Finite Group Databasebeta

← C7×#33558 #13210774

Subgroups — C7×#33558 #13210774

Order 1792.

1158 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3545C211yesmaximal, characteristic, complemented
2C7×#3026C212yesmaximal, complemented
3C2×C7×#455C211yesmaximal, characteristic, complemented
4C7×#2201C212yesmaximal, complemented
5C7×#2199C212yesmaximal, complemented
6C7×#2197C214yesmaximal, complemented
7C2×C7×#354C211yesmaximal, characteristic, complemented
8C7×#2121C212yesmaximal, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C2×C7×#137—21no—
11C7×#563C2211yescharacteristic, complemented
12C7×#562—21no—
13C7×H(3,𝔽4)C2211yescharacteristic, complemented
14C7×#555—21no—
15C7×#551—22no—
16C7×#551C2212yescomplemented
17C7×#546—22no—
18C7×#542—22no—
19C7×#538—22no—
20C7×#536—21no—
21C7×#534—22no—
22C7×#534C2212yes—
23C7×#529—21no—
24C2×C7×#124C2211yescharacteristic
25C7×#458—22no—
26C7×#458C2214yes—
27C7×B(4,𝔽2)—22no—
28C7×B(4,𝔽2)C2214yes—
29C7×#455—21no—
30C7×#455—21no—
31C7×#455C2212yes—
32C7×#455C2212yes—
33C2×C7×#104—21no—
34C2×C7×#100C2211yescharacteristic
35C2×C7×#99—22no—
36C2×C7×#99C2212yes—
37C7×#354C2212yescomplemented
38C7×#354C2212yescomplemented
39C7×#352C2214yescomplemented
40C7×AΣL(1,𝔽16)C2214yescomplemented
41#33558C711yesmaximal, characteristic, direct product factor, Sylow
42C7×H(5,𝔽2)—22no—
43C7×H(5,𝔽2)—21no—
44C7×H(5,𝔽2)—42no—
45C7×H(5,𝔽2)C2312yes—
46C2×C7×#41—21no—
47C2×C7×#41—21no—
48C2×C7×#41—41no—
49C22×C7×Q8D411yescharacteristic, complemented
50C22×C7×D4—22no—
51C22×C7×D4—41no—
52C22×C7×D4D412yescomplemented
53C22×C7×D4C2311yescharacteristic
54C7×#137—22no—
55C7×#137—22no—
56C7×#137—42no—
57C2×C7×Q16—41no—
58C2×C7×SD16—41no—
59C2×C7×M16—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#125—21no—
66C7×#124—21no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#124—21no—
72C7×#124D411yescharacteristic, complemented
73C7×#124D411yescharacteristic, complemented
74C7×#124C2311yescharacteristic
75C7×#124C2311yescharacteristic
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—22no—
81C7×#123—24no—
82C7×#123—42no—
83C7×#123D412yescomplemented
84C7×#123C2312yes—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×#121—21no—
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—22no—
94C7×#121—22no—
95C7×#121—44no—
96C7×#120—22no—
97C7×#120—22no—
98C7×#120—24no—
99C7×#120—42no—
100C7×#120D412yescomplemented
101C7×#120C2312yes—
102C4×C7×Q8—21no—
103C4×C7×D4—21no—
104C4×C7×D4—22no—
105C4×C7×D4—22no—
106C4×C7×D4—22no—
107C7×#117—21no—
108C7×#117—22no—
109C7×#117—22no—
110C2×C7×C22⋊C4—22no—
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4—22no—
113C2×C7×C22⋊C4D412yescomplemented
114C2×C42×C7D411yescharacteristic, complemented
115C7×#104—22no—
116C7×#104—22no—
117C7×#100—21no—
118C7×#100C2312yes—
119C7×#99—22no—
120C7×#99—24no—
121C7×#99—24no—
122C7×#99C2314yes—
123#3545C2×C711yescharacteristic, complemented
124#3026C2×C712yescomplemented
125C2×#455C2×C711yescharacteristic, complemented
126#2201C2×C712yescomplemented
127#2199C2×C712yescomplemented
128#2197C2×C714yescomplemented
129C2×#354C2×C711yescharacteristic, complemented
130#2121C2×C712yescomplemented
131C24×C7—22no—
132C24×C7C2×D412yes—
133C24×C7C2×D412yes—
134C7×#41—22no—
135C7×#41—22no—
136C7×#41—42no—
137C7×#41—41no—
138C7×#41—44no—
139C7×#41—42no—
140C7×#41—41no—
141C7×#41—81no—
142C7×#41—81no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—41no—
148C2×C7×Q8—41no—
149C2×C7×Q8C2×D411yescharacteristic
150C2×C7×Q8C2×D411yescharacteristic
151C2×C7×D4—21no—
152C2×C7×D4—24no—
153C2×C7×D4—22no—
154C2×C7×D4—24no—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—41no—
161C2×C7×D4—41no—
162C2×C7×D4—42no—
163C2×C7×D4—42no—
164C2×C7×D4—42no—
165C2×C7×D4—42no—
166C2×C7×D4—44no—
167C2×C7×D4—42no—
168C2×C7×D4—41no—
169C2×C7×D4—42no—
170C2×C7×D4—42no—
171C2×C7×D4—44no—
172C2×C7×D4C2×D412yes—
173C2×C7×D4C2×D411yescharacteristic
174C2×C7×D4C2411yescharacteristic
175C22×C4×C7—22no—
176C22×C4×C7—21no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—21no—
181C22×C4×C7—41no—
182C22×C4×C7—42no—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D412yes—
185C7×Q16—42no—
186C7×Q16—42no—
187C7×SD16—42no—
188C7×SD16—42no—
189C7×M16—22no—
190C7×M16—41no—
191C2×C7×C8—41no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—44no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—24no—
216C7×C22⋊C4—44no—
217C7×C22⋊C4—42no—
218C7×C22⋊C4—44no—
219C7×C22⋊C4C2×D412yes—
220C7×C22⋊C4C2×D412yes—
221C42×C7—21no—
222C42×C7—22no—
223C42×C7—21no—
224C42×C7—21no—
225C42×C7C2×D411yescharacteristic
226C42×C7C2×D411yescharacteristic
227C2×H(5,𝔽2)C22×C711yescharacteristic
228C2×#137—21no—
229#563C22×C711yescharacteristic, complemented
230#562—21no—
231H(3,𝔽4)C22×C711yescharacteristic, complemented
232#555—21no—
233#551—22no—
234#551C22×C712yescomplemented
235#546—22no—
236#542—22no—
237#538—22no—
238#536—21no—
239#534—22no—
240#534C22×C712yes—
241#529—21no—
242C2×#124C22×C711yescharacteristic
243#458—22no—
244#458C22×C714yes—
245B(4,𝔽2)—22no—
246B(4,𝔽2)C22×C714yes—
247#455—21no—
248#455—21no—
249#455C22×C712yes—
250#455C22×C712yes—
251C2×#104—21no—
252C2×#100C22×C711yescharacteristic
253C2×#99—22no—
254C2×#99C22×C712yes—
255#354C22×C712yescomplemented
256#354C22×C712yescomplemented
257#352C22×C714yescomplemented
258AΣL(1,𝔽16)C22×C714yescomplemented
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—24no—
262C23×C7—22no—
263C23×C7—42no—
264C23×C7—42no—
265C23×C7—41no—
266C23×C7—82no—
267C23×C7—82no—
268C23×C7—162no—
269C23×C7#12012yes—
270C23×C7#12011yescharacteristic
271C23×C7C22×D412yes—
272C7×Q8—22no—
273C7×Q8—41no—
274C7×Q8—41no—
275C7×Q8—41no—
276C7×Q8—81no—
277C7×Q8—81no—
278C7×Q8—81no—
279C7×D4—22no—
280C7×D4—41no—
281C7×D4—42no—
282C7×D4—41no—
283C7×D4—41no—
284C7×D4—44no—
285C7×D4—44no—
286C7×D4—42no—
287C7×D4—42no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—82no—
291C7×D4—82no—
292C7×D4—82no—
293C7×D4—81no—
294C7×D4—81no—
295C7×D4—164no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—24no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—21no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—44no—
314C2×C4×C7—82no—
315C2×C4×C7—81no—
316C2×C4×C7—82no—
317C2×C4×C7—82no—
318C2×C4×C7—162no—
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7C22×D411yescharacteristic
321C7×C8—42no—
322H(5,𝔽2)—22no—
323H(5,𝔽2)—21no—
324H(5,𝔽2)—42no—
325H(5,𝔽2)C23×C712yes—
326C2×#41—21no—
327C2×#41—21no—
328C2×#41—41no—
329C22×Q8C7×D411yescharacteristic, complemented
330C22×D4—22no—
331C22×D4—41no—
332C22×D4C7×D412yescomplemented
333C22×D4C23×C711yescharacteristic
334#137—22no—
335#137—22no—
336#137—42no—
337C2×Q16—41no—
338C2×SD16—41no—
339C2×M16—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342#126—22no—
343#126—22no—
344#126—22no—
345#125—21no—
346#124—21no—
347#124—22no—
348#124—21no—
349#124—22no—
350#124—22no—
351#124—21no—
352#124C7×D411yescharacteristic, complemented
353#124C7×D411yescharacteristic, complemented
354#124C23×C711yescharacteristic
355#124C23×C711yescharacteristic
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—24no—
362#123—42no—
363#123C7×D412yescomplemented
364#123C23×C712yes—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370#121—21no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—44no—
376#120—22no—
377#120—22no—
378#120—24no—
379#120—42no—
380#120C7×D412yescomplemented
381#120C23×C712yes—
382C4×Q8—21no—
383C4×D4—21no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—22no—
387#117—21no—
388#117—22no—
389#117—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4C7×D412yescomplemented
394C2×C42C7×D411yescharacteristic, complemented
395#104—22no—
396#104—22no—
397#100—21no—
398#100C23×C712yes—
399#99—22no—
400#99—24no—
401#99—24no—
402#99C23×C714yes—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—41no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—82no—
413C22×C7—82no—
414C22×C7—82no—
415C22×C7—84no—
416C22×C7—84no—
417C22×C7—162no—
418C22×C7B(4,𝔽2)11yescharacteristic
419C22×C7B(4,𝔽2)11yescharacteristic
420C22×C7C2×#12011yescharacteristic
421C4×C7—21no—
422C4×C7—21no—
423C4×C7—41no—
424C4×C7—42no—
425C4×C7—42no—
426C4×C7—41no—
427C4×C7—42no—
428C4×C7—41no—
429C4×C7—41no—
430C4×C7—42no—
431C4×C7—84no—
432C24—22no—
433C24C2×C7×D412yes—
434C24C2×C7×D412yes—
435#41—22no—
436#41—22no—
437#41—42no—
438#41—41no—
439#41—44no—
440#41—42no—
441#41—41no—
442#41—81no—
443#41—81no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8C2×C7×D411yescharacteristic
451C2×Q8C2×C7×D411yescharacteristic
452C2×D4—21no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4C2×C7×D412yes—
474C2×D4C2×C7×D411yescharacteristic
475C2×D4C24×C711yescharacteristic
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—41no—
483C22×C4—42no—
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D412yes—
486Q16—42no—
487Q16—42no—
488SD16—42no—
489SD16—42no—
490M16—22no—
491M16—41no—
492C2×C8—41no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4C2×C7×D412yes—
521C22⋊C4C2×C7×D412yes—
522C42—21no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C2×C7—21no—
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—42no—
532C2×C7—42no—
533C2×C7—82no—
534C2×C7—82no—
535C2×C7C2×B(4,𝔽2)11yescharacteristic
536C23—22no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—82no—
544C23—82no—
545C23—162no—
546C23C7×#12012yes—
547C23C7×#12011yescharacteristic
548C23C22×C7×D412yes—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—81no—
554Q8—81no—
555Q8—81no—
556D4—22no—
557D4—41no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—44no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—82no—
568D4—82no—
569D4—82no—
570D4—81no—
571D4—81no—
572D4—164no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—162no—
596C2×C4C7×#12011yescharacteristic
597C2×C4C22×C7×D411yescharacteristic
598C8—42no—
599C7#3355811yescharacteristic, direct product factor, Sylow
600C22—22no—
601C22—22no—
602C22—41no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—84no—
613C22—84no—
614C22—162no—
615C22C7×B(4,𝔽2)11yescharacteristic
616C22C7×B(4,𝔽2)11yescharacteristic
617C22C2×C7×#12011yescharacteristic
618C4—21no—
619C4—21no—
620C4—41no—
621C4—42no—
622C4—42no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—84no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—42no—
634C2—82no—
635C2—82no—
636C2C2×C7×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C7C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C7
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3545C2semidirect
2. C7×#3026C2semidirect
3. C2×C7×#455C2semidirect
4. C7×#2201C2semidirect
5. C7×#2199C2semidirect
6. C7×#2197C2semidirect
7. C2×C7×#354C2semidirect
8. C7×#2121C2semidirect
11. C7×#563C22semidirect
13. C7×H(3,𝔽4)C22semidirect
16. C7×#551C22semidirect
37. C7×#354C22semidirect
38. C7×#354C22semidirect
39. C7×#352C22semidirect
40. C7×AΣL(1,𝔽16)C22semidirect
41. #33558C7direct (N × H)
49. C22×C7×Q8D4semidirect
52. C22×C7×D4D4semidirect
72. C7×#124D4semidirect
73. C7×#124D4semidirect
83. C7×#123D4semidirect
100. C7×#120D4semidirect
113. C2×C7×C22⋊C4D4semidirect
114. C2×C42×C7D4semidirect
123. #3545C2×C7semidirect
124. #3026C2×C7semidirect
125. C2×#455C2×C7semidirect
126. #2201C2×C7semidirect
127. #2199C2×C7semidirect
128. #2197C2×C7semidirect
129. C2×#354C2×C7semidirect
130. #2121C2×C7semidirect
229. #563C22×C7semidirect
231. H(3,𝔽4)C22×C7semidirect
234. #551C22×C7semidirect
255. #354C22×C7semidirect
256. #354C22×C7semidirect
257. #352C22×C7semidirect
258. AΣL(1,𝔽16)C22×C7semidirect
329. C22×Q8C7×D4semidirect
332. C22×D4C7×D4semidirect
352. #124C7×D4semidirect
353. #124C7×D4semidirect
363. #123C7×D4semidirect
380. #120C7×D4semidirect
393. C2×C22⋊C4C7×D4semidirect
394. C2×C42C7×D4semidirect
599. C7#33558direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35452
C7×#30262
C2×C7×#4552
C7×#22012
C7×#21992
C7×#21972
C2×C7×#3542
C7×#21212
#335587