Finite Group Databasebeta

← C7×#10591510 #72246840

Subgroups — C7×#10591510 #72246840

Order 4032.

780 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#833C211yesmaximal, characteristic, complemented
2C7×#63568C212yesmaximal, complemented
3C3×C7×#1004C211yesmaximal, characteristic, complemented
4C7×#64149C211yesmaximal, characteristic, complemented
5C7×#63825C212yesmaximal, complemented
6C7×#5241—31nomaximal
7C7×#4747—31nomaximal
8C2×C32×C7×D4C2211yescharacteristic
9C2×C7×#638—22no—
10C2×C7×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
11C2×C3×C7×#67—22no—
12C3×C7×#237—22no—
13C3×D3×C7×D4—22no—
14C3×C7×#235C2212yes—
15C4×C7×C3⋊Dic3C2211yescharacteristic, complemented
16C3×C7×#208C2212yescomplemented
17C7×#3717—22no—
18C7×#3714—22no—
19C3×C7×H(5,𝔽2)—31no—
20C7×#1004D311yescharacteristic, complemented
21C7×#949—32no—
22C7×#940—31no—
23C7×#940—31no—
24C7×#940—61no—
25C7×#908—32no—
26C3×C7×#100—31no—
27C7×#837—32no—
28C7×#833D311yescharacteristic, complemented
29C7×#825—32no—
30#10591510C711yesmaximal, characteristic, direct product factor, Hall
31C23×C32×C7D412yes—
32C22×C7×C3⋊D3—21no—
33C22×C3×D3×C7—22no—
34C32×C7×D4—22no—
35C2×C32×C4×C7C2311yescharacteristic
36C7×#638—44no—
37C2×C7×C3⋊Dic3—21no—
38C7×C4⋊(C3⋊D3)—22no—
39C3×C7×#67—24no—
40C3×C7×#67—22no—
41C2×C3×C7×Dic3—22no—
42C3×C7×D12D412yescomplemented
43C3×C4×D3×C7—22no—
44C3×C7×Dic6D412yescomplemented
45C3×C7×C3⋊C8—22no—
46C7×#453—91noHall
47C3×C7×#41—32no—
48C3×C7×#41—62no—
49C2×C3×C7×D4—21no—
50C2×C3×C7×D4—62no—
51C2×C3×C7×D4—62no—
52C2×C3×C7×D4C2×D311yescharacteristic
53C2×C3×C7×D4C2×D311yescharacteristic
54C2×C7×#67—22no—
55C2×C7×#67—62no—
56C2×C7×#67—62no—
57C2×C7×#67—62no—
58C2×C7×#67—62no—
59C7×#237—22no—
60D3×C7×D4—22no—
61C7×#235C2×D312yes—
62C2×C7×D12—31no—
63C2×C7×D12—31no—
64C2×C7×D12—61no—
65C3×C7×M16—32no—
66C7×#215—62no—
67C7×#213—62no—
68C4×C7×Dic3—31no—
69C4×C7×Dic3—31no—
70C4×C7×Dic3—61no—
71C7×#208C2×D312yescomplemented
72C7×D24—62no—
73C7×#204—62no—
74C3×#1004C2×C711yescharacteristic, complemented
75#64149C2×C711yescharacteristic, complemented
76#63825C2×C712yescomplemented
77C3×#833C2×C711yescharacteristic, complemented
78#63568C2×C712yescomplemented
79C22×C32×C7—22no—
80C22×C32×C7—42no—
81C22×C32×C7C2×D411yescharacteristic
82C2×C7×C3⋊D3—42no—
83C2×C7×C3⋊D3—41no—
84C2×C3×D3×C7—22no—
85C2×C3×D3×C7—82no—
86C32×C4×C7C2×D412yes—
87C7×C3⋊Dic3—22no—
88C3×C7×Dic3—22no—
89C7×H(5,𝔽2)—31no—
90C7×AGL(1,ℤ/8)—92no—
91C7×#127—91no—
92C7×#104—92no—
93C7×#100—31no—
94#5241—31no—
95#4747—31no—
96C23×C3×C7—22no—
97C23×C3×C7—62no—
98C23×C3×C7D1212yes—
99C23×C3×C7#6712yes—
100C22×D3×C7—22no—
101C22×D3×C7—61no—
102C22×D3×C7—61no—
103C22×D3×C7—61no—
104C22×D3×C7—61no—
105C3×C7×Q8—32no—
106C3×C7×D4—22no—
107C3×C7×D4—22no—
108C3×C7×D4—32no—
109C3×C7×D4—42no—
110C3×C7×D4—64no—
111C3×C7×D4—62no—
112C2×C3×C4×C7—21no—
113C2×C3×C4×C7—62no—
114C2×C3×C4×C7—62no—
115C2×C3×C4×C7C22×D311yescharacteristic
116C2×C3×C4×C7C22×D311yescharacteristic
117C7×#67—24no—
118C7×#67—22no—
119C7×#67—124no—
120C7×#67—124no—
121C7×#67—124no—
122C7×#67—124no—
123C2×C7×Dic3—22no—
124C2×C7×Dic3—61no—
125C2×C7×Dic3—61no—
126C2×C7×Dic3—61no—
127C2×C7×Dic3—61no—
128C7×D12—62no—
129C7×D12—62no—
130C7×D12—122no—
131C7×D12#6712yescomplemented
132C4×D3×C7—22no—
133C7×Dic6#6712yescomplemented
134C3×C7×C8—62no—
135C7×C3⋊C8—22no—
136C2×C32×D4C22×C711yescharacteristic
137C2×#638—22no—
138C2×C4⋊(C3⋊D3)C22×C711yescharacteristic, complemented
139C2×C3×#67—22no—
140C3×#237—22no—
141C3×D3×D4—22no—
142C3×#235C22×C712yes—
143C4×C3⋊Dic3C22×C711yescharacteristic, complemented
144C3×#208C22×C712yescomplemented
145#3717—22no—
146#3714—22no—
147C2×C32×C7—21no—
148C2×C32×C7—42no—
149C2×C32×C7#12011yescharacteristic
150C7×C3⋊D3—81no—
151C3×D3×C7—42no—
152C7×#41—32no—
153C7×#41—62no—
154C2×C7×D4—62no—
155C2×C7×D4—62no—
156C2×C7×D4—91no—
157C2×C7×D4—182no—
158C2×C7×D4D3211yescharacteristic
159C7×SD16—182no—
160C7×D8—182no—
161C7×M16—32no—
162C42×C7—91no—
163C3×H(5,𝔽2)—31no—
164#1004D3×C711yescharacteristic, complemented
165#949—32no—
166#940—31no—
167#940—31no—
168#940—61no—
169#908—32no—
170C3×#100—31no—
171#837—32no—
172#833D3×C711yescharacteristic, complemented
173#825—32no—
174C22×C3×C7—22no—
175C22×C3×C7—22no—
176C22×C3×C7—21no—
177C22×C3×C7—42no—
178C22×C3×C7—42no—
179C22×C3×C7—42no—
180C22×C3×C7—42no—
181C22×C3×C7—42no—
182C22×C3×C7—62no—
183C22×C3×C7—242no—
184C22×C3×C7C2×D1211yescharacteristic
185C22×C3×C7C2×#6711yescharacteristic
186C2×D3×C7—22no—
187C2×D3×C7—82no—
188C2×D3×C7—121no—
189C2×D3×C7—122no—
190C2×D3×C7—122no—
191C2×D3×C7—122no—
192C2×D3×C7—122no—
193C2×D3×C7—121no—
194C2×D3×C7—121no—
195C2×D3×C7—121no—
196C3×C4×C7—22no—
197C3×C4×C7—62no—
198C3×C4×C7D3×D412yes—
199C3×C4×C7C2×#6712yes—
200C7×Dic3—22no—
201C7×Dic3—62no—
202C7×Dic3—62no—
203C7×Dic3—62no—
204C7×Dic3—62no—
205C23×C32C7×D412yes—
206C22×C3⋊D3—21no—
207C22×C3×D3—22no—
208C32×D4—22no—
209C2×C32×C4C23×C711yescharacteristic
210#638—44no—
211C2×C3⋊Dic3—21no—
212C4⋊(C3⋊D3)—22no—
213C3×#67—24no—
214C3×#67—22no—
215C2×C3×Dic3—22no—
216C3×D12C7×D412yescomplemented
217C3×C4×D3—22no—
218C3×Dic6C7×D412yescomplemented
219C3×C3⋊C8—22no—
220#453—91noSylow
221C32×C7#45311yescharacteristic, complemented, Hall
222C23×C7—62no—
223C23×C7—181no—
224C23×C7#62612yes—
225C7×Q8—32no—
226C7×D4—22no—
227C7×D4—32no—
228C7×D4—64no—
229C7×D4—62no—
230C7×D4—182no—
231C7×D4—364no—
232C2×C4×C7—62no—
233C2×C4×C7—62no—
234C2×C4×C7—181no—
235C2×C4×C7C2×D3211yescharacteristic
236C7×C8—62no—
237C3×#41—32no—
238C3×#41—62no—
239C2×C3×D4—21no—
240C2×C3×D4—62no—
241C2×C3×D4—62no—
242C2×C3×D4C2×D3×C711yescharacteristic
243C2×C3×D4C2×D3×C711yescharacteristic
244C2×#67—22no—
245C2×#67—62no—
246C2×#67—62no—
247C2×#67—62no—
248C2×#67—62no—
249#237—22no—
250D3×D4—22no—
251#235C2×D3×C712yes—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—61no—
255C3×M16—32no—
256#215—62no—
257#213—62no—
258C4×Dic3—31no—
259C4×Dic3—31no—
260C4×Dic3—61no—
261#208C2×D3×C712yescomplemented
262D24—62no—
263#204—62no—
264C2×C3×C7—21no—
265C2×C3×C7—21no—
266C2×C3×C7—21no—
267C2×C3×C7—21no—
268C2×C3×C7—21no—
269C2×C3×C7—42no—
270C2×C3×C7—42no—
271C2×C3×C7—82no—
272C2×C3×C7—122no—
273C2×C3×C7#88211yescharacteristic
274C2×C3×C7#95311yescharacteristic
275D3×C7—42no—
276D3×C7—241no—
277D3×C7—241no—
278D3×C7—241no—
279D3×C7—241no—
280C22×C32—22no—
281C22×C32—42no—
282C22×C32C2×C7×D411yescharacteristic
283C2×C3⋊D3—42no—
284C2×C3⋊D3—41no—
285C2×C3×D3—22no—
286C2×C3×D3—82no—
287C32×C4C2×C7×D412yes—
288C3⋊Dic3—22no—
289C3×Dic3—22no—
290H(5,𝔽2)—31no—
291AGL(1,ℤ/8)—92no—
292#127—91no—
293#104—92no—
294#100—31no—
295C22×C7—22no—
296C22×C7—42no—
297C22×C7—62no—
298C22×C7—242no—
299C22×C7—362no—
300C22×C7—361no—
301C22×C7C2×#62611yescharacteristic
302C4×C7—62no—
303C4×C7—182no—
304C4×C7D3×#6712yes—
305C23×C3—22no—
306C23×C3—62no—
307C23×C3C7×D1212yes—
308C23×C3C7×#6712yes—
309C22×D3—22no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—61no—
314C3×Q8—32no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—32no—
318C3×D4—42no—
319C3×D4—64no—
320C3×D4—62no—
321C2×C3×C4—21no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4C22×D3×C711yescharacteristic
325C2×C3×C4C22×D3×C711yescharacteristic
326#67—24no—
327#67—22no—
328#67—124no—
329#67—124no—
330#67—124no—
331#67—124no—
332C2×Dic3—22no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337D12—62no—
338D12—62no—
339D12—122no—
340D12C7×#6712yescomplemented
341C4×D3—22no—
342Dic6C7×#6712yescomplemented
343C3×C8—62no—
344C3⋊C8—22no—
345C3×C7—21no—
346C3×C7#474711yescharacteristic, complemented
347C3×C7#524111yescharacteristic, complemented
348C2×C32—21no—
349C2×C32—42no—
350C2×C32C7×#12011yescharacteristic
351C3⋊D3—81no—
352C3×D3—42no—
353#41—32no—
354#41—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—91no—
358C2×D4—182no—
359C2×D4D32×C711yescharacteristic
360SD16—182no—
361D8—182no—
362M16—32no—
363C42—91no—
364C2×C7—21no—
365C2×C7—42no—
366C2×C7—122no—
367C2×C7—721no—
368C2×C7#6397711yescharacteristic
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—62no—
378C22×C3—242no—
379C22×C3C2×C7×D1211yescharacteristic
380C22×C3C2×C7×#6711yescharacteristic
381C2×D3—22no—
382C2×D3—82no—
383C2×D3—121no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—122no—
387C2×D3—122no—
388C2×D3—121no—
389C2×D3—121no—
390C2×D3—121no—
391C3×C4—22no—
392C3×C4—62no—
393C3×C4D3×C7×D412yes—
394C3×C4C2×C7×#6712yes—
395Dic3—22no—
396Dic3—62no—
397Dic3—62no—
398Dic3—62no—
399Dic3—62no—
400C32C7×#45311yescharacteristic, complemented, Sylow
401C23—62no—
402C23—181no—
403C23C7×#62612yes—
404Q8—32no—
405D4—22no—
406D4—32no—
407D4—64no—
408D4—62no—
409D4—182no—
410D4—364no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—181no—
414C2×C4C2×D32×C711yescharacteristic
415C8—62no—
416C7#1059151011yescharacteristic, direct product factor, Sylow
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—42no—
423C2×C3—42no—
424C2×C3—82no—
425C2×C3—122no—
426C2×C3C7×#88211yescharacteristic
427C2×C3C7×#95311yescharacteristic
428D3—42no—
429D3—241no—
430D3—241no—
431D3—241no—
432D3—241no—
433C22—22no—
434C22—42no—
435C22—62no—
436C22—242no—
437C22—362no—
438C22—361no—
439C22C2×C7×#62611yescharacteristic
440C4—62no—
441C4—182no—
442C4D3×C7×#6712yes—
443C3—21no—
444C3C7×#474711yescharacteristic, complemented
445C3C7×#524111yescharacteristic, complemented
446C2—21no—
447C2—42no—
448C2—122no—
449C2—721no—
450C2C7×#6397711yescharacteristic
subgroupG/H
centerC2×C7#63977
commutator, cosocleC2×C32×C4C23×C7
FrattiniC2×C4C2×D32×C7
FittingC2×C32×C7×D4C22
socleC2×C32×C7#120
hypercenterC2×C7×D4D32
hypocenterC32C7×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×D4

Complements

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

NH ≅ G/Ntype
1. C3×C7×#833C2semidirect
2. C7×#63568C2semidirect
3. C3×C7×#1004C2semidirect
4. C7×#64149C2semidirect
5. C7×#63825C2semidirect
10. C2×C7×C4⋊(C3⋊D3)C22semidirect
15. C4×C7×C3⋊Dic3C22semidirect
16. C3×C7×#208C22semidirect
20. C7×#1004D3semidirect
28. C7×#833D3semidirect
30. #10591510C7direct (N × H)
42. C3×C7×D12D4semidirect
44. C3×C7×Dic6D4semidirect
71. C7×#208C2×D3semidirect
74. C3×#1004C2×C7semidirect
75. #64149C2×C7semidirect
76. #63825C2×C7semidirect
77. C3×#833C2×C7semidirect
78. #63568C2×C7semidirect
131. C7×D12#67semidirect
133. C7×Dic6#67semidirect
138. C2×C4⋊(C3⋊D3)C22×C7semidirect
143. C4×C3⋊Dic3C22×C7semidirect
144. C3×#208C22×C7semidirect
164. #1004D3×C7semidirect
172. #833D3×C7semidirect
216. C3×D12C7×D4semidirect
218. C3×Dic6C7×D4semidirect
221. C32×C7#453semidirect
261. #208C2×D3×C7semidirect
340. D12C7×#67semidirect
342. Dic6C7×#67semidirect
346. C3×C7#4747semidirect
347. C3×C7#5241semidirect
400. C32C7×#453semidirect
416. C7#10591510direct (N × H)
444. C3C7×#4747semidirect
445. C3C7×#5241semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#8332
C7×#635682
C3×C7×#10042
C7×#641492
C7×#638252
C7×#52413
C7×#47473
#105915107