Finite Group Databasebeta

← C7×#10240 #13187456

Subgroups — C7×#10240 #13187456

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3206C211yesmaximal, characteristic, complemented
2C7×#3036C211yesmaximal, characteristic
3C7×#2542C211yesmaximal, characteristic, complemented
4C7×#1548C211yesmaximal, characteristic, complemented
5C7×#1502C211yesmaximal, characteristic, complemented
6C7×#1478C211yesmaximal, characteristic, complemented
7C7×#1303C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—21no—
9C7×#532C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C22⋊Q8—22no—
12C2×C4×C7×D4C2211yescharacteristic
13C7×#492—21no—
14C7×#487—21no—
15C7×#463—22no—
16C7×#461—21no—
17C7×#461—22no—
18C7×#459—21no—
19C7×#448—22no—
20C7×#431C2211yescharacteristic
21C7×#431C2211yescharacteristic
22C7×#423C2211yescharacteristic
23C7×#417—21no—
24C7×#399—22no—
25C7×#390—21no—
26C7×#384—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C7×#333—22no—
29C7×#327—22no—
30C7×D4⋊C8—22no—
31#10240C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×Q8D412yes—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7—21no—
36C7×AGL(1,ℤ/8)—42no—
37C7×AGL(1,ℤ/8)—42no—
38C2×C7×SD16—22no—
39C2×C7×SD16—24no—
40C2×C7×D8—22no—
41C2×C7×M16—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8D411yescharacteristic
47C7×#127D411yescharacteristic
48C7×#124—21no—
49C7×C22⋊Q8—48no—
50C7×#121—21no—
51C7×#121—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—21no—
65C2×C42×C7—21no—
66C2×C42×C7C2311yescharacteristic
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8D412yes—
70C7×C4⋊C8D412yes—
71C7×#103—24no—
72C7×#103—22no—
73C7×#102—22no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8—21no—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78C7×#95D412yes—
79#3206C2×C711yescharacteristic, complemented
80#3036C2×C711yescharacteristic
81#2542C2×C711yescharacteristic, complemented
82#1548C2×C711yescharacteristic, complemented
83#1502C2×C711yescharacteristic, complemented
84#1478C2×C711yescharacteristic, complemented
85#1303C2×C711yescharacteristic, complemented
86C24×C7—21no—
87C7×#41—41no—
88C7×#41—41no—
89C7×#41—41no—
90C7×#41—41no—
91C2×C7×Q8—21no—
92C2×C7×Q8—22no—
93C2×C7×Q8—22no—
94C2×C7×Q8—44no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—41no—
104C2×C7×D4SD1612yescomplemented
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—44no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7—41no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C7×SD16—44no—
116C7×SD16—84no—
117C7×D8—44no—
118C7×M16—41no—
119C7×M16—41no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—41no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—48no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C4⋊C4—41no—
133C7×C22⋊C4—48no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C7×C22⋊C4—41no—
138C42×C7—41no—
139C42×C7—41no—
140C42×C7C2×D411yescharacteristic
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7#4111yescharacteristic
144C2×AGL(1,ℤ/8)—21no—
145#532C22×C711yescharacteristic
146C2×C4⋊Q8C22×C711yescharacteristic
147C2×C22⋊Q8—22no—
148C2×C4×D4C22×C711yescharacteristic
149#492—21no—
150#487—21no—
151#463—22no—
152#461—21no—
153#461—22no—
154#459—21no—
155#448—22no—
156#431C22×C711yescharacteristic
157#431C22×C711yescharacteristic
158#423C22×C711yescharacteristic
159#417—21no—
160#399—22no—
161#390—21no—
162#384—21no—
163C4×C4⋊C4C22×C711yescharacteristic
164#333—22no—
165#327—22no—
166D4⋊C8—22no—
167C23×C7—22no—
168C23×C7—42no—
169C23×C7—42no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—41no—
173C23×C7#12011yescharacteristic
174C7×Q8—41no—
175C7×Q8—41no—
176C7×Q8—44no—
177C7×Q8—82no—
178C7×D4—22no—
179C7×D4—22no—
180C7×D4—41no—
181C7×D4—41no—
182C7×D4—41no—
183C7×D4—42no—
184C7×D4—41no—
185C7×D4—81no—
186C7×D4—81no—
187C7×D4—81no—
188C2×C4×C7—22no—
189C2×C4×C7—24no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—44no—
194C2×C4×C7—42no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—42no—
198C2×C4×C7—82no—
199C2×C4×C7—82no—
200C2×C4×C7—81no—
201C2×C4×C7—81no—
202C2×C4×C7#12011yescharacteristic
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C2×C4×C7#12111yescharacteristic
206C2×C4×C7#12111yescharacteristic
207C2×C4×C7#12711yescharacteristic
208C2×C4×C7C2×SD1611yescharacteristic
209C2×C4×C7#13511yescharacteristic
210C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C7#13711yescharacteristic
212C7×C8—42no—
213C7×C8—42no—
214C7×C8—81no—
215C2×#41—21no—
216C22×Q8C7×D412yes—
217C22×D4C7×D411yescharacteristic
218C23×C4—21no—
219AGL(1,ℤ/8)—42no—
220AGL(1,ℤ/8)—42no—
221C2×SD16—22no—
222C2×SD16—24no—
223C2×D8—22no—
224C2×M16—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C7×D411yescharacteristic
230#127C7×D411yescharacteristic
231#124—21no—
232C22⋊Q8—48no—
233#121—21no—
234#121—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C7×D411yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C42—21no—
249C2×C42C23×C711yescharacteristic
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8C7×D412yes—
253C4⋊C8C7×D412yes—
254#103—22no—
255#103—24no—
256#102—22no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261#95C7×D412yes—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—44no—
265C22×C7—42no—
266C22×C7—82no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7#39211yescharacteristic
273C22×C7#44811yescharacteristic
274C22×C7#44911yescharacteristic
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—44no—
279C4×C7—81no—
280C4×C7—81no—
281C4×C7—82no—
282C4×C7#45011yescharacteristic
283C4×C7#45211yescharacteristic
284C4×C7#45312yes—
285C4×C7#46511yescharacteristic
286C4×C7#46811yescharacteristic
287C24—21no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4C7×SD1612yescomplemented
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×C7×D411yescharacteristic
314C22×C4C2×C7×D411yescharacteristic
315C22×C4C2×C7×D411yescharacteristic
316SD16—44no—
317SD16—84no—
318D8—44no—
319M16—41no—
320M16—41no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C4⋊C4—24no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—48no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C22⋊C4—48no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—41no—
340C42—41no—
341C42C2×C7×D411yescharacteristic
342C42C2×C7×D411yescharacteristic
343C42C2×C7×D411yescharacteristic
344C42C7×#4111yescharacteristic
345C2×C7—21no—
346C2×C7—21no—
347C2×C7—82no—
348C2×C7—161no—
349C2×C7#200011yescharacteristic
350C2×C7#200211yescharacteristic
351C2×C7#200911yescharacteristic
352C23—22no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C7×#12011yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—44no—
362Q8—82no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—81no—
371D4—81no—
372D4—81no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4C7×#12011yescharacteristic
388C2×C4C7×#12011yescharacteristic
389C2×C4C7×#12111yescharacteristic
390C2×C4C7×#12111yescharacteristic
391C2×C4C7×#12111yescharacteristic
392C2×C4C7×#12711yescharacteristic
393C2×C4C2×C7×SD1611yescharacteristic
394C2×C4C7×#13511yescharacteristic
395C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C7×#13711yescharacteristic
397C8—42no—
398C8—42no—
399C8—81no—
400C7#1024011yescharacteristic, direct product factor, Sylow
401C22—21no—
402C22—21no—
403C22—44no—
404C22—42no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C7×#39211yescharacteristic
412C22C7×#44811yescharacteristic
413C22C7×#44911yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—81no—
419C4—81no—
420C4—82no—
421C4C7×#45011yescharacteristic
422C4C7×#45211yescharacteristic
423C4C7×#45312yes—
424C4C7×#46511yescharacteristic
425C4C7×#46811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2C7×#200011yescharacteristic
431C2C7×#200211yescharacteristic
432C2C7×#200911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3206C2semidirect
3. C7×#2542C2semidirect
4. C7×#1548C2semidirect
5. C7×#1502C2semidirect
6. C7×#1478C2semidirect
7. C7×#1303C2semidirect
31. #10240C7direct (N × H)
79. #3206C2×C7semidirect
81. #2542C2×C7semidirect
82. #1548C2×C7semidirect
83. #1502C2×C7semidirect
84. #1478C2×C7semidirect
85. #1303C2×C7semidirect
104. C2×C7×D4SD16semidirect
305. C2×D4C7×SD16semidirect
400. C7#10240direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32062
C7×#30362
C7×#25422
C7×#15482
C7×#15022
C7×#14782
C7×#13032
#102407