Finite Group Databasebeta

← C7×#23494 #13200710

Subgroups — C7×#23494 #13200710

Order 1792.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C7×#2911C211yesmaximal, characteristic, complemented
3C2×C7×#421C211yesmaximal, characteristic
4C7×#2896C211yesmaximal, characteristic
5C22×C7×#101C211yesmaximal, characteristic, complemented
6C7×#2310C211yesmaximal, characteristic, complemented
7C7×#2038C218yesmaximal, complemented
8C7×#1733C211yesmaximal, characteristic, complemented
9C22×C7×#41C2211yescharacteristic
10C23×C7×Q8C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C7×#536—22no—
14C7×#536—22no—
15C7×#536—22no—
16C7×#536C2214yes—
17C2×C7×C4⋊Q8—21no—
18C2×C7×#124—21no—
19C2×C7×#123—22no—
20C2×C7×C22⋊Q8—22no—
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic
23C2×C7×#117C2211yescharacteristic
24C7×#428—22no—
25C7×#428C2214yescomplemented
26C7×#427—22no—
27C7×#421—24no—
28C2×C7×#104C2214yes—
29C7×#419—21no—
30C7×#418—21no—
31C7×#417C2214yes—
32C7×#416—22no—
33C2×C7×#101—24no—
34C2×C7×#101C2214yescomplemented
35C7×#391—22no—
36C7×#391C2214yescomplemented
37C7×#381—22no—
38C7×#339C2214yescomplemented
39#23494C711yesmaximal, characteristic, direct product factor, Sylow
40C2×C7×#41—21no—
41C2×C7×#41—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41D412yes—
47C2×C7×#41C2312yes—
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8—21no—
51C22×C7×Q8—21no—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—22no—
55C22×C7×Q8—22no—
56C22×C7×Q8D411yescharacteristic
57C22×C7×Q8D412yes—
58C22×C7×Q8C2312yes—
59C22×C7×D4—21no—
60C22×C7×D4D411yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7D412yes—
63C23×C4×C7C2311yescharacteristic
64C2×C7×M16—21no—
65C2×C7×M16—22no—
66C2×C7×M16—21no—
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C2×C7×M16D414yes—
70C2×C7×M16C2312yes—
71C2×C7×M16C2312yes—
72C22×C7×C8—21no—
73C22×C7×C8—21no—
74C7×C4⋊Q8—24no—
75C7×C4⋊Q8—24no—
76C7×#124—24no—
77C7×#124—24no—
78C7×#123—28no—
79C7×#123—28no—
80C7×C22⋊Q8—28no—
81C7×C22⋊Q8—28no—
82C7×#117—21no—
83C7×#117—21no—
84C7×#117—22no—
85C7×#117—22no—
86C7×#117—21no—
87C7×#117—21no—
88C7×#117—21no—
89C7×#117—21no—
90C7×#117D412yes—
91C7×#117C2312yes—
92C2×C7×C4⋊C4—24no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4D412yes—
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C22⋊C4—24no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C2×C42×C7C2312yes—
103C7×#107—24no—
104C7×#106—24no—
105C7×#104—28no—
106C7×#103—24no—
107C7×#102—24no—
108C7×#101—28no—
109C7×#95—28no—
110C2×#536C2×C711yescharacteristic
111#2911C2×C711yescharacteristic, complemented
112C2×#421C2×C711yescharacteristic
113#2896C2×C711yescharacteristic
114C22×#101C2×C711yescharacteristic, complemented
115#2310C2×C711yescharacteristic, complemented
116#2038C2×C718yescomplemented
117#1733C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7C2×D411yescharacteristic
120C7×#41—24no—
121C7×#41—24no—
122C7×#41—22no—
123C7×#41—22no—
124C7×#41—22no—
125C7×#41—22no—
126C7×#41—42no—
127C7×#41—42no—
128C7×#41—42no—
129C7×#41—42no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—21no—
133C2×C7×Q8—24no—
134C2×C7×Q8—24no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—42no—
144C2×C7×Q8—42no—
145C2×C7×Q8—42no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—42no—
149C2×C7×Q8—42no—
150C2×C7×Q8—42no—
151C2×C7×Q8C2×D412yes—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—21no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4C2×D412yes—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—21no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—24no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7C2×D414yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D412yes—
179C22×C4×C7#4112yes—
180C22×C4×C7#4112yes—
181C22×C4×C7#4112yes—
182C22×C4×C7C2411yescharacteristic
183C7×M16—24no—
184C7×M16—22no—
185C7×M16—22no—
186C7×M16—22no—
187C7×M16—42no—
188C7×M16C2×D414yes—
189C2×C7×C8—24no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C2×C7×C8—42no—
193C7×C4⋊C4—216no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—42no—
198C7×C4⋊C4C2×D414yes—
199C7×C22⋊C4—216no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—41no—
203C7×C22⋊C4—41no—
204C7×C22⋊C4—42no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7—22no—
208C42×C7—22no—
209C42×C7—41no—
210C42×C7—41no—
211C22×#41C22×C711yescharacteristic
212C23×Q8C22×C711yescharacteristic
213C22×M16C22×C711yescharacteristic
214C22×M16C22×C711yescharacteristic
215#536—22no—
216#536—22no—
217#536—22no—
218#536C22×C714yes—
219C2×C4⋊Q8—21no—
220C2×#124—21no—
221C2×#123—22no—
222C2×C22⋊Q8—22no—
223C2×#117C22×C711yescharacteristic
224C2×#117C22×C711yescharacteristic
225C2×#117C22×C711yescharacteristic
226#428—22no—
227#428C22×C714yescomplemented
228#427—22no—
229#421—24no—
230C2×#104C22×C714yes—
231#419—21no—
232#418—21no—
233#417C22×C714yes—
234#416—22no—
235C2×#101—24no—
236C2×#101C22×C714yescomplemented
237#391—22no—
238#391C22×C714yescomplemented
239#381—22no—
240#339C22×C714yescomplemented
241C23×C7—22no—
242C23×C7—42no—
243C23×C7—44no—
244C23×C7—42no—
245C23×C7#12011yescharacteristic
246C23×C7#12011yescharacteristic
247C23×C7#12112yes—
248C23×C7#12312yes—
249C23×C7C22×D411yescharacteristic
250C7×Q8—22no—
251C7×Q8—22no—
252C7×Q8—42no—
253C7×Q8—42no—
254C7×Q8—42no—
255C7×Q8—42no—
256C7×Q8—44no—
257C7×Q8—44no—
258C7×Q8—42no—
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—42no—
262C7×D4—42no—
263C7×D4—44no—
264C7×D4—42no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—44no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—44no—
277C2×C4×C7—44no—
278C2×C4×C7—44no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7#12011yescharacteristic
282C2×C4×C7#12011yescharacteristic
283C2×C4×C7#12112yes—
284C2×C4×C7#12112yes—
285C2×C4×C7#12112yes—
286C2×C4×C7#12112yes—
287C2×C4×C7#12112yes—
288C2×C4×C7#12312yes—
289C2×C4×C7#12312yes—
290C2×C4×C7#12312yes—
291C2×C4×C7#12412yes—
292C2×C4×C7#12412yes—
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C7×C8—24no—
300C7×C8—42no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41C7×D412yes—
308C2×#41C23×C712yes—
309C22×Q8—21no—
310C22×Q8—21no—
311C22×Q8—21no—
312C22×Q8—21no—
313C22×Q8—22no—
314C22×Q8—22no—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8C7×D411yescharacteristic
318C22×Q8C7×D412yes—
319C22×Q8C23×C712yes—
320C22×D4—21no—
321C22×D4C7×D411yescharacteristic
322C23×C4—21no—
323C23×C4C7×D412yes—
324C23×C4C23×C711yescharacteristic
325C2×M16—21no—
326C2×M16—22no—
327C2×M16—21no—
328C2×M16—22no—
329C2×M16—22no—
330C2×M16C7×D414yes—
331C2×M16C23×C712yes—
332C2×M16C23×C712yes—
333C22×C8—21no—
334C22×C8—21no—
335C4⋊Q8—24no—
336C4⋊Q8—24no—
337#124—24no—
338#124—24no—
339#123—28no—
340#123—28no—
341C22⋊Q8—28no—
342C22⋊Q8—28no—
343#117—21no—
344#117—21no—
345#117—22no—
346#117—22no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—21no—
351#117C7×D412yes—
352#117C23×C712yes—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4C7×D412yes—
356C2×C4⋊C4C23×C712yes—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C23×C712yes—
364#107—24no—
365#106—24no—
366#104—28no—
367#103—24no—
368#102—24no—
369#101—28no—
370#95—28no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—44no—
378C22×C7—84no—
379C22×C7#39412yes—
380C22×C7#39412yes—
381C22×C7C2×#12011yescharacteristic
382C22×C7C2×#12111yescharacteristic
383C22×C7C2×#12311yescharacteristic
384C4×C7—44no—
385C4×C7—42no—
386C4×C7—44no—
387C4×C7—82no—
388C4×C7—82no—
389C4×C7#39412yes—
390C4×C7#39412yes—
391C4×C7C2×#12111yescharacteristic
392C4×C7C2×#12111yescharacteristic
393C4×C7C2×#12311yescharacteristic
394C4×C7C2×#12411yescharacteristic
395C24—21no—
396C24C2×C7×D411yescharacteristic
397#41—24no—
398#41—24no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—42no—
404#41—42no—
405#41—42no—
406#41—42no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8C2×C7×D412yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C2×C7×D412yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4C2×C7×D414yes—
450C22×C4C2×C7×D411yescharacteristic
451C22×C4C2×C7×D411yescharacteristic
452C22×C4C2×C7×D411yescharacteristic
453C22×C4C2×C7×D411yescharacteristic
454C22×C4C2×C7×D411yescharacteristic
455C22×C4C2×C7×D412yes—
456C22×C4C7×#4112yes—
457C22×C4C7×#4112yes—
458C22×C4C7×#4112yes—
459C22×C4C24×C711yescharacteristic
460M16—24no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—42no—
465M16C2×C7×D414yes—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—42no—
470C4⋊C4—216no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4C2×C7×D414yes—
476C22⋊C4—216no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—41no—
487C42—41no—
488C2×C7—21no—
489C2×C7—22no—
490C2×C7—21no—
491C2×C7—22no—
492C2×C7—82no—
493C2×C7C2×#39411yescharacteristic
494C2×C7#302012yes—
495C23—22no—
496C23—42no—
497C23—44no—
498C23—42no—
499C23C7×#12011yescharacteristic
500C23C7×#12011yescharacteristic
501C23C7×#12112yes—
502C23C7×#12312yes—
503C23C22×C7×D411yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—44no—
511Q8—44no—
512Q8—42no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—42no—
517D4—44no—
518D4—42no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4C7×#12011yescharacteristic
536C2×C4C7×#12011yescharacteristic
537C2×C4C7×#12112yes—
538C2×C4C7×#12112yes—
539C2×C4C7×#12112yes—
540C2×C4C7×#12112yes—
541C2×C4C7×#12112yes—
542C2×C4C7×#12312yes—
543C2×C4C7×#12312yes—
544C2×C4C7×#12312yes—
545C2×C4C7×#12412yes—
546C2×C4C7×#12412yes—
547C2×C4C22×C7×D411yescharacteristic
548C2×C4C22×C7×D411yescharacteristic
549C2×C4C22×C7×D411yescharacteristic
550C2×C4C2×C7×#4111yescharacteristic
551C2×C4C2×C7×#4111yescharacteristic
552C2×C4C2×C7×#4111yescharacteristic
553C8—24no—
554C8—42no—
555C7#2349411yescharacteristic, direct product factor, Sylow
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—44no—
562C22—42no—
563C22—84no—
564C22C7×#39412yes—
565C22C7×#39412yes—
566C22C2×C7×#12011yescharacteristic
567C22C2×C7×#12111yescharacteristic
568C22C2×C7×#12311yescharacteristic
569C4—44no—
570C4—44no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4C7×#39412yes—
575C4C7×#39412yes—
576C4C2×C7×#12111yescharacteristic
577C4C2×C7×#12111yescharacteristic
578C4C2×C7×#12311yescharacteristic
579C4C2×C7×#12411yescharacteristic
580C2—21no—
581C2—22no—
582C2—21no—
583C2—22no—
584C2—82no—
585C2C2×C7×#39411yescharacteristic
586C2C7×#302012yes—
subgroupG/H
center, socleC22×C7C2×#120
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
2. C7×#2911C2semidirect
5. C22×C7×#101C2semidirect
6. C7×#2310C2semidirect
7. C7×#2038C2semidirect
8. C7×#1733C2semidirect
25. C7×#428C22semidirect
34. C2×C7×#101C22semidirect
36. C7×#391C22semidirect
38. C7×#339C22semidirect
39. #23494C7direct (N × H)
111. #2911C2×C7semidirect
114. C22×#101C2×C7semidirect
115. #2310C2×C7semidirect
116. #2038C2×C7semidirect
117. #1733C2×C7semidirect
227. #428C22×C7semidirect
236. C2×#101C22×C7semidirect
238. #391C22×C7semidirect
240. #339C22×C7semidirect
555. C7#23494direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C7×#29112
C2×C7×#4212
C7×#28962
C22×C7×#1012
C7×#23102
C7×#20382
C7×#17332
#234947