Finite Group Databasebeta

← C7×#23953 #13201169

Subgroups — C7×#23953 #13201169

Order 1792.

1166 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3493C211yesmaximal, characteristic
2C7×#3004C212yesmaximal, complemented
3C7×#2310C211yesmaximal, characteristic, complemented
4C7×#2028C212yesmaximal, complemented
5C7×#2007C212yesmaximal, complemented
6C7×#1897C212yesmaximal, complemented
7C7×#1882C214yesmaximal, complemented
8C7×#1822C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic
10C2×C7×#137C2214yes—
11C7×#556—22no—
12C7×#555—22no—
13C7×#553—22no—
14C7×#552—22no—
15C7×#544C2211yescharacteristic
16C7×#538—22no—
17C7×#537C2211yescharacteristic
18C7×#529—22no—
19C7×#525C2212yes—
20C7×#520C2212yes—
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic
23C7×#451—22no—
24C7×#451—22no—
25C7×#448—22no—
26C7×#448—22no—
27C2×C7×#104C2214yes—
28C7×#419C2214yes—
29C7×#407C2212yes—
30C7×#391—22no—
31C7×#391C2212yes—
32C7×#391C2212yes—
33C7×#381—22no—
34C7×#343C2212yes—
35C7×#343C2212yes—
36C7×#339C2214yes—
37#23953C711yesmaximal, characteristic, direct product factor, Sylow
38C2×C7×#41C2312yes—
39C22×C7×Q8—24no—
40C22×C7×Q8—21no—
41C22×C7×Q8—22no—
42C22×C7×Q8—22no—
43C22×C7×Q8—42no—
44C22×C7×Q8C2312yes—
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C7×#137—28no—
48C7×#137—44no—
49C2×C7×Q16—24no—
50C2×C7×SD16—24no—
51C2×C7×M16C2314yes—
52C7×C4⋊Q8—22no—
53C7×#126—24no—
54C7×#126—24no—
55C7×#125—22no—
56C7×#125—22no—
57C7×#124—22no—
58C7×#124—24no—
59C7×#123—24no—
60C7×#123—22no—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8D412yes—
64C7×C22⋊Q8D412yes—
65C7×#121—24no—
66C7×#121—22no—
67C7×#121—22no—
68C7×#121D412yes—
69C7×#121D412yes—
70C7×#120—22no—
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—24no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C7×#117—24no—
79C7×#117—22no—
80C7×#117—22no—
81C7×#117—22no—
82C7×#117—42no—
83C7×#117C2312yes—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4D412yes—
90C2×C42×C7D412yes—
91C2×C42×C7C2312yes—
92C7×#104—44no—
93C7×#103—24no—
94C7×#103—24no—
95C7×C22⋊C8—22no—
96C7×C22⋊C8—22no—
97C7×#95—24no—
98C7×#95—24no—
99#3493C2×C711yescharacteristic
100#3004C2×C712yescomplemented
101#2310C2×C711yescharacteristic, complemented
102#2028C2×C712yescomplemented
103#2007C2×C712yescomplemented
104#1897C2×C712yescomplemented
105#1882C2×C714yescomplemented
106#1822C2×C711yescharacteristic, complemented
107C24×C7C2×D411yescharacteristic
108C7×#41—24no—
109C7×#41—24no—
110C2×C7×Q8—24no—
111C2×C7×Q8—22no—
112C2×C7×Q8—22no—
113C2×C7×Q8—24no—
114C2×C7×Q8—24no—
115C2×C7×Q8—48no—
116C2×C7×Q8—48no—
117C2×C7×Q8—42no—
118C2×C7×Q8—41no—
119C2×C7×Q8—41no—
120C2×C7×Q8C2×D412yes—
121C2×C7×Q8#4114yes—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4C2×D412yes—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—24no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—42no—
135C22×C4×C7—42no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2411yescharacteristic
142C7×Q16—48no—
143C7×SD16—48no—
144C7×M16—24no—
145C7×M16—24no—
146C2×C7×C8—24no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—24no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—42no—
156C7×C4⋊C4C2×D412yes—
157C7×C4⋊C4C2×D412yes—
158C7×C22⋊C4—24no—
159C7×C22⋊C4—24no—
160C7×C22⋊C4—24no—
161C7×C22⋊C4—22no—
162C7×C22⋊C4—22no—
163C7×C22⋊C4—22no—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—42no—
167C42×C7—24no—
168C42×C7—22no—
169C42×C7—22no—
170C42×C7—22no—
171C42×C7—24no—
172C23×Q8C22×C711yescharacteristic
173C2×#137C22×C714yes—
174#556—22no—
175#555—22no—
176#553—22no—
177#552—22no—
178#544C22×C711yescharacteristic
179#538—22no—
180#537C22×C711yescharacteristic
181#529—22no—
182#525C22×C712yes—
183#520C22×C712yes—
184C2×#117C22×C711yescharacteristic
185C2×#117C22×C711yescharacteristic
186#451—22no—
187#451—22no—
188#448—22no—
189#448—22no—
190C2×#104C22×C714yes—
191#419C22×C714yes—
192#407C22×C712yes—
193#391—22no—
194#391C22×C712yes—
195#391C22×C712yes—
196#381—22no—
197#343C22×C712yes—
198#343C22×C712yes—
199#339C22×C714yes—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—22no—
203C23×C7—41no—
204C23×C7—41no—
205C23×C7#12012yes—
206C23×C7C22×D411yescharacteristic
207C7×Q8—28no—
208C7×Q8—24no—
209C7×Q8—44no—
210C7×Q8—44no—
211C7×Q8—44no—
212C7×Q8—82no—
213C7×D4—24no—
214C7×D4—42no—
215C7×D4—42no—
216C7×D4—82no—
217C7×D4—82no—
218C7×D4—82no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—24no—
229C2×C4×C7—48no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—44no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—84no—
239C2×C4×C7#12011yescharacteristic
240C2×C4×C7#12011yescharacteristic
241C2×C4×C7C22×D412yes—
242C2×C4×C7C2×#4112yes—
243C2×C4×C7H(5,𝔽2)12yes—
244C7×C8—44no—
245C2×#41C23×C712yes—
246C22×Q8—24no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8C23×C712yes—
252C23×C4—21no—
253C23×C4C23×C711yescharacteristic
254#137—28no—
255#137—44no—
256C2×Q16—24no—
257C2×SD16—24no—
258C2×M16C23×C714yes—
259C4⋊Q8—22no—
260#126—24no—
261#126—24no—
262#125—22no—
263#125—22no—
264#124—22no—
265#124—24no—
266#123—24no—
267#123—22no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8C7×D412yes—
271C22⋊Q8C7×D412yes—
272#121—24no—
273#121—22no—
274#121—22no—
275#121C7×D412yes—
276#121C7×D412yes—
277#120—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285#117—24no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—42no—
290#117C23×C712yes—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C23×C711yescharacteristic
294C2×C4⋊C4C23×C711yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4C7×D412yes—
297C2×C42C7×D412yes—
298C2×C42C23×C712yes—
299#104—44no—
300#103—24no—
301#103—24no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304#95—24no—
305#95—24no—
306C22×C7—21no—
307C22×C7—21no—
308C22×C7—22no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7—42no—
312C22×C7—42no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7—42no—
316C22×C7C2×#12011yescharacteristic
317C22×C7#52511yescharacteristic
318C22×C7#53511yescharacteristic
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—44no—
322C4×C7—42no—
323C4×C7—42no—
324C4×C7—42no—
325C4×C7—42no—
326C4×C7—42no—
327C4×C7—82no—
328C4×C7#54614yes—
329C24C2×C7×D411yescharacteristic
330#41—24no—
331#41—24no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—42no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C2×C7×D412yes—
343C2×Q8C7×#4114yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4C2×C7×D412yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C7×D411yescharacteristic
359C22×C4C2×C7×D412yes—
360C22×C4C2×C7×D412yes—
361C22×C4C2×C7×D412yes—
362C22×C4C2×C7×D412yes—
363C22×C4C24×C711yescharacteristic
364Q16—48no—
365SD16—48no—
366M16—24no—
367M16—24no—
368C2×C8—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4C2×C7×D412yes—
379C4⋊C4C2×C7×D412yes—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—42no—
389C42—24no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—24no—
394C2×C7—21no—
395C2×C7—21no—
396C2×C7—42no—
397C2×C7—82no—
398C2×C7#240411yescharacteristic
399C2×C7#302211yescharacteristic
400C2×C7#302211yescharacteristic
401C23—21no—
402C23—21no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23C7×#12012yes—
407C23C22×C7×D411yescharacteristic
408Q8—28no—
409Q8—24no—
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—82no—
414D4—24no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—82no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4C7×#12011yescharacteristic
441C2×C4C7×#12011yescharacteristic
442C2×C4C22×C7×D412yes—
443C2×C4C2×C7×#4112yes—
444C2×C4C7×H(5,𝔽2)12yes—
445C8—44no—
446C7#2395311yescharacteristic, direct product factor, Sylow
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22C2×C7×#12011yescharacteristic
458C22C7×#52511yescharacteristic
459C22C7×#53511yescharacteristic
460C4—21no—
461C4—21no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4C7×#54614yes—
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2C7×#240411yescharacteristic
475C2C7×#302211yescharacteristic
476C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3004C2semidirect
3. C7×#2310C2semidirect
4. C7×#2028C2semidirect
5. C7×#2007C2semidirect
6. C7×#1897C2semidirect
7. C7×#1882C2semidirect
8. C7×#1822C2semidirect
37. #23953C7direct (N × H)
100. #3004C2×C7semidirect
101. #2310C2×C7semidirect
102. #2028C2×C7semidirect
103. #2007C2×C7semidirect
104. #1897C2×C7semidirect
105. #1882C2×C7semidirect
106. #1822C2×C7semidirect
446. C7#23953direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34932
C7×#30042
C7×#23102
C7×#20282
C7×#20072
C7×#18972
C7×#18822
C7×#18222
#239537