Finite Group Databasebeta

← C7×#18432 #13195648

Subgroups — C7×#18432 #13195648

Order 1792.

940 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic
3C7×#3208C211yesmaximal, characteristic, complemented
4C7×#2641C211yesmaximal, characteristic, complemented
5C7×#1599C214yesmaximal, complemented
6C7×#1597C214yesmaximal, complemented
7C7×#1506C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic, complemented
9C7×#1470C211yesmaximal, characteristic, complemented
10C7×#564—21no—
11C7×#563—21no—
12C7×#557—21no—
13C7×#554—21no—
14C7×#552—21no—
15C7×#552—21no—
16C7×#552—22no—
17C7×#544—21no—
18C7×#544C2212yes—
19C7×#541—21no—
20C7×#541—22no—
21C7×#539C2212yes—
22C7×#533C2211yescharacteristic
23C2×C7×#127C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#496—21no—
27C7×#492—21no—
28C7×#490—21no—
29C7×#486—21no—
30C7×#460—22no—
31C7×#460—21no—
32C7×#459—21no—
33C7×#450C2214yes—
34C7×#450C2212yes—
35C7×#447C2212yes—
36C7×#432C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#394—22no—
39C7×#394C2214yes—
40C7×#390—21no—
41C7×#390—21no—
42C7×#376C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#324C2214yes—
47#18432C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4C2312yes—
51C2×C7×SD16—24no—
52C2×C7×SD16—22no—
53C2×C7×D8—22no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×#127—21no—
60C7×#127—21no—
61C7×#127—21no—
62C7×#127—21no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#126—21no—
68C7×#125—22no—
69C7×#125—22no—
70C7×#125—22no—
71C7×#125—22no—
72C7×#125—21no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—21no—
78C7×#125—21no—
79C7×#125—21no—
80C7×#125—21no—
81C7×#125—21no—
82C7×#125D411yescharacteristic
83C7×#124D411yescharacteristic
84C7×#123—22no—
85C7×#123—21no—
86C7×#123—22no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8C2311yescharacteristic
95C7×C22⋊Q8C2312yes—
96C7×#121C2311yescharacteristic
97C4×C7×Q8—21no—
98C4×C7×Q8—22no—
99C4×C7×Q8—21no—
100C4×C7×Q8—21no—
101C4×C7×Q8D411yescharacteristic
102C4×C7×Q8D412yes—
103C4×C7×D4—21no—
104C4×C7×D4—22no—
105C4×C7×D4D411yescharacteristic
106C7×#117—22no—
107C7×#117—21no—
108C7×#117—21no—
109C7×#117—22no—
110C7×#117—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4D411yescharacteristic
113C2×C7×C4⋊C4D411yescharacteristic
114C2×C7×C4⋊C4D412yes—
115C2×C7×C4⋊C4C2312yes—
116C2×C7×C22⋊C4—24no—
117C2×C7×C22⋊C4—24no—
118C2×C42×C7—21no—
119C2×C42×C7C2311yescharacteristic
120C7×C4⋊C8—21no—
121C7×C4⋊C8—21no—
122C7×#102—24no—
123C7×#102—22no—
124C7×#102—22no—
125C7×C22⋊C8C2312yes—
126C7×C22⋊C8C2312yes—
127C7×#97—21no—
128C4×C7×C8—21no—
129C7×#95—21no—
130C7×#95C2312yes—
131#3563C2×C711yescharacteristic, complemented
132#3217C2×C711yescharacteristic
133#3208C2×C711yescharacteristic, complemented
134#2641C2×C711yescharacteristic, complemented
135#1599C2×C714yescomplemented
136#1597C2×C714yescomplemented
137#1506C2×C711yescharacteristic, complemented
138#1500C2×C711yescharacteristic, complemented
139#1470C2×C711yescharacteristic, complemented
140C24×C7—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—21no—
143C2×C7×Q8C2×D412yes—
144C2×C7×Q8C2×D411yescharacteristic
145C2×C7×D4—21no—
146C2×C7×D4—22no—
147C2×C7×D4—22no—
148C2×C7×D4—44no—
149C2×C7×D4—41no—
150C2×C7×D4—41no—
151C2×C7×D4—41no—
152C2×C7×D4—41no—
153C2×C7×D4—82no—
154C2×C7×D4—82no—
155C2×C7×D4C2×D411yescharacteristic
156C22×C4×C7—22no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2411yescharacteristic
164C7×SD16—82no—
165C7×SD16—84no—
166C7×D8—82no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4C2×D411yescharacteristic
195C7×C4⋊C4C2×D411yescharacteristic
196C7×C4⋊C4C2×D412yes—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—84no—
207C7×C22⋊C4—84no—
208C42×C7—22no—
209C42×C7—22no—
210C42×C7—22no—
211C42×C7—22no—
212C42×C7—21no—
213C42×C7—42no—
214C42×C7C2×D411yescharacteristic
215C42×C7C2×D411yescharacteristic
216C42×C7C2×D411yescharacteristic
217C42×C7C2×D411yescharacteristic
218#564—21no—
219#563—21no—
220#557—21no—
221#554—21no—
222#552—21no—
223#552—21no—
224#552—22no—
225#544—21no—
226#544C22×C712yes—
227#541—21no—
228#541—22no—
229#539C22×C712yes—
230#533C22×C711yescharacteristic
231C2×#127C22×C711yescharacteristic
232#529C22×C711yescharacteristic
233C2×#125C22×C711yescharacteristic
234#496—21no—
235#492—21no—
236#490—21no—
237#486—21no—
238#460—22no—
239#460—21no—
240#459—21no—
241#450C22×C714yes—
242#450C22×C712yes—
243#447C22×C712yes—
244#432C22×C711yescharacteristic
245#431C22×C711yescharacteristic
246#394—22no—
247#394C22×C714yes—
248#390—21no—
249#390—21no—
250#376C22×C711yescharacteristic
251#328C22×C714yes—
252#328C22×C712yes—
253#327C22×C712yes—
254#324C22×C714yes—
255C23×C7—21no—
256C23×C7—44no—
257C23×C7—44no—
258C23×C7—42no—
259C23×C7—44no—
260C23×C7C22×D411yescharacteristic
261C7×Q8—41no—
262C7×Q8—42no—
263C7×Q8—42no—
264C7×Q8—82no—
265C7×D4—41no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—44no—
269C7×D4—42no—
270C7×D4—82no—
271C7×D4—82no—
272C7×D4—82no—
273C7×D4—82no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—82no—
291C2×C4×C7#12011yescharacteristic
292C2×C4×C7#12011yescharacteristic
293C2×C4×C7#12011yescharacteristic
294C2×C4×C7#12011yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C7×C8—42no—
298C7×C8—42no—
299C22×D4—21no—
300C22×D4—21no—
301C22×D4C23×C712yes—
302C2×SD16—24no—
303C2×SD16—22no—
304C2×D8—22no—
305C4⋊Q8—21no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8C7×D411yescharacteristic
309C4⋊Q8C7×D411yescharacteristic
310#127—21no—
311#127—21no—
312#127—21no—
313#127—21no—
314#126—22no—
315#126—22no—
316#126—22no—
317#126—22no—
318#126—21no—
319#125—22no—
320#125—22no—
321#125—22no—
322#125—22no—
323#125—21no—
324#125—22no—
325#125—22no—
326#125—22no—
327#125—22no—
328#125—21no—
329#125—21no—
330#125—21no—
331#125—21no—
332#125—21no—
333#125C7×D411yescharacteristic
334#124C7×D411yescharacteristic
335#123—22no—
336#123—21no—
337#123—22no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8C23×C711yescharacteristic
346C22⋊Q8C23×C712yes—
347#121C23×C711yescharacteristic
348C4×Q8—21no—
349C4×Q8—22no—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8C7×D411yescharacteristic
353C4×Q8C7×D412yes—
354C4×D4—21no—
355C4×D4—22no—
356C4×D4C7×D411yescharacteristic
357#117—22no—
358#117—21no—
359#117—21no—
360#117—22no—
361#117—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C7×D411yescharacteristic
364C2×C4⋊C4C7×D412yes—
365C2×C4⋊C4C7×D411yescharacteristic
366C2×C4⋊C4C23×C712yes—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C42—21no—
370C2×C42C23×C711yescharacteristic
371C4⋊C8—21no—
372C4⋊C8—21no—
373#102—22no—
374#102—22no—
375#102—24no—
376C22⋊C8C23×C712yes—
377C22⋊C8C23×C712yes—
378#97—21no—
379C4×C8—21no—
380#95—21no—
381#95C23×C712yes—
382C22×C7—21no—
383C22×C7—21no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—88no—
388C22×C7—82no—
389C22×C7—82no—
390C22×C7—82no—
391C22×C7C2×#12011yescharacteristic
392C22×C7#57712yes—
393C4×C7—21no—
394C4×C7—21no—
395C4×C7—21no—
396C4×C7—21no—
397C4×C7—21no—
398C4×C7—42no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—42no—
402C4×C7—42no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—82no—
406C4×C7#45412yes—
407C24—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8C2×C7×D412yes—
411C2×Q8C2×C7×D411yescharacteristic
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4C2×C7×D411yescharacteristic
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4C2×C7×D412yes—
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D412yes—
430C22×C4C24×C711yescharacteristic
431SD16—82no—
432SD16—84no—
433D8—82no—
434C2×C8—22no—
435C2×C8—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4C2×C7×D411yescharacteristic
462C4⋊C4C2×C7×D411yescharacteristic
463C4⋊C4C2×C7×D412yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—42no—
481C42C2×C7×D411yescharacteristic
482C42C2×C7×D411yescharacteristic
483C42C2×C7×D411yescharacteristic
484C42C2×C7×D411yescharacteristic
485C2×C7—22no—
486C2×C7—81no—
487C2×C7—162no—
488C2×C7#301011yescharacteristic
489C2×C7C2×#45411yescharacteristic
490C2×C7#302711yescharacteristic
491C23—21no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23C22×C7×D411yescharacteristic
497Q8—41no—
498Q8—42no—
499Q8—42no—
500Q8—82no—
501D4—41no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—42no—
506D4—82no—
507D4—82no—
508D4—82no—
509D4—82no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4C7×#12011yescharacteristic
528C2×C4C7×#12011yescharacteristic
529C2×C4C7×#12011yescharacteristic
530C2×C4C7×#12011yescharacteristic
531C2×C4C22×C7×D411yescharacteristic
532C2×C4C22×C7×D411yescharacteristic
533C8—42no—
534C8—42no—
535C7#1843211yescharacteristic, direct product factor, Sylow
536C22—21no—
537C22—21no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—88no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×C7×#12011yescharacteristic
546C22C7×#57712yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—41no—
559C4—82no—
560C4C7×#45412yes—
561C2—22no—
562C2—81no—
563C2—162no—
564C2C7×#301011yescharacteristic
565C2C2×C7×#45411yescharacteristic
566C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3217C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3563C2semidirect
3. C7×#3208C2semidirect
4. C7×#2641C2semidirect
5. C7×#1599C2semidirect
6. C7×#1597C2semidirect
7. C7×#1506C2semidirect
8. C7×#1500C2semidirect
9. C7×#1470C2semidirect
47. #18432C7direct (N × H)
131. #3563C2×C7semidirect
133. #3208C2×C7semidirect
134. #2641C2×C7semidirect
135. #1599C2×C7semidirect
136. #1597C2×C7semidirect
137. #1506C2×C7semidirect
138. #1500C2×C7semidirect
139. #1470C2×C7semidirect
535. C7#18432direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#32172
C7×#32082
C7×#26412
C7×#15992
C7×#15972
C7×#15062
C7×#15002
C7×#14702
#184327