Finite Group Databasebeta

← C7×#9343 #13186559

Subgroups — C7×#9343 #13186559

Order 1792.

576 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3192C211yesmaximal, characteristic
2C7×#2377C211yesmaximal, characteristic, complemented
3C7×#1964C211yesmaximal, characteristic, complemented
4C7×#1510C211yesmaximal, characteristic, complemented
5C7×#1507C211yesmaximal, characteristic, complemented
6C7×#1305C211yesmaximal, characteristic, complemented
7C7×#1284C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C7×#481—21no—
11C7×#481C412yescomplemented
12C7×#469—22no—
13C7×#447—21no—
14C7×#447C412yescomplemented
15C2×C7×#99—21no—
16C2×C7×#99—21no—
17C7×#407C2211yescharacteristic
18C7×#407C2211yescharacteristic
19C7×#390—21no—
20C7×#388—21no—
21C7×#387—21no—
22C7×#386—21no—
23C7×#386—21no—
24C7×#385—21no—
25C7×#381C2211yescharacteristic
26C7×#380C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#327—22no—
29C7×#327—21no—
30C7×#327—21no—
31#9343C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4C2×C411yescharacteristic, complemented
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C23×C4×C7C2311yescharacteristic
38C2×C7×D8—22no—
39C2×C7×M16—21no—
40C2×C7×M16—21no—
41C7×#121—21no—
42C7×#121—21no—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121C2×C412yes—
48C7×#121D412yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—21no—
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4D411yescharacteristic
61C2×C7×C22⋊C4D411yescharacteristic
62C2×C42×C7—21no—
63C2×C42×C7—21no—
64C7×#106—22no—
65C7×#102—22no—
66C7×#102—22no—
67C7×#99—42no—
68C7×#99—42no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8C2×C412yescomplemented
73C7×#95—21no—
74C7×#95—21no—
75C7×#95—21no—
76C7×#95—21no—
77C7×#95—21no—
78C7×#95D411yescharacteristic
79C7×#95D411yescharacteristic
80#3192C2×C711yescharacteristic
81#2377C2×C711yescharacteristic, complemented
82#1964C2×C711yescharacteristic, complemented
83#1510C2×C711yescharacteristic, complemented
84#1507C2×C711yescharacteristic, complemented
85#1305C2×C711yescharacteristic, complemented
86#1284C2×C711yescharacteristic, complemented
87C24×C7—21no—
88C24×C7—21no—
89C24×C7C2×D411yescharacteristic
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—81no—
102C2×C7×D4—81no—
103C2×C7×D4—81no—
104C2×C7×D4—81no—
105C2×C7×D4C22⋊C412yes—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—41no—
113C22×C4×C7—81no—
114C22×C4×C7—81no—
115C22×C4×C7—81no—
116C22×C4×C7—81no—
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7#4111yescharacteristic
122C22×C4×C7#4111yescharacteristic
123C7×D8—82no—
124C7×M16—81no—
125C7×M16—81no—
126C2×C7×C8—22no—
127C2×C7×C8—41no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—81no—
132C7×C4⋊C4—81no—
133C7×C4⋊C4C22⋊C412yes—
134C7×C22⋊C4—22no—
135C7×C22⋊C4—22no—
136C7×C22⋊C4—22no—
137C7×C22⋊C4—41no—
138C7×C22⋊C4—41no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—41no—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—81no—
143C7×C22⋊C4—81no—
144C7×C22⋊C4—81no—
145C7×C22⋊C4—81no—
146C42×C7—81no—
147C42×C7—81no—
148C2×#121C22×C711yescharacteristic
149C2×#121C22×C711yescharacteristic
150#481—21no—
151#481C4×C712yescomplemented
152#469—22no—
153#447—21no—
154#447C4×C712yescomplemented
155C2×#99—21no—
156C2×#99—21no—
157#407C22×C711yescharacteristic
158#407C22×C711yescharacteristic
159#390—21no—
160#388—21no—
161#387—21no—
162#386—21no—
163#386—21no—
164#385—21no—
165#381C22×C711yescharacteristic
166#380C22×C711yescharacteristic
167C4×C22⋊C4C22×C711yescharacteristic
168#327—22no—
169#327—21no—
170#327—21no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—22no—
174C23×C7—41no—
175C23×C7—41no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—41no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7—81no—
186C23×C7—81no—
187C23×C7C4×D411yescharacteristic
188C23×C7#12011yescharacteristic
189C23×C7#12111yescharacteristic
190C7×D4—42no—
191C7×D4—42no—
192C7×D4—81no—
193C7×D4—81no—
194C7×D4—82no—
195C7×D4—82no—
196C7×D4—81no—
197C7×D4—81no—
198C7×D4—81no—
199C7×D4—81no—
200C2×C4×C7—22no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—41no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—81no—
226C2×C4×C7—81no—
227C2×C4×C7—84no—
228C2×C4×C7C2×C22⋊C411yescharacteristic
229C2×C4×C7C4×D411yescharacteristic
230C2×C4×C7#12111yescharacteristic
231C2×C4×C7#12311yescharacteristic
232C7×C8—42no—
233C7×C8—81no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2×C4×C711yescharacteristic, complemented
237C22×D4C7×D411yescharacteristic
238C23×C4—21no—
239C23×C4C23×C711yescharacteristic
240C2×D8—22no—
241C2×M16—21no—
242C2×M16—21no—
243#121—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121C2×C4×C712yes—
250#121C7×D412yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
254C2×C4⋊C4C7×D411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4C7×D411yescharacteristic
263C2×C22⋊C4C7×D411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266#106—22no—
267#102—22no—
268#102—22no—
269#99—42no—
270#99—42no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8C2×C4×C712yescomplemented
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95C7×D411yescharacteristic
281#95C7×D411yescharacteristic
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—21no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—42no—
293C22×C7—81no—
294C22×C7—81no—
295C22×C7—81no—
296C22×C7—81no—
297C22×C7—81no—
298C22×C7—81no—
299C22×C7—81no—
300C22×C7—81no—
301C22×C7—81no—
302C22×C7—161no—
303C22×C7—161no—
304C22×C7#38611yescharacteristic
305C22×C7#45311yescharacteristic
306C22×C7#45411yescharacteristic
307C4×C7—21no—
308C4×C7—21no—
309C4×C7—21no—
310C4×C7—21no—
311C4×C7—42no—
312C4×C7—81no—
313C4×C7—81no—
314C4×C7—81no—
315C4×C7—81no—
316C4×C7—82no—
317C4×C7—81no—
318C24—21no—
319C24—21no—
320C24C2×C7×D411yescharacteristic
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4—81no—
335C2×D4—81no—
336C2×D4C7×C22⋊C412yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—41no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4C22×C4×C711yescharacteristic
349C22×C4C2×C7×D411yescharacteristic
350C22×C4C2×C7×D411yescharacteristic
351C22×C4C2×C7×D411yescharacteristic
352C22×C4C7×#4111yescharacteristic
353C22×C4C7×#4111yescharacteristic
354D8—82no—
355M16—81no—
356M16—81no—
357C2×C8—22no—
358C2×C8—41no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4C7×C22⋊C412yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—81no—
374C22⋊C4—81no—
375C22⋊C4—81no—
376C22⋊C4—81no—
377C42—81no—
378C42—81no—
379C2×C7—21no—
380C2×C7—21no—
381C2×C7—41no—
382C2×C7—41no—
383C2×C7—82no—
384C2×C7—161no—
385C2×C7#188311yescharacteristic
386C2×C7#190411yescharacteristic
387C2×C7#191311yescharacteristic
388C23—21no—
389C23—21no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—81no—
403C23—81no—
404C23C4×C7×D411yescharacteristic
405C23C7×#12011yescharacteristic
406C23C7×#12111yescharacteristic
407D4—42no—
408D4—42no—
409D4—81no—
410D4—81no—
411D4—82no—
412D4—82no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—84no—
445C2×C4C2×C7×C22⋊C411yescharacteristic
446C2×C4C4×C7×D411yescharacteristic
447C2×C4C7×#12111yescharacteristic
448C2×C4C7×#12311yescharacteristic
449C8—42no—
450C8—81no—
451C7#934311yescharacteristic, direct product factor, Sylow
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—161no—
473C22—161no—
474C22C7×#38611yescharacteristic
475C22C7×#45311yescharacteristic
476C22C7×#45411yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—82no—
487C4—81no—
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—161no—
494C2C7×#188311yescharacteristic
495C2C7×#190411yescharacteristic
496C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3192C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2377C2semidirect
3. C7×#1964C2semidirect
4. C7×#1510C2semidirect
5. C7×#1507C2semidirect
6. C7×#1305C2semidirect
7. C7×#1284C2semidirect
11. C7×#481C4semidirect
14. C7×#447C4semidirect
31. #9343C7direct (N × H)
34. C22×C7×D4C2×C4semidirect
51. C2×C7×C4⋊C4C2×C4semidirect
72. C7×C22⋊C8C2×C4semidirect
81. #2377C2×C7semidirect
82. #1964C2×C7semidirect
83. #1510C2×C7semidirect
84. #1507C2×C7semidirect
85. #1305C2×C7semidirect
86. #1284C2×C7semidirect
151. #481C4×C7semidirect
154. #447C4×C7semidirect
236. C22×D4C2×C4×C7semidirect
253. C2×C4⋊C4C2×C4×C7semidirect
274. C22⋊C8C2×C4×C7semidirect
451. C7#9343direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31922
C7×#23772
C7×#19642
C7×#15102
C7×#15072
C7×#13052
C7×#12842
#93437