Finite Group Databasebeta

← #13356498

Subgroups — #13356498

Order 1792.

1198 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
1C2×C7×#417C211yesmaximal, characteristic
2#11741531C211yesmaximal, characteristic
3C2×#90004C211yesmaximal, characteristic
4C2×M16×Dic7C211yesmaximal, characteristic, complemented
5C2×#89745C211yesmaximal, characteristic, complemented
6#11729739C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11728010C212yesmaximal, complemented
9#11728008C212yesmaximal, complemented
10#11727942C212yesmaximal, complemented
11#11727940C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4—21no—
18C22×C4×Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C7×#417—24no—
21C2×C7×#103C2212yes—
22C2×C7×#102C2212yes—
23#90004C418yescomplemented
24#90002C2211yescharacteristic
25#90002C2211yescharacteristic
26C2×#6409C2212yes—
27#89993C2211yescharacteristic
28#89993C2211yescharacteristic
29#89992—21no—
30#89992—21no—
31C2×#6406C2212yes—
32C4×C22⋊Dic7—21no—
33C4×C22⋊Dic7—21no—
34C2×C7⋊#95—21no—
35M16×Dic7—24no—
36C2×#6389C2212yes—
37C2×C8×Dic7C2212yes—
38#89745—21no—
39#89745—21no—
40#89745C2212yes—
41#89745C2212yes—
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21093—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41C2×C414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C23×Dic7—21no—
60C23×Dic7D411yescharacteristic
61C23×Dic7D411yescharacteristic
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C2×C7×M16D414yes—
65C22×C7×C8C2312yes—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2312yes—
70C2×C22⋊Dic7—21no—
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75#6489—22no—
76#6489—22no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7C2×C414yes—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2312yes—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—44no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7D411yescharacteristic
90C2×C4×Dic7D411yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C2×#1136—22no—
94C2×#1136—22no—
95C2×#1136C2×C414yes—
96C22×C7⋊C8C2312yes—
97C7×#103—24no—
98C7×#102—24no—
99#6409C2×C418yescomplemented
100#6406C2×C418yescomplemented
101C7⋊#95—22no—
102C7⋊#95—22no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107#6389—24no—
108C8×Dic7—24no—
109C2×#417—71no—
110C2×#417D711yescharacteristic
111C2×C4×M16—71no—
112#2297—71no—
113C2×#389—71no—
114#1936—72no—
115#1935—72no—
116#1934—72no—
117#1933—72no—
118#1736—71no—
119#1736—71no—
120C24×C7—21no—
121C24×C7C2×D411yescharacteristic
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—44no—
125C7×#41—44no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8C22×C412yes—
132C2×C7×D4—24no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4C22×C412yes—
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7#4112yes—
153C22×C4×C7#4112yes—
154C22×C4×C7#4112yes—
155C22×C4×C7#4112yes—
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—24no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—41no—
166C22×Dic7—41no—
167C22×Dic7—42no—
168C22×Dic7—42no—
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C22×Dic7C2×D411yescharacteristic
173C7×M16—28no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8C2×D414yes—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—42no—
180C22⋊Dic7—42no—
181C22⋊Dic7—42no—
182C22⋊Dic7—42no—
183C22⋊Dic7—42no—
184C22⋊Dic7—42no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7C22×C414yes—
192Dic7⋊C4—44no—
193Dic7⋊C4—44no—
194C4×Dic7—22no—
195C4×Dic7—22no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—42no—
201#1136—28no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8C22×C414yes—
205C22×#41C2×D711yescharacteristic
206C22×M16—71no—
207C22×M16C2×D711yescharacteristic
208C2×#117—71no—
209C22×C4⋊C4—71no—
210C22×C4⋊C4—141no—
211C22×C4⋊C4C2×D711yescharacteristic
212C22×C42—71no—
213#417—24no—
214#417—78no—
215C2×#103—72no—
216C2×#103C2×D712yes—
217C2×#102—72no—
218C2×#102C2×D712yes—
219C4×M16—144no—
220C2×#97—72no—
221C2×C4×C8—72no—
222#391—71no—
223#391—71no—
224#390—71no—
225#390—71no—
226#389—72no—
227#389—72no—
228#389—141no—
229#389—141no—
230#386—141no—
231#386—141no—
232C4×C22⋊C4—141no—
233C4×C22⋊C4—141no—
234C2×#95—141no—
235#340—72no—
236#340—72no—
237#340—72no—
238#340—72no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—22no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7C4×D412yes—
251C23×C7#12112yes—
252C23×C7#12412yes—
253C23×C7C22×D411yescharacteristic
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8—41no—
257C7×Q8—41no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—48no—
261C7×D4—41no—
262C7×D4—41no—
263C2×C4×C7—22no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—48no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—84no—
271C2×C4×C7#11714yes—
272C2×C4×C7C4×D412yes—
273C2×C4×C7C4×D414yes—
274C2×C4×C7#12112yes—
275C2×C4×C7#12314yes—
276C2×C4×C7#12412yes—
277C2×C4×C7#12614yes—
278C2×C4×C7C23×C411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—24no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—44no—
296C2×Dic7—44no—
297C2×Dic7—48no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7AGL(1,ℤ/8)12yes—
307C2×Dic7#13712yes—
308C7×C8—24no—
309C7⋊C8—24no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41C4×D714yes—
316C22×Q8C22×D711yescharacteristic
317C22×D4—21no—
318C22×D4C22×D711yescharacteristic
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—141no—
324C23×C4C22×D711yescharacteristic
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—74no—
328C2×M16—142no—
329C2×M16—142no—
330C2×M16C22⋊D714yes—
331C22×C8—72no—
332C22×C8C22×D712yes—
333#117—142no—
334#117—142no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—74no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—284no—
347C2×C4⋊C4C22×D712yes—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—142no—
358C2×C42—142no—
359C2×C42—141no—
360#103—24no—
361#103—78no—
362#102—24no—
363#102—78no—
364#97—144no—
365C4×C8—144no—
366#95—142no—
367#95—142no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—81no—
382C22×C7—81no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7#38814yes—
386C22×C7C2×C4×D411yescharacteristic
387C22×C7C2×#12111yescharacteristic
388C22×C7C2×#12411yescharacteristic
389C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C7C2×#13711yescharacteristic
391C22×C7#57511yescharacteristic
392C22×C7#57511yescharacteristic
393C4×C7—42no—
394C4×C7—82no—
395C4×C7#38814yes—
396C4×C7C2×#11711yescharacteristic
397C4×C7C2×C4×D411yescharacteristic
398C4×C7C2×#12311yescharacteristic
399C4×C7C2×#12611yescharacteristic
400Dic7—22no—
401Dic7—22no—
402Dic7—44no—
403Dic7—41no—
404Dic7—41no—
405Dic7—81no—
406Dic7—81no—
407C24—21no—
408C24D4×D711yescharacteristic
409#41—28no—
410#41—28no—
411#41—44no—
412#41—44no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D712yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D712yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—144no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—142no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—281no—
449C22×C4—281no—
450C22×C4—282no—
451C22×C4—282no—
452C22×C4C2×C4×D712yes—
453C22×C4#115712yes—
454C22×C4D4×D711yescharacteristic
455C22×C4#115912yes—
456C22×C4#115912yes—
457C22×C4#115912yes—
458C22×C4C2×C22⋊D712yes—
459C22×C4C23×D711yescharacteristic
460M16—28no—
461M16—148no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—74no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8C2×C22⋊D714yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—74no—
472C4⋊C4—144no—
473C4⋊C4—142no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—284no—
479C4⋊C4—284no—
480C22⋊C4—282no—
481C22⋊C4—282no—
482C22⋊C4—282no—
483C22⋊C4—282no—
484C22⋊C4—282no—
485C42—142no—
486C42—142no—
487C42—141no—
488C42—141no—
489C42—141no—
490C42—141no—
491C42—282no—
492C2×C7—24no—
493C2×C7—82no—
494C2×C7C2×#38811yescharacteristic
495C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
496C2×C7C4×#13711yescharacteristic
497C2×C7#305511yescharacteristic
498C2×C7#305611yescharacteristic
499C2×C7#314111yescharacteristic
500C2×C7#314211yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23(C22⋊D7)⋊C412yes—
513C23#644812yes—
514C23#650112yes—
515C23C2×D4×D711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—72no—
533C2×C4—72no—
534C2×C4—84no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—144no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—284no—
547C2×C4—284no—
548C2×C4—288no—
549C2×C4—282no—
550C2×C4—282no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4#644014yes—
558C2×C4#644214yes—
559C2×C4(C22⋊D7)⋊C412yes—
560C2×C4#644812yes—
561C2×C4C4×C22⋊D714yes—
562C2×C4#649814yes—
563C2×C4#650112yes—
564C2×C4C22×C4×D711yescharacteristic
565C2×C4C2×#115711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4C2×#115911yescharacteristic
568C2×C4C2×#115911yescharacteristic
569C2×C4C22×C22⋊D711yescharacteristic
570C8—24no—
571C8—144no—
572C7#2109311yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#8971614yes—
587C22C2×(C22⋊D7)⋊C411yescharacteristic
588C22C2×#644811yescharacteristic
589C22D7×AGL(1,ℤ/8)11yescharacteristic
590C22#9046411yescharacteristic
591C22D7×#13711yescharacteristic
592C22#9046811yescharacteristic
593C22C2×#650111yescharacteristic
594C4—42no—
595C4—82no—
596C4—142no—
597C4—142no—
598C4—281no—
599C4—281no—
600C4—284no—
601C4—561no—
602C4—561no—
603C4#8971614yes—
604C4C2×#644011yescharacteristic
605C4C2×#644211yescharacteristic
606C4C2×C4×C22⋊D711yescharacteristic
607C4C2×#649811yescharacteristic
608C2—24no—
609C2—82no—
610C2#1173631111yescharacteristic
611C2#1173631211yescharacteristic
612C2#1173632311yescharacteristic
613C2#1173632411yescharacteristic
614C2C2×#8971611yescharacteristic
615C2#1174139511yescharacteristic
616C2#1174143011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#21093
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×M16×Dic7C2semidirect
5. C2×#89745C2semidirect
6. #11729739C2semidirect
7. #11729312C2semidirect
8. #11728010C2semidirect
9. #11728008C2semidirect
10. #11727942C2semidirect
11. #11727940C2semidirect
23. #90004C4semidirect
99. #6409C2×C4semidirect
100. #6406C2×C4semidirect
572. C7#21093semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4172
#117415312
C2×#900042
C2×M16×Dic72
C2×#897452
#117297392
#117293122
#117280102
#117280082
#117279422
#117279402
#210937