Finite Group Databasebeta

← #13320342

Subgroups — #13320342

Order 1792.

988 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2D7×#431C211yesmaximal, characteristic
3#11735962C211yesmaximal, characteristic
4C7×#1485C211yesmaximal, characteristic, complemented
5#11729579C211yesmaximal, characteristic, complemented
6#11728523C211yesmaximal, characteristic, complemented
7#11727610C211yesmaximal, characteristic, complemented
8#11725986C214yesmaximal, complemented
9#11725985C214yesmaximal, complemented
10C7×#533C2211yescharacteristic
11#90419C411yescharacteristic
12#90419C411yescharacteristic
13D7×C4⋊Q8C411yescharacteristic
14D7×C4⋊Q8C411yescharacteristic
15#90379—21no—
16D7×#125—21no—
17D7×C22⋊Q8—21no—
18D7×C22⋊Q8C412yes—
19#90313—21no—
20#90313C412yes—
21#90269—21no—
22#90269C2212yes—
23C4×Q8×D7—21no—
24C4×Q8×D7C2212yes—
25D7×#117—21no—
26#90208—21no—
27C2×C42×D7C2211yescharacteristic
28#90161C2211yescharacteristic
29C7×#431C2211yescharacteristic
30C4×C7×M16C2211yescharacteristic
31#89898—21no—
32#89890—21no—
33#89697C2211yescharacteristic
34C4×#1136C2211yescharacteristic
35#89605C2212yes—
36D7×C4⋊C8C2212yes—
37D7×C22⋊C8—21no—
38#89490—21no—
39#89477—21no—
40#89477—21no—
41#89457—21no—
42#89457C2212yes—
43#89457C2212yes—
44C4×C8×D7—21no—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17464—71nomaximal, Sylow
50C2×Q8×D7—21no—
51C2×Q8×D7C2×C412yes—
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C2×C7×M16—21no—
56C7×C4⋊Q8C2×C411yescharacteristic
57C7×C4⋊Q8C2×C411yescharacteristic
58C7×#125—21no—
59C7×C22⋊Q8—21no—
60C7×C22⋊Q8C2×C412yes—
61C4×C7×Q8—21no—
62C4×C7×Q8C2312yes—
63C7×#117—21no—
64C2×C42×C7C2311yescharacteristic
65#6509—21no—
66#6509C2×C412yes—
67Q8×Dic7—21no—
68Q8×Dic7C2×C412yes—
69#6507—21no—
70#6507C2×C412yes—
71#6489—21no—
72#6487—21no—
73#6487C2×C412yes—
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×#1136—21no—
78C2×C8×D7—22no—
79#6460—21no—
80#6460C2×C412yes—
81#6459—22no—
82#6459—22no—
83#6455—21no—
84#6455—21no—
85D7×C4⋊C4—21no—
86D7×C4⋊C4—21no—
87D7×C4⋊C4—22no—
88D7×C4⋊C4—21no—
89D7×C4⋊C4—22no—
90D7×C4⋊C4C2×C412yes—
91#6453—21no—
92#6453—21no—
93#6452—22no—
94#6452—22no—
95#6451—21no—
96#6451—21no—
97#6451—21no—
98#6451C2×C412yes—
99#6450—22no—
100#6450—21no—
101#6450—22no—
102#6450C2×C412yes—
103D7×C22⋊C4—22no—
104#6441—22no—
105#6441—22no—
106#6440—22no—
107#6435—22no—
108C42×D7—22no—
109C42×D7C2×C411yescharacteristic
110C42×D7C2×C411yescharacteristic
111C42×D7D411yescharacteristic
112C42×D7D411yescharacteristic
113C42×D7D411yescharacteristic
114C42×D7D411yescharacteristic
115C42×D7C2311yescharacteristic
116C42×D7C2311yescharacteristic
117#6433—21no—
118#6432C2×C411yescharacteristic
119#6432C2×C411yescharacteristic
120C4×Dic14—21no—
121C4×Dic14—22no—
122C4×Dic14—22no—
123C4×Dic14C2312yes—
124C7×C4⋊C8C2312yes—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C4×C7×C8—21no—
128C4×C7×C8C2312yes—
129C22⋊(C7⋊C8)—21no—
130(C2×D7)⋊C8—22no—
131(C2×D7)⋊C8—22no—
132(C2×D7)⋊C8—22no—
133Dic7⋊C8—22no—
134Dic7⋊C8—22no—
135Dic7⋊C8—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)C2312yes—
138#6377—21no—
139C4×C7⋊C8—21no—
140C4×C7⋊C8C2312yes—
141C2×#533—71no—
142#2920—71no—
143C2×#431—71no—
144#1485—71no—
145#1485—71no—
146#1485—71no—
147#1485D711yescharacteristic, complemented
148#1481—74no—
149#1479—74no—
150C23×D7C22⋊C411yescharacteristic
151C2×C7×Q8—21no—
152C2×C7×Q8C22×C412yes—
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×Dic7C22⋊C411yescharacteristic
157C22×Dic7C22⋊C411yescharacteristic
158C22×Dic7C22⋊C411yescharacteristic
159Q8×D7—28no—
160Q8×D7—44no—
161C2×C4×D7—24no—
162C2×C4×D7—24no—
163C2×C4×D7—22no—
164C2×C4×D7—22no—
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×D7C22×C411yescharacteristic
173C2×C4×D7C22×C411yescharacteristic
174C2×C4×D7C2×D411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×Dic14—21no—
179C2×Dic14—22no—
180C2×Dic14—22no—
181C2×Dic14C22×C412yes—
182C7×M16—42no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4C22×C412yes—
191C7×C22⋊C4—22no—
192C42×C7—22no—
193C42×C7C22×C411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2411yescharacteristic
197C22⋊Dic7—22no—
198(C2×D7)⋊C4—22no—
199(C2×D7)⋊C4—22no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—22no—
204C4⋊Dic7—22no—
205C4⋊Dic7C22×C412yes—
206Dic7⋊C4—22no—
207Dic7⋊C4—22no—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214C4×Dic7—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7C22⋊C411yescharacteristic
218C4×Dic7C22⋊C411yescharacteristic
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22×C411yescharacteristic
224C4×Dic7C22×C411yescharacteristic
225C4×Dic7C2×D411yescharacteristic
226C4×Dic7C2×D411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229#1136—42no—
230C2×C7⋊C8—22no—
231C2×C7⋊C8—22no—
232C8×D7—44no—
233#533—71no—
234#533—71no—
235#533—72no—
236#533—72no—
237#533—71no—
238#533—141no—
239#533—141no—
240#533—141no—
241#533—141no—
242#533C2×D711yescharacteristic
243C2×C4⋊Q8—71no—
244C2×C4⋊Q8—71no—
245C2×#125—141no—
246C2×C22⋊Q8—72no—
247C2×C22⋊Q8—141no—
248C2×C4×Q8—72no—
249C2×C4×Q8—141no—
250C2×#117—141no—
251C22×C42—71no—
252#432—141no—
253#432—141no—
254#431—72no—
255#431—72no—
256#431—71no—
257#431—72no—
258#431—141no—
259#431—141no—
260#431C2×D711yescharacteristic
261#423—141no—
262C2×C4⋊C8—72no—
263#407—141no—
264C2×C22⋊C8—141no—
265C4×M16—71no—
266C4×M16C2×D711yescharacteristic
267C2×C4×C8—141no—
268Q8⋊C8—74no—
269Q8⋊C8—74no—
270Q8⋊C8—74no—
271Q8⋊C8C2×D714yescomplemented
272C23×C7C2×C22⋊C411yescharacteristic
273C22×D7—22no—
274C22×D7—22no—
275C22×D7—41no—
276C22×D7C2×C22⋊C411yescharacteristic
277C22×D7C2×C22⋊C411yescharacteristic
278C7×Q8—24no—
279C7×Q8—42no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7C2×C22⋊C411yescharacteristic
286C2×C4×C7C2×C22⋊C411yescharacteristic
287C2×C4×C7C2×C22⋊C411yescharacteristic
288C2×C4×C7C23×C411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—41no—
296C2×Dic7C2×C22⋊C411yescharacteristic
297C2×Dic7C2×C22⋊C411yescharacteristic
298C2×Dic7C2×C22⋊C411yescharacteristic
299C2×Dic7C2×C22⋊C411yescharacteristic
300C2×Dic7C2×C22⋊C411yescharacteristic
301C2×Dic7C2×C22⋊C411yescharacteristic
302C4×D7—22no—
303C4×D7—22no—
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—41no—
311C4×D7—41no—
312C4×D7—44no—
313C4×D7—44no—
314C4×D7#13114yes—
315C4×D7#13714yes—
316Dic14—24no—
317Dic14—44no—
318Dic14—42no—
319Dic14—44no—
320C7×C8—42no—
321C7×C8—42no—
322C7⋊C8—42no—
323C7⋊C8—42no—
324C22×Q8—72no—
325C22×Q8—141no—
326C23×C4—71no—
327C23×C4—71no—
328C23×C4—71no—
329C2×M16—21no—
330C2×M16—141no—
331C22×C8—142no—
332C4⋊Q8—72no—
333C4⋊Q8—72no—
334C4⋊Q8—71no—
335C4⋊Q8—71no—
336C4⋊Q8—141no—
337C4⋊Q8—141no—
338C4⋊Q8—141no—
339C4⋊Q8—141no—
340C4⋊Q8C4×D711yescharacteristic
341C4⋊Q8C4×D711yescharacteristic
342#125—21no—
343#125—142no—
344#125—142no—
345#125—141no—
346#125—141no—
347#125—141no—
348C22⋊Q8—21no—
349C22⋊Q8—72no—
350C22⋊Q8—72no—
351C22⋊Q8—72no—
352C22⋊Q8—142no—
353C22⋊Q8—142no—
354C22⋊Q8—142no—
355C22⋊Q8—142no—
356C22⋊Q8—141no—
357C22⋊Q8—141no—
358C22⋊Q8—141no—
359C22⋊Q8C4×D712yes—
360C4×Q8—21no—
361C4×Q8—72no—
362C4×Q8—72no—
363C4×Q8—72no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×Q8—142no—
367C4×Q8—141no—
368C4×Q8—142no—
369C4×Q8—141no—
370C4×Q8—141no—
371C4×Q8C22×D712yes—
372#117—21no—
373#117—142no—
374#117—142no—
375#117—141no—
376#117—141no—
377#117—141no—
378C2×C4⋊C4—72no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—142no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—141no—
384C2×C22⋊C4—142no—
385C2×C42—71no—
386C2×C42—71no—
387C2×C42—71no—
388C2×C42—71no—
389C2×C42—71no—
390C2×C42—71no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—142no—
397C2×C42C22×D711yescharacteristic
398C4⋊C8—72no—
399C4⋊C8—142no—
400C4⋊C8—142no—
401C4⋊C8—142no—
402C4⋊C8C22×D712yes—
403C22⋊C8—21no—
404C22⋊C8—142no—
405C22⋊C8—142no—
406C22⋊C8—142no—
407C22⋊C8—141no—
408#97—21no—
409#97—141no—
410C4×C8—21no—
411C4×C8—72no—
412C4×C8—142no—
413C4×C8—141no—
414C4×C8C22×D712yes—
415C22×C7—21no—
416C22×C7—21no—
417C22×C7—21no—
418C22×C7C22×C22⋊C411yescharacteristic
419C2×D7—21no—
420C2×D7—21no—
421C2×D7—21no—
422C2×D7—21no—
423C2×D7—44no—
424C2×D7#40812yes—
425C2×D7#41912yes—
426C4×C7—21no—
427C4×C7—21no—
428C4×C7—21no—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7#42112yes—
432C4×C7C2×#13112yes—
433C4×C7C2×#13712yes—
434Dic7—21no—
435Dic7—21no—
436Dic7—21no—
437Dic7—21no—
438Dic7—42no—
439Dic7—42no—
440Dic7#40812yes—
441Dic7#41912yes—
442C24—71no—
443C2×Q8—21no—
444C2×Q8—72no—
445C2×Q8—142no—
446C2×Q8—142no—
447C2×Q8—148no—
448C2×Q8—141no—
449C2×Q8—284no—
450C2×Q8C2×C4×D712yes—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—71no—
457C22×C4—71no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—144no—
467C22×C4—144no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—282no—
471C22×C4C2×C4×D711yescharacteristic
472C22×C4D4×D711yescharacteristic
473C22×C4D4×D711yescharacteristic
474M16—42no—
475M16—282no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—284no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—72no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C4⋊C4—142no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—142no—
500C4⋊C4C2×C4×D712yes—
501C22⋊C4—22no—
502C22⋊C4—142no—
503C22⋊C4—142no—
504C22⋊C4—142no—
505C42—22no—
506C42—71no—
507C42—71no—
508C42—71no—
509C42—71no—
510C42—71no—
511C42—71no—
512C42—71no—
513C42—71no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—142no—
519C42—142no—
520C42—142no—
521C42C2×C4×D711yescharacteristic
522C42D4×D711yescharacteristic
523C42D4×D711yescharacteristic
524C42C23×D711yescharacteristic
525C2×C7—41no—
526C2×C7C2×#40811yescharacteristic
527C2×C7C2×#41911yescharacteristic
528C2×C7C2×#42111yescharacteristic
529D7—22no—
530D7—22no—
531C23—71no—
532C23—71no—
533C23—142no—
534C23—142no—
535C23—281no—
536C23D7×C22⋊C411yescharacteristic
537Q8—24no—
538Q8—42no—
539Q8—144no—
540Q8—284no—
541Q8—284no—
542Q8—282no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—71no—
549C2×C4—71no—
550C2×C4—74no—
551C2×C4—74no—
552C2×C4—71no—
553C2×C4—71no—
554C2×C4—71no—
555C2×C4—71no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—284no—
571C2×C4—284no—
572C2×C4—281no—
573C2×C4D7×C22⋊C411yescharacteristic
574C2×C4D7×C22⋊C411yescharacteristic
575C2×C4D7×C22⋊C411yescharacteristic
576C2×C4C22×C4×D711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4C2×D4×D711yescharacteristic
579C8—42no—
580C8—42no—
581C8—282no—
582C8—282no—
583C7#1746411yescharacteristic, complemented, Sylow
584C22—21no—
585C22—21no—
586C22—21no—
587C22—72no—
588C22—72no—
589C22—141no—
590C22—141no—
591C22—141no—
592C22—141no—
593C22—284no—
594C22C2×D7×C22⋊C411yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—72no—
601C4—72no—
602C4—141no—
603C4—141no—
604C4—141no—
605C4—141no—
606C4—282no—
607C4—282no—
608C4#8959312yes—
609C4D7×#13112yes—
610C4D7×#13712yes—
611C2—41no—
612C2—142no—
613C2—142no—
614C2D7×#40811yescharacteristic
615C2D7×#41911yescharacteristic
616C2C2×#8959311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1485C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1485D7
hypocenterC7#17464
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
4. C7×#1485C2semidirect
5. #11729579C2semidirect
6. #11728523C2semidirect
7. #11727610C2semidirect
8. #11725986C2semidirect
9. #11725985C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
147. #1485D7semidirect
271. Q8⋊C8C2×D7semidirect
583. C7#17464semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
D7×#4312
#117359622
C7×#14852
#117295792
#117285232
#117276102
#117259862
#117259852
#174647