Finite Group Databasebeta

← #13058075

Subgroups — #13058075

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#10621362C211yesmaximal, characteristic, complemented
2C7×#885C211yesmaximal, characteristic, complemented
3C3×D7×C22⋊C4C211yesmaximal, characteristic, complemented
4#10621125C211yesmaximal, characteristic
5#10621117C211yesmaximal, characteristic, complemented
6D7×#217C211yesmaximal, characteristic, complemented
7#10621093C211yesmaximal, characteristic, complemented
8#10621009C211yesmaximal, characteristic, complemented
9#10621004C211yesmaximal, characteristic, complemented
10#10621002C211yesmaximal, characteristic, complemented
11#10620983C211yesmaximal, characteristic, complemented
12#10620976C211yesmaximal, characteristic, complemented
13#10620955C211yesmaximal, characteristic, complemented
14#10620921C211yesmaximal, characteristic, complemented
15#10620919C211yesmaximal, characteristic, complemented
16#90370—31nomaximal, Hall
17C23×C3×D7C2211yescharacteristic
18C2×C7×#67C2211yescharacteristic
19C2×C7×Dic6C2211yescharacteristic, complemented
20C2×C3×C4×D7C2211yescharacteristic
21C2×C3×C4×D7C2211yescharacteristic
22C2×#4245C2211yescharacteristic
23C2×#4245C2211yescharacteristic
24C2×D3×Dic7C2211yescharacteristic, complemented
25C2×Dic3×D7C2211yescharacteristic
26C2×Dic3×D7C2211yescharacteristic
27C3×C7×C22⋊C4C2211yescharacteristic
28#66636C2211yescharacteristic
29#66636C2211yescharacteristic
30C4⋊Dic21C2211yescharacteristic, complemented
31#66629C2211yescharacteristic, complemented
32#66629C2211yescharacteristic, complemented
33C4×Dic21C2211yescharacteristic, complemented
34C7×#217C2211yescharacteristic
35C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
36C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
37C4×C7×Dic3C2211yescharacteristic, complemented
38C3×C22⋊Dic7C2211yescharacteristic
39C3×(C2×D7)⋊C4C2211yescharacteristic
40C3×(C2×D7)⋊C4C2211yescharacteristic
41Dic3⋊Dic7C2211yescharacteristic, complemented
42#66577C2211yescharacteristic, complemented
43#66577C2211yescharacteristic, complemented
44Dic7⋊Dic3C2211yescharacteristic, complemented
45Dic7⋊Dic3C2211yescharacteristic, complemented
46(C2×D3)⋊Dic7C2211yescharacteristic, complemented
47(C2×D7)⋊Dic3C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic
49(C2×D7)⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51Dic3×Dic7C2211yescharacteristic, complemented
52C7×#124—31no—
53#6509—31no—
54#6509—31no—
55#6500—31no—
56#6498—31no—
57#6446—31no—
58#6446—31no—
59(C22⋊D7)⋊C4—31no—
60(C22⋊D7)⋊C4—31no—
61D7×C22⋊C4—31no—
62D7×C22⋊C4D311yescharacteristic, complemented
63#6442—31no—
64#6442—31no—
65#6435—31no—
66#6435—31no—
67#5495—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D3×C7C2311yescharacteristic
70C22×C3×D7—22no—
71C22×C3×D7—22no—
72C22×C3×D7—41no—
73C22×C3×D7C2311yescharacteristic
74C22×C3×D7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77C2×Dic21C2311yescharacteristic
78C2×Dic21C2311yescharacteristic
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C7×#67—41no—
82C2×C7×Dic3C2311yescharacteristic
83C2×C7×Dic3C2311yescharacteristic
84C2×C7×Dic3C2311yescharacteristic
85C7×Dic6—41no—
86C2×C3×Dic7C2311yescharacteristic
87C2×C3×Dic7C2311yescharacteristic
88C3×C4×D7—22no—
89C3×C4×D7—41no—
90#4245—22no—
91#4245—22no—
92D3×Dic7—41no—
93Dic3×D7—22no—
94Dic3×D7—41no—
95C23×D7C2×D311yescharacteristic
96C2×C7×Q8—31no—
97C2×C7×D4—31no—
98C2×C22⋊D7—31no—
99C2×C22⋊D7—31no—
100C22×Dic7—31no—
101C2×C4×D7—31no—
102C2×C4×D7—31no—
103C2×C4×D7C2×D311yescharacteristic
104C2×C4×D7C2×D311yescharacteristic
105C7×C22⋊C4—31no—
106C7×C22⋊C4—31no—
107C7×C22⋊C4—31no—
108C7×C22⋊C4C2×D311yescharacteristic
109C42×C7—31no—
110C22⋊Dic7—31no—
111C22⋊Dic7—31no—
112C22⋊Dic7—31no—
113C22⋊Dic7C2×D311yescharacteristic
114(C2×D7)⋊C4—31no—
115(C2×D7)⋊C4—31no—
116(C2×D7)⋊C4—31no—
117(C2×D7)⋊C4—31no—
118(C2×D7)⋊C4C2×D311yescharacteristic
119(C2×D7)⋊C4C2×D311yescharacteristic
120C4⋊Dic7—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—
130C2×C3×C22⋊C4—71no—
131#953—71no—
132C2×#217—71no—
133#929—71no—
134C4×#67—71no—
135#926—71no—
136#924—71no—
137#886—71no—
138#885D711yescharacteristic, complemented
139#883—71no—
140#67⋊C4—71no—
141#879—71no—
142#879—71no—
143#878—71no—
144#877—71no—
145C22×C3×C7—21no—
146C22×C3×C7—21no—
147C22×C3×C7—21no—
148C22×C3×C7C2411yescharacteristic
149C2×D3×C7—21no—
150C2×D3×C7—21no—
151C2×D3×C7—21no—
152C2×C3×D7—21no—
153C2×C3×D7—21no—
154C2×C3×D7—21no—
155C2×C3×D7—21no—
156C2×C3×D7—44no—
157C2×C3×D7#4112yes—
158C2×C3×D7#4112yes—
159C3×C4×C7—21no—
160C3×C4×C7—21no—
161Dic21—21no—
162Dic21—21no—
163Dic21—21no—
164Dic21—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—22no—
174C22×D7—22no—
175C22×D7—41no—
176C22×D7C22×D311yescharacteristic
177C22×D7C22×D311yescharacteristic
178C7×Q8—121no—
179C7×D4—121no—
180C2×C4×C7—31no—
181C2×C4×C7—31no—
182C2×C4×C7—31no—
183C2×C4×C7C22×D311yescharacteristic
184C2×C4×C7C22×D311yescharacteristic
185C22⋊D7—62no—
186C22⋊D7—62no—
187C2×Dic7—31no—
188C2×Dic7—31no—
189C2×Dic7—31no—
190C2×Dic7—31no—
191C2×Dic7—121no—
192C2×Dic7C22×D311yescharacteristic
193C2×Dic7C22×D311yescharacteristic
194C4×D7—22no—
195C4×D7—41no—
196C4×D7—62no—
197C4×D7—121no—
198C24×C3—71no—
199C22×C3×C4—71no—
200C22×C3×C4—71no—
201C2×#67—71no—
202C2×#67—71no—
203C2×#67C2×D711yescharacteristic
204C22×Dic3—71no—
205C22×Dic3—71no—
206C2×C4×D3—71no—
207C2×Dic6C2×D711yescharacteristic, complemented
208C3×C22⋊C4—71no—
209C3×C22⋊C4—71no—
210C3×C22⋊C4—71no—
211C3×C22⋊C4C2×D711yescharacteristic
212#217—71no—
213#217—71no—
214#217—71no—
215#217—71no—
216#217—71no—
217#217—71no—
218#217C2×D711yescharacteristic
219(C2×D3)⋊C4—71no—
220(C2×D3)⋊C4C2×D711yescharacteristic, complemented
221(C2×D3)⋊C4C2×D711yescharacteristic, complemented
222C4⋊Dic3—71no—
223C4⋊Dic3—71no—
224C4⋊Dic3—71no—
225Dic3⋊C4—71no—
226Dic3⋊C4—71no—
227Dic3⋊C4—71no—
228Dic3⋊C4—71no—
229Dic3⋊C4—71no—
230C4×Dic3—71no—
231C4×Dic3—71no—
232C4×Dic3C2×D711yescharacteristic, complemented
233C2×C3×C7—41no—
234C2×C3×C7C2×#4111yescharacteristic
235C2×C3×C7C2×#4111yescharacteristic
236C2×C3×C7H(5,𝔽2)11yescharacteristic
237C3×D7—22no—
238C3×D7—22no—
239D3×C7—41no—
240#126—211no—
241#126—211no—
242#124—31no—
243#123—211no—
244#123—211no—
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—71no—
254C2×C22⋊C4—211no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—61no—
259C22×C7—61no—
260C22×C7—61no—
261C22×C7C23×D311yescharacteristic
262C2×D7—21no—
263C2×D7—21no—
264C2×D7—21no—
265C2×D7—21no—
266C2×D7—44no—
267C2×D7#23512yes—
268C2×D7#23712yes—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—61no—
272C4×C7—61no—
273C4×C7—61no—
274Dic7—21no—
275Dic7—21no—
276Dic7—61no—
277Dic7—61no—
278Dic7—61no—
279Dic7—61no—
280C23×C3—71no—
281C23×C3—71no—
282C23×C3—142no—
283C23×C3—142no—
284C23×C3—281no—
285C23×C3C22×D711yescharacteristic
286C22×D3C22×D711yescharacteristic
287C2×C3×C4—71no—
288C2×C3×C4—71no—
289C2×C3×C4—142no—
290C2×C3×C4—281no—
291C2×C3×C4C22×D711yescharacteristic
292C2×C3×C4C22×D711yescharacteristic
293#67—41no—
294#67—142no—
295#67—142no—
296C2×Dic3—71no—
297C2×Dic3—71no—
298C2×Dic3—71no—
299C2×Dic3—71no—
300C2×Dic3—142no—
301C2×Dic3—281no—
302C2×Dic3C22×D711yescharacteristic
303C2×Dic3C22×D711yescharacteristic
304C2×Dic3C22×D711yescharacteristic
305C4×D3—281no—
306Dic6—41no—
307C3×C7#55111yescharacteristic, complemented, Hall
308C24—71no—
309C2×Q8—31no—
310C2×D4—31no—
311C2×D4—211no—
312C2×D4—211no—
313C22×C4—71no—
314C22×C4—71no—
315C22×C4—211no—
316C22×C4—211no—
317C22×C4—211no—
318C4⋊C4—211no—
319C4⋊C4—211no—
320C4⋊C4—211no—
321C4⋊C4—211no—
322C4⋊C4—211no—
323C4⋊C4—211no—
324C4⋊C4—211no—
325C4⋊C4—211no—
326C22⋊C4—31no—
327C22⋊C4—31no—
328C22⋊C4—31no—
329C22⋊C4—71no—
330C22⋊C4—71no—
331C22⋊C4—71no—
332C22⋊C4—211no—
333C22⋊C4—211no—
334C22⋊C4—211no—
335C22⋊C4—211no—
336C22⋊C4—211no—
337C22⋊C4—211no—
338C22⋊C4—211no—
339C22⋊C4D3×D711yescharacteristic
340C42—31no—
341C42—211no—
342C42—211no—
343C2×C7—41no—
344C2×C7—121no—
345C2×C7C2×#23511yescharacteristic
346C2×C7C2×#23711yescharacteristic
347C2×C7#100411yescharacteristic
348D7—22no—
349D7—22no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—72no—
354C22×C3—72no—
355C22×C3—141no—
356C22×C3—141no—
357C22×C3—141no—
358C22×C3—141no—
359C22×C3—284no—
360C22×C3C23×D711yescharacteristic
361C2×D3—21no—
362C2×D3—21no—
363C2×D3—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—141no—
367C3×C4—141no—
368Dic3—21no—
369Dic3—21no—
370Dic3—21no—
371Dic3—141no—
372Dic3—141no—
373Dic3—141no—
374Dic3—141no—
375C23—31no—
376C23—71no—
377C23—71no—
378C23—142no—
379C23—142no—
380C23—281no—
381C23C2×D3×D711yescharacteristic
382Q8—121no—
383D4—121no—
384D4—422no—
385D4—422no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—71no—
390C2×C4—71no—
391C2×C4—142no—
392C2×C4—211no—
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#549511yescharacteristic, complemented, Sylow
403C2×C3—41no—
404C2×C3—142no—
405C2×C3—142no—
406C2×C3#654811yescharacteristic
407C2×C3D7×#4111yescharacteristic
408C2×C3D7×#4111yescharacteristic
409D3—41no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—72no—
417C22—72no—
418C22—141no—
419C22—141no—
420C22—141no—
421C22—141no—
422C22—284no—
423C22C22×D3×D711yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—141no—
430C4—141no—
431C4—421no—
432C4—421no—
433C4—421no—
434C4—421no—
435C3#9037011yescharacteristic, complemented, Sylow
436C2—41no—
437C2—121no—
438C2—142no—
439C2—142no—
440C2D7×#23511yescharacteristic
441C2D7×#23711yescharacteristic
442C2#1062157211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radical#10621125C2
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. #10621362C2semidirect
2. C7×#885C2semidirect
3. C3×D7×C22⋊C4C2semidirect
5. #10621117C2semidirect
6. D7×#217C2semidirect
7. #10621093C2semidirect
8. #10621009C2semidirect
9. #10621004C2semidirect
10. #10621002C2semidirect
11. #10620983C2semidirect
12. #10620976C2semidirect
13. #10620955C2semidirect
14. #10620921C2semidirect
15. #10620919C2semidirect
19. C2×C7×Dic6C22semidirect
24. C2×D3×Dic7C22semidirect
30. C4⋊Dic21C22semidirect
31. #66629C22semidirect
32. #66629C22semidirect
33. C4×Dic21C22semidirect
35. C7×(C2×D3)⋊C4C22semidirect
36. C7×(C2×D3)⋊C4C22semidirect
37. C4×C7×Dic3C22semidirect
41. Dic3⋊Dic7C22semidirect
42. #66577C22semidirect
43. #66577C22semidirect
44. Dic7⋊Dic3C22semidirect
45. Dic7⋊Dic3C22semidirect
46. (C2×D3)⋊Dic7C22semidirect
51. Dic3×Dic7C22semidirect
62. D7×C22⋊C4D3semidirect
138. #885D7semidirect
207. C2×Dic6C2×D7semidirect
220. (C2×D3)⋊C4C2×D7semidirect
221. (C2×D3)⋊C4C2×D7semidirect
232. C4×Dic3C2×D7semidirect
307. C3×C7#551semidirect
402. C7#5495semidirect
435. C3#90370semidirect

Maximal subgroups

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

subgroupindex
#106213622
C7×#8852
C3×D7×C22⋊C42
#106211252
#106211172
D7×#2172
#106210932
#106210092
#106210042
#106210022
#106209832
#106209762
#106209552
#106209212
#106209192
#903703
#54957