Finite Group Databasebeta

← #13061647

Subgroups — #13061647

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
1C3×C4×C7×D4C211yesmaximal, characteristic, complemented
2#10621420C211yesmaximal, characteristic
3#10621418C211yesmaximal, characteristic, complemented
4#10621412C211yesmaximal, characteristic, complemented
5C4×#4268C211yesmaximal, characteristic, complemented
6C2×#66631C211yesmaximal, characteristic, complemented
7#10621407C211yesmaximal, characteristic, complemented
8#10621381C211yesmaximal, characteristic, complemented
9#10621379C211yesmaximal, characteristic, complemented
10#10621368C211yesmaximal, characteristic, complemented
11#10621366C211yesmaximal, characteristic, complemented
12C22⋊C4×D21C211yesmaximal, characteristic, complemented
13#10621360C211yesmaximal, characteristic, complemented
14#10621359C211yesmaximal, characteristic, complemented
15#10621355C211yesmaximal, characteristic, complemented
16#90247—31nomaximal, Hall
17C23×D21C2211yescharacteristic
18C2×C3×C7×D4C2211yescharacteristic
19C22×C3×C4×C7C2211yescharacteristic
20C22×C3×C4×C7C2211yescharacteristic, complemented
21C2×#4268C2211yescharacteristic
22C2×#4268C2211yescharacteristic
23C22×Dic21C2211yescharacteristic
24C2×C4×D21C2211yescharacteristic
25C2×C4×D21C2211yescharacteristic
26C2×Dic42C2211yescharacteristic, complemented
27C3×C7×C4⋊C4C2211yescharacteristic, complemented
28C3×C7×C22⋊C4C2211yescharacteristic
29C3×C7×C22⋊C4C2211yescharacteristic, complemented
30C3×C42×C7C2211yescharacteristic, complemented
31#66636C2211yescharacteristic
32#66636C2211yescharacteristic
33#66636C2211yescharacteristic, complemented
34#66636C2211yescharacteristic, complemented
35#66631C2211yescharacteristic
36#66631C2211yescharacteristic
37#66631C2211yescharacteristic
38#66631C2211yescharacteristic
39#66631C2211yescharacteristic
40#66631C2211yescharacteristic
41#66631C2211yescharacteristic
42#66631C2211yescharacteristic
43C4⋊Dic21C2211yescharacteristic, complemented
44#66629C2211yescharacteristic, complemented
45#66629C2211yescharacteristic, complemented
46#66629C2211yescharacteristic, complemented
47#66629C2211yescharacteristic, complemented
48#66629C2211yescharacteristic, complemented
49#66629C2211yescharacteristic, complemented
50C4×Dic21C2211yescharacteristic, complemented
51C4×Dic21C2211yescharacteristic, complemented
52C4×C7×D4D311yescharacteristic, complemented
53#6500—31no—
54#6498—31no—
55#6492—31no—
56C4×C22⋊D7—31no—
57C2×(C2×D7)⋊C4—31no—
58#6487—31no—
59#6461—31no—
60#6459—31no—
61#6448—31no—
62#6446—31no—
63D7×C22⋊C4—31no—
64#6440—31no—
65#6439—31no—
66#6435—31no—
67#5556—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C22×D21—22no—
71C22×D21—22no—
72C22×D21—41no—
73C22×D21C2311yescharacteristic
74C22×D21C2311yescharacteristic
75C3×C7×D4—22no—
76C2×C3×C4×C7—41no—
77C2×C3×C4×C7—41no—
78C2×C3×C4×C7C2311yescharacteristic
79C2×C3×C4×C7C2311yescharacteristic
80C2×C3×C4×C7C2311yescharacteristic
81C2×C3×C4×C7C2311yescharacteristic
82C2×C3×C4×C7C2311yescharacteristic
83#4268—22no—
84#4268—41no—
85C2×Dic21—22no—
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88C2×Dic21C2311yescharacteristic
89C2×Dic21C2311yescharacteristic
90C2×Dic21C2311yescharacteristic
91C2×Dic21C2311yescharacteristic
92C4×D21—22no—
93C4×D21—41no—
94Dic42—41no—
95C23×D7—31no—
96C2×C7×D4C2×D311yescharacteristic
97C22×C4×C7C2×D311yescharacteristic
98C22×C4×C7C2×D311yescharacteristic, complemented
99C2×C22⋊D7—31no—
100C2×C22⋊D7—31no—
101C22×Dic7—31no—
102C2×C4×D7—31no—
103C2×C4×D7—31no—
104C2×Dic14—31no—
105C7×C4⋊C4C2×D311yescharacteristic, complemented
106C7×C22⋊C4C2×D311yescharacteristic
107C7×C22⋊C4C2×D311yescharacteristic, complemented
108C42×C7C2×D311yescharacteristic, complemented
109C22⋊Dic7—31no—
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)⋊C4—31no—
120(C2×D7)⋊C4—31no—
121C4⋊Dic7—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125Dic7⋊C4—31no—
126Dic7⋊C4—31no—
127Dic7⋊C4—31no—
128C4×Dic7—31no—
129C4×Dic7—31no—
130C3×C4×D4D711yescharacteristic, complemented
131#937—71no—
132#935—71no—
133#929—71no—
134C4×#67—71no—
135C2×(C2×D3)⋊C4—71no—
136#924—71no—
137#898—71no—
138#896—71no—
139#885—71no—
140#883—71no—
141D3×C22⋊C4—71no—
142#877—71no—
143#876—71no—
144#872—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7#4112yes—
151C22×C3×C7C2411yescharacteristic
152C2×D21—21no—
153C2×D21—21no—
154C2×D21—21no—
155C2×D21—21no—
156C2×D21—21no—
157C2×D21—44no—
158C2×D21#4112yes—
159C3×C4×C7—21no—
160C3×C4×C7—21no—
161C3×C4×C7—21no—
162C3×C4×C7—21no—
163C3×C4×C7—21no—
164Dic21—21no—
165Dic21—21no—
166Dic21—21no—
167Dic21—21no—
168Dic21—21no—
169Dic21—21no—
170#551—211noSylow
171C23×C7C22×D311yescharacteristic
172C23×C7C22×D311yescharacteristic
173C22×D7—31no—
174C22×D7—31no—
175C22×D7—62no—
176C22×D7—62no—
177C22×D7—121no—
178C7×D4—22no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7C22×D311yescharacteristic
182C2×C4×C7C22×D311yescharacteristic
183C2×C4×C7C22×D311yescharacteristic
184C2×C4×C7C22×D311yescharacteristic
185C2×C4×C7C22×D311yescharacteristic
186C22⋊D7—62no—
187C22⋊D7—121no—
188C2×Dic7—31no—
189C2×Dic7—31no—
190C2×Dic7—31no—
191C2×Dic7—31no—
192C2×Dic7—31no—
193C2×Dic7—31no—
194C2×Dic7—62no—
195C4×D7—62no—
196C4×D7—121no—
197Dic14—121no—
198C23×D3—71no—
199C2×C3×D4C2×D711yescharacteristic
200C22×C3×C4C2×D711yescharacteristic
201C22×C3×C4C2×D711yescharacteristic, complemented
202C2×#67—71no—
203C2×#67—71no—
204C22×Dic3—71no—
205C2×C4×D3—71no—
206C2×C4×D3—71no—
207C2×Dic6—71no—
208C3×C4⋊C4C2×D711yescharacteristic, complemented
209C3×C22⋊C4C2×D711yescharacteristic
210C3×C22⋊C4C2×D711yescharacteristic, complemented
211C3×C42C2×D711yescharacteristic, complemented
212#217—71no—
213#217—71no—
214#217—71no—
215#217—71no—
216(C2×D3)⋊C4—71no—
217(C2×D3)⋊C4—71no—
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)⋊C4—71no—
224C4⋊Dic3—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4—71no—
229Dic3⋊C4—71no—
230Dic3⋊C4—71no—
231C4×Dic3—71no—
232C4×Dic3—71no—
233C2×C3×C7—22no—
234C2×C3×C7—41no—
235C2×C3×C7C2×#4111yescharacteristic
236C2×C3×C7C2×#4111yescharacteristic
237C2×C3×C7H(5,𝔽2)11yescharacteristic
238D21—22no—
239D21—41no—
240#126—211no—
241#126—211no—
242#124—211no—
243#123—211no—
244#123—211no—
245#123—211no—
246C22⋊Q8—211no—
247C22⋊Q8—211no—
248#120—211no—
249C4×D4—211no—
250C4×D4D2111yescharacteristic, complemented
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—21no—
259C22×C7—21no—
260C22×C7#23512yes—
261C22×C7C23×D311yescharacteristic
262C2×D7—32no—
263C2×D7—61no—
264C2×D7—61no—
265C2×D7—61no—
266C2×D7—61no—
267C2×D7—61no—
268C2×D7—124no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274Dic7—61no—
275Dic7—61no—
276Dic7—61no—
277Dic7—61no—
278Dic7—61no—
279Dic7—61no—
280C23×C3C22×D711yescharacteristic
281C23×C3C22×D711yescharacteristic
282C22×D3—71no—
283C22×D3—71no—
284C22×D3—142no—
285C22×D3—142no—
286C22×D3—281no—
287C3×D4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4C22×D711yescharacteristic
291C2×C3×C4C22×D711yescharacteristic
292C2×C3×C4C22×D711yescharacteristic
293C2×C3×C4C22×D711yescharacteristic
294C2×C3×C4C22×D711yescharacteristic
295#67—142no—
296#67—281no—
297C2×Dic3—71no—
298C2×Dic3—71no—
299C2×Dic3—71no—
300C2×Dic3—71no—
301C2×Dic3—71no—
302C2×Dic3—71no—
303C2×Dic3—142no—
304C4×D3—142no—
305C4×D3—281no—
306Dic6—281no—
307C3×C7#55111yescharacteristic, complemented, Hall
308C24—211no—
309C2×Q8—211no—
310C2×D4—211no—
311C2×D4—211no—
312C2×D4C2×D2111yescharacteristic
313C22×C4—211no—
314C22×C4—211no—
315C22×C4—211no—
316C22×C4C2×D2111yescharacteristic
317C22×C4C2×D2111yescharacteristic, complemented
318C4⋊C4—211no—
319C4⋊C4—211no—
320C4⋊C4—211no—
321C4⋊C4—211no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4C2×D2111yescharacteristic, complemented
326C22⋊C4—211no—
327C22⋊C4—211no—
328C22⋊C4—211no—
329C22⋊C4—211no—
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⋊C4C2×D2111yescharacteristic
339C22⋊C4C2×D2111yescharacteristic, complemented
340C42—211no—
341C42—211no—
342C42C2×D2111yescharacteristic, complemented
343C2×C7—22no—
344C2×C7—41no—
345C2×C7C2×#23511yescharacteristic
346C2×C7#100411yescharacteristic
347C2×C7D3×#4111yescharacteristic
348D7—62no—
349D7—121no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3#115712yes—
356C22×C3C23×D711yescharacteristic
357C2×D3—72no—
358C2×D3—141no—
359C2×D3—141no—
360C2×D3—141no—
361C2×D3—141no—
362C2×D3—141no—
363C2×D3—284no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369Dic3—141no—
370Dic3—141no—
371Dic3—141no—
372Dic3—141no—
373Dic3—141no—
374Dic3—141no—
375C23—211no—
376C23—211no—
377C23—422no—
378C23—422no—
379C23—841no—
380C23C22×D2111yescharacteristic
381C23C22×D2111yescharacteristic
382Q8—841no—
383D4—22no—
384D4—422no—
385D4—841no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—211no—
389C2×C4—211no—
390C2×C4—211no—
391C2×C4—211no—
392C2×C4—211no—
393C2×C4—211no—
394C2×C4—422no—
395C2×C4—422no—
396C2×C4—841no—
397C2×C4C22×D2111yescharacteristic
398C2×C4C22×D2111yescharacteristic
399C2×C4C22×D2111yescharacteristic
400C2×C4C22×D2111yescharacteristic
401C2×C4C22×D2111yescharacteristic
402C7#555611yescharacteristic, complemented, Sylow
403C2×C3—22no—
404C2×C3—41no—
405C2×C3C2×#115711yescharacteristic
406C2×C3#654811yescharacteristic
407C2×C3D7×#4111yescharacteristic
408D3—142no—
409D3—281no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—212no—
416C22—421no—
417C22—421no—
418C22—421no—
419C22—421no—
420C22—421no—
421C22—844no—
422C22#6672812yes—
423C22C23×D2111yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—421no—
430C4—421no—
431C4—421no—
432C4—421no—
433C4—421no—
434C4—421no—
435C3#9024711yescharacteristic, complemented, Sylow
436C2—22no—
437C2—41no—
438C2—422no—
439C2—841no—
440C2C2×#6672811yescharacteristic
441C2#1062165711yescharacteristic
442C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C4×C7×D4C2
prime radical#10621420C2
hypercenterC4×D4D21
hypocenterC3×C7#551
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C7×D4C2semidirect
3. #10621418C2semidirect
4. #10621412C2semidirect
5. C4×#4268C2semidirect
6. C2×#66631C2semidirect
7. #10621407C2semidirect
8. #10621381C2semidirect
9. #10621379C2semidirect
10. #10621368C2semidirect
11. #10621366C2semidirect
12. C22⋊C4×D21C2semidirect
13. #10621360C2semidirect
14. #10621359C2semidirect
15. #10621355C2semidirect
20. C22×C3×C4×C7C22semidirect
26. C2×Dic42C22semidirect
27. C3×C7×C4⋊C4C22semidirect
29. C3×C7×C22⋊C4C22semidirect
30. C3×C42×C7C22semidirect
33. #66636C22semidirect
34. #66636C22semidirect
43. C4⋊Dic21C22semidirect
44. #66629C22semidirect
45. #66629C22semidirect
46. #66629C22semidirect
47. #66629C22semidirect
48. #66629C22semidirect
49. #66629C22semidirect
50. C4×Dic21C22semidirect
51. C4×Dic21C22semidirect
52. C4×C7×D4D3semidirect
98. C22×C4×C7C2×D3semidirect
105. C7×C4⋊C4C2×D3semidirect
107. C7×C22⋊C4C2×D3semidirect
108. C42×C7C2×D3semidirect
130. C3×C4×D4D7semidirect
201. C22×C3×C4C2×D7semidirect
208. C3×C4⋊C4C2×D7semidirect
210. C3×C22⋊C4C2×D7semidirect
211. C3×C42C2×D7semidirect
250. C4×D4D21semidirect
307. C3×C7#551semidirect
317. C22×C4C2×D21semidirect
325. C4⋊C4C2×D21semidirect
339. C22⋊C4C2×D21semidirect
342. C42C2×D21semidirect
402. C7#5556semidirect
435. C3#90247semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C7×D42
#106214202
#106214182
#106214122
C4×#42682
C2×#666312
#106214072
#106213812
#106213792
#106213682
#106213662
C22⋊C4×D212
#106213602
#106213592
#106213552
#902473
#55567