Finite Group Databasebeta

← C7×#60176 #13237392

Subgroups — C7×#60176 #13237392

Order 1792.

2818 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#576C211yesmaximal, characteristic
2C4×C7×H(5,𝔽2)C212yesmaximal, complemented
3C7×#2952C213yesmaximal, complemented
4C7×#2948C213yesmaximal, complemented
5C7×#2946C216yesmaximal, complemented
6C4×C7×AGL(1,ℤ/8)C216yesmaximal, complemented
7C7×#2943C213yesmaximal, complemented
8C2×C7×#442C213yesmaximal, complemented
9C7×#2912C211yesmaximal, characteristic, complemented
10C7×#2897C212yesmaximal, complemented
11C7×#2875C211yesmaximal, characteristic, complemented
12C2×C7×H(5,𝔽2)C2212yes—
13C7×#576C4116yescomplemented
14C2×C7×AGL(1,ℤ/8)C2216yes—
15C2×C7×#135C2213yes—
16C22×C7×D8C2213yes—
17C2×C7×#131C2211yescharacteristic
18C2×C7×#131C2211yescharacteristic
19C7×#520—22no—
20C7×#520C2212yes—
21C7×#520C2216yescomplemented
22C7×#518—26no—
23C7×#518C2216yescomplemented
24C4×C7×#41—22no—
25C4×C7×#41C2211yescharacteristic
26C4×C7×#41C2216yescomplemented
27C2×C4×C7×D4—26no—
28C2×C4×C7×D4C2216yescomplemented
29C7×#442C2213yescomplemented
30C7×#442C2216yescomplemented
31C7×#442C2216yescomplemented
32C7×#442C2216yescomplemented
33C7×#442C2213yescomplemented
34C7×#441C2211yescharacteristic, complemented
35C7×#441C2213yescomplemented
36C7×#440C2216yescomplemented
37C7×#440C2216yescomplemented
38C4×C7×SD16C22112yescomplemented
39C4×C7×D8C22112yescomplemented
40C7×#428C2216yescomplemented
41C7×#427C2213yescomplemented
42C2×C7×#106C2213yescomplemented
43C7×#418C2216yescomplemented
44C7×#417C2216yescomplemented
45C7×#416C2216yescomplemented
46C2×C7×#102C2216yescomplemented
47C7×#405C2213yescomplemented
48C7×#405C2213yescomplemented
49C4×C7×M16C2213yescomplemented
50C2×C7×#97C2213yescomplemented
51#60176C711yesmaximal, characteristic, direct product factor, Sylow
52C7×H(5,𝔽2)—28no—
53C7×H(5,𝔽2)C2×C4116yescomplemented
54C2×C7×#41—22no—
55C2×C7×#41C2316yes—
56C2×C7×#41C2311yescharacteristic
57C22×C7×D4—26no—
58C22×C7×D4C2316yes—
59C23×C4×C7—26no—
60C7×AGL(1,ℤ/8)C2×C4148yescomplemented
61C7×#135C2×C4124yescomplemented
62C2×C7×Q16C2311yescharacteristic
63C2×C7×SD16C2316yes—
64C2×C7×D8C2×C4124yescomplemented
65C2×C7×D8C2313yes—
66C2×C7×D8C2316yes—
67C7×#131—24no—
68C7×#131C2×C418yescomplemented
69C2×C7×M16C2313yes—
70C2×C7×M16C2313yes—
71C22×C7×C8C2313yes—
72C22×C7×C8C2313yes—
73C4×C7×Q8—22no—
74C4×C7×Q8D412yes—
75C4×C7×Q8C2312yes—
76C4×C7×Q8C2316yes—
77C4×C7×D4—26no—
78C4×C7×D4—26no—
79C4×C7×D4—26no—
80C4×C7×D4—26no—
81C4×C7×D4—26no—
82C4×C7×D4—212no—
83C4×C7×D4—26no—
84C4×C7×D4—26no—
85C4×C7×D4D416yes—
86C4×C7×D4C2316yes—
87C4×C7×D4C23112yes—
88C4×C7×D4C2316yes—
89C7×#117—26no—
90C7×#117—26no—
91C7×#117C2316yes—
92C7×#117C2313yes—
93C2×C7×C4⋊C4—26no—
94C2×C7×C4⋊C4C2316yes—
95C2×C7×C22⋊C4—26no—
96C2×C7×C22⋊C4—212no—
97C2×C42×C7—26no—
98C2×C42×C7C2313yes—
99C7×#107C2316yes—
100C7×#106C2313yes—
101C7×#106C2316yes—
102C7×#106C2311yescharacteristic
103C7×#103C2312yes—
104C7×#103C2316yes—
105C7×#102C2316yes—
106C7×#102C23112yes—
107C7×#102C2316yes—
108C7×#97C2311yescharacteristic
109C7×#97C2313yes—
110C7×#97C2313yes—
111C7×#97C2313yes—
112C4×C7×C8C2316yes—
113C2×#576C2×C711yescharacteristic
114C4×H(5,𝔽2)C2×C712yescomplemented
115#2952C2×C713yescomplemented
116#2948C2×C713yescomplemented
117#2946C2×C716yescomplemented
118C4×AGL(1,ℤ/8)C2×C716yescomplemented
119#2943C2×C713yescomplemented
120C2×#442C2×C713yescomplemented
121#2912C2×C711yescharacteristic, complemented
122#2897C2×C712yescomplemented
123#2875C2×C711yescharacteristic, complemented
124C24×C7—26no—
125C7×#41—216no—
126C7×#41—212no—
127C7×#41C22×C414yes—
128C7×#41C22×C4124yescomplemented
129C7×#41#4114yes—
130C2×C7×Q8C2×D411yescharacteristic
131C2×C7×Q8C2412yes—
132C2×C7×D4—26no—
133C2×C7×D4—248no—
134C2×C7×D4—26no—
135C2×C7×D4—212no—
136C2×C7×D4C22×C4124yescomplemented
137C2×C7×D4C2×D413yes—
138C2×C7×D4C2416yes—
139C22×C4×C7—26no—
140C22×C4×C7—26no—
141C22×C4×C7—26no—
142C22×C4×C7—26no—
143C22×C4×C7—212no—
144C22×C4×C7—812no—
145C22×C4×C7C2×D416yes—
146C22×C4×C7C2413yes—
147C7×Q16C22×C414yescomplemented
148C7×SD16C22×C4124yescomplemented
149C7×D8C22×C4112yescomplemented
150C7×D8C22×C4124yescomplemented
151C7×M16—26no—
152C7×M16C22×C4112yescomplemented
153C2×C7×C8—26no—
154C2×C7×C8C22×C4112yescomplemented
155C2×C7×C8C2413yes—
156C2×C7×C8C2411yescharacteristic
157C2×C7×C8C2413yes—
158C2×C7×C8C2411yescharacteristic
159C7×C4⋊C4—26no—
160C7×C4⋊C4—26no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4C2×D416yes—
163C7×C4⋊C4C2416yes—
164C7×C4⋊C4C2412yes—
165C7×C22⋊C4—26no—
166C7×C22⋊C4—26no—
167C7×C22⋊C4—26no—
168C7×C22⋊C4—212no—
169C7×C22⋊C4C2×D416yes—
170C42×C7—22no—
171C42×C7—26no—
172C42×C7—26no—
173C42×C7C2×D416yes—
174C42×C7C2413yes—
175C42×C7C2411yescharacteristic
176C2×H(5,𝔽2)C22×C712yes—
177#576C4×C7116yescomplemented
178C2×AGL(1,ℤ/8)C22×C716yes—
179C2×#135C22×C713yes—
180C22×D8C22×C713yes—
181C2×#131C22×C711yescharacteristic
182C2×#131C22×C711yescharacteristic
183#520—22no—
184#520C22×C712yes—
185#520C22×C716yescomplemented
186#518—26no—
187#518C22×C716yescomplemented
188C4×#41—22no—
189C4×#41C22×C711yescharacteristic
190C4×#41C22×C716yescomplemented
191C2×C4×D4—26no—
192C2×C4×D4C22×C716yescomplemented
193#442C22×C713yescomplemented
194#442C22×C716yescomplemented
195#442C22×C716yescomplemented
196#442C22×C716yescomplemented
197#442C22×C713yescomplemented
198#441C22×C711yescharacteristic, complemented
199#441C22×C713yescomplemented
200#440C22×C716yescomplemented
201#440C22×C716yescomplemented
202C4×SD16C22×C7112yescomplemented
203C4×D8C22×C7112yescomplemented
204#428C22×C716yescomplemented
205#427C22×C713yescomplemented
206C2×#106C22×C713yescomplemented
207#418C22×C716yescomplemented
208#417C22×C716yescomplemented
209#416C22×C716yescomplemented
210C2×#102C22×C716yescomplemented
211#405C22×C713yescomplemented
212#405C22×C713yescomplemented
213C4×M16C22×C713yescomplemented
214C2×#97C22×C713yescomplemented
215C23×C7—26no—
216C23×C7—224no—
217C23×C7—86no—
218C23×C7—86no—
219C23×C7C22×D413yes—
220C7×Q8—22no—
221C7×Q8C4×D414yes—
222C7×Q8C23×C414yes—
223C7×D4—224no—
224C7×D4—224no—
225C7×D4—26no—
226C7×D4C4×D4112yes—
227C7×D4C23×C4112yes—
228C2×C4×C7—22no—
229C2×C4×C7—224no—
230C2×C4×C7—22no—
231C2×C4×C7—26no—
232C2×C4×C7—46no—
233C2×C4×C7—412no—
234C2×C4×C7—86no—
235C2×C4×C7—812no—
236C2×C4×C7—86no—
237C2×C4×C7C23×C416yes—
238C2×C4×C7C22×D412yes—
239C2×C4×C7C22×D416yes—
240C2×C4×C7C22×D413yes—
241C2×C4×C7C2×#4116yes—
242C2×C4×C7C2511yescharacteristic
243C7×C8—21no—
244C7×C8—23no—
245C7×C8C23×C412yes—
246C7×C8C23×C416yes—
247H(5,𝔽2)—28no—
248H(5,𝔽2)C2×C4×C7116yescomplemented
249C2×#41—22no—
250C2×#41C23×C716yes—
251C2×#41C23×C711yescharacteristic
252C22×D4—26no—
253C22×D4C23×C716yes—
254C23×C4—26no—
255AGL(1,ℤ/8)C2×C4×C7148yescomplemented
256#135C2×C4×C7124yescomplemented
257C2×Q16C23×C711yescharacteristic
258C2×SD16C23×C716yes—
259C2×D8C2×C4×C7124yescomplemented
260C2×D8C23×C713yes—
261C2×D8C23×C716yes—
262#131—24no—
263#131C2×C4×C718yescomplemented
264C2×M16C23×C713yes—
265C2×M16C23×C713yes—
266C22×C8C23×C713yes—
267C22×C8C23×C713yes—
268C4×Q8—22no—
269C4×Q8C7×D412yes—
270C4×Q8C23×C712yes—
271C4×Q8C23×C716yes—
272C4×D4—26no—
273C4×D4—26no—
274C4×D4—26no—
275C4×D4—26no—
276C4×D4—26no—
277C4×D4—212no—
278C4×D4—26no—
279C4×D4—26no—
280C4×D4C7×D416yes—
281C4×D4C23×C716yes—
282C4×D4C23×C7112yes—
283C4×D4C23×C716yes—
284#117—26no—
285#117—26no—
286#117C23×C716yes—
287#117C23×C713yes—
288C2×C4⋊C4—26no—
289C2×C4⋊C4C23×C716yes—
290C2×C22⋊C4—26no—
291C2×C22⋊C4—212no—
292C2×C42—26no—
293C2×C42C23×C713yes—
294#107C23×C716yes—
295#106C23×C713yes—
296#106C23×C716yes—
297#106C23×C711yescharacteristic
298#103C23×C712yes—
299#103C23×C716yes—
300#102C23×C716yes—
301#102C23×C7112yes—
302#102C23×C716yes—
303#97C23×C711yescharacteristic
304#97C23×C713yes—
305#97C23×C713yes—
306#97C23×C713yes—
307C4×C8C23×C716yes—
308C22×C7—26no—
309C22×C7—212no—
310C22×C7—46no—
311C22×C7—46no—
312C22×C7—824no—
313C22×C7C2×C4×D416yes—
314C22×C7C23×D411yescharacteristic
315C4×C7—22no—
316C4×C7—26no—
317C4×C7—24no—
318C4×C7—46no—
319C4×C7—42no—
320C4×C7C2×C4×D416yes—
321C4×C7C24×C411yescharacteristic
322C4×C7C22×#4111yescharacteristic
323C24—26no—
324#41—216no—
325#41—212no—
326#41C22×C4×C714yes—
327#41C22×C4×C7124yescomplemented
328#41C7×#4114yes—
329C2×Q8C2×C7×D411yescharacteristic
330C2×Q8C24×C712yes—
331C2×D4—26no—
332C2×D4—248no—
333C2×D4—26no—
334C2×D4—212no—
335C2×D4C22×C4×C7124yescomplemented
336C2×D4C2×C7×D413yes—
337C2×D4C24×C716yes—
338C22×C4—26no—
339C22×C4—26no—
340C22×C4—26no—
341C22×C4—26no—
342C22×C4—212no—
343C22×C4—812no—
344C22×C4C2×C7×D416yes—
345C22×C4C24×C713yes—
346Q16C22×C4×C714yescomplemented
347SD16C22×C4×C7124yescomplemented
348D8C22×C4×C7112yescomplemented
349D8C22×C4×C7124yescomplemented
350M16—26no—
351M16C22×C4×C7112yescomplemented
352C2×C8—26no—
353C2×C8C22×C4×C7112yescomplemented
354C2×C8C24×C713yes—
355C2×C8C24×C711yescharacteristic
356C2×C8C24×C713yes—
357C2×C8C24×C711yescharacteristic
358C4⋊C4—26no—
359C4⋊C4—26no—
360C4⋊C4—22no—
361C4⋊C4C2×C7×D416yes—
362C4⋊C4C24×C716yes—
363C4⋊C4C24×C712yes—
364C22⋊C4—26no—
365C22⋊C4—26no—
366C22⋊C4—26no—
367C22⋊C4—212no—
368C22⋊C4C2×C7×D416yes—
369C42—22no—
370C42—26no—
371C42—26no—
372C42C2×C7×D416yes—
373C42C24×C713yes—
374C42C24×C711yescharacteristic
375C2×C7—26no—
376C2×C7—412no—
377C2×C7C22×C4×D411yescharacteristic
378C2×C7#358512yes—
379C23—26no—
380C23—224no—
381C23—86no—
382C23—86no—
383C23C22×C7×D413yes—
384Q8—22no—
385Q8C4×C7×D414yes—
386Q8C23×C4×C714yes—
387D4—224no—
388D4—26no—
389D4—224no—
390D4C4×C7×D4112yes—
391D4C23×C4×C7112yes—
392C2×C4—22no—
393C2×C4—224no—
394C2×C4—22no—
395C2×C4—26no—
396C2×C4—46no—
397C2×C4—412no—
398C2×C4—86no—
399C2×C4—812no—
400C2×C4—86no—
401C2×C4C23×C4×C716yes—
402C2×C4C22×C7×D412yes—
403C2×C4C22×C7×D416yes—
404C2×C4C22×C7×D413yes—
405C2×C4C2×C7×#4116yes—
406C2×C4C25×C711yescharacteristic
407C8—21no—
408C8—23no—
409C8C23×C4×C712yes—
410C8C23×C4×C716yes—
411C7#6017611yescharacteristic, direct product factor, Sylow
412C22—26no—
413C22—212no—
414C22—46no—
415C22—46no—
416C22—824no—
417C22C2×C4×C7×D416yes—
418C22C23×C7×D411yescharacteristic
419C4—22no—
420C4—26no—
421C4—24no—
422C4—46no—
423C4—42no—
424C4C2×C4×C7×D416yes—
425C4C24×C4×C711yescharacteristic
426C4C22×C7×#4111yescharacteristic
427C2—26no—
428C2—412no—
429C2C22×C4×C7×D411yescharacteristic
430C2C7×#358512yes—
subgroupG/H
center, socleC22×C7C23×D4
commutatorC4C24×C4×C7
Frattini, cosocleC2×C4C25×C7
prime radicalC2×C7×#576C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C7×H(5,𝔽2)C2semidirect
3. C7×#2952C2semidirect
4. C7×#2948C2semidirect
5. C7×#2946C2semidirect
6. C4×C7×AGL(1,ℤ/8)C2semidirect
7. C7×#2943C2semidirect
8. C2×C7×#442C2semidirect
9. C7×#2912C2semidirect
10. C7×#2897C2semidirect
11. C7×#2875C2semidirect
13. C7×#576C4semidirect
21. C7×#520C22semidirect
23. C7×#518C22semidirect
26. C4×C7×#41C22semidirect
28. C2×C4×C7×D4C22semidirect
29. C7×#442C22semidirect
30. C7×#442C22semidirect
31. C7×#442C22semidirect
32. C7×#442C22semidirect
33. C7×#442C22semidirect
34. C7×#441C22semidirect
35. C7×#441C22semidirect
36. C7×#440C22semidirect
37. C7×#440C22semidirect
38. C4×C7×SD16C22semidirect
39. C4×C7×D8C22semidirect
40. C7×#428C22semidirect
41. C7×#427C22semidirect
42. C2×C7×#106C22semidirect
43. C7×#418C22semidirect
44. C7×#417C22semidirect
45. C7×#416C22semidirect
46. C2×C7×#102C22semidirect
47. C7×#405C22semidirect
48. C7×#405C22semidirect
49. C4×C7×M16C22semidirect
50. C2×C7×#97C22semidirect
51. #60176C7direct (N × H)
53. C7×H(5,𝔽2)C2×C4semidirect
60. C7×AGL(1,ℤ/8)C2×C4semidirect
61. C7×#135C2×C4semidirect
64. C2×C7×D8C2×C4semidirect
68. C7×#131C2×C4semidirect
114. C4×H(5,𝔽2)C2×C7semidirect
115. #2952C2×C7semidirect
116. #2948C2×C7semidirect
117. #2946C2×C7semidirect
118. C4×AGL(1,ℤ/8)C2×C7semidirect
119. #2943C2×C7semidirect
120. C2×#442C2×C7semidirect
121. #2912C2×C7semidirect
122. #2897C2×C7semidirect
123. #2875C2×C7semidirect
128. C7×#41C22×C4semidirect
136. C2×C7×D4C22×C4semidirect
147. C7×Q16C22×C4semidirect
148. C7×SD16C22×C4semidirect
149. C7×D8C22×C4semidirect
150. C7×D8C22×C4semidirect
152. C7×M16C22×C4semidirect
154. C2×C7×C8C22×C4semidirect
177. #576C4×C7semidirect
185. #520C22×C7semidirect
187. #518C22×C7semidirect
190. C4×#41C22×C7semidirect
192. C2×C4×D4C22×C7semidirect
193. #442C22×C7semidirect
194. #442C22×C7semidirect
195. #442C22×C7semidirect
196. #442C22×C7semidirect
197. #442C22×C7semidirect
198. #441C22×C7semidirect
199. #441C22×C7semidirect
200. #440C22×C7semidirect
201. #440C22×C7semidirect
202. C4×SD16C22×C7semidirect
203. C4×D8C22×C7semidirect
204. #428C22×C7semidirect
205. #427C22×C7semidirect
206. C2×#106C22×C7semidirect
207. #418C22×C7semidirect
208. #417C22×C7semidirect
209. #416C22×C7semidirect
210. C2×#102C22×C7semidirect
211. #405C22×C7semidirect
212. #405C22×C7semidirect
213. C4×M16C22×C7semidirect
214. C2×#97C22×C7semidirect
248. H(5,𝔽2)C2×C4×C7semidirect
255. AGL(1,ℤ/8)C2×C4×C7semidirect
256. #135C2×C4×C7semidirect
259. C2×D8C2×C4×C7semidirect
263. #131C2×C4×C7semidirect
327. #41C22×C4×C7semidirect
335. C2×D4C22×C4×C7semidirect
346. Q16C22×C4×C7semidirect
347. SD16C22×C4×C7semidirect
348. D8C22×C4×C7semidirect
349. D8C22×C4×C7semidirect
351. M16C22×C4×C7semidirect
353. C2×C8C22×C4×C7semidirect
411. C7#60176direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5762
C4×C7×H(5,𝔽2)2
C7×#29522
C7×#29482
C7×#29462
C4×C7×AGL(1,ℤ/8)2
C7×#29432
C2×C7×#4422
C7×#29122
C7×#28972
C7×#28752
#601767