Finite Group Databasebeta

← #13372813

Subgroups — #13372813

Order 1792.

1052 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C4⋊C8C211yesmaximal, characteristic, complemented
2C42×C22⋊D7C211yesmaximal, characteristic
3C2×#89605C211yesmaximal, characteristic, complemented
4#11729729C212yesmaximal
5C8×C22⋊Dic7C212yesmaximal, complemented
6#11729284C211yesmaximal, characteristic, complemented
7#11728229C211yesmaximal, characteristic, complemented
8#11728203C212yesmaximal, complemented
9#11728201C212yesmaximal, complemented
10#11728197C211yesmaximal, characteristic, complemented
11Dic7×C4⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×C8C2212yes—
13C22×C42×C7C2211yescharacteristic
14C2×C4×C22⋊D7C411yescharacteristic
15C2×C4×C22⋊D7C411yescharacteristic
16C2×C4×C22⋊D7C2211yescharacteristic
17C2×C42×D7C2211yescharacteristic
18C2×C7×C4⋊C8—24no—
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C4⋊C8C2212yes—
21C2×C7×C4⋊C8C2211yescharacteristic
22C4×C22⋊Dic7C411yescharacteristic
23C4×C22⋊Dic7C411yescharacteristic
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2212yes—
26C2×(C2×D7)⋊C8C2212yes—
27C2×(C2×D7)⋊C8C2212yes—
28C2×C8×Dic7C2212yes—
29C2×C8×Dic7C2212yes—
30C4×(C2×D7)⋊C4C411yescharacteristic
31C4×(C2×D7)⋊C4C411yescharacteristic
32C4×(C2×D7)⋊C4C2211yescharacteristic
33C4×Dic7⋊C4C411yescharacteristic
34C4×Dic7⋊C4C411yescharacteristic
35C4×Dic7⋊C4C2211yescharacteristic
36C42×Dic7C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C2×C4⋊(C7⋊C8)C2211yescharacteristic
39#89605—24no—
40#89345C2212yes—
41#89345C2212yes—
42#89345C2212yes—
43#89345C2212yes—
44#89319C2212yes—
45#22748—71nomaximal, Sylow
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C411yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C22×C22⋊D7C2×C411yescharacteristic
50C22×C4×D7C2×C411yescharacteristic
51C22×C4×D7C2×C411yescharacteristic
52C22×C4×D7C2311yescharacteristic
53C22×C7×C8—24no—
54C22×C7×C8—24no—
55C22×C7×C8—42no—
56C22×C7×C8C2312yes—
57C22×C7×C8C2312yes—
58C2×C42×C7—22no—
59C2×C42×C7—22no—
60C2×C42×C7C2×C411yescharacteristic
61C2×C42×C7C2×C411yescharacteristic
62C2×C42×C7C2311yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×C22⋊Dic7C2×C411yescharacteristic
65C2×C22⋊Dic7C2×C411yescharacteristic
66C2×C22⋊Dic7C2×C411yescharacteristic
67C4×C22⋊D7—22no—
68C4×C22⋊D7—22no—
69C4×C22⋊D7—24no—
70C4×C22⋊D7C818yescomplemented
71C2×(C2×D7)⋊C4C2×C411yescharacteristic
72C2×(C2×D7)⋊C4C2×C411yescharacteristic
73C2×(C2×D7)⋊C4C2×C411yescharacteristic
74C2×Dic7⋊C4C2×C411yescharacteristic
75C2×Dic7⋊C4C2×C411yescharacteristic
76C2×Dic7⋊C4C2×C411yescharacteristic
77C2×C4×Dic7C2×C411yescharacteristic
78C2×C4×Dic7C2×C411yescharacteristic
79C2×C4×Dic7C2×C411yescharacteristic
80C2×C4×Dic7C2×C411yescharacteristic
81C2×C4×Dic7C2×C411yescharacteristic
82C2×C4×Dic7C2×C411yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C2×C4×Dic7C2311yescharacteristic
85C2×C4×Dic7C2311yescharacteristic
86C22×C7⋊C8C2312yes—
87C22×C7⋊C8C2312yes—
88C42×D7—22no—
89C42×D7—22no—
90C7×C4⋊C8—24no—
91C7×C4⋊C8—22no—
92C7×C4⋊C8D414yescomplemented
93C7⋊#95C2×C411yescharacteristic
94C7⋊#95C2×C411yescharacteristic
95C7⋊#95C2×C411yescharacteristic
96C7⋊#95C2×C411yescharacteristic
97C7⋊#95C2×C411yescharacteristic
98C7⋊#95C2×C411yescharacteristic
99C22⋊(C7⋊C8)—24no—
100(C2×D7)⋊C8—24no—
101(C2×D7)⋊C8—42no—
102C8×Dic7—24no—
103C8×Dic7—42no—
104C4⋊(C7⋊C8)—22no—
105C4⋊(C7⋊C8)—22no—
106C2×#431—71no—
107C22×C4⋊C8D711yescharacteristic, complemented
108C42×D4—71no—
109#1923—72no—
110#1916—72no—
111#1869—72no—
112#1831—71no—
113#1797—71no—
114#1797—71no—
115C4×C4⋊C8—71no—
116C8×C22⋊C4—72no—
117C24×C7C22×C411yescharacteristic
118C23×D7C22×C411yescharacteristic
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—41no—
128C22×C4×C7C2×C814yes—
129C22×C4×C7C22×C411yescharacteristic
130C22×C4×C7C22×C411yescharacteristic
131C22×C4×C7C22×C411yescharacteristic
132C22×C4×C7C22×C411yescharacteristic
133C22×C4×C7C22×C411yescharacteristic
134C22×C4×C7#4112yes—
135C22×C4×C7#4112yes—
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—22no—
138C2×C22⋊D7—22no—
139C2×C22⋊D7C2×C814yes—
140C22×Dic7C22×C411yescharacteristic
141C22×Dic7C22×C411yescharacteristic
142C22×Dic7C22×C411yescharacteristic
143C22×Dic7C22×C411yescharacteristic
144C22×Dic7C22×C411yescharacteristic
145C22×Dic7C22×C411yescharacteristic
146C22×Dic7C22×C411yescharacteristic
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—22no—
150C2×C4×D7—21no—
151C2×C4×D7—21no—
152C2×C4×D7—21no—
153C2×C4×D7—21no—
154C2×C4×D7—21no—
155C2×C4×D7—21no—
156C2×C4×D7—22no—
157C2×C4×D7—41no—
158C2×C4×D7C2×C814yescomplemented
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—48no—
165C2×C7×C8C2×D414yes—
166C42×C7—22no—
167C42×C7—22no—
168C42×C7—21no—
169C42×C7—21no—
170C42×C7C2×D412yes—
171C42×C7#4112yes—
172C22⋊Dic7—22no—
173C22⋊Dic7C2×C814yes—
174C22⋊Dic7M1614yes—
175(C2×D7)⋊C4—22no—
176(C2×D7)⋊C4—22no—
177(C2×D7)⋊C4C2×C814yescomplemented
178Dic7⋊C4—22no—
179Dic7⋊C4—22no—
180Dic7⋊C4C2×C814yescomplemented
181C4×Dic7—22no—
182C4×Dic7—22no—
183C4×Dic7—22no—
184C4×Dic7—21no—
185C4×Dic7—21no—
186C4×Dic7—21no—
187C4×Dic7—21no—
188C4×Dic7—21no—
189C4×Dic7—21no—
190C4×Dic7—22no—
191C4×Dic7—41no—
192C4×Dic7C2×C814yescomplemented
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C23×C8C2×D712yes—
200C2×C4×D4—71no—
201C2×C4×D4—71no—
202C2×C4×D4—71no—
203C22×C42—71no—
204C22×C42C2×D711yescharacteristic
205#431—144no—
206C2×C4⋊C8—24no—
207C2×C4⋊C8—71no—
208C2×C4⋊C8—71no—
209C2×C4⋊C8C2×D711yescharacteristic
210C2×C4⋊C8C2×D712yes—
211C2×C4⋊C8C2×D711yescharacteristic
212C2×C22⋊C8—72no—
213C2×C22⋊C8—72no—
214C2×C22⋊C8—72no—
215C2×C4×C8—72no—
216C2×C4×C8—72no—
217C4×C4⋊C4—71no—
218C4×C4⋊C4—71no—
219C4×C4⋊C4—71no—
220C4×C22⋊C4—71no—
221C4×C22⋊C4—71no—
222C4×C22⋊C4—71no—
223C4×C22⋊C4—71no—
224C4×C22⋊C4—71no—
225C4×C22⋊C4—71no—
226C43—71no—
227#336—72no—
228#336—72no—
229#336—72no—
230#336—72no—
231#336—72no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7#11712yes—
237C23×C7C22×C812yes—
238C23×C7C2×M1612yes—
239C23×C7C23×C411yescharacteristic
240C22×D7—21no—
241C22×D7—21no—
242C22×D7—21no—
243C22×D7—21no—
244C22×D7—21no—
245C22×D7—21no—
246C22×D7C22×C812yes—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—28no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—44no—
262C2×C4×C7#11712yes—
263C2×C4×C7C4×D412yes—
264C2×C4×C7C4×D412yes—
265C2×C4×C7C22×C812yes—
266C2×C4×C7C22×C812yes—
267C2×C4×C7#13112yes—
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7H(5,𝔽2)11yescharacteristic
273C2×C4×C72−1+411yescharacteristic
274C22⋊D7—28no—
275C2×Dic7—21no—
276C2×Dic7—21no—
277C2×Dic7—21no—
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7C22×C812yes—
292C2×Dic7C22×C812yes—
293C2×Dic7C22×C812yes—
294C2×Dic7C2×M1612yes—
295C2×Dic7C2×M1612yes—
296C4×D7—28no—
297C4×D7—44no—
298C4×D7—41no—
299C4×D7—41no—
300C4×D7—41no—
301C4×D7—41no—
302C7×C8—24no—
303C7×C8—42no—
304C7⋊C8—42no—
305C7⋊C8—42no—
306C22×D4—71no—
307C23×C4—71no—
308C23×C4—71no—
309C23×C4—71no—
310C23×C4C4×D711yescharacteristic
311C23×C4C4×D711yescharacteristic
312C23×C4C22×D711yescharacteristic
313C22×C8—24no—
314C22×C8—24no—
315C22×C8—42no—
316C22×C8—72no—
317C22×C8—72no—
318C22×C8C22×D712yes—
319C22×C8C22×D712yes—
320C4×D4—78no—
321C4×D4—144no—
322C4×D4—142no—
323C4×D4—142no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C22⋊C4—71no—
328C2×C22⋊C4—71no—
329C2×C22⋊C4—71no—
330C2×C22⋊C4—71no—
331C2×C22⋊C4—71no—
332C2×C22⋊C4—71no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—71no—
336C2×C42—71no—
337C2×C42—71no—
338C2×C42—71no—
339C2×C42—71no—
340C2×C42—71no—
341C2×C42—71no—
342C2×C42—71no—
343C2×C42—71no—
344C2×C42—142no—
345C2×C42—142no—
346C2×C42C4×D711yescharacteristic
347C2×C42C4×D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C4⋊C8—24no—
351C4⋊C8—22no—
352C4⋊C8—142no—
353C4⋊C8—142no—
354C4⋊C8C22⋊D714yescomplemented
355C22⋊C8—144no—
356C22⋊C8—144no—
357C22⋊C8—282no—
358C4×C8—144no—
359C4×C8—282no—
360#95—71no—
361#95—71no—
362#95—71no—
363#95—71no—
364#95—71no—
365#95—71no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7#43114yescomplemented
373C22×C7C2×#11711yescharacteristic
374C22×C7C2×C4×D411yescharacteristic
375C22×C7#52011yescharacteristic
376C22×C7C23×C811yescharacteristic
377C22×C7C22×M1611yescharacteristic
378C22×C7C2×#13111yescharacteristic
379C22×C7#56811yescharacteristic
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—22no—
383C2×D7—41no—
384C2×D7—41no—
385C2×D7—41no—
386C2×D7—41no—
387C4×C7—22no—
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—41no—
391C4×C7C8×D414yes—
392C4×C7#54611yescharacteristic
393C4×C7#54811yescharacteristic
394C4×C7#55211yescharacteristic
395C4×C7#55311yescharacteristic
396Dic7—22no—
397Dic7—22no—
398Dic7—22no—
399Dic7—22no—
400Dic7—22no—
401Dic7—41no—
402Dic7—41no—
403C24—71no—
404C24C2×C4×D711yescharacteristic
405C2×D4—74no—
406C2×D4—142no—
407C2×D4—142no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—74no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—141no—
430C22×C4—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—281no—
436C22×C4C8×D714yes—
437C22×C4C2×C4×D711yescharacteristic
438C22×C4C2×C4×D711yescharacteristic
439C22×C4C2×C4×D711yescharacteristic
440C22×C4C2×C4×D711yescharacteristic
441C22×C4C2×C4×D711yescharacteristic
442C22×C4#115912yes—
443C22×C4#116112yes—
444C22×C4C23×D711yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—48no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—142no—
454C2×C8—142no—
455C2×C8—142no—
456C2×C8—142no—
457C2×C8C2×C22⋊D714yes—
458C4⋊C4—74no—
459C4⋊C4—142no—
460C4⋊C4—142no—
461C22⋊C4—74no—
462C22⋊C4—74no—
463C22⋊C4—74no—
464C22⋊C4—142no—
465C22⋊C4—142no—
466C22⋊C4—142no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—74no—
472C42—142no—
473C42—142no—
474C42—142no—
475C42—142no—
476C42—141no—
477C42—141no—
478C42—141no—
479C42—141no—
480C42—141no—
481C42—141no—
482C42—281no—
483C42#115712yes—
484C42C2×C22⋊D712yes—
485C2×C7—24no—
486C2×C7#235211yescharacteristic
487C2×C7C2×#43111yescharacteristic
488C2×C7#292511yescharacteristic
489C2×C7C2×C8×D411yescharacteristic
490C2×C7D4×M1611yescharacteristic
491C2×C7#298811yescharacteristic
492C2×C7#299011yescharacteristic
493D7—42no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—72no—
499C23—141no—
500C23—141no—
501C23—141no—
502C23—141no—
503C23—141no—
504C23—141no—
505C23#645512yes—
506C23C2×C8×D712yes—
507C23D7×M1612yes—
508C23C22×C4×D711yescharacteristic
509D4—148no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—72no—
526C2×C4—72no—
527C2×C4—72no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—148no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—284no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4#645512yes—
553C2×C4C2×C8×D712yes—
554C2×C4C2×C8×D712yes—
555C2×C4#646412yes—
556C2×C4C4×C22⋊D712yes—
557C2×C4C4×C22⋊D712yes—
558C2×C4C2×#115711yescharacteristic
559C2×C4C2×#115911yescharacteristic
560C2×C4#654811yescharacteristic
561C2×C4C2×#116111yescharacteristic
562C2×C4#655111yescharacteristic
563C2×C4C22×C22⋊D711yescharacteristic
564C8—24no—
565C8—42no—
566C8—282no—
567C8—282no—
568C7#2274811yescharacteristic, complemented, Sylow
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—142no—
576C22—142no—
577C22—142no—
578C22—281no—
579C22—281no—
580C22—281no—
581C22—281no—
582C22#8960514yescomplemented
583C22C2×#645511yescharacteristic
584C22#9019511yescharacteristic
585C22C22×C8×D711yescharacteristic
586C22C2×#646411yescharacteristic
587C22C2×D7×M1611yescharacteristic
588C22#9043611yescharacteristic
589C22C2×C4×C22⋊D711yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—142no—
595C4—142no—
596C4—142no—
597C4—142no—
598C4—142no—
599C4—281no—
600C4—281no—
601C4C8×C22⋊D714yes—
602C4#9020411yescharacteristic
603C4#9020511yescharacteristic
604C4#9049711yescharacteristic
605C4#9050811yescharacteristic
606C2—24no—
607C2—282no—
608C2C2×#8960511yescharacteristic
609C2#1173644511yescharacteristic
610C2#1173645811yescharacteristic
611C2#1173646011yescharacteristic
612C2#1173960611yescharacteristic
613C2C2×C8×C22⋊D711yescharacteristic
614C2M16×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutatorC22×C7C23×C8
FittingC22×C7×C4⋊C8C2
socleC23×C7C23×C4
prime radicalC22×C22⋊D7C2×C4
cosocleC22×C4×C7C24
hypercenterC22×C4⋊C8D7
hypocenterC7#22748
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C7×C4⋊C8C2semidirect
3. C2×#89605C2semidirect
5. C8×C22⋊Dic7C2semidirect
6. #11729284C2semidirect
7. #11728229C2semidirect
8. #11728203C2semidirect
9. #11728201C2semidirect
10. #11728197C2semidirect
11. Dic7×C4⋊C8C2semidirect
70. C4×C22⋊D7C8semidirect
92. C7×C4⋊C8D4semidirect
107. C22×C4⋊C8D7semidirect
158. C2×C4×D7C2×C8semidirect
177. (C2×D7)⋊C4C2×C8semidirect
180. Dic7⋊C4C2×C8semidirect
192. C4×Dic7C2×C8semidirect
354. C4⋊C8C22⋊D7semidirect
372. C22×C7#431semidirect
568. C7#22748semidirect
582. C22#89605semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊C82
C42×C22⋊D72
C2×#896052
#117297292
C8×C22⋊Dic72
#117292842
#117282292
#117282032
#117282012
#117281972
Dic7×C4⋊C82
#227487