Finite Group Databasebeta

← #13370711

Subgroups — #13370711

Order 1792.

1280 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊C8C211yesmaximal, characteristic
2C2×C4×C22⋊Dic7C211yesmaximal, characteristic
3C2×#89490C211yesmaximal, characteristic, complemented
4#11729880C211yesmaximal, characteristic
5#11729728C212yesmaximal, complemented
6C8×C22⋊Dic7C212yesmaximal, complemented
7#11727667C212yesmaximal, complemented
8#11727666C212yesmaximal, complemented
9#11727664C211yesmaximal, characteristic, complemented
10#11727662C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C23×C7×C8C2212yes—
14C22×C22⋊Dic7C411yescharacteristic
15C22×C22⋊Dic7C411yescharacteristic
16C22×C4×Dic7C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C2×C7×C22⋊C8—24no—
19C2×C7×C22⋊C8C2211yescharacteristic
20C2×C7×C22⋊C8C2212yes—
21C2×C7×C22⋊C8C2211yescharacteristic
22C4×C22⋊Dic7C411yescharacteristic
23C4×C22⋊Dic7C411yescharacteristic
24C4×C22⋊Dic7C411yescharacteristic
25C4×C22⋊Dic7C411yescharacteristic
26C4×C22⋊Dic7C2211yescharacteristic
27C4×C22⋊Dic7C2211yescharacteristic
28C4×C22⋊Dic7C2211yescharacteristic
29C4×C22⋊Dic7C2211yescharacteristic
30C2×C7⋊#95C411yescharacteristic
31C2×C7⋊#95C411yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2212yes—
35C2×C22⋊(C7⋊C8)C2212yes—
36C2×Dic7⋊C8C2212yescomplemented
37C2×Dic7⋊C8C2212yescomplemented
38C2×C8×Dic7C2212yescomplemented
39C2×C8×Dic7C2212yescomplemented
40#89490—24no—
41#89345C2212yescomplemented
42#89345C2212yescomplemented
43#89345C2212yescomplemented
44#89345C2212yescomplemented
45#22552—71nomaximal, Sylow
46C25×C7C2×C411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—22no—
49C23×C4×C7—24no—
50C23×C4×C7—42no—
51C23×C4×C7C2×C411yescharacteristic
52C23×C4×C7C2×C411yescharacteristic
53C23×C4×C7C2×C411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C23×Dic7C2×C411yescharacteristic
58C23×Dic7C2×C411yescharacteristic
59C23×Dic7C2×C411yescharacteristic
60C23×Dic7C2×C411yescharacteristic
61C22×C7×C8—24no—
62C22×C7×C8—24no—
63C22×C7×C8—42no—
64C22×C7×C8C2312yes—
65C22×C7×C8C2312yes—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7C818yescomplemented
68C2×C22⋊Dic7C2×C411yescharacteristic
69C2×C22⋊Dic7C2×C411yescharacteristic
70C2×C22⋊Dic7C2×C411yescharacteristic
71C2×C22⋊Dic7C2×C411yescharacteristic
72C2×C22⋊Dic7C2×C411yescharacteristic
73C2×C22⋊Dic7C2×C411yescharacteristic
74C2×C22⋊Dic7C2×C411yescharacteristic
75C2×C22⋊Dic7C2×C411yescharacteristic
76C2×C4×Dic7—22no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—42no—
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
86C2×C4×Dic7C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88C22×C7⋊C8C2312yes—
89C7×C22⋊C8—22no—
90C7×C22⋊C8—24no—
91C7×C22⋊C8D414yes—
92C7⋊#95C2×C411yescharacteristic
93C7⋊#95C2×C411yescharacteristic
94C7⋊#95C2×C411yescharacteristic
95C7⋊#95C2×C411yescharacteristic
96C7⋊#95C2×C411yescharacteristic
97C7⋊#95C2×C411yescharacteristic
98C7⋊#95C2×C411yescharacteristic
99C7⋊#95C2×C411yescharacteristic
100C22⋊(C7⋊C8)—22no—
101C22⋊(C7⋊C8)—22no—
102C22⋊(C7⋊C8)—24no—
103C22⋊(C7⋊C8)—42no—
104Dic7⋊C8—24no—
105Dic7⋊C8—42no—
106C8×Dic7—24no—
107C8×Dic7—42no—
108C2×#431—71no—
109C22×C22⋊C8D711yescharacteristic
110C2×C4×C22⋊C4—71no—
111#1923—72no—
112#1923—72no—
113#1850—72no—
114#1817—71no—
115#1797—71no—
116#1779—71no—
117C8×C22⋊C4—72no—
118C4×C22⋊C8—71no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7—22no—
122C24×C7—22no—
123C24×C7—41no—
124C24×C7—41no—
125C24×C7C2×C814yes—
126C24×C7C22×C411yescharacteristic
127C24×C7C22×C411yescharacteristic
128C24×C7C22×C411yescharacteristic
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7—41no—
147C22×C4×C7—48no—
148C22×C4×C7—48no—
149C22×C4×C7C22×C411yescharacteristic
150C22×C4×C7C22×C411yescharacteristic
151C22×C4×C7C22×C411yescharacteristic
152C22×C4×C7C2×D412yes—
153C22×C4×C7#4114yes—
154C22×C4×C7#4112yes—
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—41no—
163C22×Dic7—41no—
164C22×Dic7—41no—
165C22×Dic7—41no—
166C22×Dic7C2×C814yescomplemented
167C22×Dic7C2×C814yescomplemented
168C22×Dic7C22×C411yescharacteristic
169C22×Dic7C22×C411yescharacteristic
170C22×Dic7C22×C411yescharacteristic
171C22×Dic7C22×C411yescharacteristic
172C22×Dic7C22×C411yescharacteristic
173C22×Dic7C22×C411yescharacteristic
174C22×Dic7C22×C411yescharacteristic
175C22×Dic7C22×C411yescharacteristic
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—48no—
182C2×C7×C8C2×D414yes—
183C22⋊Dic7—22no—
184C22⋊Dic7—22no—
185C22⋊Dic7—22no—
186C22⋊Dic7C2×C814yes—
187C22⋊Dic7C2×C814yes—
188C22⋊Dic7C2×C814yescomplemented
189C22⋊Dic7C2×C814yescomplemented
190C22⋊Dic7M1614yes—
191C4×Dic7—22no—
192C4×Dic7—22no—
193C4×Dic7—21no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—21no—
197C4×Dic7—22no—
198C4×Dic7—22no—
199C4×Dic7—41no—
200C4×Dic7—41no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C24×C4C2×D711yescharacteristic
208C23×C8C2×D712yes—
209C22×C22⋊C4—71no—
210C22×C22⋊C4—71no—
211C22×C42—71no—
212C22×C42—71no—
213#431—144no—
214C2×C4⋊C8—72no—
215C2×C4⋊C8—72no—
216C2×C22⋊C8—24no—
217C2×C22⋊C8—71no—
218C2×C22⋊C8—71no—
219C2×C22⋊C8—72no—
220C2×C22⋊C8—72no—
221C2×C22⋊C8C2×D711yescharacteristic
222C2×C22⋊C8C2×D712yes—
223C2×C22⋊C8C2×D711yescharacteristic
224C2×C4×C8—72no—
225C2×C4×C8—72no—
226C4×C22⋊C4—71no—
227C4×C22⋊C4—71no—
228C4×C22⋊C4—71no—
229C4×C22⋊C4—71no—
230C4×C22⋊C4—71no—
231C4×C22⋊C4—71no—
232C4×C22⋊C4—71no—
233C4×C22⋊C4—71no—
234C2×#95—71no—
235C2×#95—71no—
236#336—72no—
237#336—72no—
238#336—72no—
239#336—72no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—21no—
249C23×C7—21no—
250C23×C7—21no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—28no—
255C23×C7—28no—
256C23×C7—44no—
257C23×C7—44no—
258C23×C7—44no—
259C23×C7—44no—
260C23×C7—44no—
261C23×C7#11712yes—
262C23×C7C4×D412yes—
263C23×C7C22×C812yes—
264C23×C7C22×C812yes—
265C23×C7C22×C812yes—
266C23×C7C2×M1612yes—
267C23×C7#13112yes—
268C23×C7C23×C411yescharacteristic
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7—44no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7#11712yes—
286C2×C4×C7C4×D412yes—
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4112yes—
290C2×C4×C7H(5,𝔽2)12yes—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—28no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—44no—
311C2×Dic7—44no—
312C2×Dic7—44no—
313C2×Dic7C22×C812yes—
314C2×Dic7C22×C812yes—
315C2×Dic7C22×C812yes—
316C2×Dic7C22×C812yes—
317C2×Dic7C2×M1612yes—
318C2×Dic7C2×M1612yes—
319C7×C8—24no—
320C7×C8—42no—
321C7⋊C8—42no—
322C7⋊C8—42no—
323C25C4×D711yescharacteristic
324C23×C4—22no—
325C23×C4—24no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—71no—
329C23×C4—71no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4C4×D711yescharacteristic
333C23×C4C4×D711yescharacteristic
334C23×C4C4×D711yescharacteristic
335C23×C4C22×D711yescharacteristic
336C23×C4C22×D711yescharacteristic
337C23×C4C22×D711yescharacteristic
338C22×C8—24no—
339C22×C8—24no—
340C22×C8—42no—
341C22×C8—72no—
342C22×C8—72no—
343C22×C8C22×D712yes—
344C22×C8C22×D712yes—
345C2×C22⋊C4—78no—
346C2×C22⋊C4—71no—
347C2×C22⋊C4—71no—
348C2×C22⋊C4—71no—
349C2×C22⋊C4—71no—
350C2×C22⋊C4—71no—
351C2×C22⋊C4—71no—
352C2×C22⋊C4—71no—
353C2×C22⋊C4—71no—
354C2×C22⋊C4—144no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—71no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—142no—
364C2×C42—142no—
365C2×C42—282no—
366C4⋊C8—144no—
367C4⋊C8—282no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370C22⋊C8—142no—
371C22⋊C8—144no—
372C22⋊C8—142no—
373C22⋊C8—282no—
374C22⋊C8C22⋊D714yes—
375C4×C8—144no—
376C4×C8—282no—
377#95—71no—
378#95—71no—
379#95—71no—
380#95—71no—
381#95—71no—
382#95—71no—
383#95—71no—
384#95—71no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—22no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—22no—
400C22×C7—416no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7#43114yescomplemented
406C22×C7C8×D414yes—
407C22×C7C2×#11711yescharacteristic
408C22×C7C2×C4×D411yescharacteristic
409C22×C7#51811yescharacteristic
410C22×C7C23×C811yescharacteristic
411C22×C7C22×M1611yescharacteristic
412C22×C7C2×#13111yescharacteristic
413C22×C7#56811yescharacteristic
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7#54612yes—
418C4×C7#55112yes—
419Dic7—22no—
420Dic7—22no—
421Dic7—22no—
422Dic7—22no—
423Dic7—22no—
424Dic7—22no—
425Dic7—41no—
426Dic7—41no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—41no—
432C24—41no—
433C24C8×D714yes—
434C24C2×C4×D711yescharacteristic
435C24C2×C4×D711yescharacteristic
436C24C2×C4×D711yescharacteristic
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—74no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—71no—
464C22×C4—74no—
465C22×C4—71no—
466C22×C4—71no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—281no—
474C22×C4—281no—
475C22×C4—281no—
476C22×C4—281no—
477C22×C4C2×C4×D711yescharacteristic
478C22×C4C2×C4×D711yescharacteristic
479C22×C4C2×C4×D711yescharacteristic
480C22×C4#115712yes—
481C22×C4#115914yes—
482C22×C4C2×C22⋊D712yes—
483C22×C4C23×D711yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—48no—
490C2×C8—142no—
491C2×C8—142no—
492C2×C8—142no—
493C2×C8—142no—
494C2×C8—142no—
495C2×C8—142no—
496C2×C8C2×C22⋊D714yes—
497C22⋊C4—74no—
498C22⋊C4—74no—
499C22⋊C4—74no—
500C22⋊C4—74no—
501C22⋊C4—74no—
502C22⋊C4—142no—
503C22⋊C4—142no—
504C22⋊C4—142no—
505C42—142no—
506C42—142no—
507C42—141no—
508C42—141no—
509C42—142no—
510C42—142no—
511C42—141no—
512C42—141no—
513C42—281no—
514C42—281no—
515C2×C7—24no—
516C2×C7—24no—
517C2×C7—42no—
518C2×C7#234211yescharacteristic
519C2×C7C2×#43111yescharacteristic
520C2×C7#292411yescharacteristic
521C2×C7C2×C8×D411yescharacteristic
522C2×C7D4×M1611yescharacteristic
523C2×C7#297211yescharacteristic
524C2×C7#297411yescharacteristic
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—28no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23#644012yes—
547C23C2×C8×D712yes—
548C23C2×C8×D712yes—
549C23C2×C8×D712yes—
550C23#646412yes—
551C23D7×M1612yes—
552C23C4×C22⋊D712yes—
553C23C22×C4×D711yescharacteristic
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—72no—
571C2×C4—72no—
572C2×C4—72no—
573C2×C4—72no—
574C2×C4—72no—
575C2×C4—72no—
576C2×C4—148no—
577C2×C4—141no—
578C2×C4—141no—
579C2×C4—141no—
580C2×C4—141no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—141no—
595C2×C4—284no—
596C2×C4—284no—
597C2×C4—284no—
598C2×C4#644012yes—
599C2×C4C4×C22⋊D712yes—
600C2×C4C2×#115711yescharacteristic
601C2×C4C2×#115912yes—
602C2×C4#654812yes—
603C2×C4C22×C22⋊D711yescharacteristic
604C8—24no—
605C8—42no—
606C8—282no—
607C8—282no—
608C7#2255211yescharacteristic, complemented, Sylow
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—416no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22#8949014yescomplemented
630C22C8×C22⋊D714yes—
631C22C2×#644011yescharacteristic
632C22#9017711yescharacteristic
633C22C22×C8×D711yescharacteristic
634C22C2×#646411yescharacteristic
635C22C2×D7×M1611yescharacteristic
636C22#9043611yescharacteristic
637C22C2×C4×C22⋊D711yescharacteristic
638C4—22no—
639C4—22no—
640C4—41no—
641C4—142no—
642C4—142no—
643C4—142no—
644C4—142no—
645C4—142no—
646C4—142no—
647C4—281no—
648C4—281no—
649C4#9018612yes—
650C4#9049712yes—
651C2—24no—
652C2—24no—
653C2—42no—
654C2C2×#8949011yescharacteristic
655C2#1173602911yescharacteristic
656C2#1173604611yescharacteristic
657C2#1173604811yescharacteristic
658C2#1173954311yescharacteristic
659C2C2×C8×C22⋊D711yescharacteristic
660C2M16×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutatorC22×C7C23×C8
FittingC22×C7×C22⋊C8C2
socleC23×C7C23×C4
prime radicalC25×C7C2×C4
cosocleC22×C4×C7C24
hypercenterC22×C22⋊C8D7
hypocenterC7#22552
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#89490C2semidirect
5. #11729728C2semidirect
6. C8×C22⋊Dic7C2semidirect
7. #11727667C2semidirect
8. #11727666C2semidirect
9. #11727664C2semidirect
10. #11727662C2semidirect
11. Dic7×C22⋊C8C2semidirect
36. C2×Dic7⋊C8C22semidirect
37. C2×Dic7⋊C8C22semidirect
38. C2×C8×Dic7C22semidirect
39. C2×C8×Dic7C22semidirect
41. #89345C22semidirect
42. #89345C22semidirect
43. #89345C22semidirect
44. #89345C22semidirect
67. C2×C22⋊Dic7C8semidirect
166. C22×Dic7C2×C8semidirect
167. C22×Dic7C2×C8semidirect
188. C22⋊Dic7C2×C8semidirect
189. C22⋊Dic7C2×C8semidirect
405. C22×C7#431semidirect
608. C7#22552semidirect
629. C22#89490semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊C82
C2×C4×C22⋊Dic72
C2×#894902
#117298802
#117297282
C8×C22⋊Dic72
#117276672
#117276662
#117276642
#117276622
Dic7×C22⋊C82
#225527