Finite Group Databasebeta

← C7×#23604 #13200820

Subgroups — C7×#23604 #13200820

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#430C211yesmaximal, characteristic, complemented
2C2×C7×#428C211yesmaximal, characteristic, complemented
3C7×#2905C211yesmaximal, characteristic, complemented
4C7×#2896C211yesmaximal, characteristic
5C2×C7×#417C211yesmaximal, characteristic
6C2×C7×#413C211yesmaximal, characteristic
7C7×#2064C214yesmaximal, complemented
8C7×#2063C214yesmaximal, complemented
9C2×C7×#343C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C7×#430—24no—
18C2×C7×#108C2214yes—
19C7×#428—22no—
20C7×#428C2214yes—
21C2×C7×#107—21no—
22C2×C7×#106—21no—
23C7×#424—22no—
24C7×#424C2214yes—
25C7×#423—22no—
26C7×#419C2212yes—
27C7×#418C2212yes—
28C7×#417—22no—
29C7×#417—22no—
30C7×#417—21no—
31C7×#417—21no—
32C7×#416—22no—
33C2×C7×#103C2212yes—
34C2×C7×#102C2212yes—
35C7×#413—24no—
36C2×C7×#101C2212yes—
37C2×C7×#100C2212yes—
38C7×#343C2214yes—
39#23604C711yesmaximal, characteristic, direct product factor, Sylow
40C2×C7×#41—22no—
41C2×C7×#41—21no—
42C2×C7×#41—21no—
43C2×C7×#41—21no—
44C2×C7×#41—21no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41Q814yes—
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C2×C7×M16—21no—
55C2×C7×M16—22no—
56C2×C7×M16—22no—
57C2×C7×M16—22no—
58C2×C7×M16—21no—
59C2×C7×M16—21no—
60C2×C7×M16—22no—
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C2×C7×M16D414yes—
65C2×C7×M16D412yes—
66C2×C7×M16D412yes—
67C2×C7×M16C2312yes—
68C2×C7×M16C2312yes—
69C2×C7×M16C2312yes—
70C22×C7×C8—21no—
71C22×C7×C8—21no—
72C22×C7×C8—21no—
73C22×C7×C8C2312yes—
74C7×#117—21no—
75C7×#117—21no—
76C7×#117D412yes—
77C7×#117C2312yes—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—21no—
88C7×#108—28no—
89C7×#107—24no—
90C7×#106—24no—
91C7×C4⋊C8—28no—
92C7×#103—24no—
93C7×#103—22no—
94C7×#103—22no—
95C7×#102—24no—
96C7×#102—22no—
97C7×#102—22no—
98C7×#101—24no—
99C7×#100—24no—
100C2×#430C2×C711yescharacteristic, complemented
101C2×#428C2×C711yescharacteristic, complemented
102#2905C2×C711yescharacteristic, complemented
103#2896C2×C711yescharacteristic
104C2×#417C2×C711yescharacteristic
105C2×#413C2×C711yescharacteristic
106#2064C2×C714yescomplemented
107#2063C2×C714yescomplemented
108C2×#343C2×C711yescharacteristic, complemented
109C24×C7—21no—
110C24×C7#4111yescharacteristic
111C7×#41—24no—
112C7×#41—24no—
113C7×#41—24no—
114C7×#41—24no—
115C7×#41—42no—
116C7×#41—42no—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8C2×Q812yes—
125C2×C7×D4—24no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—41no—
132C2×C7×D4—41no—
133C2×C7×D4—42no—
134C2×C7×D4C2×Q812yes—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—24no—
139C22×C4×C7—22no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7—42no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7—82no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×Q812yes—
152C22×C4×C7#4111yescharacteristic
153C22×C4×C7#4112yes—
154C22×C4×C7#4112yes—
155C22×C4×C7C2411yescharacteristic
156C7×M16—24no—
157C7×M16—24no—
158C7×M16—24no—
159C7×M16—22no—
160C7×M16—22no—
161C7×M16—22no—
162C7×M16—22no—
163C7×M16—22no—
164C7×M16—42no—
165C7×M16—42no—
166C7×M16C2×D414yes—
167C2×C7×C8—24no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C2×C7×C8—22no—
173C2×C7×C8—42no—
174C2×C7×C8—42no—
175C2×C7×C8C2×D414yes—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—41no—
181C7×C4⋊C4—41no—
182C7×C4⋊C4C2×D412yes—
183C7×C4⋊C4C2×D412yes—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C42×C7—22no—
187C42×C7—22no—
188C22×#41C22×C711yescharacteristic
189C22×M16C22×C711yescharacteristic
190C22×M16C22×C711yescharacteristic
191C22×M16C22×C711yescharacteristic
192C22×M16C22×C711yescharacteristic
193C2×#117C22×C711yescharacteristic
194C22×C4⋊C4C22×C711yescharacteristic
195#430—24no—
196C2×#108C22×C714yes—
197#428—22no—
198#428C22×C714yes—
199C2×#107—21no—
200C2×#106—21no—
201#424—22no—
202#424C22×C714yes—
203#423—22no—
204#419C22×C712yes—
205#418C22×C712yes—
206#417—22no—
207#417—22no—
208#417—21no—
209#417—21no—
210#416—22no—
211C2×#103C22×C712yes—
212C2×#102C22×C712yes—
213#413—24no—
214C2×#101C22×C712yes—
215C2×#100C22×C712yes—
216#343C22×C714yes—
217C23×C7—22no—
218C23×C7—41no—
219C23×C7—41no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—81no—
223C23×C7—81no—
224C23×C7#12111yescharacteristic
225C23×C7#12111yescharacteristic
226C23×C7C22⋊Q812yes—
227C23×C7#12612yes—
228C23×C7C2×#4111yescharacteristic
229C7×Q8—22no—
230C7×Q8—22no—
231C7×Q8—41no—
232C7×Q8—41no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—44no—
236C7×D4—44no—
237C7×D4—41no—
238C7×D4—41no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—44no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—81no—
248C2×C4×C7—81no—
249C2×C4×C7—82no—
250C2×C4×C7—82no—
251C2×C4×C7—82no—
252C2×C4×C7#12112yes—
253C2×C4×C7#12112yes—
254C2×C4×C7#12112yes—
255C2×C4×C7#12112yes—
256C2×C4×C7#12111yescharacteristic
257C2×C4×C7#12111yescharacteristic
258C2×C4×C7C22⋊Q812yes—
259C2×C4×C7C22⋊Q812yes—
260C2×C4×C7C22⋊Q812yes—
261C2×C4×C7C22⋊Q812yes—
262C2×C4×C7C22⋊Q812yes—
263C2×C4×C7#12612yes—
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C22×Q811yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C7×C8—24no—
271C7×C8—24no—
272C7×C8—42no—
273C7×C8—42no—
274C2×#41—22no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41—21no—
278C2×#41—21no—
279C2×#41—22no—
280C2×#41—22no—
281C2×#41C7×Q814yes—
282C22×Q8C23×C711yescharacteristic
283C22×D4—21no—
284C22×D4C23×C711yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C23×C711yescharacteristic
288C2×M16—21no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—21no—
293C2×M16—21no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16C7×D414yes—
299C2×M16C7×D412yes—
300C2×M16C7×D412yes—
301C2×M16C23×C712yes—
302C2×M16C23×C712yes—
303C2×M16C23×C712yes—
304C22×C8—21no—
305C22×C8—21no—
306C22×C8—21no—
307C22×C8C23×C712yes—
308#117—21no—
309#117—21no—
310#117C7×D412yes—
311#117C23×C712yes—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C7×D411yescharacteristic
318C2×C4⋊C4C7×D411yescharacteristic
319C2×C4⋊C4C23×C712yes—
320C2×C22⋊C4—21no—
321C2×C42—21no—
322#108—28no—
323#107—24no—
324#106—24no—
325C4⋊C8—28no—
326#103—24no—
327#103—22no—
328#103—22no—
329#102—24no—
330#102—22no—
331#102—22no—
332#101—24no—
333#100—24no—
334C22×C7—21no—
335C22×C7—21no—
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—21no—
339C22×C7—21no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—82no—
344C22×C7—82no—
345C22×C7—81no—
346C22×C7—81no—
347C22×C7#39612yes—
348C22×C7#39612yes—
349C22×C7C2×#12111yescharacteristic
350C22×C7C2×C22⋊Q811yescharacteristic
351C22×C7C2×#12611yescharacteristic
352C4×C7—42no—
353C4×C7—42no—
354C4×C7—42no—
355C4×C7—82no—
356C4×C7#39612yes—
357C4×C7#39612yes—
358C4×C7C2×#12111yescharacteristic
359C4×C7C2×#12111yescharacteristic
360C4×C7C2×C22⋊Q811yescharacteristic
361C4×C7C2×C22⋊Q811yescharacteristic
362C24—21no—
363C24C7×#4111yescharacteristic
364#41—24no—
365#41—24no—
366#41—24no—
367#41—24no—
368#41—42no—
369#41—42no—
370#41—42no—
371#41—42no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8C2×C7×Q812yes—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—42no—
387C2×D4C2×C7×Q812yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—82no—
399C22×C4C2×C7×D412yes—
400C22×C4C2×C7×D411yescharacteristic
401C22×C4C2×C7×D411yescharacteristic
402C22×C4C2×C7×D411yescharacteristic
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×Q812yes—
405C22×C4C7×#4111yescharacteristic
406C22×C4C7×#4112yes—
407C22×C4C7×#4112yes—
408C22×C4C24×C711yescharacteristic
409M16—24no—
410M16—24no—
411M16—24no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—42no—
418M16—42no—
419M16C2×C7×D414yes—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8C2×C7×D414yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4C2×C7×D412yes—
436C4⋊C4C2×C7×D412yes—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C42—22no—
440C42—22no—
441C2×C7—21no—
442C2×C7—21no—
443C2×C7—22no—
444C2×C7—22no—
445C2×C7—82no—
446C2×C7C2×#39611yescharacteristic
447C2×C7#306811yescharacteristic
448C2×C7#306911yescharacteristic
449C23—22no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—81no—
455C23—81no—
456C23C7×#12111yescharacteristic
457C23C7×#12111yescharacteristic
458C23C7×C22⋊Q812yes—
459C23C7×#12612yes—
460C23C2×C7×#4111yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—41no—
464Q8—41no—
465D4—22no—
466D4—22no—
467D4—44no—
468D4—44no—
469D4—41no—
470D4—41no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4C7×#12112yes—
485C2×C4C7×#12112yes—
486C2×C4C7×#12112yes—
487C2×C4C7×#12112yes—
488C2×C4C7×#12111yescharacteristic
489C2×C4C7×#12111yescharacteristic
490C2×C4C7×C22⋊Q812yes—
491C2×C4C7×C22⋊Q812yes—
492C2×C4C7×C22⋊Q812yes—
493C2×C4C7×C22⋊Q812yes—
494C2×C4C7×C22⋊Q812yes—
495C2×C4C7×#12612yes—
496C2×C4C22×C7×D411yescharacteristic
497C2×C4C22×C7×D411yescharacteristic
498C2×C4C22×C7×D411yescharacteristic
499C2×C4C22×C7×Q811yescharacteristic
500C2×C4C2×C7×#4111yescharacteristic
501C2×C4C2×C7×#4111yescharacteristic
502C8—24no—
503C8—24no—
504C8—42no—
505C8—42no—
506C7#2360411yescharacteristic, direct product factor, Sylow
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—82no—
518C22—81no—
519C22—81no—
520C22C7×#39612yes—
521C22C7×#39612yes—
522C22C2×C7×#12111yescharacteristic
523C22C2×C7×C22⋊Q811yescharacteristic
524C22C2×C7×#12611yescharacteristic
525C4—42no—
526C4—42no—
527C4—42no—
528C4—82no—
529C4C7×#39612yes—
530C4C7×#39612yes—
531C4C2×C7×#12111yescharacteristic
532C4C2×C7×#12111yescharacteristic
533C4C2×C7×C22⋊Q811yescharacteristic
534C4C2×C7×C22⋊Q811yescharacteristic
535C2—21no—
536C2—21no—
537C2—22no—
538C2—22no—
539C2—82no—
540C2C2×C7×#39611yescharacteristic
541C2C7×#306811yescharacteristic
542C2C7×#306911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#121
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#430C2semidirect
2. C2×C7×#428C2semidirect
3. C7×#2905C2semidirect
7. C7×#2064C2semidirect
8. C7×#2063C2semidirect
9. C2×C7×#343C2semidirect
39. #23604C7direct (N × H)
100. C2×#430C2×C7semidirect
101. C2×#428C2×C7semidirect
102. #2905C2×C7semidirect
106. #2064C2×C7semidirect
107. #2063C2×C7semidirect
108. C2×#343C2×C7semidirect
506. C7#23604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4302
C2×C7×#4282
C7×#29052
C7×#28962
C2×C7×#4172
C2×C7×#4132
C7×#20642
C7×#20632
C2×C7×#3432
#236047