Finite Group Databasebeta

← #13061702

Subgroups — #13061702

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
1C2×#66730C211yesmaximal, characteristic, complemented
2C2×D4×D21C211yesmaximal, characteristic, complemented
3C3×C7×#121C211yesmaximal, characteristic, complemented
4#10621422C211yesmaximal, characteristic, complemented
5#10621420C211yesmaximal, characteristic, complemented
6#10621420C211yesmaximal, characteristic, complemented
7#10621418C211yesmaximal, characteristic, complemented
8C4×#4268C211yesmaximal, characteristic, complemented
9C2×#66631C211yesmaximal, characteristic, complemented
10#10621379C211yesmaximal, characteristic, complemented
11#10621377C211yesmaximal, characteristic, complemented
12#10621368C211yesmaximal, characteristic, complemented
13#10621367C211yesmaximal, characteristic, complemented
14#10621364C211yesmaximal, characteristic, complemented
15C22⋊C4×D21C211yesmaximal, characteristic, complemented
16#90301—31nomaximal, Hall
17C23×D21C2211yescharacteristic
18C23×D21C2211yescharacteristic
19C2×C3×C7×D4C2211yescharacteristic
20C2×C3×C7×D4C2211yescharacteristic, complemented
21C2×C3×C7×D4C2211yescharacteristic, complemented
22C22×C3×C4×C7C2211yescharacteristic, complemented
23C2×#4268C2211yescharacteristic
24C2×#4268C2211yescharacteristic
25C2×#4268C2211yescharacteristic
26C2×#4268C2211yescharacteristic, complemented
27C2×#4268C2211yescharacteristic, complemented
28C22×Dic21C2211yescharacteristic, complemented
29C22×Dic21C2211yescharacteristic, complemented
30#66730—24no—
31D4×D21—24no—
32C2×D84C2211yescharacteristic
33C2×C4×D21C2211yescharacteristic
34C2×C4×D21C2211yescharacteristic, complemented
35C2×C4×D21C2211yescharacteristic, complemented
36C2×Dic42C2211yescharacteristic, complemented
37C3×C7×C4⋊C4C2211yescharacteristic, complemented
38C3×C7×C22⋊C4C2211yescharacteristic, complemented
39C3×C7×C22⋊C4C2211yescharacteristic, complemented
40#66636C2211yescharacteristic, complemented
41#66636C2211yescharacteristic, complemented
42#66636C2211yescharacteristic, complemented
43#66631C2211yescharacteristic
44#66631C2211yescharacteristic
45#66631C2211yescharacteristic
46#66631C2211yescharacteristic, complemented
47#66631C2211yescharacteristic, complemented
48#66631C2211yescharacteristic, complemented
49#66631C2211yescharacteristic, complemented
50#66629C2211yescharacteristic, complemented
51#66629C2211yescharacteristic, complemented
52#66629C2211yescharacteristic, complemented
53C4×Dic21C2211yescharacteristic, complemented
54C2×#1159—31no—
55C2×D4×D7—31no—
56C7×#121D311yescharacteristic, complemented
57#6502—31no—
58#6500—31no—
59#6500—31no—
60#6498—31no—
61C4×C22⋊D7—31no—
62C2×(C2×D7)⋊C4—31no—
63#6459—31no—
64#6457—31no—
65#6448—31no—
66#6447—31no—
67(C22⋊D7)⋊C4—31no—
68D7×C22⋊C4—31no—
69#5610—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C23×C3×C7C2311yescharacteristic
73C22×D21—22no—
74C22×D21—22no—
75C22×D21—22no—
76C22×D21—22no—
77C22×D21—41no—
78C22×D21—41no—
79C22×D21C2311yescharacteristic
80C22×D21C2311yescharacteristic
81C22×D21C2311yescharacteristic
82C3×C7×D4—22no—
83C3×C7×D4—22no—
84C3×C7×D4—41no—
85C2×C3×C4×C7—41no—
86C2×C3×C4×C7C2311yescharacteristic
87C2×C3×C4×C7C2311yescharacteristic
88C2×C3×C4×C7C2311yescharacteristic
89C2×C3×C4×C7C2311yescharacteristic
90#4268—22no—
91#4268—22no—
92#4268—22no—
93#4268—41no—
94#4268D414yescomplemented
95C2×Dic21—22no—
96C2×Dic21—22no—
97C2×Dic21C2311yescharacteristic
98C2×Dic21C2311yescharacteristic
99C2×Dic21C2311yescharacteristic
100C2×Dic21C2311yescharacteristic
101C2×Dic21C2311yescharacteristic
102D84—22no—
103C4×D21—22no—
104C4×D21—22no—
105C4×D21—41no—
106Dic42—22no—
107C23×D7—31no—
108C23×D7—31no—
109C2×C7×D4C2×D311yescharacteristic
110C2×C7×D4C2×D311yescharacteristic, complemented
111C2×C7×D4C2×D311yescharacteristic, complemented
112C22×C4×C7C2×D311yescharacteristic, complemented
113C2×C22⋊D7—31no—
114C2×C22⋊D7—31no—
115C2×C22⋊D7—31no—
116C2×C22⋊D7—31no—
117C2×C22⋊D7—31no—
118C22×Dic7—31no—
119C22×Dic7—31no—
120#1159—64no—
121D4×D7—64no—
122C2×D28—31no—
123C2×C4×D7—31no—
124C2×C4×D7—31no—
125C2×C4×D7—31no—
126C2×Dic14—31no—
127C7×C4⋊C4C2×D311yescharacteristic, complemented
128C7×C22⋊C4C2×D311yescharacteristic, complemented
129C7×C22⋊C4C2×D311yescharacteristic, complemented
130C22⋊Dic7—31no—
131C22⋊Dic7—31no—
132C22⋊Dic7—31no—
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)⋊C4—31no—
140Dic7⋊C4—31no—
141Dic7⋊C4—31no—
142Dic7⋊C4—31no—
143C4×Dic7—31no—
144C2×#237—71no—
145C2×D3×D4—71no—
146C3×#121D711yescharacteristic, complemented
147#939—71no—
148#937—71no—
149#937—71no—
150#935—71no—
151C4×#67—71no—
152C2×(C2×D3)⋊C4—71no—
153#896—71no—
154#894—71no—
155#885—71no—
156#884—71no—
157#67⋊C4—71no—
158D3×C22⋊C4—71no—
159C22×C3×C7—21no—
160C22×C3×C7—21no—
161C22×C3×C7—21no—
162C22×C3×C7—21no—
163C22×C3×C7—21no—
164C22×C3×C7—21no—
165C22×C3×C7—21no—
166C22×C3×C7—21no—
167C22×C3×C7C2×D412yes—
168C22×C3×C7C2411yescharacteristic
169C2×D21—21no—
170C2×D21—21no—
171C2×D21—21no—
172C2×D21—21no—
173C2×D21—21no—
174C2×D21—21no—
175C2×D21—21no—
176C2×D21—44no—
177C2×D21—44no—
178C2×D21C2×D412yes—
179C2×D21#4112yescomplemented
180C3×C4×C7—21no—
181C3×C4×C7—21no—
182C3×C4×C7—21no—
183C3×C4×C7—21no—
184Dic21—21no—
185Dic21—21no—
186Dic21—21no—
187Dic21—21no—
188Dic21C2×D412yes—
189#546—211noSylow
190C23×C7C22×D311yescharacteristic
191C23×C7C22×D311yescharacteristic
192C23×C7C22×D311yescharacteristic
193C22×D7—31no—
194C22×D7—31no—
195C22×D7—31no—
196C22×D7—62no—
197C22×D7—62no—
198C22×D7—62no—
199C22×D7—62no—
200C22×D7—121no—
201C22×D7—121no—
202C7×D4—22no—
203C7×D4—22no—
204C7×D4—41no—
205C2×C4×C7—41no—
206C2×C4×C7C22×D311yescharacteristic
207C2×C4×C7C22×D311yescharacteristic
208C2×C4×C7C22×D311yescharacteristic
209C2×C4×C7C22×D311yescharacteristic
210C22⋊D7—34no—
211C22⋊D7—62no—
212C22⋊D7—62no—
213C22⋊D7—62no—
214C22⋊D7—121no—
215C2×Dic7—31no—
216C2×Dic7—31no—
217C2×Dic7—31no—
218C2×Dic7—31no—
219C2×Dic7—31no—
220C2×Dic7—62no—
221C2×Dic7—62no—
222D28—62no—
223C4×D7—62no—
224C4×D7—62no—
225C4×D7—121no—
226Dic14—62no—
227C23×D3—71no—
228C23×D3—71no—
229C2×C3×D4C2×D711yescharacteristic
230C2×C3×D4C2×D711yescharacteristic, complemented
231C2×C3×D4C2×D711yescharacteristic, complemented
232C22×C3×C4C2×D711yescharacteristic, complemented
233C2×#67—71no—
234C2×#67—71no—
235C2×#67—71no—
236C2×#67—71no—
237C2×#67—71no—
238C22×Dic3—71no—
239C22×Dic3—71no—
240#237—144no—
241D3×D4—144no—
242C2×D12—71no—
243C2×C4×D3—71no—
244C2×C4×D3—71no—
245C2×C4×D3—71no—
246C2×Dic6—71no—
247C3×C4⋊C4C2×D711yescharacteristic, complemented
248C3×C22⋊C4C2×D711yescharacteristic, complemented
249C3×C22⋊C4C2×D711yescharacteristic, complemented
250#217—71no—
251#217—71no—
252#217—71no—
253(C2×D3)⋊C4—71no—
254(C2×D3)⋊C4—71no—
255(C2×D3)⋊C4—71no—
256(C2×D3)⋊C4—71no—
257(C2×D3)⋊C4—71no—
258(C2×D3)⋊C4—71no—
259(C2×D3)⋊C4—71no—
260Dic3⋊C4—71no—
261Dic3⋊C4—71no—
262Dic3⋊C4—71no—
263C4×Dic3—71no—
264C2×C3×C7—22no—
265C2×C3×C7—41no—
266C2×C3×C7—41no—
267C2×C3×C7C22×D411yescharacteristic
268C2×C3×C7C2×#4111yescharacteristic
269C2×C3×C7H(5,𝔽2)11yescharacteristic
270D21—22no—
271D21—22no—
272D21—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#121D2111yescharacteristic, complemented
282#120—211no—
283#120—211no—
284C4×D4—211no—
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—21no—
294C22×C7—21no—
295C22×C7—21no—
296C22×C7D3×D412yes—
297C22×C7C23×D311yescharacteristic
298C2×D7—32no—
299C2×D7—32no—
300C2×D7—61no—
301C2×D7—61no—
302C2×D7—61no—
303C2×D7—61no—
304C2×D7—61no—
305C2×D7—61no—
306C2×D7—61no—
307C2×D7—124no—
308C2×D7—124no—
309C4×C7—21no—
310C4×C7—21no—
311C4×C7—21no—
312C4×C7—21no—
313Dic7—32no—
314Dic7—61no—
315Dic7—61no—
316Dic7—61no—
317Dic7—61no—
318C23×C3C22×D711yescharacteristic
319C23×C3C22×D711yescharacteristic
320C23×C3C22×D711yescharacteristic
321C22×D3—71no—
322C22×D3—71no—
323C22×D3—71no—
324C22×D3—142no—
325C22×D3—142no—
326C22×D3—142no—
327C22×D3—142no—
328C22×D3—281no—
329C22×D3—281no—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—41no—
333C2×C3×C4—41no—
334C2×C3×C4C22×D711yescharacteristic
335C2×C3×C4C22×D711yescharacteristic
336C2×C3×C4C22×D711yescharacteristic
337C2×C3×C4C22×D711yescharacteristic
338#67—74no—
339#67—142no—
340#67—142no—
341#67—142no—
342#67—281no—
343C2×Dic3—71no—
344C2×Dic3—71no—
345C2×Dic3—71no—
346C2×Dic3—71no—
347C2×Dic3—71no—
348C2×Dic3—142no—
349C2×Dic3—142no—
350D12—142no—
351C4×D3—142no—
352C4×D3—142no—
353C4×D3—281no—
354Dic6—142no—
355C3×C7#54611yescharacteristic, complemented, Hall
356C24—211no—
357C24—211no—
358#41—424no—
359C2×Q8—211no—
360C2×D4—211no—
361C2×D4—211no—
362C2×D4—211no—
363C2×D4—211no—
364C2×D4—211no—
365C2×D4—211no—
366C2×D4—424no—
367C2×D4C2×D2111yescharacteristic
368C2×D4C2×D2111yescharacteristic, complemented
369C2×D4C2×D2111yescharacteristic, complemented
370C22×C4—211no—
371C22×C4—211no—
372C22×C4—211no—
373C22×C4—211no—
374C22×C4—211no—
375C22×C4C2×D2111yescharacteristic, complemented
376C4⋊C4—211no—
377C4⋊C4—211no—
378C4⋊C4—211no—
379C4⋊C4C2×D2111yescharacteristic, complemented
380C22⋊C4—211no—
381C22⋊C4—211no—
382C22⋊C4—211no—
383C22⋊C4—211no—
384C22⋊C4—211no—
385C22⋊C4—211no—
386C22⋊C4—211no—
387C22⋊C4—211no—
388C22⋊C4—211no—
389C22⋊C4—211no—
390C22⋊C4C2×D2111yescharacteristic, complemented
391C22⋊C4C2×D2111yescharacteristic, complemented
392C42—211no—
393C2×C7—22no—
394C2×C7—41no—
395C2×C7—41no—
396C2×C7C2×D3×D411yescharacteristic
397C2×C7#100411yescharacteristic
398C2×C7D3×#4111yescharacteristic
399D7—62no—
400D7—62no—
401D7—121no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3D4×D712yes—
411C22×C3C23×D711yescharacteristic
412C2×D3—72no—
413C2×D3—72no—
414C2×D3—141no—
415C2×D3—141no—
416C2×D3—141no—
417C2×D3—141no—
418C2×D3—141no—
419C2×D3—141no—
420C2×D3—141no—
421C2×D3—284no—
422C2×D3—284no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427Dic3—72no—
428Dic3—141no—
429Dic3—141no—
430Dic3—141no—
431Dic3—141no—
432C23—211no—
433C23—211no—
434C23—211no—
435C23—422no—
436C23—422no—
437C23—422no—
438C23—422no—
439C23—841no—
440C23—841no—
441C23C22×D2111yescharacteristic
442C23C22×D2111yescharacteristic
443C23C22×D2111yescharacteristic
444Q8—422no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—214no—
449D4—422no—
450D4—422no—
451D4—422no—
452D4—422no—
453D4—841no—
454C2×C4—41no—
455C2×C4—211no—
456C2×C4—211no—
457C2×C4—211no—
458C2×C4—211no—
459C2×C4—211no—
460C2×C4—422no—
461C2×C4—422no—
462C2×C4—422no—
463C2×C4—422no—
464C2×C4—841no—
465C2×C4C22×D2111yescharacteristic
466C2×C4C22×D2111yescharacteristic
467C2×C4C22×D2111yescharacteristic
468C2×C4C22×D2111yescharacteristic
469C7#561011yescharacteristic, complemented, Sylow
470C2×C3—22no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3C2×D4×D711yescharacteristic
474C2×C3#654811yescharacteristic
475C2×C3D7×#4111yescharacteristic
476D3—142no—
477D3—142no—
478D3—281no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—212no—
488C22—212no—
489C22—421no—
490C22—421no—
491C22—421no—
492C22—421no—
493C22—421no—
494C22—421no—
495C22—421no—
496C22—844no—
497C22—844no—
498C22D4×D2112yes—
499C22C23×D2111yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—212no—
505C4—421no—
506C4—421no—
507C4—421no—
508C4—421no—
509C3#9030111yescharacteristic, complemented, Sylow
510C2—22no—
511C2—41no—
512C2—41no—
513C2—422no—
514C2—422no—
515C2—841no—
516C2C2×D4×D2111yescharacteristic
517C2#1062165711yescharacteristic
518C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#546
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#66730C2semidirect
2. C2×D4×D21C2semidirect
3. C3×C7×#121C2semidirect
4. #10621422C2semidirect
5. #10621420C2semidirect
6. #10621420C2semidirect
7. #10621418C2semidirect
8. C4×#4268C2semidirect
9. C2×#66631C2semidirect
10. #10621379C2semidirect
11. #10621377C2semidirect
12. #10621368C2semidirect
13. #10621367C2semidirect
14. #10621364C2semidirect
15. C22⋊C4×D21C2semidirect
20. C2×C3×C7×D4C22semidirect
21. C2×C3×C7×D4C22semidirect
22. C22×C3×C4×C7C22semidirect
26. C2×#4268C22semidirect
27. C2×#4268C22semidirect
28. C22×Dic21C22semidirect
29. C22×Dic21C22semidirect
34. C2×C4×D21C22semidirect
35. C2×C4×D21C22semidirect
36. C2×Dic42C22semidirect
37. C3×C7×C4⋊C4C22semidirect
38. C3×C7×C22⋊C4C22semidirect
39. C3×C7×C22⋊C4C22semidirect
40. #66636C22semidirect
41. #66636C22semidirect
42. #66636C22semidirect
46. #66631C22semidirect
47. #66631C22semidirect
48. #66631C22semidirect
49. #66631C22semidirect
50. #66629C22semidirect
51. #66629C22semidirect
52. #66629C22semidirect
53. C4×Dic21C22semidirect
56. C7×#121D3semidirect
94. #4268D4semidirect
110. C2×C7×D4C2×D3semidirect
111. C2×C7×D4C2×D3semidirect
112. C22×C4×C7C2×D3semidirect
127. C7×C4⋊C4C2×D3semidirect
128. C7×C22⋊C4C2×D3semidirect
129. C7×C22⋊C4C2×D3semidirect
146. C3×#121D7semidirect
179. C2×D21#41semidirect
230. C2×C3×D4C2×D7semidirect
231. C2×C3×D4C2×D7semidirect
232. C22×C3×C4C2×D7semidirect
247. C3×C4⋊C4C2×D7semidirect
248. C3×C22⋊C4C2×D7semidirect
249. C3×C22⋊C4C2×D7semidirect
281. #121D21semidirect
355. C3×C7#546semidirect
368. C2×D4C2×D21semidirect
369. C2×D4C2×D21semidirect
375. C22×C4C2×D21semidirect
379. C4⋊C4C2×D21semidirect
390. C22⋊C4C2×D21semidirect
391. C22⋊C4C2×D21semidirect
469. C7#5610semidirect
509. C3#90301semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×D4×D212
C3×C7×#1212
#106214222
#106214202
#106214182
C4×#42682
C2×#666312
#106213792
#106213772
#106213682
#106213672
#106213642
C22⋊C4×D212
#903013
#56107