Finite Group Databasebeta

← C7×#18333 #13195549

Subgroups — C7×#18333 #13195549

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3458C211yesmaximal, characteristic, complemented
2C7×#3215C212yesmaximal, complemented
3C7×#2610C211yesmaximal, characteristic
4C7×#1605C212yesmaximal, complemented
5C7×#1602C212yesmaximal, complemented
6C7×#1599C212yesmaximal, complemented
7C7×#1596C212yesmaximal
8C7×#1506C212yesmaximal, complemented
9C7×#1464C211yesmaximal, characteristic, complemented
10C7×#544—22no—
11C7×#544—22no—
12C7×#544—21no—
13C7×#544C2212yes—
14C7×#543—21no—
15C7×#543C2212yes—
16C7×#533—24no—
17C7×#533—21no—
18C7×#533C2212yes—
19C2×C7×#124C2211yescharacteristic
20C2×C7×#117—21no—
21C2×C7×#117C2211yescharacteristic
22C7×#497—22no—
23C7×#489—22no—
24C7×#464—24no—
25C7×#451C2214yes—
26C7×#450C2214yes—
27C7×#432C2212yes—
28C7×#395C2211yescharacteristic
29C7×#394C2211yescharacteristic
30C7×#393C2211yescharacteristic
31C7×#392C2211yescharacteristic
32C7×#390—21no—
33C7×#388—21no—
34C7×#388—21no—
35C7×#386—21no—
36C7×#376C2211yescharacteristic
37C7×#328C2214yes—
38C7×#328C2214yes—
39C7×#324C2212yes—
40C7×#323C2212yes—
41#18333C711yesmaximal, characteristic, direct product factor, Sylow
42C22×C7×Q8C2311yescharacteristic
43C22×C7×D4—21no—
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7D411yescharacteristic
46C2×C7×Q16—24no—
47C2×C7×SD16—24no—
48C7×C4⋊Q8—24no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—24no—
52C7×C4⋊Q8D412yescomplemented
53C7×C4⋊Q8D412yescomplemented
54C7×#125—24no—
55C7×#125—22no—
56C7×#125—24no—
57C7×#125—22no—
58C7×#124—22no—
59C7×#124—22no—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8C2314yes—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—22no—
71C4×C7×Q8D414yescomplemented
72C7×#117—22no—
73C7×#117—22no—
74C7×#117—22no—
75C7×#117—22no—
76C7×#117—22no—
77C7×#117—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4D411yescharacteristic
91C2×C7×C22⋊C4C2311yescharacteristic
92C2×C42×C7—22no—
93C2×C42×C7—21no—
94C2×C42×C7D411yescharacteristic
95C2×C42×C7C2311yescharacteristic
96C7×C4⋊C8—22no—
97C7×#103—24no—
98C7×#102—24no—
99C7×C22⋊C8C2314yes—
100C7×#97—22no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95C2311yescharacteristic
105C7×#95C2311yescharacteristic
106#3458C2×C711yescharacteristic, complemented
107#3215C2×C712yescomplemented
108#2610C2×C711yescharacteristic
109#1605C2×C712yescomplemented
110#1602C2×C712yescomplemented
111#1599C2×C712yescomplemented
112#1596C2×C712yes—
113#1506C2×C712yescomplemented
114#1464C2×C711yescharacteristic, complemented
115C24×C7—21no—
116C24×C7C2×D411yescharacteristic
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—81no—
122C2×C7×Q8C2×D414yes—
123C2×C7×D4—22no—
124C2×C7×D4—42no—
125C2×C7×D4—42no—
126C2×C7×D4—81no—
127C2×C7×D4—81no—
128C22×C4×C7—22no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C7×Q16—84no—
144C7×SD16—84no—
145C2×C7×C8—24no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—24no—
150C7×C4⋊C4—24no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—81no—
160C7×C4⋊C4C2×D414yes—
161C7×C22⋊C4—24no—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—22no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—42no—
166C7×C22⋊C4—41no—
167C7×C22⋊C4—41no—
168C7×C22⋊C4—42no—
169C7×C22⋊C4—42no—
170C7×C22⋊C4—81no—
171C7×C22⋊C4—81no—
172C42×C7—24no—
173C42×C7—24no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—42no—
177C42×C7C2×D412yes—
178C42×C7C2×D412yes—
179#544—22no—
180#544—22no—
181#544—21no—
182#544C22×C712yes—
183#543—21no—
184#543C22×C712yes—
185#533—24no—
186#533—21no—
187#533C22×C712yes—
188C2×#124C22×C711yescharacteristic
189C2×#117—21no—
190C2×#117C22×C711yescharacteristic
191#497—22no—
192#489—22no—
193#464—24no—
194#451C22×C714yes—
195#450C22×C714yes—
196#432C22×C712yes—
197#395C22×C711yescharacteristic
198#394C22×C711yescharacteristic
199#393C22×C711yescharacteristic
200#392C22×C711yescharacteristic
201#390—21no—
202#388—21no—
203#388—21no—
204#386—21no—
205#376C22×C711yescharacteristic
206#328C22×C714yes—
207#328C22×C714yes—
208#324C22×C712yes—
209#323C22×C712yes—
210C23×C7—21no—
211C23×C7—21no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7#13712yes—
222C23×C7C22×D411yescharacteristic
223C7×Q8—44no—
224C7×Q8—44no—
225C7×Q8—82no—
226C7×Q8—82no—
227C7×Q8—82no—
228C7×D4—44no—
229C7×D4—84no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—22no—
237C2×C4×C7—24no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—24no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—82no—
254C2×C4×C7—82no—
255C2×C4×C7#12011yescharacteristic
256C2×C4×C7#12011yescharacteristic
257C2×C4×C7#12012yes—
258C2×C4×C7#13712yes—
259C2×C4×C7C22×D412yes—
260C7×C8—44no—
261C22×Q8C23×C711yescharacteristic
262C22×D4—21no—
263C22×D4C23×C711yescharacteristic
264C23×C4C7×D411yescharacteristic
265C2×Q16—24no—
266C2×SD16—24no—
267C4⋊Q8—24no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—24no—
271C4⋊Q8C7×D412yescomplemented
272C4⋊Q8C7×D412yescomplemented
273#125—24no—
274#125—22no—
275#125—24no—
276#125—22no—
277#124—22no—
278#124—22no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—24no—
286C22⋊Q8C23×C714yes—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8C7×D414yescomplemented
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C7×D411yescharacteristic
301C2×C4⋊C4C23×C711yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4C7×D411yescharacteristic
310C2×C22⋊C4C23×C711yescharacteristic
311C2×C42—22no—
312C2×C42—21no—
313C2×C42C7×D411yescharacteristic
314C2×C42C23×C711yescharacteristic
315C4⋊C8—22no—
316#103—24no—
317#102—24no—
318C22⋊C8C23×C714yes—
319#97—22no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C23×C711yescharacteristic
324#95C23×C711yescharacteristic
325C22×C7—21no—
326C22×C7—21no—
327C22×C7—22no—
328C22×C7—42no—
329C22×C7—41no—
330C22×C7—41no—
331C22×C7—81no—
332C22×C7—81no—
333C22×C7—81no—
334C22×C7—81no—
335C22×C7—81no—
336C22×C7—81no—
337C22×C7—81no—
338C22×C7—81no—
339C22×C7C2×#12011yescharacteristic
340C22×C7C2×#13711yescharacteristic
341C22×C7C2×#13711yescharacteristic
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—22no—
345C4×C7—22no—
346C4×C7—22no—
347C4×C7—42no—
348C4×C7—41no—
349C4×C7—44no—
350C4×C7—41no—
351C4×C7—44no—
352C4×C7—81no—
353C4×C7—81no—
354C24—21no—
355C24C2×C7×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—81no—
361C2×Q8C2×C7×D414yes—
362C2×D4—22no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—81no—
366C2×D4—81no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4C2×C7×D411yescharacteristic
377C22×C4C2×C7×D411yescharacteristic
378C22×C4C2×C7×D411yescharacteristic
379C22×C4C2×C7×D411yescharacteristic
380C22×C4C2×C7×D411yescharacteristic
381C22×C4C24×C711yescharacteristic
382Q16—84no—
383SD16—84no—
384C2×C8—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—81no—
399C4⋊C4C2×C7×D414yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—81no—
410C22⋊C4—81no—
411C42—24no—
412C42—24no—
413C42—21no—
414C42—21no—
415C42—42no—
416C42C2×C7×D412yes—
417C42C2×C7×D412yes—
418C2×C7—21no—
419C2×C7—21no—
420C2×C7—42no—
421C2×C7—161no—
422C2×C7#300411yescharacteristic
423C2×C7#301811yescharacteristic
424C2×C7#302711yescharacteristic
425C23—21no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C7×#13712yes—
437C23C22×C7×D411yescharacteristic
438Q8—44no—
439Q8—44no—
440Q8—82no—
441Q8—82no—
442Q8—82no—
443D4—44no—
444D4—84no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4C7×#12011yescharacteristic
471C2×C4C7×#12011yescharacteristic
472C2×C4C7×#12012yes—
473C2×C4C7×#13712yes—
474C2×C4C22×C7×D412yes—
475C8—44no—
476C7#1833311yescharacteristic, direct product factor, Sylow
477C22—21no—
478C22—21no—
479C22—22no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22C2×C7×#12011yescharacteristic
492C22C2×C7×#13711yescharacteristic
493C22C2×C7×#13711yescharacteristic
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—42no—
500C4—41no—
501C4—44no—
502C4—41no—
503C4—44no—
504C4—81no—
505C4—81no—
506C2—21no—
507C2—21no—
508C2—42no—
509C2—161no—
510C2C7×#300411yescharacteristic
511C2C7×#301811yescharacteristic
512C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#392C22
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
1. C7×#3458C2semidirect
2. C7×#3215C2semidirect
4. C7×#1605C2semidirect
5. C7×#1602C2semidirect
6. C7×#1599C2semidirect
8. C7×#1506C2semidirect
9. C7×#1464C2semidirect
41. #18333C7direct (N × H)
52. C7×C4⋊Q8D4semidirect
53. C7×C4⋊Q8D4semidirect
71. C4×C7×Q8D4semidirect
106. #3458C2×C7semidirect
107. #3215C2×C7semidirect
109. #1605C2×C7semidirect
110. #1602C2×C7semidirect
111. #1599C2×C7semidirect
113. #1506C2×C7semidirect
114. #1464C2×C7semidirect
271. C4⋊Q8C7×D4semidirect
272. C4⋊Q8C7×D4semidirect
290. C4×Q8C7×D4semidirect
476. C7#18333direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34582
C7×#32152
C7×#26102
C7×#16052
C7×#16022
C7×#15992
C7×#15962
C7×#15062
C7×#14642
#183337