Finite Group Databasebeta

← #13320498

Subgroups — #13320498

Order 1792.

1026 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2D7×#423C211yesmaximal, characteristic
3C2×#89457C211yesmaximal, characteristic
4C7×#1493C211yesmaximal, characteristic, complemented
5#11729580C211yesmaximal, characteristic, complemented
6#11728255C211yesmaximal, characteristic, complemented
7#11727492C211yesmaximal, characteristic, complemented
8#11725990C214yesmaximal, complemented
9#11725986C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11C2×D7×M16—21no—
12#90419C411yescharacteristic
13#90419C411yescharacteristic
14D7×C4⋊Q8C411yescharacteristic
15D7×C4⋊Q8C411yescharacteristic
16#90379—21no—
17D7×#125—21no—
18D7×C22⋊Q8—21no—
19D7×C22⋊Q8C412yes—
20#90313—21no—
21#90313C412yes—
22#90269—21no—
23#90269C2212yes—
24C4×Q8×D7—21no—
25C4×Q8×D7C2212yes—
26D7×#117—21no—
27#90208—21no—
28C2×C42×D7C2211yescharacteristic
29#90161C2211yescharacteristic
30C7×#423C2211yescharacteristic
31C2×C4×C7×C8C2211yescharacteristic
32#89890—21no—
33C2×(C2×D7)⋊C8—21no—
34C2×Dic7⋊C8—21no—
35#89696C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89609—21no—
38#89609C2212yes—
39D7×C4⋊C8—21no—
40D7×C4⋊C8C2212yes—
41#89457—21no—
42#89457—21no—
43#89457C2212yes—
44#89457C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17486—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7C2×C412yes—
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C2×C7×M16—21no—
56C22×C7×C8—21no—
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×C4⋊Q8C2×C411yescharacteristic
59C7×#125—21no—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8C2×C412yes—
62C4×C7×Q8—21no—
63C4×C7×Q8C2312yes—
64C7×#117—21no—
65C2×C42×C7C2311yescharacteristic
66#6509—21no—
67#6509C2×C412yes—
68Q8×Dic7—21no—
69Q8×Dic7C2×C412yes—
70#6507—21no—
71#6507C2×C412yes—
72#6489—21no—
73#6487—21no—
74#6487C2×C412yes—
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×#1136—21no—
79C22×C7⋊C8—21no—
80D7×M16—44no—
81C2×#1131—22no—
82C2×C8×D7—22no—
83#6460—21no—
84#6460C2×C412yes—
85#6459—22no—
86#6459—22no—
87#6455—21no—
88#6455—21no—
89D7×C4⋊C4—21no—
90D7×C4⋊C4—21no—
91D7×C4⋊C4—22no—
92D7×C4⋊C4—21no—
93D7×C4⋊C4—22no—
94D7×C4⋊C4C2×C412yes—
95#6453—21no—
96#6453—21no—
97#6452—22no—
98#6452—22no—
99#6451—21no—
100#6451—21no—
101#6451—21no—
102#6451C2×C412yes—
103#6450—22no—
104#6450—21no—
105#6450—22no—
106#6450C2×C412yes—
107D7×C22⋊C4—22no—
108#6441—22no—
109#6441—22no—
110#6440—22no—
111#6435—22no—
112C42×D7—22no—
113C42×D7C2×C411yescharacteristic
114C42×D7C2×C411yescharacteristic
115C42×D7D411yescharacteristic
116C42×D7D411yescharacteristic
117C42×D7D411yescharacteristic
118C42×D7D411yescharacteristic
119C42×D7C2311yescharacteristic
120C42×D7C2311yescharacteristic
121#6433—21no—
122#6432C2×C411yescharacteristic
123#6432C2×C411yescharacteristic
124C4×Dic14—21no—
125C4×Dic14—22no—
126C4×Dic14—22no—
127C4×Dic14C2312yes—
128C7×C4⋊C8—21no—
129C7×C4⋊C8C2312yes—
130C4×C7×C8—21no—
131C4×C7×C8C2312yes—
132(C2×D7)⋊C8—22no—
133(C2×D7)⋊C8—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136Dic7⋊C8—22no—
137Dic7⋊C8—22no—
138C4⋊(C7⋊C8)—21no—
139C4⋊(C7⋊C8)C2312yes—
140C4×C7⋊C8—21no—
141C4×C7⋊C8C2312yes—
142C2×#533—71no—
143C2×#431—71no—
144C2×#423—71no—
145#1493—71no—
146#1493—71no—
147#1493—71no—
148#1493D711yescharacteristic, complemented
149#1487—74no—
150#1481—74no—
151C23×D7C22⋊C411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8C22×C412yes—
154C22×C4×C7C22×C411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×Dic7C22⋊C411yescharacteristic
158C22×Dic7C22⋊C411yescharacteristic
159C22×Dic7C22⋊C411yescharacteristic
160Q8×D7—44no—
161Q8×D7—44no—
162C2×C4×D7—22no—
163C2×C4×D7—22no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7C22⋊C411yescharacteristic
167C2×C4×D7C22⋊C411yescharacteristic
168C2×C4×D7C22⋊C411yescharacteristic
169C2×C4×D7C22⋊C411yescharacteristic
170C2×C4×D7C22⋊C411yescharacteristic
171C2×C4×D7C22⋊C411yescharacteristic
172C2×C4×D7SD1614yes—
173C2×C4×D7Q1614yes—
174C2×C4×D7C22×C411yescharacteristic
175C2×C4×D7C22×C411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×Dic14—21no—
181C2×Dic14—22no—
182C2×Dic14—22no—
183C2×Dic14C22×C412yes—
184C7×M16—42no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C2×C7×C8—42no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4C22×C412yes—
194C7×C22⋊C4—22no—
195C42×C7—22no—
196C42×C7C22×C411yescharacteristic
197C42×C7C2×D411yescharacteristic
198C42×C7C2×D411yescharacteristic
199C42×C7C2411yescharacteristic
200C22⋊Dic7—22no—
201(C2×D7)⋊C4—22no—
202(C2×D7)⋊C4—22no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7C22×C412yes—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217C4×Dic7—22no—
218C4×Dic7—22no—
219C4×Dic7—22no—
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22⋊C411yescharacteristic
225C4×Dic7C22⋊C411yescharacteristic
226C4×Dic7C22×C411yescharacteristic
227C4×Dic7C22×C411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230C4×Dic7C2×D411yescharacteristic
231C4×Dic7C2×D411yescharacteristic
232#1136—42no—
233C2×C7⋊C8—22no—
234C2×C7⋊C8—22no—
235C2×C7⋊C8—42no—
236#1131—44no—
237C8×D7—44no—
238C22×M16—141no—
239#533—71no—
240#533—71no—
241#533—72no—
242#533—72no—
243#533—71no—
244#533—141no—
245#533—141no—
246#533—141no—
247#533—141no—
248#533C2×D711yescharacteristic
249C2×C4⋊Q8—71no—
250C2×C4⋊Q8—71no—
251C2×#125—141no—
252C2×C22⋊Q8—72no—
253C2×C22⋊Q8—141no—
254C2×C4×Q8—72no—
255C2×C4×Q8—141no—
256C2×#117—141no—
257C22×C42—71no—
258#431—72no—
259#431—72no—
260#431—141no—
261#431—141no—
262#423—71no—
263#423—72no—
264#423—141no—
265#423—141no—
266#423C2×D711yescharacteristic
267C2×C4⋊C8—72no—
268C2×C4⋊C8—141no—
269C2×C4⋊C8—141no—
270C2×C22⋊C8—141no—
271C2×C4×C8—71no—
272C2×C4×C8C2×D711yescharacteristic
273Q8⋊C8—74no—
274Q8⋊C8—74no—
275Q8⋊C8—74no—
276Q8⋊C8C2×D714yescomplemented
277C23×C7C2×C22⋊C411yescharacteristic
278C22×D7—41no—
279C22×D7—41no—
280C22×D7#10314yes—
281C22×D7C2×C22⋊C411yescharacteristic
282C22×D7C2×C22⋊C411yescharacteristic
283C7×Q8—24no—
284C7×Q8—42no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C2×C22⋊C411yescharacteristic
292C2×C4×C7C2×SD1612yes—
293C2×C4×C7C2×Q1612yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7#10314yes—
302C2×Dic7C2×C22⋊C411yescharacteristic
303C2×Dic7C2×C22⋊C411yescharacteristic
304C2×Dic7C2×C22⋊C411yescharacteristic
305C2×Dic7C2×C22⋊C411yescharacteristic
306C2×Dic7C2×C22⋊C411yescharacteristic
307C2×Dic7C2×C22⋊C411yescharacteristic
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—41no—
313C4×D7—41no—
314C4×D7—41no—
315C4×D7—41no—
316C4×D7—44no—
317C4×D7—44no—
318C4×D7#10314yes—
319C4×D7#10314yes—
320C4×D7C2×SD1612yes—
321C4×D7C2×SD1612yes—
322C4×D7C2×Q1612yes—
323C4×D7C2×Q1612yes—
324Dic14—24no—
325Dic14—44no—
326Dic14—44no—
327Dic14—42no—
328C7×C8—42no—
329C7×C8—42no—
330C7⋊C8—42no—
331C7⋊C8—42no—
332C22×Q8—72no—
333C22×Q8—141no—
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—71no—
337C2×M16—21no—
338C2×M16—142no—
339C2×M16—141no—
340C2×M16—284no—
341C22×C8—21no—
342C22×C8—142no—
343C22×C8—141no—
344C4⋊Q8—72no—
345C4⋊Q8—72no—
346C4⋊Q8—71no—
347C4⋊Q8—71no—
348C4⋊Q8—141no—
349C4⋊Q8—141no—
350C4⋊Q8—141no—
351C4⋊Q8—141no—
352C4⋊Q8C4×D711yescharacteristic
353C4⋊Q8C4×D711yescharacteristic
354#125—21no—
355#125—142no—
356#125—142no—
357#125—141no—
358#125—141no—
359#125—141no—
360C22⋊Q8—21no—
361C22⋊Q8—72no—
362C22⋊Q8—72no—
363C22⋊Q8—72no—
364C22⋊Q8—142no—
365C22⋊Q8—142no—
366C22⋊Q8—142no—
367C22⋊Q8—142no—
368C22⋊Q8—141no—
369C22⋊Q8—141no—
370C22⋊Q8—141no—
371C22⋊Q8C4×D712yes—
372C4×Q8—21no—
373C4×Q8—72no—
374C4×Q8—72no—
375C4×Q8—72no—
376C4×Q8—142no—
377C4×Q8—142no—
378C4×Q8—142no—
379C4×Q8—141no—
380C4×Q8—142no—
381C4×Q8—141no—
382C4×Q8—141no—
383C4×Q8C22×D712yes—
384#117—21no—
385#117—142no—
386#117—142no—
387#117—141no—
388#117—141no—
389#117—141no—
390C2×C4⋊C4—72no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—141no—
393C2×C4⋊C4—142no—
394C2×C4⋊C4—142no—
395C2×C4⋊C4—141no—
396C2×C22⋊C4—142no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—71no—
403C2×C42—71no—
404C2×C42—71no—
405C2×C42—71no—
406C2×C42—71no—
407C2×C42—71no—
408C2×C42—142no—
409C2×C42C22×D711yescharacteristic
410C4⋊C8—21no—
411C4⋊C8—72no—
412C4⋊C8—142no—
413C4⋊C8—142no—
414C4⋊C8—142no—
415C4⋊C8—142no—
416C4⋊C8—141no—
417C4⋊C8C22×D712yes—
418C22⋊C8—142no—
419C22⋊C8—142no—
420C4×C8—21no—
421C4×C8—72no—
422C4×C8—141no—
423C4×C8C22×D712yes—
424C22×C7—21no—
425C22×C7—21no—
426C22×C7C2×#10312yes—
427C22×C7C22×C22⋊C411yescharacteristic
428C2×D7—21no—
429C2×D7—21no—
430C2×D7—21no—
431C2×D7—21no—
432C2×D7—44no—
433C2×D7C2×#10312yes—
434C2×D7C2×#10312yes—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—42no—
438C4×C7—42no—
439C4×C7C2×#10312yes—
440C4×C7#42112yes—
441C4×C7#56812yes—
442C4×C7C22×SD1611yescharacteristic
443C4×C7C22×Q1611yescharacteristic
444Dic7—21no—
445Dic7—21no—
446Dic7—21no—
447Dic7—21no—
448Dic7—42no—
449Dic7—42no—
450Dic7C2×#10312yes—
451Dic7C2×#10312yes—
452C24—71no—
453C2×Q8—21no—
454C2×Q8—72no—
455C2×Q8—142no—
456C2×Q8—142no—
457C2×Q8—141no—
458C2×Q8—284no—
459C2×Q8—284no—
460C2×Q8C2×C4×D712yes—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—74no—
465C22×C4—74no—
466C22×C4—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—71no—
471C22×C4—71no—
472C22×C4—71no—
473C22×C4—71no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4C2×C4×D711yescharacteristic
483C22×C4D4×D711yescharacteristic
484C22×C4D4×D711yescharacteristic
485M16—42no—
486M16—284no—
487M16—282no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—42no—
491C2×C8—142no—
492C2×C8—142no—
493C2×C8—284no—
494C2×C8—282no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—72no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C4⋊C4—142no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—142no—
514C4⋊C4C2×C4×D712yes—
515C22⋊C4—22no—
516C22⋊C4—142no—
517C22⋊C4—142no—
518C22⋊C4—142no—
519C42—22no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—71no—
525C42—71no—
526C42—71no—
527C42—71no—
528C42—71no—
529C42—71no—
530C42—71no—
531C42—71no—
532C42—142no—
533C42—142no—
534C42—142no—
535C42C2×C4×D711yescharacteristic
536C42D4×D711yescharacteristic
537C42D4×D711yescharacteristic
538C42C23×D711yescharacteristic
539C2×C7—22no—
540C2×C7#287911yescharacteristic
541C2×C7C22×#10311yescharacteristic
542C2×C7C2×#42111yescharacteristic
543D7—22no—
544D7—22no—
545C23—71no—
546C23—71no—
547C23—74no—
548C23—281no—
549C23—281no—
550C23D7×C22⋊C411yescharacteristic
551Q8—24no—
552Q8—42no—
553Q8—144no—
554Q8—284no—
555Q8—284no—
556Q8—282no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—71no—
562C2×C4—71no—
563C2×C4—74no—
564C2×C4—72no—
565C2×C4—72no—
566C2×C4—72no—
567C2×C4—72no—
568C2×C4—74no—
569C2×C4—74no—
570C2×C4—71no—
571C2×C4—71no—
572C2×C4—71no—
573C2×C4—71no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—284no—
585C2×C4—284no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4D7×C22⋊C411yescharacteristic
589C2×C4D7×C22⋊C411yescharacteristic
590C2×C4D7×C22⋊C411yescharacteristic
591C2×C4D7×SD1612yes—
592C2×C4D7×Q1612yes—
593C2×C4C22×C4×D711yescharacteristic
594C2×C4C2×D4×D711yescharacteristic
595C2×C4C2×D4×D711yescharacteristic
596C8—42no—
597C8—42no—
598C8—282no—
599C8—282no—
600C7#1748611yescharacteristic, complemented, Sylow
601C22—21no—
602C22—21no—
603C22—72no—
604C22—72no—
605C22—141no—
606C22—141no—
607C22—141no—
608C22—141no—
609C22—284no—
610C22D7×#10312yes—
611C22C2×D7×C22⋊C411yescharacteristic
612C4—21no—
613C4—21no—
614C4—42no—
615C4—42no—
616C4—72no—
617C4—72no—
618C4—141no—
619C4—141no—
620C4—141no—
621C4—141no—
622C4—282no—
623C4—282no—
624C4D7×#10312yes—
625C4#8959312yes—
626C4#9044112yes—
627C4C2×D7×SD1611yescharacteristic
628C4C2×D7×Q1611yescharacteristic
629C2—22no—
630C2—142no—
631C2—142no—
632C2#1173610711yescharacteristic
633C2C2×D7×#10311yescharacteristic
634C2C2×#8959311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1493C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1493D7
hypocenterC7#17486
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
4. C7×#1493C2semidirect
5. #11729580C2semidirect
6. #11728255C2semidirect
7. #11727492C2semidirect
8. #11725990C2semidirect
9. #11725986C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
148. #1493D7semidirect
276. Q8⋊C8C2×D7semidirect
600. C7#17486semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
D7×#4232
C2×#894572
C7×#14932
#117295802
#117282552
#117274922
#117259902
#117259862
#174867