Finite Group Databasebeta

← C7×#18996 #13196212

Subgroups — C7×#18996 #13196212

Order 1792.

948 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3529C211yesmaximal, characteristic
2C7×#3239C212yesmaximal, complemented
3C7×#3152C211yesmaximal, characteristic, complemented
4C7×#1661C214yesmaximal, complemented
5C7×#1660C212yesmaximal, complemented
6C7×#1659C212yesmaximal, complemented
7C7×#1536C211yesmaximal, characteristic, complemented
8C7×#1529C212yesmaximal, complemented
9C2×C7×#137—21no—
10C2×C7×#137—21no—
11C7×#563—21no—
12C7×#563C2211yescharacteristic
13C7×#562—22no—
14C7×#562C2212yes—
15C7×H(3,𝔽4)C2211yescharacteristic
16C7×#560—21no—
17C7×#560—22no—
18C7×#532—21no—
19C7×#532C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic, complemented
21C7×#529—22no—
22C7×#529C2212yes—
23C7×#525—21no—
24C7×#525—22no—
25C7×#501C2214yescomplemented
26C7×#497C2212yes—
27C7×#497C2212yescomplemented
28C7×#494—21no—
29C7×#492—21no—
30C7×#489C2214yes—
31C7×#484—22no—
32C7×#481—22no—
33C7×#475—22no—
34C7×#474—22no—
35C7×#432C2212yescomplemented
36C4×C7×M16C2211yescharacteristic, complemented
37C7×#330C2214yescomplemented
38C7×#329C2214yes—
39C7×#329C2212yes—
40C7×#329C2212yescomplemented
41#18996C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C2×C7×#41D411yescharacteristic
44C2×C7×#41D411yescharacteristic
45C22×C7×Q8—21no—
46C23×C4×C7—21no—
47C7×#137—44no—
48C7×#137—44no—
49C2×C7×Q16—22no—
50C2×C7×Q16—22no—
51C2×C7×SD16—22no—
52C2×C7×SD16—22no—
53C2×C7×M16—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D411yescharacteristic
58C7×C4⋊Q8C2312yes—
59C7×#127—21no—
60C7×#127D411yescharacteristic
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#125—22no—
65C7×#125C2312yes—
66C7×#124—21no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#124C2312yes—
70C7×#124C2311yescharacteristic
71C7×#124C2311yescharacteristic
72C7×#123—22no—
73C7×#123—21no—
74C7×#123—22no—
75C7×#123—21no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8D412yes—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121D412yes—
92C7×#120—21no—
93C7×#120—22no—
94C7×#120—21no—
95C7×#120—22no—
96C4×C7×Q8D412yes—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4D412yes—
102C7×#117—21no—
103C7×#117—22no—
104C2×C7×C4⋊C4—22no—
105C2×C42×C7—21no—
106C2×C42×C7C2311yescharacteristic
107C7×#107—24no—
108C7×#106—24no—
109C7×C4⋊C8—22no—
110C7×#103—24no—
111C7×#102—24no—
112C7×C22⋊C8—22no—
113C7×#97C2314yes—
114C7×#97C2312yes—
115C4×C7×C8—21no—
116#3529C2×C711yescharacteristic
117#3239C2×C712yescomplemented
118#3152C2×C711yescharacteristic, complemented
119#1661C2×C714yescomplemented
120#1660C2×C712yescomplemented
121#1659C2×C712yescomplemented
122#1536C2×C711yescharacteristic, complemented
123#1529C2×C712yescomplemented
124C24×C7—21no—
125C7×#41—42no—
126C7×#41—42no—
127C7×#41—82no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—42no—
131C2×C7×Q8—42no—
132C2×C7×Q8—41no—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×D411yescharacteristic
135C2×C7×Q8C2×D411yescharacteristic
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—21no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—22no—
144C2×C7×D4C2×D412yes—
145C2×C7×D4C2×D411yescharacteristic
146C2×C7×D4C2×D411yescharacteristic
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—41no—
154C22×C4×C7—82no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C7×Q16—44no—
158C7×Q16—44no—
159C7×SD16—44no—
160C7×SD16—44no—
161C7×M16—24no—
162C2×C7×C8—22no—
163C2×C7×C8—24no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4C2×D412yes—
175C7×C4⋊C4C2×D412yes—
176C7×C22⋊C4—21no—
177C7×C22⋊C4—21no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—22no—
192C42×C7C2×D411yescharacteristic
193C42×C7C2×D412yes—
194C42×C7C2411yescharacteristic
195C2×#137—21no—
196C2×#137—21no—
197#563—21no—
198#563C22×C711yescharacteristic
199#562—22no—
200#562C22×C712yes—
201H(3,𝔽4)C22×C711yescharacteristic
202#560—21no—
203#560—22no—
204#532—21no—
205#532C22×C711yescharacteristic
206C2×C4⋊Q8C22×C711yescharacteristic, complemented
207#529—22no—
208#529C22×C712yes—
209#525—21no—
210#525—22no—
211#501C22×C714yescomplemented
212#497C22×C712yes—
213#497C22×C712yescomplemented
214#494—21no—
215#492—21no—
216#489C22×C714yes—
217#484—22no—
218#481—22no—
219#475—22no—
220#474—22no—
221#432C22×C712yescomplemented
222C4×M16C22×C711yescharacteristic, complemented
223#330C22×C714yescomplemented
224#329C22×C712yes—
225#329C22×C714yes—
226#329C22×C712yescomplemented
227C23×C7—21no—
228C23×C7—22no—
229C23×C7—21no—
230C23×C7—81no—
231C23×C7—82no—
232C23×C7#12011yescharacteristic
233C7×Q8—41no—
234C7×Q8—41no—
235C7×Q8—42no—
236C7×Q8—42no—
237C7×Q8—42no—
238C7×Q8—81no—
239C7×Q8—82no—
240C7×D4—41no—
241C7×D4—41no—
242C7×D4—42no—
243C7×D4—82no—
244C7×D4—82no—
245C7×D4—81no—
246C7×D4—81no—
247C7×D4—82no—
248C7×D4—82no—
249C7×D4—82no—
250C7×D4—81no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—22no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—44no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—82no—
263C2×C4×C7—82no—
264C2×C4×C7—81no—
265C2×C4×C7—82no—
266C2×C4×C7#12012yes—
267C2×C4×C7#12011yescharacteristic
268C2×C4×C7C22×D412yes—
269C2×C4×C7C22×D411yescharacteristic
270C7×C8—24no—
271C7×C8—44no—
272C2×#41—21no—
273C2×#41C7×D411yescharacteristic
274C2×#41C7×D411yescharacteristic
275C22×Q8—21no—
276C23×C4—21no—
277#137—44no—
278#137—44no—
279C2×Q16—22no—
280C2×Q16—22no—
281C2×SD16—22no—
282C2×SD16—22no—
283C2×M16—21no—
284C4⋊Q8—21no—
285C4⋊Q8—22no—
286C4⋊Q8—21no—
287C4⋊Q8C7×D411yescharacteristic
288C4⋊Q8C23×C712yes—
289#127—21no—
290#127C7×D411yescharacteristic
291#126—22no—
292#126—22no—
293#126—22no—
294#125—22no—
295#125C23×C712yes—
296#124—21no—
297#124—22no—
298#124—21no—
299#124C23×C712yes—
300#124C23×C711yescharacteristic
301#124C23×C711yescharacteristic
302#123—22no—
303#123—21no—
304#123—22no—
305#123—21no—
306#123—22no—
307#123—22no—
308#123—22no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8C7×D412yes—
314#121—22no—
315#121—22no—
316#121—22no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—22no—
321#121C7×D412yes—
322#120—21no—
323#120—22no—
324#120—21no—
325#120—22no—
326C4×Q8C7×D412yes—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4C7×D412yes—
332#117—21no—
333#117—22no—
334C2×C4⋊C4—22no—
335C2×C42—21no—
336C2×C42C23×C711yescharacteristic
337#107—24no—
338#106—24no—
339C4⋊C8—22no—
340#103—24no—
341#102—24no—
342C22⋊C8—22no—
343#97C23×C714yes—
344#97C23×C712yes—
345C4×C8—21no—
346C22×C7—21no—
347C22×C7—22no—
348C22×C7—41no—
349C22×C7—42no—
350C22×C7—41no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—84no—
356C22×C7C2×#12011yescharacteristic
357C4×C7—21no—
358C4×C7—22no—
359C4×C7—22no—
360C4×C7—41no—
361C4×C7—42no—
362C4×C7—41no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—82no—
366C4×C7#45312yes—
367C24—21no—
368#41—42no—
369#41—42no—
370#41—82no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—41no—
376C2×Q8C2×C7×D412yes—
377C2×Q8C2×C7×D411yescharacteristic
378C2×Q8C2×C7×D411yescharacteristic
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4C2×C7×D412yes—
388C2×D4C2×C7×D411yescharacteristic
389C2×D4C2×C7×D411yescharacteristic
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—82no—
398C22×C4C2×C7×D411yescharacteristic
399C22×C4C2×C7×D412yes—
400Q16—44no—
401Q16—44no—
402SD16—44no—
403SD16—44no—
404M16—24no—
405C2×C8—22no—
406C2×C8—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4C2×C7×D412yes—
418C4⋊C4C2×C7×D412yes—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42—21no—
433C42—21no—
434C42—22no—
435C42C2×C7×D411yescharacteristic
436C42C2×C7×D412yes—
437C42C24×C711yescharacteristic
438C2×C7—41no—
439C2×C7—81no—
440C2×C7—81no—
441C2×C7—82no—
442C2×C7C2×#45311yescharacteristic
443C2×C7#302212yes—
444C23—21no—
445C23—22no—
446C23—21no—
447C23—81no—
448C23—82no—
449C23C7×#12011yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—81no—
456Q8—82no—
457D4—41no—
458D4—41no—
459D4—42no—
460D4—82no—
461D4—82no—
462D4—81no—
463D4—81no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—81no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4C7×#12012yes—
484C2×C4C7×#12011yescharacteristic
485C2×C4C22×C7×D412yes—
486C2×C4C22×C7×D411yescharacteristic
487C8—24no—
488C8—44no—
489C7#1899611yescharacteristic, direct product factor, Sylow
490C22—21no—
491C22—22no—
492C22—41no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—84no—
500C22C2×C7×#12011yescharacteristic
501C4—21no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4C7×#45312yes—
511C2—41no—
512C2—81no—
513C2—82no—
514C2—81no—
515C2C2×C7×#45311yescharacteristic
516C2C7×#302212yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3529C2
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×#3239C2semidirect
3. C7×#3152C2semidirect
4. C7×#1661C2semidirect
5. C7×#1660C2semidirect
6. C7×#1659C2semidirect
7. C7×#1536C2semidirect
8. C7×#1529C2semidirect
20. C2×C7×C4⋊Q8C22semidirect
25. C7×#501C22semidirect
27. C7×#497C22semidirect
35. C7×#432C22semidirect
36. C4×C7×M16C22semidirect
37. C7×#330C22semidirect
40. C7×#329C22semidirect
41. #18996C7direct (N × H)
117. #3239C2×C7semidirect
118. #3152C2×C7semidirect
119. #1661C2×C7semidirect
120. #1660C2×C7semidirect
121. #1659C2×C7semidirect
122. #1536C2×C7semidirect
123. #1529C2×C7semidirect
206. C2×C4⋊Q8C22×C7semidirect
211. #501C22×C7semidirect
213. #497C22×C7semidirect
221. #432C22×C7semidirect
222. C4×M16C22×C7semidirect
223. #330C22×C7semidirect
226. #329C22×C7semidirect
489. C7#18996direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35292
C7×#32392
C7×#31522
C7×#16612
C7×#16602
C7×#16592
C7×#15362
C7×#15292
#189967