51. #66575C22semidirect
52. (C2×D7)⋊Dic3C22semidirect
53. (C2×D7)⋊Dic3C22semidirect
58. #6500D3semidirect
102. #4246D4semidirect
114. C2×C22⋊D7C2×D3semidirect
115. C2×C22⋊D7C2×D3semidirect
139. (C2×D7)⋊C4C2×D3semidirect
140. (C2×D7)⋊C4C2×D3semidirect
152. D4×Dic3D7semidirect
164. C22×C3×C7#41semidirect
242. C22×Dic3C2×D7semidirect
254. #217C2×D7semidirect
260. C4⋊Dic3C2×D7semidirect
263. C4×Dic3C2×D7semidirect
293. C22×C7#237semidirect
355. C3×C7#546semidirect
414. C22×C3#1157semidirect
469. C7#5591semidirect
498. C22#66683semidirect
509. C3#90241semidirect

Maximal subgroups

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

subgroupindex
C22×#42462
C2×#666832
#106214202
C7×D4×Dic32
C3×#65002
#106211372
#106211312
#106211232
#106211122
D7×#2172
#106210952
#106209752
#106209572
#106209182
#106209152
#902413
#55917