Finite Group Databasebeta

← #13058521

Subgroups — #13058521

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621366C211yesmaximal, characteristic, complemented
2C7×#883C211yesmaximal, characteristic, complemented
3C3×#6449C211yesmaximal, characteristic, complemented
4#10621139C211yesmaximal, characteristic
5#10621114C211yesmaximal, characteristic, complemented
6#67×Dic7C211yesmaximal, characteristic, complemented
7#10621096C211yesmaximal, characteristic, complemented
8#10621035C211yesmaximal, characteristic, complemented
9D3×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
10#10620998C211yesmaximal, characteristic, complemented
11#10620991C211yesmaximal, characteristic, complemented
12#10620924C211yesmaximal, characteristic, complemented
13#10620923C211yesmaximal, characteristic, complemented
14#10620909C211yesmaximal, characteristic, complemented
15#10620900C211yesmaximal, characteristic, complemented
16#90349—31nomaximal, Hall
17C22×D3×D7C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×C7×#67C2211yescharacteristic
21C2×C4×D3×C7C2211yescharacteristic
22C2×C3×C22⋊D7C2211yescharacteristic
23C22×C3×Dic7C2211yescharacteristic, complemented
24C2×#4248C2211yescharacteristic, complemented
25C2×D21⋊C4C2211yescharacteristic
26C2×D3×Dic7C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic, complemented
28#66636C2211yescharacteristic, complemented
29#66631C2211yescharacteristic
30C4⋊Dic21C2211yescharacteristic, complemented
31#66629C2211yescharacteristic, complemented
32C7×#217C2211yescharacteristic, complemented
33C7×(C2×D3)⋊C4C2211yescharacteristic
34C7×C4⋊Dic3C2211yescharacteristic, complemented
35C7×Dic3⋊C4C2211yescharacteristic, complemented
36C3×(C2×D7)⋊C4C2211yescharacteristic
37C3×(C2×D7)⋊C4C2211yescharacteristic
38C3×C4⋊Dic7C2211yescharacteristic, complemented
39C3×C4⋊Dic7C2211yescharacteristic, complemented
40Dic3⋊Dic7C2211yescharacteristic, complemented
41#66577C2211yescharacteristic, complemented
42#66575C2211yescharacteristic
43#66575C2211yescharacteristic
44(C2×D3)⋊Dic7C2211yescharacteristic
45(C2×D3)⋊Dic7C2211yescharacteristic
46(C2×D7)⋊Dic3C2211yescharacteristic
47(C2×D7)⋊Dic3C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49Dic3×Dic7C2211yescharacteristic, complemented
50Dic3×Dic7C2211yescharacteristic, complemented
51Dic3×Dic7C2211yescharacteristic, complemented
52C7×#123—31no—
53#6500—31no—
54D4×Dic7—31no—
55C2×(C2×D7)⋊C4—31no—
56#6489—31no—
57#6461—31no—
58#6461—31no—
59#6460—31no—
60#6455—31no—
61#6449D311yescharacteristic, complemented
62#6448—31no—
63#6446—31no—
64(C22⋊D7)⋊C4—31no—
65D7×C22⋊C4—31no—
66#6441—31no—
67#5659—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C22×D3×C7C2311yescharacteristic
71C22×C3×D7C2311yescharacteristic
72C2×D3×D7—22no—
73C2×D3×D7—22no—
74C2×D3×D7—41no—
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77#4268—22no—
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C4×D21—22no—
82C7×#67—22no—
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C2×C7×Dic3C2311yescharacteristic
86C4×D3×C7—22no—
87C3×C22⋊D7—41no—
88C2×C3×Dic7—41no—
89C2×C3×Dic7C2311yescharacteristic
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92#4248—41no—
93D21⋊C4—41no—
94D3×Dic7—41no—
95C23×D7—31no—
96C2×C7×D4—31no—
97C22×C4×C7—31no—
98C2×C22⋊D7—31no—
99C2×C22⋊D7C2×D311yescharacteristic
100C22×Dic7—31no—
101C22×Dic7C2×D311yescharacteristic, complemented
102C2×C4×D7—31no—
103C2×C4×D7—31no—
104C2×Dic14—31no—
105C7×C4⋊C4—31no—
106C7×C4⋊C4—31no—
107C7×C22⋊C4—31no—
108C7×C22⋊C4—31no—
109C7×C22⋊C4C2×D311yescharacteristic, complemented
110C22⋊Dic7—31no—
111C22⋊Dic7—31no—
112C22⋊Dic7—31no—
113(C2×D7)⋊C4—31no—
114(C2×D7)⋊C4—31no—
115(C2×D7)⋊C4—31no—
116(C2×D7)⋊C4—31no—
117(C2×D7)⋊C4—31no—
118(C2×D7)⋊C4—31no—
119(C2×D7)⋊C4C2×D311yescharacteristic
120(C2×D7)⋊C4C2×D311yescharacteristic
121C4⋊Dic7—31no—
122C4⋊Dic7—31no—
123C4⋊Dic7C2×D311yescharacteristic, complemented
124C4⋊Dic7C2×D311yescharacteristic, complemented
125Dic7⋊C4—31no—
126Dic7⋊C4—31no—
127C4×Dic7—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C3×#123—71no—
131#937—71no—
132#936—71no—
133C4×#67—71no—
134C4×#67—71no—
135#896—71no—
136#896—71no—
137#892—71no—
138#892—71no—
139#883—71no—
140#883D711yescharacteristic, complemented
141D3×C22⋊C4—71no—
142D3×C22⋊C4—71no—
143#879—71no—
144#879—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7C2411yescharacteristic
149C2×D21—21no—
150C2×D21—21no—
151C2×D21#4112yes—
152C2×D3×C7—21no—
153C2×D3×C7—21no—
154C2×D3×C7#4112yes—
155C2×C3×D7—21no—
156C2×C3×D7—21no—
157C2×C3×D7—21no—
158D3×D7—44no—
159C3×C4×C7—21no—
160C3×C4×C7—21no—
161Dic21—21no—
162Dic21—21no—
163Dic21—21no—
164C3×Dic7—21no—
165C3×Dic7—21no—
166C3×Dic7—21no—
167C7×Dic3—21no—
168C7×Dic3—21no—
169C7×Dic3—21no—
170#551—211noSylow
171C23×C7—31no—
172C23×C7C22×D311yescharacteristic
173C22×D7—31no—
174C22×D7—62no—
175C22×D7—62no—
176C22×D7—121no—
177C22×D7C22×D311yescharacteristic
178C7×D4—62no—
179C2×C4×C7—31no—
180C2×C4×C7—31no—
181C2×C4×C7—31no—
182C2×C4×C7—62no—
183C2×C4×C7C22×D311yescharacteristic
184C2×C4×C7C22×D311yescharacteristic
185C22⋊D7—41no—
186C22⋊D7—62no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—31no—
190C2×Dic7—41no—
191C2×Dic7—121no—
192C2×Dic7C22×D311yescharacteristic
193C2×Dic7C22×D311yescharacteristic
194C2×Dic7C22×D311yescharacteristic
195C4×D7—62no—
196C4×D7—121no—
197Dic14—121no—
198C23×D3—71no—
199C2×C3×D4—71no—
200C22×C3×C4—71no—
201C2×#67—71no—
202C2×#67C2×D711yescharacteristic
203C2×C4×D3—71no—
204C2×C4×D3—71no—
205C2×C4×D3—71no—
206C2×C4×D3C2×D711yescharacteristic
207C2×Dic6—71no—
208C3×C4⋊C4—71no—
209C3×C4⋊C4—71no—
210C3×C22⋊C4—71no—
211C3×C22⋊C4—71no—
212C3×C22⋊C4C2×D711yescharacteristic, complemented
213#217—71no—
214#217—71no—
215#217—71no—
216#217—71no—
217#217C2×D711yescharacteristic, complemented
218(C2×D3)⋊C4—71no—
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4—71no—
221(C2×D3)⋊C4—71no—
222(C2×D3)⋊C4—71no—
223(C2×D3)⋊C4C2×D711yescharacteristic
224C4⋊Dic3—71no—
225C4⋊Dic3C2×D711yescharacteristic, complemented
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4—71no—
229Dic3⋊C4C2×D711yescharacteristic, complemented
230C4×Dic3—71no—
231C4×Dic3—71no—
232C4×Dic3—71no—
233C2×C3×C7—41no—
234C2×C3×C7C2×#4111yescharacteristic
235C2×C3×C7C2×#4111yescharacteristic
236C2×C3×C7H(5,𝔽2)11yescharacteristic
237D21—22no—
238C3×D7—41no—
239D3×C7—22no—
240#126—211no—
241#126—211no—
242#124—211no—
243#123—31no—
244#123—71no—
245#123—211no—
246C22⋊Q8—211no—
247C22⋊Q8—211no—
248#120—211no—
249C4×D4—211no—
250C4×D4—211no—
251#117—211no—
252#117—211no—
253C2×C22⋊C4—211no—
254C2×C22⋊C4—211no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—32no—
259C22×C7—61no—
260C22×C7—61no—
261C22×C7C23×D311yescharacteristic
262C2×D7—21no—
263C2×D7—21no—
264C2×D7—21no—
265C2×D7—32no—
266C2×D7—61no—
267C2×D7—61no—
268C2×D7—124no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—61no—
272C4×C7—61no—
273C4×C7—61no—
274Dic7—21no—
275Dic7—21no—
276Dic7—21no—
277Dic7—61no—
278Dic7—61no—
279Dic7—61no—
280C23×C3—71no—
281C23×C3C22×D711yescharacteristic
282C22×D3—71no—
283C22×D3—142no—
284C22×D3—142no—
285C22×D3—281no—
286C22×D3C22×D711yescharacteristic
287C3×D4—281no—
288C2×C3×C4—71no—
289C2×C3×C4—71no—
290C2×C3×C4—71no—
291C2×C3×C4—281no—
292C2×C3×C4C22×D711yescharacteristic
293C2×C3×C4C22×D711yescharacteristic
294#67—22no—
295#67—142no—
296C2×Dic3—71no—
297C2×Dic3—71no—
298C2×Dic3—71no—
299C2×Dic3C22×D711yescharacteristic
300C2×Dic3C22×D711yescharacteristic
301C2×Dic3C22×D711yescharacteristic
302C4×D3—22no—
303C4×D3—142no—
304C4×D3—281no—
305C4×D3—281no—
306Dic6—281no—
307C3×C7#55111yescharacteristic, complemented, Hall
308C24—211no—
309C2×Q8—211no—
310C2×D4—31no—
311C2×D4—71no—
312C2×D4—211no—
313C22×C4—31no—
314C22×C4—71no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—71no—
321C4⋊C4—71no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4—211no—
326C22⋊C4—31no—
327C22⋊C4—31no—
328C22⋊C4—71no—
329C22⋊C4—71no—
330C22⋊C4—211no—
331C22⋊C4—211no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C22⋊C4—211no—
338C22⋊C4—211no—
339C22⋊C4D3×D711yescharacteristic, complemented
340C42—211no—
341C42—211no—
342C42—211no—
343C2×C7—41no—
344C2×C7—62no—
345C2×C7#100411yescharacteristic
346C2×C7D3×#4111yescharacteristic
347C2×C7D3×#4111yescharacteristic
348D7—41no—
349D7—62no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—141no—
354C22×C3—141no—
355C22×C3—141no—
356C22×C3C23×D711yescharacteristic
357C2×D3—21no—
358C2×D3—21no—
359C2×D3—72no—
360C2×D3—141no—
361C2×D3—141no—
362C2×D3—284no—
363C2×D3#115912yes—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—141no—
367C3×C4—141no—
368C3×C4—141no—
369Dic3—21no—
370Dic3—21no—
371Dic3—21no—
372Dic3—141no—
373Dic3—141no—
374Dic3—141no—
375C23—31no—
376C23—71no—
377C23—211no—
378C23—422no—
379C23—422no—
380C23—841no—
381C23C2×D3×D711yescharacteristic
382Q8—841no—
383D4—62no—
384D4—281no—
385D4—422no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—62no—
390C2×C4—71no—
391C2×C4—71no—
392C2×C4—71no—
393C2×C4—211no—
394C2×C4—211no—
395C2×C4—211no—
396C2×C4—281no—
397C2×C4—422no—
398C2×C4—841no—
399C2×C4—841no—
400C2×C4C2×D3×D711yescharacteristic
401C2×C4C2×D3×D711yescharacteristic
402C7#565911yescharacteristic, complemented, Sylow
403C2×C3—41no—
404C2×C3—281no—
405C2×C3C2×#115911yescharacteristic
406C2×C3D7×#4111yescharacteristic
407C2×C3#655311yescharacteristic
408D3—22no—
409D3—142no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—32no—
414C22—61no—
415C22—61no—
416C22—141no—
417C22—141no—
418C22—141no—
419C22—212no—
420C22—421no—
421C22—421no—
422C22—844no—
423C22C22×D3×D711yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—141no—
430C4—141no—
431C4—141no—
432C4—421no—
433C4—421no—
434C4—421no—
435C3#9034911yescharacteristic, complemented, Sylow
436C2—41no—
437C2—62no—
438C2—281no—
439C2—422no—
440C2#1062156411yescharacteristic
441C2D3×#115911yescharacteristic
442C2#1062157111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radical#10621139C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#551
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. #10621366C2semidirect
2. C7×#883C2semidirect
3. C3×#6449C2semidirect
5. #10621114C2semidirect
6. #67×Dic7C2semidirect
7. #10621096C2semidirect
8. #10621035C2semidirect
9. D3×(C2×D7)⋊C4C2semidirect
10. #10620998C2semidirect
11. #10620991C2semidirect
12. #10620924C2semidirect
13. #10620923C2semidirect
14. #10620909C2semidirect
15. #10620900C2semidirect
23. C22×C3×Dic7C22semidirect
24. C2×#4248C22semidirect
27. C3×C7×C22⋊C4C22semidirect
28. #66636C22semidirect
30. C4⋊Dic21C22semidirect
31. #66629C22semidirect
32. C7×#217C22semidirect
34. C7×C4⋊Dic3C22semidirect
35. C7×Dic3⋊C4C22semidirect
38. C3×C4⋊Dic7C22semidirect
39. C3×C4⋊Dic7C22semidirect
40. Dic3⋊Dic7C22semidirect
41. #66577C22semidirect
49. Dic3×Dic7C22semidirect
50. Dic3×Dic7C22semidirect
51. Dic3×Dic7C22semidirect
61. #6449D3semidirect
101. C22×Dic7C2×D3semidirect
109. C7×C22⋊C4C2×D3semidirect
123. C4⋊Dic7C2×D3semidirect
124. C4⋊Dic7C2×D3semidirect
140. #883D7semidirect
212. C3×C22⋊C4C2×D7semidirect
217. #217C2×D7semidirect
225. C4⋊Dic3C2×D7semidirect
229. Dic3⋊C4C2×D7semidirect
307. C3×C7#551semidirect
339. C22⋊C4D3×D7semidirect
402. C7#5659semidirect
435. C3#90349semidirect

Maximal subgroups

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

subgroupindex
#106213662
C7×#8832
C3×#64492
#106211392
#106211142
#67×Dic72
#106210962
#106210352
D3×(C2×D7)⋊C42
#106209982
#106209912
#106209242
#106209232
#106209092
#106209002
#903493
#56597