Finite Group Databasebeta

← C7×#18749 #13195965

Subgroups — C7×#18749 #13195965

Order 1792.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 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×#3104C211yesmaximal, characteristic, complemented
4C7×#3077C211yesmaximal, characteristic
5C7×#1644C214yesmaximal, complemented
6C7×#1632C214yesmaximal, complemented
7C7×#1548C211yesmaximal, characteristic, complemented
8C7×#1487C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic
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×#484—21no—
26C7×#478—22no—
27C7×#477C2212yescomplemented
28C7×#476C2212yes—
29C7×#474C2212yes—
30C7×#462—22no—
31C7×#462C2212yescomplemented
32C7×#451—22no—
33C7×#451—21no—
34C4×C7×Q16—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#428—21no—
38C7×#423C2211yescharacteristic
39C7×#418—21no—
40C7×#333C2214yescomplemented
41C7×Q8⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yes—
43#18749C711yesmaximal, 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—22no—
50C2×C7×Q16—24no—
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×#107—22no—
95C7×#106—22no—
96C7×C4⋊C8—21no—
97C7×C4⋊C8—21no—
98C7×C4⋊C8C2312yes—
99C7×C4⋊C8C2312yes—
100C7×#103—22no—
101C7×#103—24no—
102C7×#103—22no—
103C7×#102—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#3104C2×C711yescharacteristic, complemented
112#3077C2×C711yescharacteristic
113#1644C2×C714yescomplemented
114#1632C2×C714yescomplemented
115#1548C2×C711yescharacteristic, complemented
116#1487C2×C711yescharacteristic, complemented
117#1478C2×C711yescharacteristic
118C24×C7—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—42no—
124C2×C7×Q8—82no—
125C2×C7×Q8C2×D412yes—
126C2×C7×Q8C2×D414yes—
127C2×C7×D4—21no—
128C2×C7×D4—22no—
129C2×C7×D4Q1614yescomplemented
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—44no—
139C22×C4×C7—84no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C7×Q16—44no—
144C7×Q16—84no—
145C7×M16—42no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8—22no—
149C7×C4⋊C4—28no—
150C7×C4⋊C4—28no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—41no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4C2×D412yes—
166C7×C4⋊C4C2×D412yes—
167C7×C22⋊C4—28no—
168C7×C22⋊C4—22no—
169C7×C22⋊C4—22no—
170C7×C22⋊C4—41no—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—41no—
174C42×C7C2×D411yescharacteristic
175C42×C7C2×D411yescharacteristic
176C42×C7C2×D411yescharacteristic
177C42×C7C2411yescharacteristic
178#554—22no—
179#554—22no—
180#554—22no—
181#554C22×C714yes—
182#533C22×C711yescharacteristic
183C2×C4⋊Q8C22×C711yescharacteristic
184C2×#125—21no—
185C2×C22⋊Q8—21no—
186C2×C22⋊Q8—22no—
187C2×C4×Q8C22×C711yescharacteristic
188C2×C4×D4—21no—
189C2×C4×D4C22×C711yescharacteristic
190C22×C4⋊C4—21no—
191#494C22×C712yescomplemented
192#487C22×C712yescomplemented
193#484—21no—
194#478—22no—
195#477C22×C712yescomplemented
196#476C22×C712yes—
197#474C22×C712yes—
198#462—22no—
199#462C22×C712yescomplemented
200#451—21no—
201#451—22no—
202C4×Q16—22no—
203#431C22×C711yescharacteristic
204#431C22×C711yescharacteristic
205#428—21no—
206#423C22×C711yescharacteristic
207#418—21no—
208#333C22×C714yescomplemented
209Q8⋊C8C22×C714yescomplemented
210D4⋊C8C22×C714yes—
211C23×C7—22no—
212C23×C7—24no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7#12011yescharacteristic
216C7×Q8—22no—
217C7×Q8—22no—
218C7×Q8—44no—
219C7×Q8—42no—
220C7×Q8—42no—
221C7×Q8—42no—
222C7×Q8—82no—
223C7×D4—22no—
224C7×D4—24no—
225C7×D4C2×Q1614yes—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—24no—
231C2×C4×C7—41no—
232C2×C4×C7—44no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—44no—
236C2×C4×C7—42no—
237C2×C4×C7—44no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—88no—
241C2×C4×C7#12011yescharacteristic
242C2×C4×C7#12011yescharacteristic
243C2×C4×C7#12011yescharacteristic
244C2×C4×C7C2×Q1612yes—
245C2×C4×C7#13712yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C7×C8—42no—
250C7×C8—42no—
251C7×C8—42no—
252C22×Q8C7×D411yescharacteristic
253C22×Q8C7×D412yes—
254C22×D4C7×D411yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C2×Q16—22no—
258C2×Q16—24no—
259C2×M16—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8C7×D412yes—
263C4⋊Q8C23×C711yescharacteristic
264C4⋊Q8C23×C711yescharacteristic
265C4⋊Q8C23×C712yes—
266#125—24no—
267#125—24no—
268#125—21no—
269C22⋊Q8—24no—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8—28no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8C7×D412yes—
280C4×Q8C23×C712yes—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4C7×D412yes—
286C4×D4C23×C712yes—
287#117—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4C7×D411yescharacteristic
296C2×C4⋊C4C7×D411yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C42—21no—
300C2×C42C23×C711yescharacteristic
301#107—22no—
302#107—22no—
303#106—22no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8C23×C712yes—
307C4⋊C8C23×C712yes—
308#103—22no—
309#103—24no—
310#103—22no—
311#102—22no—
312#102—22no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C4×C8—21no—
316C4×C8C23×C712yes—
317C22×C7—21no—
318C22×C7—21no—
319C22×C7—24no—
320C22×C7—48no—
321C22×C7—42no—
322C22×C7—42no—
323C22×C7#45112yes—
324C22×C7C2×#12011yescharacteristic
325C4×C7—21no—
326C4×C7—21no—
327C4×C7—21no—
328C4×C7—42no—
329C4×C7—42no—
330C4×C7—42no—
331C4×C7—44no—
332C4×C7—81no—
333C4×C7—81no—
334C4×C7#45112yes—
335C4×C7C22×Q1611yescharacteristic
336C4×C7C2×#13711yescharacteristic
337C4×C7#57512yes—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—42no—
344C2×Q8—82no—
345C2×Q8C2×C7×D412yes—
346C2×Q8C2×C7×D414yes—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4C7×Q1614yescomplemented
350C2×D4C2×C7×D412yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—84no—
360C22×C4C2×C7×D411yescharacteristic
361C22×C4C2×C7×D411yescharacteristic
362C22×C4C2×C7×D411yescharacteristic
363Q16—44no—
364Q16—84no—
365M16—42no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4C2×C7×D412yes—
386C4⋊C4C2×C7×D412yes—
387C22⋊C4—28no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—41no—
391C42—22no—
392C42—22no—
393C42—41no—
394C42C2×C7×D411yescharacteristic
395C42C2×C7×D411yescharacteristic
396C42C2×C7×D411yescharacteristic
397C42C24×C711yescharacteristic
398C2×C7—22no—
399C2×C7—44no—
400C2×C7C2×#45111yescharacteristic
401C2×C7#300511yescharacteristic
402C2×C7#301911yescharacteristic
403C23—22no—
404C23—24no—
405C23—42no—
406C23—42no—
407C23C7×#12011yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—44no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—82no—
415D4—22no—
416D4—24no—
417D4C2×C7×Q1614yes—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—88no—
433C2×C4C7×#12011yescharacteristic
434C2×C4C7×#12011yescharacteristic
435C2×C4C7×#12011yescharacteristic
436C2×C4C2×C7×Q1612yes—
437C2×C4C7×#13712yes—
438C2×C4C22×C7×D411yescharacteristic
439C2×C4C22×C7×D411yescharacteristic
440C2×C4C22×C7×D411yescharacteristic
441C8—42no—
442C8—42no—
443C8—42no—
444C7#1874911yescharacteristic, direct product factor, Sylow
445C22—21no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—42no—
450C22—42no—
451C22C7×#45112yes—
452C22C2×C7×#12011yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—44no—
460C4—81no—
461C4—81no—
462C4C7×#45112yes—
463C4C22×C7×Q1611yescharacteristic
464C4C2×C7×#13711yescharacteristic
465C4C7×#57512yes—
466C2—22no—
467C2—44no—
468C2C2×C7×#45111yescharacteristic
469C2C7×#300511yescharacteristic
470C2C7×#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
3. C7×#3104C2semidirect
5. C7×#1644C2semidirect
6. C7×#1632C2semidirect
7. C7×#1548C2semidirect
8. C7×#1487C2semidirect
23. C7×#494C22semidirect
24. C7×#487C22semidirect
27. C7×#477C22semidirect
31. C7×#462C22semidirect
40. C7×#333C22semidirect
41. C7×Q8⋊C8C22semidirect
43. #18749C7direct (N × H)
110. #3209C2×C7semidirect
111. #3104C2×C7semidirect
113. #1644C2×C7semidirect
114. #1632C2×C7semidirect
115. #1548C2×C7semidirect
116. #1487C2×C7semidirect
129. C2×C7×D4Q16semidirect
191. #494C22×C7semidirect
192. #487C22×C7semidirect
195. #477C22×C7semidirect
199. #462C22×C7semidirect
208. #333C22×C7semidirect
209. Q8⋊C8C22×C7semidirect
349. C2×D4C7×Q16semidirect
444. C7#18749direct (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×#31042
C7×#30772
C7×#16442
C7×#16322
C7×#15482
C7×#14872
C7×#14782
#187497