Finite Group Databasebeta

← #13059907

Subgroups — #13059907

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
1#10621413C211yesmaximal, characteristic, complemented
2C4×C7×#67C211yesmaximal, characteristic, complemented
3C3×#6492C211yesmaximal, characteristic, complemented
4#10621139C211yesmaximal, characteristic, complemented
5#10621120C211yesmaximal, characteristic, complemented
6#10621095C211yesmaximal, characteristic, complemented
7#10621088C211yesmaximal, characteristic, complemented
8#10621038C211yesmaximal, characteristic, complemented
9D3×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
10#10621008C211yesmaximal, characteristic, complemented
11#10620995C211yesmaximal, characteristic, complemented
12#10620979C211yesmaximal, characteristic, complemented
13#10620942C211yesmaximal, characteristic, complemented
14#10620924C211yesmaximal, characteristic, complemented
15#10620915C211yesmaximal, characteristic, complemented
16#90251—31nomaximal, Hall
17C22×D3×D7C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic, complemented
19C2×#4268C2211yescharacteristic
20C2×#4268C2211yescharacteristic, complemented
21C2×D84C2211yescharacteristic
22C2×C7×#67C2211yescharacteristic
23C2×C4×D3×C7C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic
25C2×#4248C2211yescharacteristic, complemented
26C2×#4247C2211yescharacteristic
27C2×#4246C2211yescharacteristic
28C2×D21⋊C4C2211yescharacteristic, complemented
29C2×D3×Dic7C2211yescharacteristic
30#66631C2211yescharacteristic
31#66631C2211yescharacteristic, complemented
32C4⋊Dic21C2211yescharacteristic, complemented
33C7×#217C2211yescharacteristic, complemented
34C7×(C2×D3)⋊C4C2211yescharacteristic
35C7×Dic3⋊C4C2211yescharacteristic, complemented
36C4×C7×Dic3C2211yescharacteristic, complemented
37C3×C22⋊Dic7C2211yescharacteristic, complemented
38C3×(C2×D7)⋊C4C2211yescharacteristic
39C3×(C2×D7)⋊C4C2211yescharacteristic
40C3×Dic7⋊C4C2211yescharacteristic, complemented
41C3×Dic7⋊C4C2211yescharacteristic, complemented
42#66577C2211yescharacteristic, complemented
43Dic7⋊Dic3C2211yescharacteristic, complemented
44#66575C2211yescharacteristic
45#66575C2211yescharacteristic
46#66575C2211yescharacteristic
47#66575C2211yescharacteristic, complemented
48(C2×D3)⋊Dic7C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic, complemented
52C4×C7×D4—31no—
53#6502—31no—
54#6500—31no—
55#6493—31no—
56#6492D311yescharacteristic, complemented
57C4×C22⋊D7—31no—
58C2×(C2×D7)⋊C4—31no—
59#6461—31no—
60#6457—31no—
61#6448—31no—
62#6447—31no—
63#6445—31no—
64#6441—31no—
65#6439—31no—
66#6438—31no—
67#5657—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×D21C2311yescharacteristic
71C22×D3×C7C2311yescharacteristic
72C22×C3×D7C2311yescharacteristic
73C2×D3×D7—22no—
74C2×D3×D7—41no—
75C2×D3×D7—41no—
76C2×C3×C4×C7—41no—
77C2×C3×C4×C7C2311yescharacteristic
78C2×C3×C4×C7C2311yescharacteristic
79#4268—41no—
80#4268—41no—
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83D84—41no—
84C7×#67—22no—
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C2×C7×Dic3C2311yescharacteristic
88C4×D3×C7—41no—
89C3×C22⋊D7—41no—
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92C2×C3×Dic7C2311yescharacteristic
93#4248—41no—
94#4247—22no—
95#4246—41no—
96D21⋊C4—41no—
97D3×Dic7—41no—
98C23×D7—31no—
99C2×C7×D4—31no—
100C22×C4×C7—31no—
101C22×C4×C7C2×D311yescharacteristic, complemented
102C2×C22⋊D7—31no—
103C2×C22⋊D7—31no—
104C2×C22⋊D7—31no—
105C2×C22⋊D7C2×D311yescharacteristic
106C22×Dic7—31no—
107C2×D28—31no—
108C2×D28—31no—
109C2×C4×D7—31no—
110C2×Dic14—31no—
111C7×C4⋊C4—31no—
112C7×C22⋊C4—31no—
113C7×C22⋊C4—31no—
114C42×C7—31no—
115C22⋊Dic7—31no—
116C22⋊Dic7C2×D311yescharacteristic, complemented
117(C2×D7)⋊C4—31no—
118(C2×D7)⋊C4—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
127C4⋊Dic7—31no—
128Dic7⋊C4—31no—
129Dic7⋊C4—31no—
130Dic7⋊C4C2×D311yescharacteristic, complemented
131Dic7⋊C4C2×D311yescharacteristic, complemented
132C4×Dic7—31no—
133C3×#123—71no—
134#938—71no—
135#937—71no—
136#930—71no—
137C4×#67D711yescharacteristic, complemented
138#898—71no—
139#897—71no—
140#894—71no—
141#893—71no—
142#885—71no—
143#885—71no—
144#884—71no—
145#882—71no—
146D3×C22⋊C4—71no—
147#879—71no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7—21no—
151C22×C3×C7C2411yescharacteristic
152C2×D21—21no—
153C2×D21—21no—
154C2×D21—21no—
155C2×D21—21no—
156C2×D21—21no—
157C2×D21—21no—
158C2×D3×C7—21no—
159C2×D3×C7—21no—
160C2×D3×C7#4112yes—
161C2×C3×D7—21no—
162C2×C3×D7—21no—
163C2×C3×D7—21no—
164D3×D7—44no—
165C3×C4×C7—21no—
166C3×C4×C7—21no—
167Dic21—21no—
168Dic21—21no—
169C3×Dic7—21no—
170C3×Dic7—21no—
171C3×Dic7—21no—
172C7×Dic3—21no—
173C7×Dic3—21no—
174C7×Dic3—21no—
175#538—211noSylow
176C23×C7—31no—
177C23×C7C22×D311yescharacteristic
178C22×D7—31no—
179C22×D7—31no—
180C22×D7—62no—
181C22×D7—121no—
182C22×D7—121no—
183C22×D7C22×D311yescharacteristic
184C7×D4—62no—
185C2×C4×C7—31no—
186C2×C4×C7—31no—
187C2×C4×C7—31no—
188C2×C4×C7—41no—
189C2×C4×C7—121no—
190C2×C4×C7C22×D311yescharacteristic
191C2×C4×C7C22×D311yescharacteristic
192C22⋊D7—41no—
193C22⋊D7—62no—
194C22⋊D7—121no—
195C22⋊D7—121no—
196C2×Dic7—31no—
197C2×Dic7—31no—
198C2×Dic7—121no—
199C2×Dic7C22×D311yescharacteristic
200C2×Dic7C22×D311yescharacteristic
201C2×Dic7C22×D311yescharacteristic
202D28—121no—
203D28—121no—
204C4×D7—121no—
205Dic14—121no—
206C23×D3—71no—
207C2×C3×D4—71no—
208C22×C3×C4C2×D711yescharacteristic, complemented
209C2×#67—71no—
210C2×#67—71no—
211C2×#67—71no—
212C2×#67C2×D711yescharacteristic
213C2×D12—71no—
214C2×D12—71no—
215C2×C4×D3—71no—
216C2×C4×D3—71no—
217C2×C4×D3C2×D711yescharacteristic
218C2×Dic6—71no—
219C3×C4⋊C4—71no—
220C3×C4⋊C4—71no—
221C3×C22⋊C4—71no—
222C3×C22⋊C4—71no—
223C3×C22⋊C4—71no—
224#217—71no—
225#217—71no—
226#217C2×D711yescharacteristic, complemented
227(C2×D3)⋊C4—71no—
228(C2×D3)⋊C4—71no—
229(C2×D3)⋊C4—71no—
230(C2×D3)⋊C4—71no—
231(C2×D3)⋊C4—71no—
232(C2×D3)⋊C4—71no—
233(C2×D3)⋊C4—71no—
234(C2×D3)⋊C4C2×D711yescharacteristic
235C4⋊Dic3—71no—
236C4⋊Dic3—71no—
237Dic3⋊C4—71no—
238Dic3⋊C4C2×D711yescharacteristic, complemented
239C4×Dic3—71no—
240C4×Dic3C2×D711yescharacteristic, complemented
241C2×C3×C7—41no—
242C2×C3×C7C2×#4111yescharacteristic
243C2×C3×C7H(5,𝔽2)11yescharacteristic
244C2×C3×C7H(5,𝔽2)11yescharacteristic
245D21—41no—
246D21—41no—
247C3×D7—41no—
248D3×C7—22no—
249#126—211no—
250#126—211no—
251#124—211no—
252#124—211no—
253#123—71no—
254#123—211no—
255C22⋊Q8—211no—
256#121—211no—
257#121—211no—
258#121—211no—
259#120—211no—
260#120—211no—
261C4×D4—31no—
262C4×D4—211no—
263C2×C22⋊C4—211no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—32no—
268C22×C7—61no—
269C22×C7—61no—
270C22×C7C23×D311yescharacteristic
271C2×D7—21no—
272C2×D7—21no—
273C2×D7—21no—
274C2×D7—61no—
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—61no—
284C4×C7—61no—
285C4×C7—61no—
286Dic7—21no—
287Dic7—21no—
288Dic7—21no—
289Dic7—61no—
290Dic7—61no—
291C23×C3—71no—
292C23×C3C22×D711yescharacteristic
293C22×D3—71no—
294C22×D3—71no—
295C22×D3—142no—
296C22×D3—281no—
297C22×D3—281no—
298C22×D3C22×D711yescharacteristic
299C3×D4—281no—
300C2×C3×C4—41no—
301C2×C3×C4—71no—
302C2×C3×C4—71no—
303C2×C3×C4—71no—
304C2×C3×C4C22×D711yescharacteristic
305C2×C3×C4C22×D711yescharacteristic
306#67—22no—
307#67—281no—
308#67—281no—
309#67—281no—
310C2×Dic3—71no—
311C2×Dic3—71no—
312C2×Dic3C22×D711yescharacteristic
313C2×Dic3C22×D711yescharacteristic
314C2×Dic3C22×D711yescharacteristic
315D12—142no—
316D12—281no—
317C4×D3—41no—
318C4×D3—281no—
319C4×D3—281no—
320Dic6—281no—
321C3×C7#53811yescharacteristic, complemented, Hall
322C24—211no—
323C2×Q8—211no—
324C2×D4—31no—
325C2×D4—71no—
326C2×D4—211no—
327C2×D4—211no—
328C2×D4—211no—
329C2×D4—211no—
330C2×D4—211no—
331C22×C4—31no—
332C22×C4—211no—
333C22×C4—211no—
334C22×C4D3×D711yescharacteristic, complemented
335C4⋊C4—31no—
336C4⋊C4—71no—
337C4⋊C4—71no—
338C4⋊C4—211no—
339C4⋊C4—211no—
340C4⋊C4—211no—
341C22⋊C4—31no—
342C22⋊C4—31no—
343C22⋊C4—71no—
344C22⋊C4—71no—
345C22⋊C4—71no—
346C22⋊C4—211no—
347C22⋊C4—211no—
348C22⋊C4—211no—
349C22⋊C4—211no—
350C22⋊C4—211no—
351C22⋊C4—211no—
352C22⋊C4—211no—
353C22⋊C4—211no—
354C22⋊C4—211no—
355C42—31no—
356C42—211no—
357C2×C7—41no—
358C2×C7—62no—
359C2×C7#100411yescharacteristic
360C2×C7D3×#4111yescharacteristic
361C2×C7#100911yescharacteristic
362D7—41no—
363D7—121no—
364D7—121no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—141no—
369C22×C3—141no—
370C22×C3—141no—
371C22×C3C23×D711yescharacteristic
372C2×D3—21no—
373C2×D3—21no—
374C2×D3—141no—
375C2×D3—141no—
376C2×D3—141no—
377C2×D3—141no—
378C2×D3—141no—
379C2×D3—141no—
380C2×D3—284no—
381C2×D3#115712yes—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—141no—
385C3×C4—141no—
386C3×C4—141no—
387Dic3—21no—
388Dic3—21no—
389Dic3—21no—
390Dic3—141no—
391Dic3—141no—
392C23—31no—
393C23—71no—
394C23—211no—
395C23—211no—
396C23—422no—
397C23—841no—
398C23—841no—
399C23C2×D3×D711yescharacteristic
400Q8—841no—
401D4—62no—
402D4—281no—
403D4—422no—
404D4—841no—
405D4—841no—
406D4—841no—
407D4—841no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—41no—
412C2×C4—71no—
413C2×C4—71no—
414C2×C4—71no—
415C2×C4—121no—
416C2×C4—211no—
417C2×C4—211no—
418C2×C4—841no—
419C2×C4—841no—
420C2×C4C2×D3×D711yescharacteristic
421C2×C4C2×D3×D711yescharacteristic
422C7#565711yescharacteristic, complemented, Sylow
423C2×C3—41no—
424C2×C3—281no—
425C2×C3C2×#115711yescharacteristic
426C2×C3#654811yescharacteristic
427C2×C3#655311yescharacteristic
428D3—22no—
429D3—281no—
430D3—281no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—32no—
435C22—61no—
436C22—61no—
437C22—141no—
438C22—141no—
439C22—141no—
440C22—421no—
441C22—421no—
442C22—421no—
443C22—421no—
444C22—421no—
445C22—421no—
446C22—844no—
447C22C22×D3×D711yescharacteristic
448C4—21no—
449C4—21no—
450C4—61no—
451C4—61no—
452C4—61no—
453C4—141no—
454C4—141no—
455C4—141no—
456C4—421no—
457C4—421no—
458C3#9025111yescharacteristic, complemented, Sylow
459C2—41no—
460C2—62no—
461C2—281no—
462C2—841no—
463C2—841no—
464C2D3×#115711yescharacteristic
465C2#1062156611yescharacteristic
466C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#538
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #10621413C2semidirect
2. C4×C7×#67C2semidirect
3. C3×#6492C2semidirect
4. #10621139C2semidirect
5. #10621120C2semidirect
6. #10621095C2semidirect
7. #10621088C2semidirect
8. #10621038C2semidirect
9. D3×(C2×D7)⋊C4C2semidirect
10. #10621008C2semidirect
11. #10620995C2semidirect
12. #10620979C2semidirect
13. #10620942C2semidirect
14. #10620924C2semidirect
15. #10620915C2semidirect
18. C22×C3×C4×C7C22semidirect
20. C2×#4268C22semidirect
25. C2×#4248C22semidirect
28. C2×D21⋊C4C22semidirect
31. #66631C22semidirect
32. C4⋊Dic21C22semidirect
33. C7×#217C22semidirect
35. C7×Dic3⋊C4C22semidirect
36. C4×C7×Dic3C22semidirect
37. C3×C22⋊Dic7C22semidirect
40. C3×Dic7⋊C4C22semidirect
41. C3×Dic7⋊C4C22semidirect
42. #66577C22semidirect
43. Dic7⋊Dic3C22semidirect
47. #66575C22semidirect
51. Dic3×Dic7C22semidirect
56. #6492D3semidirect
101. C22×C4×C7C2×D3semidirect
116. C22⋊Dic7C2×D3semidirect
130. Dic7⋊C4C2×D3semidirect
131. Dic7⋊C4C2×D3semidirect
137. C4×#67D7semidirect
208. C22×C3×C4C2×D7semidirect
226. #217C2×D7semidirect
238. Dic3⋊C4C2×D7semidirect
240. C4×Dic3C2×D7semidirect
321. C3×C7#538semidirect
334. C22×C4D3×D7semidirect
422. C7#5657semidirect
458. C3#90251semidirect

Maximal subgroups

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

subgroupindex
#106214132
C4×C7×#672
C3×#64922
#106211392
#106211202
#106210952
#106210882
#106210382
D3×(C2×D7)⋊C42
#106210082
#106209952
#106209792
#106209422
#106209242
#106209152
#902513
#56577