Finite Group Databasebeta

← #13320595

Subgroups — #13320595

Order 1792.

1044 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#533C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3C4×D7×M16C211yesmaximal, characteristic
4C7×#1485C211yesmaximal, characteristic, complemented
5#11729579C211yesmaximal, characteristic, complemented
6#11728523C211yesmaximal, characteristic, complemented
7#11727610C211yesmaximal, characteristic, complemented
8#11725994C214yesmaximal, complemented
9#11725993C214yesmaximal, 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×#431C2211yescharacteristic
31C4×C7×M16C2211yescharacteristic
32M16×Dic7—21no—
33#89697C2211yescharacteristic
34C4×#1136C2211yescharacteristic
35#89609C2212yes—
36#89607C2212yes—
37#89498—21no—
38#89491—21no—
39#89479—21no—
40D7×#97—21no—
41C4×#1131—21no—
42C4×#1131C2212yes—
43#89457—21no—
44C4×C8×D7C2212yes—
45C7×Q8⋊C8C2214yescomplemented
46Q8⋊(C7⋊C8)C2214yescomplemented
47#89279C2214yescomplemented
48#89251C2214yescomplemented
49#17501—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—
78D7×M16—28no—
79C2×#1131—22no—
80C2×#1131—22no—
81C2×C8×D7—22no—
82#6460—21no—
83#6460C2×C412yes—
84#6459—22no—
85#6459—22no—
86#6455—21no—
87#6455—21no—
88D7×C4⋊C4—21no—
89D7×C4⋊C4—21no—
90D7×C4⋊C4—22no—
91D7×C4⋊C4—21no—
92D7×C4⋊C4—22no—
93D7×C4⋊C4C2×C412yes—
94#6453—21no—
95#6453—21no—
96#6452—22no—
97#6452—22no—
98#6451—21no—
99#6451—21no—
100#6451—21no—
101#6451C2×C412yes—
102#6450—22no—
103#6450—21no—
104#6450—22no—
105#6450C2×C412yes—
106D7×C22⋊C4—22no—
107#6441—22no—
108#6441—22no—
109#6440—22no—
110#6435—22no—
111C42×D7—22no—
112C42×D7C2×C411yescharacteristic
113C42×D7C2×C411yescharacteristic
114C42×D7D411yescharacteristic
115C42×D7D411yescharacteristic
116C42×D7D411yescharacteristic
117C42×D7D411yescharacteristic
118C42×D7C2311yescharacteristic
119C42×D7C2311yescharacteristic
120#6433—21no—
121#6432C2×C411yescharacteristic
122#6432C2×C411yescharacteristic
123C4×Dic14—21no—
124C4×Dic14—22no—
125C4×Dic14—22no—
126C4×Dic14C2312yes—
127C7×C4⋊C8C2312yes—
128C7×C22⋊C8—21no—
129C7×#97—21no—
130C4×C7×C8—21no—
131C4×C7×C8C2312yes—
132C22⋊(C7⋊C8)—21no—
133(C2×D7)⋊C8—22no—
134#6389—22no—
135#6389—22no—
136Dic7⋊C8—22no—
137C8×Dic7—22no—
138C4⋊(C7⋊C8)C2312yes—
139#6377—21no—
140C4×C7⋊C8—21no—
141C4×C7⋊C8C2312yes—
142C2×#533—71no—
143#2920—71no—
144C2×C4×M16—71no—
145#1493—74no—
146#1491—74no—
147#1485—71no—
148#1485—71no—
149#1485—71no—
150#1485D711yescharacteristic, complemented
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—24no—
163C2×C4×D7—24no—
164C2×C4×D7—22no—
165C2×C4×D7—22no—
166C2×C4×D7—42no—
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×D7C22×C411yescharacteristic
175C2×C4×D7C2×D411yescharacteristic
176C2×C4×D7C2×D411yescharacteristic
177C2×C4×D7C2×D411yescharacteristic
178C2×C4×D7C2×D411yescharacteristic
179C2×Dic14—21no—
180C2×Dic14—22no—
181C2×Dic14—22no—
182C2×Dic14C22×C412yes—
183C7×M16—24no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4C22×C412yes—
192C7×C22⋊C4—22no—
193C42×C7—22no—
194C42×C7C22×C411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D411yescharacteristic
197C42×C7C2411yescharacteristic
198C22⋊Dic7—22no—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—22no—
205C4⋊Dic7—22no—
206C4⋊Dic7C22×C412yes—
207Dic7⋊C4—22no—
208Dic7⋊C4—22no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7—22no—
218C4×Dic7C22⋊C411yescharacteristic
219C4×Dic7C22⋊C411yescharacteristic
220C4×Dic7C22⋊C411yescharacteristic
221C4×Dic7C22⋊C411yescharacteristic
222C4×Dic7C22⋊C411yescharacteristic
223C4×Dic7C22⋊C411yescharacteristic
224C4×Dic7C22×C411yescharacteristic
225C4×Dic7C22×C411yescharacteristic
226C4×Dic7C2×D411yescharacteristic
227C4×Dic7C2×D411yescharacteristic
228C4×Dic7C2×D411yescharacteristic
229C4×Dic7C2×D411yescharacteristic
230#1136—24no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—22no—
233#1131—28no—
234#1131—44no—
235C8×D7—28no—
236C22×M16—141no—
237#533—71no—
238#533—71no—
239#533—72no—
240#533—72no—
241#533—71no—
242#533—141no—
243#533—141no—
244#533—141no—
245#533—141no—
246#533C2×D711yescharacteristic
247C2×C4⋊Q8—71no—
248C2×C4⋊Q8—71no—
249C2×#125—141no—
250C2×C22⋊Q8—72no—
251C2×C22⋊Q8—141no—
252C2×C4×Q8—72no—
253C2×C4×Q8—141no—
254C2×#117—141no—
255C22×C42—71no—
256#432—72no—
257#432—141no—
258#431—71no—
259#431—141no—
260#431C2×D711yescharacteristic
261#423—72no—
262#407—141no—
263C4×M16—71no—
264C4×M16—72no—
265C4×M16—141no—
266C4×M16—141no—
267C4×M16—141no—
268C4×M16C2×D711yescharacteristic
269C2×#97—141no—
270C2×C4×C8—72no—
271Q8⋊C8—74no—
272Q8⋊C8—74no—
273Q8⋊C8—74no—
274Q8⋊C8C2×D714yescomplemented
275C23×C7C2×C22⋊C411yescharacteristic
276C22×D7—22no—
277C22×D7—22no—
278C22×D7—41no—
279C22×D7C2×C22⋊C411yescharacteristic
280C22×D7C2×C22⋊C411yescharacteristic
281C7×Q8—24no—
282C7×Q8—42no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7C2×C22⋊C411yescharacteristic
289C2×C4×C7C2×C22⋊C411yescharacteristic
290C2×C4×C7C2×C22⋊C411yescharacteristic
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×Dic7—22no—
295C2×Dic7—22no—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—41no—
299C2×Dic7C2×C22⋊C411yescharacteristic
300C2×Dic7C2×C22⋊C411yescharacteristic
301C2×Dic7C2×C22⋊C411yescharacteristic
302C2×Dic7C2×C22⋊C411yescharacteristic
303C2×Dic7C2×C22⋊C411yescharacteristic
304C2×Dic7C2×C22⋊C411yescharacteristic
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—41no—
314C4×D7—41no—
315C4×D7—44no—
316C4×D7—44no—
317C4×D7#10414yes—
318C4×D7#13714yes—
319Dic14—24no—
320Dic14—44no—
321Dic14—42no—
322Dic14—44no—
323C7×C8—24no—
324C7×C8—42no—
325C7⋊C8—24no—
326C7⋊C8—42no—
327C22×Q8—72no—
328C22×Q8—141no—
329C23×C4—71no—
330C23×C4—71no—
331C23×C4—71no—
332C2×M16—21no—
333C2×M16—142no—
334C2×M16—142no—
335C2×M16—148no—
336C2×M16—141no—
337C22×C8—142no—
338C4⋊Q8—72no—
339C4⋊Q8—72no—
340C4⋊Q8—71no—
341C4⋊Q8—71no—
342C4⋊Q8—141no—
343C4⋊Q8—141no—
344C4⋊Q8—141no—
345C4⋊Q8—141no—
346C4⋊Q8C4×D711yescharacteristic
347C4⋊Q8C4×D711yescharacteristic
348#125—21no—
349#125—142no—
350#125—142no—
351#125—141no—
352#125—141no—
353#125—141no—
354C22⋊Q8—21no—
355C22⋊Q8—72no—
356C22⋊Q8—72no—
357C22⋊Q8—72no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360C22⋊Q8—142no—
361C22⋊Q8—142no—
362C22⋊Q8—141no—
363C22⋊Q8—141no—
364C22⋊Q8—141no—
365C22⋊Q8C4×D712yes—
366C4×Q8—21no—
367C4×Q8—72no—
368C4×Q8—72no—
369C4×Q8—72no—
370C4×Q8—142no—
371C4×Q8—142no—
372C4×Q8—142no—
373C4×Q8—141no—
374C4×Q8—142no—
375C4×Q8—141no—
376C4×Q8—141no—
377C4×Q8C22×D712yes—
378#117—21no—
379#117—142no—
380#117—142no—
381#117—141no—
382#117—141no—
383#117—141no—
384C2×C4⋊C4—72no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4—142no—
389C2×C4⋊C4—141no—
390C2×C22⋊C4—142no—
391C2×C42—71no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—71no—
401C2×C42—71no—
402C2×C42—142no—
403C2×C42C22×D711yescharacteristic
404C4⋊C8—72no—
405C4⋊C8—142no—
406C4⋊C8C22×D712yes—
407C22⋊C8—21no—
408C22⋊C8—142no—
409C22⋊C8—141no—
410#97—21no—
411#97—142no—
412#97—142no—
413#97—141no—
414C4×C8—21no—
415C4×C8—72no—
416C4×C8—142no—
417C4×C8—141no—
418C4×C8C22×D712yes—
419C22×C7—21no—
420C22×C7—21no—
421C22×C7—21no—
422C22×C7C22×C22⋊C411yescharacteristic
423C2×D7—21no—
424C2×D7—21no—
425C2×D7—21no—
426C2×D7—21no—
427C2×D7—44no—
428C2×D7#41912yes—
429C2×D7C2×#10412yes—
430C4×C7—21no—
431C4×C7—21no—
432C4×C7—21no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7C2×#10412yes—
436C4×C7#56812yes—
437C4×C7C2×#13712yes—
438Dic7—21no—
439Dic7—21no—
440Dic7—21no—
441Dic7—21no—
442Dic7—42no—
443Dic7—42no—
444Dic7#41912yes—
445Dic7C2×#10412yes—
446C24—71no—
447C2×Q8—21no—
448C2×Q8—72no—
449C2×Q8—142no—
450C2×Q8—142no—
451C2×Q8—141no—
452C2×Q8—284no—
453C2×Q8—284no—
454C2×Q8C2×C4×D712yes—
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—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—144no—
471C22×C4—144no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—282no—
475C22×C4C2×C4×D711yescharacteristic
476C22×C4D4×D711yescharacteristic
477C22×C4D4×D711yescharacteristic
478M16—24no—
479M16—148no—
480M16—144no—
481M16—284no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—148no—
485C2×C8—142no—
486C2×C8—142no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—72no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—142no—
498C4⋊C4—142no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—141no—
503C4⋊C4—141no—
504C4⋊C4—141no—
505C4⋊C4—142no—
506C4⋊C4C2×C4×D712yes—
507C22⋊C4—22no—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C22⋊C4—142no—
511C42—22no—
512C42—71no—
513C42—71no—
514C42—71no—
515C42—71no—
516C42—71no—
517C42—71no—
518C42—71no—
519C42—71no—
520C42—71no—
521C42—71no—
522C42—71no—
523C42—71no—
524C42—142no—
525C42—142no—
526C42—142no—
527C42C2×C4×D711yescharacteristic
528C42D4×D711yescharacteristic
529C42D4×D711yescharacteristic
530C42C23×D711yescharacteristic
531C2×C7—41no—
532C2×C7#287911yescharacteristic
533C2×C7C2×#41911yescharacteristic
534C2×C7C22×#10411yescharacteristic
535D7—22no—
536D7—22no—
537C23—71no—
538C23—71no—
539C23—142no—
540C23—142no—
541C23—281no—
542C23D7×C22⋊C411yescharacteristic
543Q8—24no—
544Q8—42no—
545Q8—144no—
546Q8—284no—
547Q8—284no—
548Q8—282no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—71no—
555C2×C4—71no—
556C2×C4—74no—
557C2×C4—74no—
558C2×C4—71no—
559C2×C4—71no—
560C2×C4—71no—
561C2×C4—71no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—284no—
577C2×C4—284no—
578C2×C4—281no—
579C2×C4D7×C22⋊C411yescharacteristic
580C2×C4D7×C22⋊C411yescharacteristic
581C2×C4D7×C22⋊C411yescharacteristic
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C2×D4×D711yescharacteristic
584C2×C4C2×D4×D711yescharacteristic
585C8—24no—
586C8—42no—
587C8—144no—
588C8—282no—
589C7#1750111yescharacteristic, complemented, Sylow
590C22—21no—
591C22—21no—
592C22—21no—
593C22—72no—
594C22—72no—
595C22—141no—
596C22—141no—
597C22—141no—
598C22—141no—
599C22—284no—
600C22C2×D7×C22⋊C411yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—42no—
605C4—42no—
606C4—72no—
607C4—72no—
608C4—141no—
609C4—141no—
610C4—141no—
611C4—141no—
612C4—282no—
613C4—282no—
614C4D7×#10412yes—
615C4#9044112yes—
616C4D7×#13712yes—
617C2—41no—
618C2—142no—
619C2—142no—
620C2#1173610711yescharacteristic
621C2D7×#41911yescharacteristic
622C2C2×D7×#10411yescharacteristic
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#17501
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. #11725994C2semidirect
9. #11725993C2semidirect
45. C7×Q8⋊C8C22semidirect
46. Q8⋊(C7⋊C8)C22semidirect
47. #89279C22semidirect
48. #89251C22semidirect
150. #1485D7semidirect
274. Q8⋊C8C2×D7semidirect
589. C7#17501semidirect

Maximal subgroups

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

subgroupindex
D7×#5332
#117367272
C4×D7×M162
C7×#14852
#117295792
#117285232
#117276102
#117259942
#117259932
#175017