Finite Group Databasebeta

← #13375525

Subgroups — #13375525

Order 1792.

860 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89890C211yesmaximal, characteristic
2C2×#89473C211yesmaximal, characteristic, complemented
3C4×#6441C211yesmaximal, characteristic
4C7×#1753C211yesmaximal, characteristic
5#11729749C211yesmaximal, characteristic
6#11727703C211yesmaximal, characteristic
7Dic7×C22⋊C8C211yesmaximal, characteristic
8#11727588C211yesmaximal, characteristic, complemented
9#11727540C211yesmaximal, characteristic, complemented
10C8×Dic7⋊C4C211yesmaximal, characteristic, complemented
11#11727204C211yesmaximal, characteristic
12#11726373C211yesmaximal, characteristic, complemented
13#11726371C211yesmaximal, characteristic, complemented
14#11726371C211yesmaximal, characteristic, complemented
15#11726367C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6441C411yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25C4×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊(C7⋊C8)C2211yescharacteristic
27#89890—24no—
28C2×#6389C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36C2×C8×Dic7C2211yescharacteristic
37C2×C8×Dic7C2211yescharacteristic
38#89722C411yescharacteristic
39#89718C411yescharacteristic
40#89714C411yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42C4×Dic7⋊C4C2211yescharacteristic
43C4×Dic7⋊C4C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46#89473—24no—
47#89427C411yescharacteristic
48#89426C411yescharacteristic
49#89419C411yescharacteristic
50#89418C411yescharacteristic
51C7×#336C2211yescharacteristic
52C7×#336C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#22957—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7C2×C411yescharacteristic
64C23×Dic7C2×C411yescharacteristic
65C22×Dic14C2×C411yescharacteristic
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16—41no—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C22×C7×C8C2311yescharacteristic
73C2×C7×C22⋊C4C2×C411yescharacteristic
74C2×C7×C22⋊C4C2×C411yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7C2×C411yescharacteristic
78C2×C22⋊Dic7C2×C411yescharacteristic
79C2×C4⋊Dic7C2×C411yescharacteristic
80C2×C4⋊Dic7C2×C411yescharacteristic
81C2×C4⋊Dic7C2×C411yescharacteristic
82C2×Dic7⋊C4C2×C411yescharacteristic
83C2×Dic7⋊C4C2×C411yescharacteristic
84C2×Dic7⋊C4C2×C411yescharacteristic
85C2×Dic7⋊C4C2×C411yescharacteristic
86C2×Dic7⋊C4C2×C411yescharacteristic
87C2×Dic7⋊C4C2×C411yescharacteristic
88C2×C4×Dic7—22no—
89C2×C4×Dic7—22no—
90C2×C4×Dic7C2×C411yescharacteristic
91C2×C4×Dic7C2×C411yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C2×#1136—22no—
97C2×#1136—22no—
98C2×#1136—41no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103#6441—24no—
104C4×Dic14—22no—
105C4×Dic14—22no—
106C7×C22⋊C8Q814yescomplemented
107C7×#97—22no—
108C4×C7×C8—41no—
109C7×#95C2×C411yescharacteristic
110C7×#95C2×C411yescharacteristic
111C7⋊#95C2×C411yescharacteristic
112C7⋊#95C2×C411yescharacteristic
113C7⋊#95C2×C411yescharacteristic
114C7⋊#95C2×C411yescharacteristic
115C7⋊#95C2×C411yescharacteristic
116C7⋊#95C2×C411yescharacteristic
117C7⋊#95C2×C411yescharacteristic
118C7⋊#95C2×C411yescharacteristic
119C22⋊(C7⋊C8)—22no—
120#6389—22no—
121Dic7⋊C8—22no—
122Dic7⋊C8—22no—
123Dic7⋊C8—22no—
124Dic7⋊C8—22no—
125Dic7⋊C8—41no—
126Dic7⋊C8D414yes—
127C8×Dic7—22no—
128C8×Dic7—22no—
129C8×Dic7—41no—
130C4⋊(C7⋊C8)—22no—
131C4⋊(C7⋊C8)—41no—
132C2×#446—71no—
133C2×#423—71no—
134C4×C22⋊Q8—71no—
135#1984—71no—
136#1939—71no—
137#1939—71no—
138#1939—71no—
139#1917—71no—
140#1917—71no—
141#1851—71no—
142#1851—71no—
143C8×C4⋊C4—71no—
144#1753—71no—
145#1753D711yescharacteristic
146C4×C22⋊C8—71no—
147C24×C7C22×C411yescharacteristic
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C22×C411yescharacteristic
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C22×C411yescharacteristic
155C22×C4×C7C22×C411yescharacteristic
156C22×C4×C7C22×C411yescharacteristic
157C22×C4×C7C2×Q812yes—
158C22×C4×C7#4112yes—
159C22×C4×C7C2411yescharacteristic
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—41no—
166C22×Dic7C22×C411yescharacteristic
167C22×Dic7C22×C411yescharacteristic
168C22×Dic7C22×C411yescharacteristic
169C22×Dic7C22×C411yescharacteristic
170C22×Dic7C22×C411yescharacteristic
171C22×Dic7C22×C411yescharacteristic
172C22×Dic7C22×C411yescharacteristic
173C22×Dic7C22×C411yescharacteristic
174C2×Dic14—22no—
175C2×Dic14—22no—
176C2×Dic14—22no—
177C7×M16—44no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8C2×D412yes—
188C2×C7×C8C2×Q812yes—
189C2×C7×C8C2×Q812yes—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—41no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C22⋊Dic7—22no—
197C22⋊Dic7—41no—
198C4⋊Dic7—22no—
199C4⋊Dic7—22no—
200C4⋊Dic7—22no—
201Dic7⋊C4—22no—
202Dic7⋊C4—22no—
203Dic7⋊C4—22no—
204Dic7⋊C4—22no—
205Dic7⋊C4—22no—
206Dic7⋊C4M1614yescomplemented
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—22no—
214C4×Dic7—22no—
215C4×Dic7C2×D412yes—
216C4×Dic7#4112yes—
217#1136—44no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C2×D412yes—
230C22×M16—71no—
231C22×M16C2×D711yescharacteristic
232C2×C22⋊Q8—71no—
233C2×C4×Q8—71no—
234C22×C42—71no—
235#446—144no—
236#423—144no—
237C2×C4⋊C8—71no—
238C2×C4⋊C8—71no—
239C2×C4⋊C8—71no—
240C2×C4⋊C8—71no—
241C2×C4⋊C8—71no—
242C2×C4⋊C8—71no—
243C2×C4⋊C8—71no—
244C2×C4⋊C8—71no—
245C2×C22⋊C8—71no—
246C2×C22⋊C8C2×D711yescharacteristic
247C2×#97—71no—
248C2×#97C2×D711yescharacteristic
249C2×C4×C8—71no—
250C2×C4×C8—71no—
251C2×C4×C8—71no—
252C2×C4×C8C2×D711yescharacteristic
253#391—71no—
254#389—71no—
255#387—71no—
256#387—71no—
257#385—71no—
258#385—71no—
259#380—71no—
260C4×C4⋊C4—71no—
261C4×C4⋊C4—71no—
262C4×C4⋊C4—71no—
263C4×C22⋊C4—71no—
264C4×C22⋊C4C2×D711yescharacteristic
265#336—71no—
266#336—71no—
267#336—71no—
268#336—71no—
269#336—71no—
270#336—71no—
271#336—71no—
272#336—71no—
273#336C2×D711yescharacteristic
274#336C2×D711yescharacteristic
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—21no—
281C23×C7C4×Q812yes—
282C23×C7C23×C411yescharacteristic
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7C4×Q812yes—
301C2×C4×C7C2×M1612yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×Q811yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7H(5,𝔽2)11yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—44no—
327C2×Dic7—44no—
328C2×Dic7C4×D412yes—
329C2×Dic7C4×D412yes—
330C2×Dic7C2×M1612yes—
331C2×Dic7C2×M1612yes—
332C2×Dic7#13112yes—
333C2×Dic7#13112yes—
334Dic14—44no—
335C7×C8—22no—
336C7×C8—22no—
337C7×C8—22no—
338C7×C8—41no—
339C7⋊C8—22no—
340C7⋊C8—41no—
341C7⋊C8—41no—
342C7⋊C8—41no—
343C22×Q8—71no—
344C23×C4—71no—
345C23×C4—71no—
346C23×C4C22×D711yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16—41no—
350C2×M16—142no—
351C2×M16—142no—
352C2×M16—281no—
353C22×C8—71no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8C22×D711yescharacteristic
358C22×C8C22×D711yescharacteristic
359C22×C8C22×D711yescharacteristic
360C22×C8C22×D711yescharacteristic
361C22⋊Q8—144no—
362C4×Q8—142no—
363C4×Q8—142no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C22⋊C4—71no—
374C2×C22⋊C4—71no—
375C2×C22⋊C4C4×D711yescharacteristic
376C2×C22⋊C4C4×D711yescharacteristic
377C2×C42—71no—
378C2×C42—71no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—71no—
383C2×C42—142no—
384C2×C42—142no—
385C2×C42C22×D711yescharacteristic
386C2×C42C22×D711yescharacteristic
387C4⋊C8—74no—
388C4⋊C8—142no—
389C4⋊C8—142no—
390C4⋊C8—142no—
391C4⋊C8—142no—
392C4⋊C8—142no—
393C4⋊C8—281no—
394C4⋊C8—281no—
395C22⋊C8—142no—
396C22⋊C8Dic1414yescomplemented
397#97—22no—
398#97—142no—
399C4×C8—41no—
400C4×C8—142no—
401C4×C8—142no—
402C4×C8—281no—
403#95—71no—
404#95—71no—
405#95—71no—
406#95—71no—
407#95—71no—
408#95—71no—
409#95—71no—
410#95—71no—
411#95C4×D711yescharacteristic
412#95C4×D711yescharacteristic
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7C2×C4×D411yescharacteristic
421C22×C7C2×C4×Q811yescharacteristic
422C22×C7#52011yescharacteristic
423C22×C7C22×M1611yescharacteristic
424C22×C7C2×#13111yescharacteristic
425C22×C7C2×#13111yescharacteristic
426C22×C7#56811yescharacteristic
427C4×C7—22no—
428C4×C7—22no—
429C4×C7—41no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7#54611yescharacteristic
433C4×C7D4×Q811yescharacteristic
434C4×C7#55211yescharacteristic
435C4×C7#55411yescharacteristic
436Dic7—22no—
437Dic7—22no—
438Dic7—22no—
439Dic7—41no—
440Dic7—41no—
441Dic7—41no—
442Dic7—41no—
443Dic7#43614yes—
444C24C2×C4×D711yescharacteristic
445C2×Q8—142no—
446C2×Q8—142no—
447C2×Q8—142no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—71no—
456C22×C4—71no—
457C22×C4—71no—
458C22×C4—71no—
459C22×C4—71no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—281no—
466C22×C4C2×Dic1412yes—
467C22×C4C2×C4×D711yescharacteristic
468C22×C4C2×C4×D711yescharacteristic
469C22×C4C2×C4×D711yescharacteristic
470C22×C4C2×C4×D711yescharacteristic
471C22×C4C2×C4×D711yescharacteristic
472C22×C4#115712yes—
473C22×C4C23×D711yescharacteristic
474M16—44no—
475M16—284no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—72no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8C2×Dic1412yes—
498C2×C8C2×Dic1412yes—
499C2×C8D4×D712yes—
500C4⋊C4—74no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C4⋊C4—142no—
504C4⋊C4—142no—
505C4⋊C4—142no—
506C4⋊C4—142no—
507C4⋊C4—142no—
508C4⋊C4—142no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—142no—
512C22⋊C4—281no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—72no—
518C42—72no—
519C42—142no—
520C42—142no—
521C42—141no—
522C42—141no—
523C42—141no—
524C42—141no—
525C42—141no—
526C42—141no—
527C2×C7—42no—
528C2×C7#236011yescharacteristic
529C2×C7C2×#43611yescharacteristic
530C2×C7#293311yescharacteristic
531C2×C7#296011yescharacteristic
532C2×C7Q8×M1611yescharacteristic
533C2×C7#299011yescharacteristic
534C2×C7#299311yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23C4×Dic1412yes—
542C23C22×C4×D711yescharacteristic
543Q8—284no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—72no—
562C2×C4—72no—
563C2×C4—72no—
564C2×C4—72no—
565C2×C4—72no—
566C2×C4—72no—
567C2×C4—141no—
568C2×C4—141no—
569C2×C4—141no—
570C2×C4—141no—
571C2×C4—141no—
572C2×C4—141no—
573C2×C4—141no—
574C2×C4—141no—
575C2×C4—141no—
576C2×C4—141no—
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—284no—
586C2×C4—284no—
587C2×C4C4×Dic1412yes—
588C2×C4D7×M1612yes—
589C2×C4C22×Dic1411yescharacteristic
590C2×C4C2×#115711yescharacteristic
591C2×C4C2×D4×D711yescharacteristic
592C2×C4#654811yescharacteristic
593C2×C4D7×#4111yescharacteristic
594C2×C4#655411yescharacteristic
595C8—22no—
596C8—22no—
597C8—22no—
598C8—41no—
599C8—142no—
600C8—281no—
601C8—281no—
602C8—281no—
603C7#2295711yescharacteristic, complemented, Sylow
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×C4×Dic1411yescharacteristic
612C22C4×D4×D711yescharacteristic
613C22#9023711yescharacteristic
614C22C2×D7×M1611yescharacteristic
615C22#9043611yescharacteristic
616C22D7×#13111yescharacteristic
617C22D7×#13111yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—74no—
624C4—142no—
625C4—142no—
626C4—142no—
627C4—281no—
628C4—281no—
629C4—281no—
630C4—281no—
631C4#9018111yescharacteristic
632C4#9022211yescharacteristic
633C4D4×Dic1411yescharacteristic
634C4#9023011yescharacteristic
635C2—42no—
636C2#1173091111yescharacteristic
637C2M16×Dic1411yescharacteristic
638C2#1173598711yescharacteristic
639C2#1173690711yescharacteristic
640C2#1173691011yescharacteristic
641C2D7×#43611yescharacteristic
642C2#1173698411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutator, socleC23×C7C23×C4
FittingC7×#1753C2
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1753D7
hypocenterC7#22957
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. C2×#89473C2semidirect
8. #11727588C2semidirect
9. #11727540C2semidirect
10. C8×Dic7⋊C4C2semidirect
12. #11726373C2semidirect
13. #11726371C2semidirect
14. #11726371C2semidirect
15. #11726367C2semidirect
106. C7×C22⋊C8Q8semidirect
206. Dic7⋊C4M16semidirect
396. C22⋊C8Dic14semidirect
603. C7#22957semidirect

Maximal subgroups

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

subgroupindex
C2×#898902
C2×#894732
C4×#64412
C7×#17532
#117297492
#117277032
Dic7×C22⋊C82
#117275882
#117275402
C8×Dic7⋊C42
#117272042
#117263732
#117263712
#117263672
#229577