Finite Group Databasebeta

← C7×#23489 #13200705

Subgroups — C7×#23489 #13200705

Order 1792.

1482 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C22×C7×#106C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C2×C7×#416C211yesmaximal, characteristic, complemented
5C2×C7×#408C211yesmaximal, characteristic, complemented
6C7×#2311C211yesmaximal, characteristic, complemented
7C7×#2035C214yesmaximal, complemented
8C7×#2034C214yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×#41C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C7×#537—24no—
15C7×#537—24no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×#121—21no—
19C2×C7×#121—22no—
20C2×C7×#121C2212yes—
21C2×C7×#117C2211yescharacteristic
22C22×C7×C4⋊C4C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C2×C7×#106—24no—
25C2×C7×#106C2214yes—
26C7×#417—24no—
27C7×#416—24no—
28C2×C7×#103C2212yes—
29C2×C7×#103C2212yes—
30C2×C7×#102C2212yes—
31C2×C7×#102C2212yes—
32C7×#408—24no—
33C2×C7×C22⋊C8C2214yes—
34C7×#391—21no—
35C7×#390—21no—
36C7×#386—22no—
37C7×#380C2214yes—
38C7×#340C2214yes—
39#23489C711yesmaximal, characteristic, direct product factor, Sylow
40C2×C7×#41—22no—
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41D414yes—
52C22×C7×Q8D411yescharacteristic
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4—22no—
56C22×C7×D4D412yes—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7—21no—
62C23×C4×C7C2312yes—
63C23×C4×C7C2311yescharacteristic
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8—22no—
68C22×C7×C8—22no—
69C22×C7×C8D414yescomplemented
70C22×C7×C8C2312yes—
71C22×C7×C8C2312yes—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8—24no—
74C7×C22⋊Q8—24no—
75C7×#121—216no—
76C7×#121—28no—
77C7×#121—24no—
78C7×#121—24no—
79C7×#117—22no—
80C7×#117—22no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4D414yes—
88C2×C7×C4⋊C4C2312yes—
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2312yes—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—24no—
95C2×C42×C7—21no—
96C7×#106—28no—
97C7×#103—24no—
98C7×#103—24no—
99C7×#102—24no—
100C7×#102—24no—
101C7×C22⋊C8—28no—
102C7×#95—24no—
103C2×#537C2×C711yescharacteristic
104C22×#106C2×C711yescharacteristic, complemented
105C2×#417C2×C711yescharacteristic, complemented
106C2×#416C2×C711yescharacteristic, complemented
107C2×#408C2×C711yescharacteristic, complemented
108#2311C2×C711yescharacteristic, complemented
109#2035C2×C714yescomplemented
110#2034C2×C714yescomplemented
111#1736C2×C711yescharacteristic, complemented
112C24×C7—21no—
113C24×C7C2×D412yes—
114C24×C7C2×D411yescharacteristic
115C7×#41—28no—
116C7×#41—28no—
117C7×#41—44no—
118C7×#41—44no—
119C7×#41—44no—
120C7×#41—44no—
121C7×#41—44no—
122C7×#41—44no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—28no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4—84no—
148C2×C7×D4C2×D412yes—
149C22×C4×C7—22no—
150C22×C4×C7—21no—
151C22×C4×C7—24no—
152C22×C4×C7—24no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—82no—
165C22×C4×C7—82no—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7#4112yes—
172C22×C4×C7#4112yes—
173C22×C4×C7#4112yes—
174C22×C4×C7C2411yescharacteristic
175C7×M16—44no—
176C2×C7×C8—28no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8C2×D414yes—
183C7×C4⋊C4—28no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4C2×D414yes—
192C7×C22⋊C4—28no—
193C7×C22⋊C4—28no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—84no—
196C42×C7—42no—
197C22×#41C22×C711yescharacteristic
198C22×#41C22×C711yescharacteristic
199C22×M16C22×C711yescharacteristic
200C23×C8C22×C711yescharacteristic
201#537—24no—
202#537—24no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8C22×C712yes—
205C2×#121—21no—
206C2×#121—22no—
207C2×#121C22×C712yes—
208C2×#117C22×C711yescharacteristic
209C22×C4⋊C4C22×C711yescharacteristic
210C22×C4⋊C4C22×C711yescharacteristic
211C2×#106—24no—
212C2×#106C22×C714yes—
213#417—24no—
214#416—24no—
215C2×#103C22×C712yes—
216C2×#103C22×C712yes—
217C2×#102C22×C712yes—
218C2×#102C22×C712yes—
219#408—24no—
220C2×C22⋊C8C22×C714yes—
221#391—21no—
222#390—21no—
223#386—22no—
224#380C22×C714yes—
225#340C22×C714yes—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—24no—
233C23×C7—24no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7#12012yes—
239C23×C7#12112yes—
240C23×C7#12312yes—
241C23×C7C22×D411yescharacteristic
242C7×Q8—22no—
243C7×Q8—22no—
244C7×Q8—41no—
245C7×Q8—41no—
246C7×Q8—44no—
247C7×D4—22no—
248C7×D4—22no—
249C7×D4—48no—
250C7×D4—48no—
251C7×D4—41no—
252C7×D4—41no—
253C7×D4—44no—
254C7×D4—88no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—44no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—48no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—44no—
272C2×C4×C7—42no—
273C2×C4×C7—84no—
274C2×C4×C7—84no—
275C2×C4×C7#12012yes—
276C2×C4×C7#12114yes—
277C2×C4×C7#12114yes—
278C2×C4×C7#12112yes—
279C2×C4×C7#12314yes—
280C2×C4×C7#12312yes—
281C2×C4×C7#12414yes—
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C7×C8—24no—
289C7×C8—42no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41C7×D414yes—
302C22×Q8C7×D411yescharacteristic
303C22×Q8C23×C711yescharacteristic
304C22×D4—21no—
305C22×D4—22no—
306C22×D4C7×D412yes—
307C22×D4C7×D411yescharacteristic
308C22×D4C23×C711yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C23×C712yes—
313C23×C4C23×C711yescharacteristic
314C2×M16—22no—
315C2×M16—22no—
316C2×M16—22no—
317C22×C8—22no—
318C22×C8—22no—
319C22×C8C7×D414yescomplemented
320C22×C8C23×C712yes—
321C22×C8C23×C712yes—
322C22⋊Q8—28no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325#121—216no—
326#121—28no—
327#121—24no—
328#121—24no—
329#117—22no—
330#117—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4C7×D414yes—
338C2×C4⋊C4C23×C712yes—
339C2×C4⋊C4C23×C712yes—
340C2×C4⋊C4C23×C712yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C42—21no—
346#106—28no—
347#103—24no—
348#103—24no—
349#102—24no—
350#102—24no—
351C22⋊C8—28no—
352#95—24no—
353C22×C7—24no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—24no—
357C22×C7—44no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—44no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7#39414yes—
368C22×C7C2×#12011yescharacteristic
369C22×C7C2×#12111yescharacteristic
370C22×C7C2×#12311yescharacteristic
371C22×C7#57712yes—
372C22×C7#57712yes—
373C4×C7—42no—
374C4×C7—42no—
375C4×C7—44no—
376C4×C7—82no—
377C4×C7—82no—
378C4×C7#39414yes—
379C4×C7C2×#12111yescharacteristic
380C4×C7C2×#12111yescharacteristic
381C4×C7C2×#12311yescharacteristic
382C4×C7C2×#12411yescharacteristic
383C24—21no—
384C24C2×C7×D412yes—
385C24C2×C7×D411yescharacteristic
386#41—28no—
387#41—28no—
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392#41—44no—
393#41—44no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C7×D412yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—84no—
419C2×D4C2×C7×D412yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4C2×C7×D412yes—
438C22×C4C2×C7×D411yescharacteristic
439C22×C4C2×C7×D412yes—
440C22×C4C2×C7×D412yes—
441C22×C4C2×C7×D412yes—
442C22×C4C7×#4112yes—
443C22×C4C7×#4112yes—
444C22×C4C7×#4112yes—
445C22×C4C24×C711yescharacteristic
446M16—44no—
447C2×C8—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8C2×C7×D414yes—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4C2×C7×D414yes—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—42no—
466C22⋊C4—84no—
467C42—42no—
468C2×C7—24no—
469C2×C7—82no—
470C2×C7—82no—
471C2×C7C2×#39411yescharacteristic
472C2×C7#300911yescharacteristic
473C2×C7#301011yescharacteristic
474C2×C7#306012yes—
475C2×C7#309912yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23C7×#12012yes—
489C23C7×#12112yes—
490C23C7×#12312yes—
491C23C22×C7×D411yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—44no—
497D4—22no—
498D4—22no—
499D4—48no—
500D4—48no—
501D4—41no—
502D4—41no—
503D4—44no—
504D4—88no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4C7×#12012yes—
526C2×C4C7×#12114yes—
527C2×C4C7×#12114yes—
528C2×C4C7×#12112yes—
529C2×C4C7×#12314yes—
530C2×C4C7×#12312yes—
531C2×C4C7×#12414yes—
532C2×C4C22×C7×D411yescharacteristic
533C2×C4C22×C7×D411yescharacteristic
534C2×C4C22×C7×D411yescharacteristic
535C2×C4C2×C7×#4111yescharacteristic
536C2×C4C2×C7×#4111yescharacteristic
537C2×C4C2×C7×#4111yescharacteristic
538C8—24no—
539C8—42no—
540C7#2348911yescharacteristic, direct product factor, Sylow
541C22—22no—
542C22—22no—
543C22—24no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22C7×#39414yes—
556C22C2×C7×#12011yescharacteristic
557C22C2×C7×#12111yescharacteristic
558C22C2×C7×#12311yescharacteristic
559C22C7×#57712yes—
560C22C7×#57712yes—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4C7×#39414yes—
567C4C2×C7×#12111yescharacteristic
568C4C2×C7×#12111yescharacteristic
569C4C2×C7×#12311yescharacteristic
570C4C2×C7×#12411yescharacteristic
571C2—24no—
572C2—82no—
573C2—82no—
574C2C2×C7×#39411yescharacteristic
575C2C7×#300911yescharacteristic
576C2C7×#301011yescharacteristic
577C2C7×#306012yes—
578C2C7×#309912yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C7×#106C2semidirect
3. C2×C7×#417C2semidirect
4. C2×C7×#416C2semidirect
5. C2×C7×#408C2semidirect
6. C7×#2311C2semidirect
7. C7×#2035C2semidirect
8. C7×#2034C2semidirect
9. C7×#1736C2semidirect
39. #23489C7direct (N × H)
69. C22×C7×C8D4semidirect
104. C22×#106C2×C7semidirect
105. C2×#417C2×C7semidirect
106. C2×#416C2×C7semidirect
107. C2×#408C2×C7semidirect
108. #2311C2×C7semidirect
109. #2035C2×C7semidirect
110. #2034C2×C7semidirect
111. #1736C2×C7semidirect
319. C22×C8C7×D4semidirect
540. C7#23489direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C22×C7×#1062
C2×C7×#4172
C2×C7×#4162
C2×C7×#4082
C7×#23112
C7×#20352
C7×#20342
C7×#17362
#234897