Finite Group Databasebeta

← #13320674

Subgroups — #13320674

Order 1792.

994 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#1157C211yesmaximal, characteristic
2#11736721C211yesmaximal, characteristic
3#11736494C211yesmaximal, characteristic
4C7×#1487C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#11728522C211yesmaximal, characteristic, complemented
7#11728255C211yesmaximal, characteristic, complemented
8#11725994C212yesmaximal, complemented
9#11725990C212yesmaximal, complemented
10#11725986C212yesmaximal, complemented
11D7×Q8⋊C8C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13Q8×C22⋊D7C412yes—
14C2×#6470—21no—
15#90270—21no—
16#90270C411yescharacteristic
17#90269—22no—
18#90269C2212yes—
19Q8×D28C411yescharacteristic
20C4×Q8×D7C2212yes—
21#90258—21no—
22#90258C411yescharacteristic
23Q8×Dic14C411yescharacteristic
24#90202—22no—
25#90202C412yes—
26C4×#1157—21no—
27C4×#1157C2211yescharacteristic
28#90161—21no—
29#90161C2211yescharacteristic
30C7×#431C2211yescharacteristic
31C7×#423C2211yescharacteristic
32#89891—21no—
33C2×C4⋊(C7⋊C8)C2211yescharacteristic
34C2×C4×C7⋊C8C2211yescharacteristic
35#89611—22no—
36#89609C2212yes—
37#89606—21no—
38D7×C4⋊C8C2212yes—
39#89602—21no—
40(C22⋊D7)⋊C8—22no—
41#89457C2212yes—
42C4×C8×D7C2212yes—
43C7×Q8⋊C8C2212yes—
44C7×Q8⋊C8C2212yes—
45Q8⋊(C7⋊C8)C2211yescharacteristic
46Q8⋊(C7⋊C8)C2211yescharacteristic
47Q8⋊(C7⋊C8)C2211yescharacteristic
48Q8⋊(C7⋊C8)C2211yescharacteristic
49#89279C2212yes—
50#89279C2212yes—
51#89251C2212yes—
52#89251C2212yes—
53#17509—71nomaximal, Sylow
54C22×C7×Q8C2×C411yescharacteristic
55C2×Q8×D7C2×C412yes—
56C2×#1157D411yescharacteristic
57C2×C7×M16—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8—21no—
60C4×C7×Q8C2×C411yescharacteristic
61C4×C7×Q8C2×C411yescharacteristic
62C4×C7×Q8C2311yescharacteristic
63C4×C7×Q8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4C2×C411yescharacteristic
66C2×C42×C7—21no—
67C2×C42×C7C2311yescharacteristic
68#6509—22no—
69#6509C2×C412yes—
70Q8×Dic7C2×C412yes—
71#6507—22no—
72#6507C2×C412yes—
73C4×C22⋊D7—22no—
74C4×C22⋊D7—22no—
75C4×C22⋊D7C2×C412yes—
76C4×C22⋊D7D412yes—
77#6489—21no—
78#6489D411yescharacteristic
79#6487—21no—
80#6487—22no—
81#6487—22no—
82#6487C2×C412yes—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85#6470—44no—
86C2×#1131—22no—
87C2×C8×D7—22no—
88C2×C8×D7—22no—
89#6460—22no—
90#6460C2×C412yes—
91#6459—22no—
92#6459—22no—
93#6459—22no—
94D7×C4⋊C4—22no—
95D7×C4⋊C4C2×C412yes—
96#6452—22no—
97#6452—22no—
98#6452—22no—
99#6451—22no—
100#6451C2×C412yes—
101#6450—22no—
102#6450C2×C412yes—
103C4×D28—21no—
104C4×D28—21no—
105C4×D28C2×C411yescharacteristic
106C4×D28D411yescharacteristic
107#6435—22no—
108#6435D412yes—
109C42×D7—22no—
110C42×D7C2312yes—
111#6433—21no—
112#6433—21no—
113#6433—21no—
114#6432—21no—
115#6432—21no—
116#6432C2×C411yescharacteristic
117#6432C2×C411yescharacteristic
118C4×Dic14—21no—
119C4×Dic14—21no—
120C4×Dic14—21no—
121C4×Dic14—22no—
122C4×Dic14—22no—
123C4×Dic14C2×C411yescharacteristic
124C4×Dic14D411yescharacteristic
125C4×Dic14C2312yes—
126C7×C4⋊C8—21no—
127C7×C4⋊C8—21no—
128C7×C4⋊C8C2312yes—
129C7×C22⋊C8—21no—
130C4×C7×C8C2312yes—
131(C2×D7)⋊C8—22no—
132Dic7⋊C8—22no—
133Dic7⋊C8—22no—
134Dic7⋊C8—22no—
135C8×Dic7—22no—
136C4⋊(C7⋊C8)—21no—
137C4⋊(C7⋊C8)C2311yescharacteristic
138C4⋊(C7⋊C8)C2311yescharacteristic
139C4×C7⋊C8—21no—
140C4×C7⋊C8C2311yescharacteristic
141C4×C7⋊C8C2311yescharacteristic
142Q8×#41—71no—
143C8×#41—71no—
144#2906—71no—
145#1493—71no—
146#1493—72no—
147#1487—72no—
148#1487D711yescharacteristic, complemented
149#1481—71no—
150#1481—72no—
151C2×Q8⋊C8—72no—
152C2×Q8⋊C8—71no—
153C2×C7×Q8—21no—
154C2×C7×Q8—22no—
155C2×C7×Q8—41no—
156C2×C7×Q8C22×C411yescharacteristic
157C2×C7×Q8C22×C411yescharacteristic
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7C22×C411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C2×C22⋊D7C22⋊C412yes—
164Q8×D7—24no—
165Q8×D7—42no—
166#1157SD1614yes—
167#1157Q1614yes—
168C2×D28C22⋊C411yescharacteristic
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C2×D412yes—
173C2×C4×D7C2×D412yes—
174C2×Dic14—22no—
175C2×Dic14—21no—
176C2×Dic14—22no—
177C2×Dic14C22⋊C411yescharacteristic
178C2×Dic14C22×C412yes—
179C7×M16—42no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4C22×C411yescharacteristic
188C7×C4⋊C4C22×C411yescharacteristic
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7C22×C411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2411yescharacteristic
197C22⋊Dic7—22no—
198C22⋊Dic7C22⋊C412yes—
199(C2×D7)⋊C4—22no—
200(C2×D7)⋊C4—22no—
201(C2×D7)⋊C4C22⋊C412yes—
202(C2×D7)⋊C4C22⋊C412yes—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7C22⋊C411yescharacteristic
209C4⋊Dic7C22⋊C411yescharacteristic
210C4⋊Dic7C22×C412yes—
211Dic7⋊C4—22no—
212Dic7⋊C4—22no—
213Dic7⋊C4—22no—
214Dic7⋊C4—22no—
215Dic7⋊C4—22no—
216Dic7⋊C4—22no—
217Dic7⋊C4—22no—
218Dic7⋊C4—22no—
219Dic7⋊C4C22⋊C412yes—
220Dic7⋊C4C22⋊C412yes—
221C4×Dic7—22no—
222C4×Dic7—22no—
223C4×Dic7—22no—
224C4×Dic7C22×C412yes—
225C4×Dic7C2×D412yes—
226C4×Dic7C2×D412yes—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—42no—
232C2×C7⋊C8—42no—
233#1131—44no—
234C8×D7—44no—
235C8×D7—44no—
236C2×#131—141no—
237Q82—71no—
238#557—71no—
239#557—141no—
240#554—71no—
241#554—72no—
242#554—142no—
243#554—141no—
244D4×Q8—72no—
245D4×Q8—71no—
246#533—72no—
247#533—71no—
248#533—141no—
249#533—142no—
250C4×#41—71no—
251C4×#41—141no—
252C2×C4×Q8—72no—
253C2×C4×Q8C2×D711yescharacteristic
254C8×Q8—141no—
255C8×D4—142no—
256C8×D4—141no—
257#431—72no—
258#431C2×D711yescharacteristic
259#424—142no—
260#424—141no—
261#423—72no—
262#423C2×D711yescharacteristic
263C2×C4⋊C8—71no—
264C2×C4⋊C8—72no—
265C2×C4×C8—71no—
266C2×C4×C8—72no—
267Q8⋊C8—71no—
268Q8⋊C8—71no—
269Q8⋊C8—71no—
270Q8⋊C8—71no—
271Q8⋊C8—72no—
272Q8⋊C8—72no—
273Q8⋊C8—72no—
274Q8⋊C8—72no—
275Q8⋊C8C2×D712yes—
276Q8⋊C8C2×D712yes—
277C23×C7C2×C22⋊C411yescharacteristic
278C22×D7C2×C22⋊C412yes—
279C7×Q8—21no—
280C7×Q8—21no—
281C7×Q8—21no—
282C7×Q8—42no—
283C7×Q8#13112yescomplemented
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—81no—
293C2×C4×C7C2×C22⋊C411yescharacteristic
294C2×C4×C7C2×C22⋊C411yescharacteristic
295C2×C4×C7C2×C22⋊C411yescharacteristic
296C2×C4×C7C2×SD1612yes—
297C2×C4×C7C2×Q1612yes—
298C2×C4×C7C23×C411yescharacteristic
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C22⋊D7#10318yes—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7C2×C22⋊C412yes—
305C2×Dic7C2×C22⋊C412yes—
306C2×Dic7C2×C22⋊C412yes—
307D28#10314yes—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—42no—
311C4×D7—44no—
312C4×D7—82no—
313C4×D7C2×SD1614yes—
314C4×D7C2×Q1614yes—
315Dic14—22no—
316Dic14—22no—
317Dic14—44no—
318Dic14—82no—
319Dic14—81no—
320Dic14#10314yes—
321C7×C8—42no—
322C7×C8—42no—
323C7⋊C8—22no—
324C7⋊C8—41no—
325C7⋊C8—41no—
326C7⋊C8—41no—
327C2×#41—71no—
328C22×Q8—72no—
329C22×Q8C4×D711yescharacteristic
330#131—284no—
331C2×M16—21no—
332C2×M16—142no—
333C22×C8—142no—
334C22×C8—142no—
335C22×C8—141no—
336C22×C8—141no—
337C4⋊Q8—72no—
338C4⋊Q8—72no—
339C4⋊Q8—71no—
340C4⋊Q8—71no—
341C4⋊Q8—141no—
342C4⋊Q8—141no—
343C4⋊Q8—142no—
344C4⋊Q8—142no—
345#125—141no—
346#125—141no—
347#125—142no—
348#125—142no—
349#125—142no—
350#125—141no—
351C22⋊Q8—72no—
352C22⋊Q8—72no—
353C22⋊Q8—72no—
354C22⋊Q8—142no—
355C22⋊Q8—142no—
356C22⋊Q8—142no—
357C22⋊Q8—142no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360C22⋊Q8—142no—
361C22⋊Q8—141no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—71no—
365C4×Q8—71no—
366C4×Q8—72no—
367C4×Q8—72no—
368C4×Q8—72no—
369C4×Q8—142no—
370C4×Q8—142no—
371C4×Q8—141no—
372C4×Q8—141no—
373C4×Q8—142no—
374C4×Q8—141no—
375C4×Q8C4×D711yescharacteristic
376C4×Q8C4×D711yescharacteristic
377C4×Q8C22×D711yescharacteristic
378C4×Q8C22×D711yescharacteristic
379C4×D4—72no—
380C4×D4—72no—
381C4×D4—71no—
382C4×D4—71no—
383C4×D4—141no—
384C4×D4—141no—
385C4×D4—142no—
386C4×D4—142no—
387#117—71no—
388#117—72no—
389#117—142no—
390#117—141no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—72no—
393C2×C4⋊C4—142no—
394C2×C4⋊C4C4×D711yescharacteristic
395C2×C42—21no—
396C2×C42—72no—
397C2×C42—142no—
398C2×C42C22×D711yescharacteristic
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—71no—
402C4⋊C8—71no—
403C4⋊C8—142no—
404C4⋊C8—142no—
405C4⋊C8—142no—
406C4⋊C8—141no—
407C4⋊C8C22×D712yes—
408C22⋊C8—21no—
409C22⋊C8—142no—
410C4×C8—71no—
411C4×C8—71no—
412C4×C8—142no—
413C4×C8—141no—
414C4×C8C22×D712yes—
415C22×C7—21no—
416C22×C7—21no—
417C22×C7C2×#10312yes—
418C22×C7C22×C22⋊C411yescharacteristic
419C2×D7—22no—
420C2×D7—22no—
421C2×D7C2×#10314yes—
422C4×C7—22no—
423C4×C7—21no—
424C4×C7—21no—
425C4×C7—21no—
426C4×C7—41no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7C2×#10312yes—
430C4×C7C2×#13111yescharacteristic
431C4×C7#56811yescharacteristic
432C4×C7C22×SD1611yescharacteristic
433C4×C7C22×Q1611yescharacteristic
434Dic7—22no—
435Dic7—22no—
436Dic7—42no—
437Dic7—42no—
438Dic7C2×#10314yes—
439#41—74no—
440#41—74no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—41no—
444C2×Q8—71no—
445C2×Q8—72no—
446C2×Q8—142no—
447C2×Q8—142no—
448C2×Q8—141no—
449C2×Q8—144no—
450C2×Q8—282no—
451C2×Q8C2×C4×D711yescharacteristic
452C2×Q8C2×C4×D711yescharacteristic
453C2×D4—72no—
454C2×D4—71no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—72no—
458C22×C4—72no—
459C22×C4—72no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4C2×C4×D711yescharacteristic
463C22×C4D4×D711yescharacteristic
464C22×C4D4×D711yescharacteristic
465M16—42no—
466M16—284no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—284no—
474C2×C8—284no—
475C2×C8—282no—
476C2×C8—282no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—72no—
483C4⋊C4—72no—
484C4⋊C4—71no—
485C4⋊C4—71no—
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×D711yescharacteristic
501C4⋊C4C2×C4×D711yescharacteristic
502C22⋊C4—72no—
503C22⋊C4—72no—
504C22⋊C4—72no—
505C22⋊C4—142no—
506C22⋊C4—142no—
507C22⋊C4—142no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—72no—
513C42—72no—
514C42—72no—
515C42—142no—
516C42—142no—
517C42—142no—
518C42C2×C4×D711yescharacteristic
519C42D4×D711yescharacteristic
520C42D4×D711yescharacteristic
521C42C23×D711yescharacteristic
522C2×C7—22no—
523C2×C7#288011yescharacteristic
524C2×C7C22×#10311yescharacteristic
525C2×C7#290111yescharacteristic
526D7—24no—
527C23—72no—
528C23D7×C22⋊C411yescharacteristic
529Q8—21no—
530Q8—21no—
531Q8—21no—
532Q8—42no—
533Q8—74no—
534Q8—142no—
535Q8—142no—
536Q8—284no—
537Q8—561no—
538Q8—562no—
539Q8#647012yescomplemented
540D4—78no—
541D4—74no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—72no—
551C2×C4—74no—
552C2×C4—74no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—81no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—282no—
561C2×C4—284no—
562C2×C4—562no—
563C2×C4D7×C22⋊C411yescharacteristic
564C2×C4D7×C22⋊C411yescharacteristic
565C2×C4D7×C22⋊C411yescharacteristic
566C2×C4D7×SD1612yes—
567C2×C4D7×Q1612yes—
568C2×C4C22×C4×D711yescharacteristic
569C2×C4C2×D4×D711yescharacteristic
570C2×C4C2×D4×D711yescharacteristic
571C8—42no—
572C8—42no—
573C8—142no—
574C8—281no—
575C8—281no—
576C8—281no—
577C7#1750911yescharacteristic, complemented, Sylow
578C22—21no—
579C22—21no—
580C22—74no—
581C22—142no—
582C22—142no—
583C22D7×#10312yes—
584C22C2×D7×C22⋊C411yescharacteristic
585C4—21no—
586C4—22no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—74no—
593C4—142no—
594C4—142no—
595C4—282no—
596C4—282no—
597C4D7×#10312yes—
598C4C2×#647011yescharacteristic
599C4#9043611yescharacteristic
600C4C2×D7×SD1611yescharacteristic
601C4C2×D7×Q1611yescharacteristic
602C2—22no—
603C2—144no—
604C2#1173607611yescharacteristic
605C2C2×D7×#10311yescharacteristic
606C2#1173643711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1487C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#1157D4
cosocleC42×C7C24
hypercenter#1487D7
hypocenterC7#17509
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. C7×#1487C2semidirect
5. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11728522C2semidirect
7. #11728255C2semidirect
8. #11725994C2semidirect
9. #11725990C2semidirect
10. #11725986C2semidirect
11. D7×Q8⋊C8C2semidirect
148. #1487D7semidirect
283. C7×Q8#131semidirect
539. Q8#6470semidirect
577. C7#17509semidirect

Maximal subgroups

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

subgroupindex
Q8×#11572
#117367212
#117364942
C7×#14872
C2×Q8⋊(C7⋊C8)2
#117285222
#117282552
#117259942
#117259902
#117259862
D7×Q8⋊C82
#175097