Finite Group Databasebeta

← C7×#18338 #13195554

Subgroups — C7×#18338 #13195554

Order 1792.

928 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3461C211yesmaximal, characteristic
2C7×#3218C212yesmaximal, complemented
3C7×#2567C211yesmaximal, characteristic, complemented
4C7×#1605C212yesmaximal, complemented
5C7×#1604C212yesmaximal
6C7×#1603C214yesmaximal, complemented
7C7×#1509C212yesmaximal, complemented
8C7×#1467C211yesmaximal, characteristic, complemented
9C7×#544C2211yescharacteristic
10C7×#542—22no—
11C7×#541—22no—
12C7×#539—21no—
13C7×#539C2212yes—
14C7×#537C2211yescharacteristic
15C7×#536—21no—
16C7×#533—22no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C7×#529—22no—
19C7×#529—21no—
20C7×#529C2212yes—
21C4×C7×#41—21no—
22C2×C4×C7×Q8C2211yescharacteristic
23C7×#497—22no—
24C7×#491—22no—
25C7×#462—22no—
26C7×#461—22no—
27C7×#451C2214yes—
28C7×#448C2214yes—
29C7×#432C2212yes—
30C7×#395C2212yes—
31C7×#395C2212yes—
32C7×#391—21no—
33C7×#389—21no—
34C7×#389—21no—
35C7×#384—21no—
36C4×C7×C4⋊C4C2211yescharacteristic
37C7×#328C2214yes—
38C7×#327C2214yes—
39C7×#324C2212yes—
40C7×#324C2212yes—
41#18338C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8D411yescharacteristic
44C22×C7×Q8C2312yes—
45C2×C7×Q16—24no—
46C2×C7×SD16—24no—
47C7×C4⋊Q8—22no—
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8—22no—
51C7×#126—22no—
52C7×#126—22no—
53C7×#126—22no—
54C7×#126—22no—
55C7×#125—21no—
56C7×#125—22no—
57C7×#125—22no—
58C7×#125—22no—
59C7×#125D412yes—
60C7×#124—22no—
61C7×#124—21no—
62C7×#124D412yes—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—22no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8C2312yes—
74C7×#121—22no—
75C7×#121—21no—
76C7×#121C2312yes—
77C4×C7×Q8—21no—
78C4×C7×Q8—21no—
79C4×C7×Q8—21no—
80C4×C7×Q8—22no—
81C4×C7×Q8—22no—
82C4×C7×Q8D412yes—
83C4×C7×D4—21no—
84C4×C7×D4—21no—
85C4×C7×D4—21no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4D412yes—
89C7×#117—22no—
90C7×#117—22no—
91C7×#117—21no—
92C7×#117—22no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4—21no—
96C2×C7×C4⋊C4—22no—
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4D411yescharacteristic
99C2×C7×C4⋊C4C2311yescharacteristic
100C2×C7×C4⋊C4C2311yescharacteristic
101C2×C42×C7—22no—
102C2×C42×C7—21no—
103C2×C42×C7D411yescharacteristic
104C2×C42×C7D411yescharacteristic
105C2×C42×C7C2311yescharacteristic
106C7×C4⋊C8—22no—
107C7×#103—24no—
108C7×#103—24no—
109C7×C22⋊C8C2314yes—
110C7×#97—22no—
111C7×#95—22no—
112C7×#95—22no—
113C7×#95C2312yes—
114#3461C2×C711yescharacteristic
115#3218C2×C712yescomplemented
116#2567C2×C711yescharacteristic, complemented
117#1605C2×C712yescomplemented
118#1604C2×C712yes—
119#1603C2×C714yescomplemented
120#1509C2×C712yescomplemented
121#1467C2×C711yescharacteristic, complemented
122C7×#41—42no—
123C7×#41—42no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—24no—
129C2×C7×Q8—41no—
130C2×C7×Q8—82no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—21no—
133C2×C7×D4—22no—
134C2×C7×D4—21no—
135C2×C7×D4C2×D412yes—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C7×Q16—84no—
149C7×SD16—84no—
150C2×C7×C8—24no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—41no—
168C7×C4⋊C4—42no—
169C7×C4⋊C4—41no—
170C7×C4⋊C4—82no—
171C7×C4⋊C4—82no—
172C7×C4⋊C4C2×D412yes—
173C7×C4⋊C4C2×D412yes—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7—21no—
186C42×C7—22no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—42no—
191C42×C7C2×D412yes—
192C42×C7C2×D412yes—
193#544C22×C711yescharacteristic
194#542—22no—
195#541—22no—
196#539—21no—
197#539C22×C712yes—
198#537C22×C711yescharacteristic
199#536—21no—
200#533—22no—
201C2×C4⋊Q8C22×C711yescharacteristic
202#529—22no—
203#529—21no—
204#529C22×C712yes—
205C4×#41—21no—
206C2×C4×Q8C22×C711yescharacteristic
207#497—22no—
208#491—22no—
209#462—22no—
210#461—22no—
211#451C22×C714yes—
212#448C22×C714yes—
213#432C22×C712yes—
214#395C22×C712yes—
215#395C22×C712yes—
216#391—21no—
217#389—21no—
218#389—21no—
219#384—21no—
220C4×C4⋊C4C22×C711yescharacteristic
221#328C22×C714yes—
222#327C22×C714yes—
223#324C22×C712yes—
224#324C22×C712yes—
225C23×C7—22no—
226C23×C7C22×D411yescharacteristic
227C7×Q8—44no—
228C7×Q8—42no—
229C7×Q8—42no—
230C7×Q8—44no—
231C7×Q8—44no—
232C7×Q8—81no—
233C7×D4—42no—
234C7×D4—42no—
235C7×D4—42no—
236C7×D4—42no—
237C7×D4—81no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—41no—
253C2×C4×C7—44no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—44no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—82no—
263C2×C4×C7#12011yescharacteristic
264C2×C4×C7#12012yes—
265C2×C4×C7#12011yescharacteristic
266C2×C4×C7#13712yes—
267C2×C4×C7C22×D412yes—
268C7×C8—44no—
269C2×#41—21no—
270C22×Q8C7×D411yescharacteristic
271C22×Q8C23×C712yes—
272C2×Q16—24no—
273C2×SD16—24no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#126—22no—
279#126—22no—
280#126—22no—
281#126—22no—
282#125—21no—
283#125—22no—
284#125—22no—
285#125—22no—
286#125C7×D412yes—
287#124—22no—
288#124—21no—
289#124C7×D412yes—
290#123—22no—
291#123—22no—
292#123—22no—
293#123—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—21no—
299C22⋊Q8—22no—
300C22⋊Q8C23×C712yes—
301#121—22no—
302#121—21no—
303#121C23×C712yes—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8C7×D412yes—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4C7×D412yes—
316#117—22no—
317#117—22no—
318#117—21no—
319#117—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4C7×D411yescharacteristic
326C2×C4⋊C4C23×C711yescharacteristic
327C2×C4⋊C4C23×C711yescharacteristic
328C2×C42—22no—
329C2×C42—21no—
330C2×C42C7×D411yescharacteristic
331C2×C42C7×D411yescharacteristic
332C2×C42C23×C711yescharacteristic
333C4⋊C8—22no—
334#103—24no—
335#103—24no—
336C22⋊C8C23×C714yes—
337#97—22no—
338#95—22no—
339#95—22no—
340#95C23×C712yes—
341C22×C7—21no—
342C22×C7—21no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7C2×#12011yescharacteristic
347C22×C7C2×#13711yescharacteristic
348C22×C7#57511yescharacteristic
349C4×C7—22no—
350C4×C7—22no—
351C4×C7—21no—
352C4×C7—22no—
353C4×C7—44no—
354C4×C7—41no—
355C4×C7—42no—
356C4×C7—42no—
357C4×C7—41no—
358C4×C7—41no—
359C4×C7—42no—
360C4×C7—82no—
361C4×C7#45612yes—
362#41—42no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—41no—
370C2×Q8—82no—
371C2×Q8C2×C7×D412yes—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4C2×C7×D412yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4C2×C7×D411yescharacteristic
382C22×C4C2×C7×D411yescharacteristic
383C22×C4C2×C7×D411yescharacteristic
384C22×C4C2×C7×D411yescharacteristic
385C22×C4C2×C7×D411yescharacteristic
386C22×C4C2×C7×D411yescharacteristic
387C22×C4C24×C711yescharacteristic
388Q16—84no—
389SD16—84no—
390C2×C8—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4C2×C7×D412yes—
413C4⋊C4C2×C7×D412yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C42—22no—
424C42—22no—
425C42—21no—
426C42—22no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—42no—
431C42C2×C7×D412yes—
432C42C2×C7×D412yes—
433C2×C7—21no—
434C2×C7—21no—
435C2×C7—82no—
436C2×C7#300711yescharacteristic
437C2×C7C2×#45611yescharacteristic
438C2×C7#302711yescharacteristic
439C23—22no—
440C23C22×C7×D411yescharacteristic
441Q8—44no—
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—44no—
446Q8—81no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—81no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4C7×#12011yescharacteristic
478C2×C4C7×#12012yes—
479C2×C4C7×#12011yescharacteristic
480C2×C4C7×#13712yes—
481C2×C4C22×C7×D412yes—
482C8—44no—
483C7#1833811yescharacteristic, direct product factor, Sylow
484C22—21no—
485C22—21no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22C2×C7×#12011yescharacteristic
490C22C2×C7×#13711yescharacteristic
491C22C7×#57511yescharacteristic
492C4—22no—
493C4—22no—
494C4—21no—
495C4—22no—
496C4—44no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—82no—
504C4C7×#45612yes—
505C2—21no—
506C2—21no—
507C2—82no—
508C2C7×#300711yescharacteristic
509C2C2×C7×#45611yescharacteristic
510C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#537C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3218C2semidirect
3. C7×#2567C2semidirect
4. C7×#1605C2semidirect
6. C7×#1603C2semidirect
7. C7×#1509C2semidirect
8. C7×#1467C2semidirect
41. #18338C7direct (N × H)
115. #3218C2×C7semidirect
116. #2567C2×C7semidirect
117. #1605C2×C7semidirect
119. #1603C2×C7semidirect
120. #1509C2×C7semidirect
121. #1467C2×C7semidirect
483. C7#18338direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34612
C7×#32182
C7×#25672
C7×#16052
C7×#16042
C7×#16032
C7×#15092
C7×#14672
#183387