Finite Group Databasebeta

← #13361499

Subgroups — #13361499

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739762C211yesmaximal, characteristic
2C2×#89781C211yesmaximal, characteristic
3#11731767C211yesmaximal, characteristic, complemented
4C7×#1802C211yesmaximal, characteristic
5#11728215C211yesmaximal, characteristic, complemented
6#11728196C211yesmaximal, characteristic, complemented
7#11727923C211yesmaximal, characteristic
8#11727923C211yesmaximal, characteristic
9#11727906C211yesmaximal, characteristic
10Dic7×#102C211yesmaximal, characteristic
11#11727322C211yesmaximal, characteristic, complemented
12#11726542C211yesmaximal, characteristic, complemented
13#11726541C211yesmaximal, characteristic, complemented
14#11726540C211yesmaximal, characteristic, complemented
15#11726538C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C2×#6433C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C7×#384C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C4×C22⋊Dic7—21no—
27C2×C7⋊#95—21no—
28C2×#6389C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89781C418yescomplemented
31#89750—21no—
32#89750—21no—
33C4⋊C4×Dic7C2211yescharacteristic
34C4⋊C4×Dic7C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40#89744—21no—
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C22⋊C4×Dic7—21no—
46C22⋊C4×Dic7—21no—
47C4×C4⋊Dic7C2211yescharacteristic
48#89704C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21619—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×Dic7—21no—
64C23×Dic7—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C4×C7×D4—22no—
68C4×C7×D4C2×C414yes—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78D4×Dic7—22no—
79D4×Dic7—22no—
80D4×Dic7—22no—
81D4×Dic7Q814yes—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6433—22no—
100#6433C2×C414yescomplemented
101C7×C4⋊C8—22no—
102C7×#102—22no—
103C7×#102—22no—
104#6406C2×C414yes—
105#6406C2×C414yes—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115C7⋊#95D411yescharacteristic
116C7⋊#95D411yescharacteristic
117#6389—22no—
118C8×Dic7—22no—
119#6382—22no—
120#6382—22no—
121#6382C2×C414yescomplemented
122#6381C2×C414yescomplemented
123C4⋊(C7⋊C8)C2×C414yescomplemented
124C2×#478—71no—
125#2330—71no—
126#2275—71no—
127#1989—71no—
128#1948—71no—
129#1919—71no—
130#1918—71no—
131#1864—71no—
132#1864—71no—
133#1802D711yescharacteristic
134#1776—71no—
135C4×#106—71no—
136#1768—71no—
137#1762—71no—
138C4×#102—71no—
139C24×C7—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4C22×C412yes—
144C2×C7×D4C2×Q812yes—
145C2×C7×D4#4112yes—
146C22×C4×C7—21no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—81no—
151C22×C4×C7—81no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—24no—
160C22×Dic7—24no—
161C22×Dic7—81no—
162C22×Dic7—81no—
163C22×Dic7—81no—
164C22×Dic7—81no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4—81no—
182C7×C4⋊C4C22×C412yes—
183C7×C22⋊C4—24no—
184C42×C7C22×C412yes—
185C42×C7#4112yes—
186C22⋊Dic7—24no—
187C22⋊Dic7—24no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7C22×C412yes—
193C4⋊Dic7C22×C412yes—
194C4⋊Dic7C2×Q812yes—
195C4⋊Dic7#4112yes—
196Dic7⋊C4—24no—
197Dic7⋊C4—24no—
198Dic7⋊C4—81no—
199Dic7⋊C4—81no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—81no—
203C4×Dic7—81no—
204C4×Dic7—81no—
205C4×Dic7—81no—
206C4×Dic7—81no—
207C4×Dic7C2×Q812yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8C22×C412yes—
214C2×C7⋊C8C22×C412yes—
215C2×#125—71no—
216C2×C4×D4—71no—
217C2×C4×D4—71no—
218C2×C4×D4C2×D711yescharacteristic
219#478—78no—
220C2×#107—71no—
221C2×#106—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8C2×D711yescharacteristic
226C2×#102—71no—
227C2×#102—71no—
228C2×#102C2×D711yescharacteristic
229C2×#102C2×D711yescharacteristic
230C2×#97—71no—
231C2×C4×C8—71no—
232#389—71no—
233#389—71no—
234#389—141no—
235#389—141no—
236#387—141no—
237#387—141no—
238#384C2×D711yescharacteristic
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C22⋊C4—141no—
245C4×C22⋊C4—141no—
246C4×C22⋊C4—141no—
247#376—71no—
248C2×#95—141no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340C2×D711yescharacteristic
256#340C2×D711yescharacteristic
257#336—71no—
258#336—71no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7C22×D411yescharacteristic
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—22no—
270C7×D4C4×Q814yes—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—84no—
285C2×C4×C7C4×D412yes—
286C2×C4×C7C4×D412yes—
287C2×C4×C7C23×C411yescharacteristic
288C2×C4×C7C22×Q811yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—22no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—84no—
316C2×Dic7—84no—
317C2×Dic7C22⋊Q812yes—
318C2×Dic7C22⋊Q812yes—
319C7×C8—41no—
320C7×C8—41no—
321C7⋊C8—22no—
322C7⋊C8—22no—
323C22×D4C22×D711yescharacteristic
324C23×C4—21no—
325C23×C4—141no—
326C23×C4—141no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8C22×D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331#125—74no—
332#125—142no—
333C4×D4—22no—
334C4×D4—74no—
335C4×D4—142no—
336C4×D4—142no—
337C4×D4—142no—
338C4×D4C4×D714yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—141no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42—141no—
361C2×C42—141no—
362C2×C42C22×D711yescharacteristic
363#107—74no—
364#106—74no—
365#106—142no—
366#106—142no—
367C4⋊C8—22no—
368C4⋊C8—74no—
369#102—22no—
370#102—22no—
371#102—74no—
372#102—74no—
373#97—142no—
374C4×C8—142no—
375#95—71no—
376#95—71no—
377#95—71no—
378#95—71no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386C22×C7—24no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7C2×C4×D411yescharacteristic
394C22×C7C2×C22⋊Q811yescharacteristic
395C22×C7#52511yescharacteristic
396C22×C7#57611yescharacteristic
397C22×C7#57711yescharacteristic
398C22×C7#57711yescharacteristic
399C22×C7#57711yescharacteristic
400C4×C7—22no—
401C4×C7—22no—
402C4×C7—42no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7C2×C4×Q811yescharacteristic
406C4×C7C4×#4111yescharacteristic
407C4×C7#52911yescharacteristic
408C4×C7#53311yescharacteristic
409Dic7—22no—
410Dic7—22no—
411Dic7—42no—
412Dic7—41no—
413Dic7—41no—
414Dic7—42no—
415Dic7—42no—
416Dic7—42no—
417C24—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C4×D712yes—
422C2×D4#115912yes—
423C2×D4Q8×D712yes—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—144no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—144no—
440C22×C4—561no—
441C22×C4—561no—
442C22×C4—561no—
443C22×C4—561no—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—72no—
454C2×C8—72no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—72no—
467C4⋊C4—72no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—144no—
471C4⋊C4—144no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—561no—
477C4⋊C4—561no—
478C4⋊C4C2×C4×D712yes—
479C22⋊C4—24no—
480C22⋊C4—144no—
481C22⋊C4—144no—
482C42—72no—
483C42—72no—
484C42—141no—
485C42—141no—
486C42—561no—
487C42—561no—
488C42—561no—
489C42—561no—
490C42—561no—
491C42C2×C4×D712yes—
492C42#115912yes—
493C2×C7—44no—
494C2×C7C4×C22⋊Q811yescharacteristic
495C2×C7#294611yescharacteristic
496C2×C7#295211yescharacteristic
497C2×C7#308111yescharacteristic
498C2×C7#308411yescharacteristic
499C2×C7#311911yescharacteristic
500C2×C7#312311yescharacteristic
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C2×D4×D711yescharacteristic
509D4—22no—
510D4—22no—
511D4—22no—
512D4#645014yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—72no—
527C2×C4—72no—
528C2×C4—84no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—564no—
552C2×C4—564no—
553C2×C4(C22⋊D7)⋊C412yes—
554C2×C4(C22⋊D7)⋊C412yes—
555C2×C4C22×C4×D711yescharacteristic
556C2×C4C2×#115911yescharacteristic
557C2×C4C2×#115911yescharacteristic
558C2×C4C2×Q8×D711yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4D7×#4111yescharacteristic
561C8—41no—
562C8—41no—
563C8—142no—
564C8—142no—
565C7#2161911yescharacteristic, complemented, Sylow
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×(C22⋊D7)⋊C411yescharacteristic
574C22#9029711yescharacteristic
575C22D7×C22⋊Q811yescharacteristic
576C22#9044811yescharacteristic
577C22#9045311yescharacteristic
578C22#9046611yescharacteristic
579C22#9046911yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—142no—
586C4—142no—
587C4—281no—
588C4—282no—
589C4—282no—
590C4—282no—
591C4—281no—
592C4—282no—
593C4C2×#645011yescharacteristic
594C4#9021311yescharacteristic
595C4#9029111yescharacteristic
596C4#9031311yescharacteristic
597C2—44no—
598C2#1173131111yescharacteristic
599C2#1173617111yescharacteristic
600C2#1173631411yescharacteristic
601C2#1173776111yescharacteristic
602C2#1173785811yescharacteristic
603C2#1173841211yescharacteristic
604C2#1173850811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1802D7
hypocenterC7#21619
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
3. #11731767C2semidirect
5. #11728215C2semidirect
6. #11728196C2semidirect
11. #11727322C2semidirect
12. #11726542C2semidirect
13. #11726541C2semidirect
14. #11726540C2semidirect
15. #11726538C2semidirect
30. #89781C4semidirect
100. #6433C2×C4semidirect
121. #6382C2×C4semidirect
122. #6381C2×C4semidirect
123. C4⋊(C7⋊C8)C2×C4semidirect
565. C7#21619semidirect

Maximal subgroups

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

subgroupindex
#117397622
C2×#897812
#117317672
C7×#18022
#117282152
#117281962
#117279232
#117279062
Dic7×#1022
#117273222
#117265422
#117265412
#117265402
#117265382
#216197