Finite Group Databasebeta

← #13057999

Subgroups — #13057999

Order 1344.

500 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D84C211yesmaximal, characteristic, complemented
2C7×#876C211yesmaximal, characteristic, complemented
3C3×#6438C211yesmaximal, characteristic, complemented
4#10621038C211yesmaximal, characteristic, complemented
5#10621036C211yesmaximal, characteristic, complemented
6#10621035C211yesmaximal, characteristic, complemented
7#10621023C211yesmaximal, characteristic, complemented
8#10621021C211yesmaximal, characteristic, complemented
9#10620976C211yesmaximal, characteristic, complemented
10#10620973C211yesmaximal, characteristic, complemented
11#10620971C211yesmaximal, characteristic, complemented
12#10620958C211yesmaximal, characteristic, complemented
13#10620924C211yesmaximal, characteristic, complemented
14#10620916C211yesmaximal, characteristic, complemented
15#10620888C211yesmaximal, characteristic, complemented
16#90395—31nomaximal, Hall
17C22×D3×D7C2211yescharacteristic
18C2×D84C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×C4×D21C2211yescharacteristic, complemented
21C2×C3×D28C2211yescharacteristic
22C2×C3×Dic14C2211yescharacteristic, complemented
23C2×#4247C2211yescharacteristic
24C2×#4246C2211yescharacteristic
25C2×#4246C2211yescharacteristic
26C2×#4246C2211yescharacteristic, complemented
27C2×#4245C2211yescharacteristic
28C2×D21⋊C4C2211yescharacteristic
29C2×Dic3×D7C2211yescharacteristic, complemented
30C3×C42×C7C2211yescharacteristic, complemented
31#66631C2211yescharacteristic
32#66631C2211yescharacteristic, complemented
33C4⋊Dic21C2211yescharacteristic, complemented
34C7×(C2×D3)⋊C4C2211yescharacteristic
35C7×(C2×D3)⋊C4C2211yescharacteristic
36C7×(C2×D3)⋊C4C2211yescharacteristic
37C7×Dic3⋊C4C2211yescharacteristic, complemented
38C7×Dic3⋊C4C2211yescharacteristic, complemented
39C7×Dic3⋊C4C2211yescharacteristic, complemented
40C3×(C2×D7)⋊C4C2211yescharacteristic
41C3×(C2×D7)⋊C4C2211yescharacteristic
42C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
43C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
44#66577C2211yescharacteristic, complemented
45Dic7⋊Dic3C2211yescharacteristic, complemented
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic, complemented
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic, complemented
52C7×#126—31no—
53#6461—31no—
54#6459—31no—
55#6458—31no—
56#6458—31no—
57#6457—31no—
58#6456—31no—
59#6448—31no—
60#6447—31no—
61#6446—31no—
62#6445—31no—
63#6445—31no—
64D7×C22⋊C4—31no—
65#6438D311yescharacteristic, complemented
66C4×D28—31no—
67#5678—71nomaximal, Hall
68C22×D21C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×D3×C7C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C22×C3×D7C2311yescharacteristic
73C2×D3×D7—22no—
74C2×D3×D7—41no—
75C2×D3×D7—41no—
76C2×C3×C4×C7C2311yescharacteristic
77C2×C3×C4×C7C2311yescharacteristic
78C2×C3×C4×C7C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81D84—22no—
82C4×D21—41no—
83C4×D21—41no—
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C2×C3×Dic7C2311yescharacteristic
89C3×D28—41no—
90C3×Dic14—41no—
91#4247—41no—
92#4246—22no—
93#4246—41no—
94#4246—41no—
95#4245—41no—
96D21⋊C4—41no—
97Dic3×D7—41no—
98C23×D7—31no—
99C2×C22⋊D7—31no—
100C2×C22⋊D7—31no—
101C2×D28—31no—
102C2×D28—31no—
103C2×D28—31no—
104C2×D28—31no—
105C2×D28C2×D311yescharacteristic
106C2×C4×D7—31no—
107C2×C4×D7—31no—
108C2×C4×D7—31no—
109C2×C4×D7—31no—
110C2×Dic14C2×D311yescharacteristic, complemented
111C7×C4⋊C4—31no—
112C7×C4⋊C4—31no—
113C7×C4⋊C4—31no—
114C7×C22⋊C4—31no—
115C7×C22⋊C4—31no—
116C7×C22⋊C4—31no—
117C42×C7C2×D311yescharacteristic, complemented
118C22⋊Dic7—31no—
119(C2×D7)⋊C4—31no—
120(C2×D7)⋊C4—31no—
121(C2×D7)⋊C4—31no—
122(C2×D7)⋊C4—31no—
123(C2×D7)⋊C4—31no—
124(C2×D7)⋊C4—31no—
125(C2×D7)⋊C4C2×D311yescharacteristic
126(C2×D7)⋊C4C2×D311yescharacteristic
127(C2×D7)⋊C4C2×D311yescharacteristic, complemented
128(C2×D7)⋊C4C2×D311yescharacteristic, complemented
129C4⋊Dic7—31no—
130Dic7⋊C4—31no—
131Dic7⋊C4—31no—
132C4×Dic7—31no—
133C3×#124—71no—
134#947—71no—
135#946—71no—
136#939—71no—
137#937—71no—
138#886—71no—
139#884—71no—
140#884—71no—
141#883—71no—
142#882—71no—
143#67⋊C4—71no—
144D3×C22⋊C4—71no—
145#879—71no—
146#876D711yescharacteristic, complemented
147C4×D12—71no—
148C22×C3×C7C2411yescharacteristic
149C2×D21—21no—
150C2×D21—21no—
151C2×D21—21no—
152C2×D21—21no—
153C2×D21—21no—
154C2×D21#4112yes—
155C2×D3×C7—21no—
156C2×D3×C7—21no—
157C2×D3×C7—21no—
158C2×C3×D7—21no—
159C2×C3×D7—21no—
160C2×C3×D7—21no—
161C2×C3×D7—21no—
162C2×C3×D7—21no—
163C2×C3×D7—21no—
164D3×D7—44no—
165C3×C4×C7—21no—
166C3×C4×C7—21no—
167C3×C4×C7—21no—
168Dic21—21no—
169Dic21—21no—
170C3×Dic7—21no—
171C3×Dic7—21no—
172C7×Dic3—21no—
173C7×Dic3—21no—
174C7×Dic3—21no—
175#538—211noSylow
176C23×C7—31no—
177C22×D7—31no—
178C22×D7—31no—
179C22×D7—62no—
180C22×D7—121no—
181C22×D7—121no—
182C22×D7C22×D311yescharacteristic
183C22×D7C22×D311yescharacteristic
184C2×C4×C7—31no—
185C2×C4×C7—31no—
186C2×C4×C7—31no—
187C2×C4×C7C22×D311yescharacteristic
188C2×C4×C7C22×D311yescharacteristic
189C2×C4×C7C22×D311yescharacteristic
190C22⋊D7—121no—
191C22⋊D7—121no—
192C2×Dic7—31no—
193C2×Dic7—31no—
194C2×Dic7C22×D311yescharacteristic
195C2×Dic7C22×D311yescharacteristic
196D28—41no—
197D28—62no—
198D28—62no—
199D28—121no—
200D28—121no—
201C4×D7—121no—
202C4×D7—121no—
203C4×D7—121no—
204C4×D7—121no—
205Dic14—41no—
206C23×D3—71no—
207C2×C3×Q8—71no—
208C2×C3×D4—71no—
209C2×#67—71no—
210C2×#67—71no—
211C2×#67—71no—
212C2×#67—71no—
213C22×Dic3—71no—
214C2×D12—71no—
215C2×D12—71no—
216C2×C4×D3—71no—
217C2×C4×D3—71no—
218C2×C4×D3—71no—
219C3×C22⋊C4—71no—
220C3×C22⋊C4—71no—
221C3×C22⋊C4—71no—
222C3×C22⋊C4—71no—
223C3×C42C2×D711yescharacteristic, complemented
224#217—71no—
225#217—71no—
226(C2×D3)⋊C4—71no—
227(C2×D3)⋊C4—71no—
228(C2×D3)⋊C4—71no—
229(C2×D3)⋊C4—71no—
230(C2×D3)⋊C4—71no—
231(C2×D3)⋊C4C2×D711yescharacteristic
232(C2×D3)⋊C4C2×D711yescharacteristic
233(C2×D3)⋊C4C2×D711yescharacteristic
234C4⋊Dic3—71no—
235C4⋊Dic3—71no—
236Dic3⋊C4—71no—
237Dic3⋊C4C2×D711yescharacteristic, complemented
238Dic3⋊C4C2×D711yescharacteristic, complemented
239Dic3⋊C4C2×D711yescharacteristic, complemented
240C4×Dic3—71no—
241C2×C3×C7C2×#4111yescharacteristic
242C2×C3×C7H(5,𝔽2)11yescharacteristic
243C2×C3×C7H(5,𝔽2)11yescharacteristic
244D21—22no—
245D21—41no—
246C3×D7—41no—
247C3×D7—41no—
248D3×C7—41no—
249#126—31no—
250#126—211no—
251#124—71no—
252#124—211no—
253#123—211no—
254#123—211no—
255C22⋊Q8—211no—
256#121—211no—
257#121—211no—
258#121—211no—
259#120—211no—
260#120—211no—
261C4×D4—211no—
262C4×D4—211no—
263C2×C22⋊C4—211no—
264C22×C7—61no—
265C22×C7—61no—
266C22×C7—61no—
267C22×C7C23×D311yescharacteristic
268C2×D7—21no—
269C2×D7—21no—
270C2×D7—21no—
271C2×D7—21no—
272C2×D7—21no—
273C2×D7—21no—
274C2×D7—32no—
275C2×D7—61no—
276C2×D7—61no—
277C2×D7—61no—
278C2×D7—61no—
279C2×D7—61no—
280C2×D7—124no—
281C4×C7—21no—
282C4×C7—21no—
283C4×C7—21no—
284C4×C7—61no—
285C4×C7—61no—
286C4×C7—61no—
287Dic7—21no—
288Dic7—21no—
289Dic7—61no—
290Dic7—61no—
291C23×C3—71no—
292C23×C3—71no—
293C22×D3—71no—
294C22×D3—71no—
295C22×D3—142no—
296C22×D3—281no—
297C22×D3—281no—
298C22×D3C22×D711yescharacteristic
299C3×Q8—281no—
300C3×D4—281no—
301C2×C3×C4—71no—
302C2×C3×C4—71no—
303C2×C3×C4C22×D711yescharacteristic
304C2×C3×C4C22×D711yescharacteristic
305C2×C3×C4C22×D711yescharacteristic
306#67—142no—
307#67—281no—
308#67—281no—
309#67—281no—
310C2×Dic3—71no—
311C2×Dic3—71no—
312C2×Dic3—281no—
313C2×Dic3C22×D711yescharacteristic
314C2×Dic3C22×D711yescharacteristic
315C2×Dic3C22×D711yescharacteristic
316D12—142no—
317D12—281no—
318C4×D3—281no—
319C4×D3—281no—
320C4×D3—281no—
321C3×C7#53811yescharacteristic, complemented, Hall
322C24—211no—
323C2×Q8—71no—
324C2×D4—71no—
325C2×D4—211no—
326C2×D4—211no—
327C2×D4—211no—
328C2×D4—211no—
329C2×D4—211no—
330C2×D4—211no—
331C22×C4—211no—
332C22×C4—211no—
333C22×C4—211no—
334C22×C4—211no—
335C4⋊C4—31no—
336C4⋊C4—31no—
337C4⋊C4—31no—
338C4⋊C4—211no—
339C4⋊C4—211no—
340C4⋊C4—211no—
341C22⋊C4—31no—
342C22⋊C4—31no—
343C22⋊C4—31no—
344C22⋊C4—71no—
345C22⋊C4—71no—
346C22⋊C4—71no—
347C22⋊C4—71no—
348C22⋊C4—211no—
349C22⋊C4—211no—
350C22⋊C4—211no—
351C22⋊C4—211no—
352C22⋊C4—211no—
353C22⋊C4—211no—
354C22⋊C4—211no—
355C42—211no—
356C42D3×D711yescharacteristic, complemented
357C2×C7—121no—
358C2×C7#100411yescharacteristic
359C2×C7D3×#4111yescharacteristic
360C2×C7#100911yescharacteristic
361D7—41no—
362D7—41no—
363D7—62no—
364D7—121no—
365C22×C3—141no—
366C22×C3—141no—
367C22×C3—141no—
368C22×C3—141no—
369C22×C3—141no—
370C22×C3—141no—
371C22×C3C23×D711yescharacteristic
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—21no—
375C2×D3—72no—
376C2×D3—141no—
377C2×D3—141no—
378C2×D3—141no—
379C2×D3—141no—
380C2×D3—141no—
381C2×D3—284no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—141no—
386C3×C4—141no—
387Dic3—21no—
388Dic3—21no—
389Dic3—21no—
390Dic3—141no—
391Dic3—141no—
392C23—31no—
393C23—71no—
394C23—71no—
395C23—211no—
396C23—211no—
397C23—422no—
398C23—841no—
399C23—841no—
400Q8—281no—
401D4—281no—
402D4—422no—
403D4—422no—
404D4—841no—
405D4—841no—
406D4—841no—
407D4—841no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—71no—
412C2×C4—71no—
413C2×C4—211no—
414C2×C4—211no—
415C2×C4—841no—
416C2×C4—841no—
417C2×C4—841no—
418C2×C4—841no—
419C2×C4C2×D3×D711yescharacteristic
420C2×C4C2×D3×D711yescharacteristic
421C2×C4C2×D3×D711yescharacteristic
422C7#567811yescharacteristic, complemented, Sylow
423C2×C3—281no—
424C2×C3—281no—
425C2×C3D7×#4111yescharacteristic
426C2×C3#655311yescharacteristic
427C2×C3#655311yescharacteristic
428D3—41no—
429D3—142no—
430D3—281no—
431C22—61no—
432C22—61no—
433C22—61no—
434C22—141no—
435C22—141no—
436C22—141no—
437C22—141no—
438C22—141no—
439C22—141no—
440C22—212no—
441C22—421no—
442C22—421no—
443C22—421no—
444C22—421no—
445C22—421no—
446C22—844no—
447C22C22×D3×D711yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—61no—
452C4—61no—
453C4—61no—
454C4—141no—
455C4—141no—
456C4—421no—
457C4—421no—
458C3#9039511yescharacteristic, complemented, Sylow
459C2—121no—
460C2—281no—
461C2—281no—
462C2—422no—
463C2—841no—
464C2#1062156311yescharacteristic
465C2#1062156411yescharacteristic
466C2#1062156611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C42×C7C22
hypercenterC42D3×D7
hypocenterC3×C7#538
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×D84C2semidirect
2. C7×#876C2semidirect
3. C3×#6438C2semidirect
4. #10621038C2semidirect
5. #10621036C2semidirect
6. #10621035C2semidirect
7. #10621023C2semidirect
8. #10621021C2semidirect
9. #10620976C2semidirect
10. #10620973C2semidirect
11. #10620971C2semidirect
12. #10620958C2semidirect
13. #10620924C2semidirect
14. #10620916C2semidirect
15. #10620888C2semidirect
20. C2×C4×D21C22semidirect
22. C2×C3×Dic14C22semidirect
26. C2×#4246C22semidirect
29. C2×Dic3×D7C22semidirect
30. C3×C42×C7C22semidirect
32. #66631C22semidirect
33. C4⋊Dic21C22semidirect
37. C7×Dic3⋊C4C22semidirect
38. C7×Dic3⋊C4C22semidirect
39. C7×Dic3⋊C4C22semidirect
42. C3×(C2×D7)⋊C4C22semidirect
43. C3×(C2×D7)⋊C4C22semidirect
44. #66577C22semidirect
45. Dic7⋊Dic3C22semidirect
47. #66575C22semidirect
51. Dic3×Dic7C22semidirect
65. #6438D3semidirect
110. C2×Dic14C2×D3semidirect
117. C42×C7C2×D3semidirect
127. (C2×D7)⋊C4C2×D3semidirect
128. (C2×D7)⋊C4C2×D3semidirect
146. #876D7semidirect
223. C3×C42C2×D7semidirect
237. Dic3⋊C4C2×D7semidirect
238. Dic3⋊C4C2×D7semidirect
239. Dic3⋊C4C2×D7semidirect
321. C3×C7#538semidirect
356. C42D3×D7semidirect
422. C7#5678semidirect
458. C3#90395semidirect

Maximal subgroups

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

subgroupindex
C4×D842
C7×#8762
C3×#64382
#106210382
#106210362
#106210352
#106210232
#106210212
#106209762
#106209732
#106209712
#106209582
#106209242
#106209162
#106208882
#903953
#56787