Finite Group Databasebeta

← C7×#18753 #13195969

Subgroups — C7×#18753 #13195969

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2C7×#3209C211yesmaximal, characteristic, complemented
3C7×#3101C211yesmaximal, characteristic
4C7×#3078C211yesmaximal, characteristic, complemented
5C7×#1643C214yesmaximal, complemented
6C7×#1632C214yesmaximal, complemented
7C7×#1548C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic
9C7×#1479C211yesmaximal, characteristic, complemented
10C7×#554—22no—
11C7×#554—22no—
12C7×#554—22no—
13C7×#554C2214yes—
14C7×#533C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C22×C7×C4⋊C4—21no—
23C7×#494C2212yescomplemented
24C7×#487C2212yescomplemented
25C7×#483—21no—
26C7×#479—22no—
27C7×#477C2212yescomplemented
28C7×#476C2212yes—
29C7×#475C2212yescomplemented
30C7×#462—22no—
31C7×#461C2212yes—
32C7×#451—22no—
33C7×#450—21no—
34C4×C7×SD16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#419—21no—
40C7×#333C2214yescomplemented
41C7×Q8⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yes—
43#18753C711yesmaximal, characteristic, direct product factor, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D412yes—
46C22×C7×D4D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—21no—
49C2×C7×Q16—24no—
50C2×C7×SD16—22no—
51C2×C7×M16—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8D412yes—
55C7×C4⋊Q8C2311yescharacteristic
56C7×C4⋊Q8C2311yescharacteristic
57C7×C4⋊Q8C2312yes—
58C7×#125—24no—
59C7×#125—24no—
60C7×#125—21no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—28no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—21no—
67C4×C7×Q8—21no—
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×Q8—21no—
71C4×C7×Q8D412yes—
72C4×C7×Q8C2312yes—
73C4×C7×D4—24no—
74C4×C7×D4—24no—
75C4×C7×D4—21no—
76C4×C7×D4—21no—
77C4×C7×D4D412yes—
78C4×C7×D4C2312yes—
79C7×#117—21no—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—24no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—24no—
87C2×C7×C4⋊C4D411yescharacteristic
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—21no—
91C2×C42×C7—21no—
92C2×C42×C7C2311yescharacteristic
93C7×#107—22no—
94C7×#106—22no—
95C7×#106—22no—
96C7×C4⋊C8—21no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8C2312yes—
99C7×C4⋊C8C2312yes—
100C7×#103—22no—
101C7×#103—22no—
102C7×#103—24no—
103C7×#103—22no—
104C7×#102—22no—
105C7×C22⋊C8—21no—
106C7×C22⋊C8—21no—
107C4×C7×C8—21no—
108C4×C7×C8C2312yes—
109C2×#554C2×C711yescharacteristic
110#3209C2×C711yescharacteristic, complemented
111#3101C2×C711yescharacteristic
112#3078C2×C711yescharacteristic, complemented
113#1643C2×C714yescomplemented
114#1632C2×C714yescomplemented
115#1548C2×C711yescharacteristic, complemented
116#1486C2×C711yescharacteristic
117#1479C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—82no—
126C2×C7×Q8C2×D412yes—
127C2×C7×Q8C2×D414yes—
128C2×C7×D4—24no—
129C2×C7×D4—21no—
130C2×C7×D4C2×D412yes—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—22no—
135C22×C4×C7—28no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—84no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C7×Q16—84no—
145C7×SD16—44no—
146C7×M16—42no—
147C2×C7×C8—22no—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C7×C4⋊C4—28no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4C2×D412yes—
167C7×C4⋊C4C2×D412yes—
168C7×C22⋊C4—28no—
169C7×C22⋊C4—22no—
170C7×C22⋊C4—22no—
171C7×C22⋊C4—41no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7—41no—
175C42×C7C2×D411yescharacteristic
176C42×C7C2×D411yescharacteristic
177C42×C7C2×D411yescharacteristic
178C42×C7C2411yescharacteristic
179#554—22no—
180#554—22no—
181#554—22no—
182#554C22×C714yes—
183#533C22×C711yescharacteristic
184C2×C4⋊Q8C22×C711yescharacteristic
185C2×#125—21no—
186C2×C22⋊Q8—21no—
187C2×C22⋊Q8—22no—
188C2×C4×Q8C22×C711yescharacteristic
189C2×C4×D4—21no—
190C2×C4×D4C22×C711yescharacteristic
191C22×C4⋊C4—21no—
192#494C22×C712yescomplemented
193#487C22×C712yescomplemented
194#483—21no—
195#479—22no—
196#477C22×C712yescomplemented
197#476C22×C712yes—
198#475C22×C712yescomplemented
199#462—22no—
200#461C22×C712yes—
201#451—22no—
202#450—21no—
203C4×SD16—22no—
204#431C22×C711yescharacteristic
205#431C22×C711yescharacteristic
206#428—21no—
207#423C22×C711yescharacteristic
208#419—21no—
209#333C22×C714yescomplemented
210Q8⋊C8C22×C714yescomplemented
211D4⋊C8C22×C714yes—
212C23×C7—24no—
213C23×C7—22no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7#12011yescharacteristic
217C7×Q8—22no—
218C7×Q8—22no—
219C7×Q8—44no—
220C7×Q8—42no—
221C7×Q8—42no—
222C7×Q8—42no—
223C7×Q8—82no—
224C7×D4—24no—
225C7×D4—22no—
226C7×D4#13714yes—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—41no—
233C2×C4×C7—42no—
234C2×C4×C7—44no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—44no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—88no—
243C2×C4×C7#12011yescharacteristic
244C2×C4×C7#12011yescharacteristic
245C2×C4×C7#12011yescharacteristic
246C2×C4×C7#13512yes—
247C2×C4×C7#13712yes—
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C7×C8—42no—
252C7×C8—42no—
253C7×C8—42no—
254C22×Q8C7×D411yescharacteristic
255C22×Q8C7×D412yes—
256C22×D4C7×D411yescharacteristic
257C23×C4—22no—
258C23×C4—21no—
259C2×Q16—24no—
260C2×SD16—22no—
261C2×M16—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C7×D412yes—
265C4⋊Q8C23×C711yescharacteristic
266C4⋊Q8C23×C711yescharacteristic
267C4⋊Q8C23×C712yes—
268#125—24no—
269#125—24no—
270#125—21no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—28no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×Q8C7×D412yes—
282C4×Q8C23×C712yes—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4C7×D412yes—
288C4×D4C23×C712yes—
289#117—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4C7×D411yescharacteristic
298C2×C4⋊C4C7×D411yescharacteristic
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42C23×C711yescharacteristic
303#107—22no—
304#106—22no—
305#106—22no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8C23×C712yes—
309C4⋊C8C23×C712yes—
310#103—22no—
311#103—22no—
312#103—24no—
313#103—22no—
314#102—22no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C4×C8—21no—
318C4×C8C23×C712yes—
319C22×C7—21no—
320C22×C7—21no—
321C22×C7—24no—
322C22×C7—48no—
323C22×C7—42no—
324C22×C7—42no—
325C22×C7#45212yes—
326C22×C7C2×#12011yescharacteristic
327C4×C7—21no—
328C4×C7—21no—
329C4×C7—21no—
330C4×C7—42no—
331C4×C7—42no—
332C4×C7—42no—
333C4×C7—44no—
334C4×C7—81no—
335C4×C7—81no—
336C4×C7#45212yes—
337C4×C7C2×#13511yescharacteristic
338C4×C7C2×#13711yescharacteristic
339C4×C7C2×#13712yes—
340C24—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—82no—
348C2×Q8C2×C7×D412yes—
349C2×Q8C2×C7×D414yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4C2×C7×D412yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—84no—
363C22×C4C2×C7×D411yescharacteristic
364C22×C4C2×C7×D411yescharacteristic
365C22×C4C2×C7×D411yescharacteristic
366Q16—84no—
367SD16—44no—
368M16—42no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C4⋊C4—28no—
373C4⋊C4—28no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C2×C7×D412yes—
389C4⋊C4C2×C7×D412yes—
390C22⋊C4—28no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—41no—
394C42—22no—
395C42—22no—
396C42—41no—
397C42C2×C7×D411yescharacteristic
398C42C2×C7×D411yescharacteristic
399C42C2×C7×D411yescharacteristic
400C42C24×C711yescharacteristic
401C2×C7—22no—
402C2×C7—44no—
403C2×C7C2×#45211yescharacteristic
404C2×C7#300411yescharacteristic
405C2×C7#301911yescharacteristic
406C23—24no—
407C23—22no—
408C23—42no—
409C23—42no—
410C23C7×#12011yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—44no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—82no—
418D4—24no—
419D4—22no—
420D4C7×#13714yes—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—88no—
437C2×C4C7×#12011yescharacteristic
438C2×C4C7×#12011yescharacteristic
439C2×C4C7×#12011yescharacteristic
440C2×C4C7×#13512yes—
441C2×C4C7×#13712yes—
442C2×C4C22×C7×D411yescharacteristic
443C2×C4C22×C7×D411yescharacteristic
444C2×C4C22×C7×D411yescharacteristic
445C8—42no—
446C8—42no—
447C8—42no—
448C7#1875311yescharacteristic, direct product factor, Sylow
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22C7×#45212yes—
456C22C2×C7×#12011yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—44no—
464C4—81no—
465C4—81no—
466C4C7×#45212yes—
467C4C2×C7×#13511yescharacteristic
468C4C2×C7×#13711yescharacteristic
469C4C2×C7×#13712yes—
470C2—22no—
471C2—44no—
472C2C2×C7×#45211yescharacteristic
473C2C7×#300411yescharacteristic
474C2C7×#301911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC22×C7×D4D4
LCSG ▷ C42 ▷ 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×#3209C2semidirect
4. C7×#3078C2semidirect
5. C7×#1643C2semidirect
6. C7×#1632C2semidirect
7. C7×#1548C2semidirect
9. C7×#1479C2semidirect
23. C7×#494C22semidirect
24. C7×#487C22semidirect
27. C7×#477C22semidirect
29. C7×#475C22semidirect
40. C7×#333C22semidirect
41. C7×Q8⋊C8C22semidirect
43. #18753C7direct (N × H)
110. #3209C2×C7semidirect
112. #3078C2×C7semidirect
113. #1643C2×C7semidirect
114. #1632C2×C7semidirect
115. #1548C2×C7semidirect
117. #1479C2×C7semidirect
192. #494C22×C7semidirect
193. #487C22×C7semidirect
196. #477C22×C7semidirect
198. #475C22×C7semidirect
209. #333C22×C7semidirect
210. Q8⋊C8C22×C7semidirect
448. C7#18753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C7×#32092
C7×#31012
C7×#30782
C7×#16432
C7×#16322
C7×#15482
C7×#14862
C7×#14792
#187537