Finite Group Databasebeta

← #13245839

Subgroups — #13245839

Order 1792.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×#41C211yesmaximal, characteristic
2#11729589C211yesmaximal, characteristic, complemented
3#11729580C211yesmaximal, characteristic, complemented
4#11729400C211yesmaximal, characteristic
5#11729392C211yesmaximal, characteristic
6C4×#6409C211yesmaximal, characteristic, complemented
7#11729376C211yesmaximal, characteristic, complemented
8C7×Q82C411yescharacteristic, complemented
9C7×#557—21no—
10C7×#557C411yescharacteristic, complemented
11C7×#554—22no—
12C7×#554—21no—
13C7×#554—21no—
14C7×#554C411yescharacteristic, complemented
15C7×D4×Q8—21no—
16C7×D4×Q8C411yescharacteristic, complemented
17C7×#533—22no—
18C7×#533—21no—
19C7×#533—21no—
20C7×#533C2211yescharacteristic
21C4×C7×#41—21no—
22C4×C7×#41C2211yescharacteristic
23C2×C4×C7×Q8—21no—
24C2×C4×C7×Q8C2211yescharacteristic
25C2×#6411—21no—
26C2×#6409—21no—
27C4×C4⋊Dic7C2211yescharacteristic
28#89697C2211yescharacteristic
29#89696C2211yescharacteristic
30C2×C4×C7⋊C8C2211yescharacteristic
31#89333—22no—
32Q8⋊(C7⋊C8)—22no—
33Q8⋊(C7⋊C8)—21no—
34Q8⋊(C7⋊C8)—21no—
35Q8⋊(C7⋊C8)—21no—
36D4⋊(C7⋊C8)—21no—
37#89327—21no—
38#89324—21no—
39#8439—71nomaximal, Sylow
40C2×C7×#41D411yescharacteristic
41C22×C7×Q8—21no—
42C22×C7×Q8D411yescharacteristic
43C7×C4⋊Q8—22no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—22no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8C2×C411yescharacteristic
50C7×C4⋊Q8C2×C411yescharacteristic
51C7×#125—22no—
52C7×#125—22no—
53C7×#125—22no—
54C7×#125—21no—
55C7×#125—21no—
56C7×#125—21no—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8—22no—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8—21no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8D412yes—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×Q8—22no—
71C4×C7×Q8—21no—
72C4×C7×Q8—21no—
73C4×C7×Q8—21no—
74C4×C7×Q8—21no—
75C4×C7×Q8—21no—
76C4×C7×Q8—21no—
77C4×C7×Q8—21no—
78C4×C7×Q8C2×C411yescharacteristic
79C4×C7×Q8C2×C411yescharacteristic
80C4×C7×Q8C2×C411yescharacteristic
81C4×C7×Q8D411yescharacteristic
82C4×C7×Q8D411yescharacteristic
83C4×C7×Q8D411yescharacteristic
84C4×C7×Q8D412yes—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—21no—
88C4×C7×D4—21no—
89C4×C7×D4—21no—
90C4×C7×D4—21no—
91C4×C7×D4C2×C411yescharacteristic
92C4×C7×D4D411yescharacteristic
93C7×#117—22no—
94C7×#117—21no—
95C7×#117—21no—
96C7×#117D411yescharacteristic
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4D411yescharacteristic
101C2×C42×C7—22no—
102C2×C42×C7—21no—
103C2×C42×C7—21no—
104C2×C42×C7C2311yescharacteristic
105C2×C4⋊Dic7—21no—
106C2×C4×Dic7—21no—
107C2×#1136—21no—
108C22×C7⋊C8—21no—
109#6411—42no—
110#6409—24no—
111C7⋊#95—21no—
112C22⋊(C7⋊C8)—21no—
113C4⋊(C7⋊C8)—21no—
114C4⋊(C7⋊C8)—21no—
115C4⋊(C7⋊C8)—21no—
116C4×C7⋊C8—21no—
117C4×C7⋊C8—21no—
118C4×C7⋊C8—21no—
119Q8×#41D711yescharacteristic
120#1806—71no—
121C4×#103—71no—
122#1503—71no—
123#1493—71no—
124#1487—71no—
125#1476—71no—
126C7×#41—21no—
127C7×#41—21no—
128C7×#41SD1612yes—
129C7×#41Q1612yes—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—22no—
135C2×C7×Q8—41no—
136C2×C7×Q8—42no—
137C2×C7×Q8—42no—
138C2×C7×Q8C22⋊C411yescharacteristic
139C2×C7×Q8C22⋊C411yescharacteristic
140C2×C7×Q8C22⋊C411yescharacteristic
141C2×C7×Q8C22⋊C412yes—
142C2×C7×D4—21no—
143C2×C7×D4C22⋊C411yescharacteristic
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×Dic7—41no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4C22⋊C411yescharacteristic
176C7×C4⋊C4C22⋊C411yescharacteristic
177C7×C4⋊C4C22⋊C411yescharacteristic
178C7×C4⋊C4C22⋊C411yescharacteristic
179C7×C4⋊C4C22⋊C412yes—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—21no—
184C7×C22⋊C4—21no—
185C7×C22⋊C4—21no—
186C42×C7—22no—
187C42×C7—22no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—22no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7C22×C411yescharacteristic
197C42×C7C2×D411yescharacteristic
198C42×C7C2×D411yescharacteristic
199C42×C7C2×D411yescharacteristic
200C4⋊Dic7—22no—
201C4⋊Dic7—41no—
202C4×Dic7—41no—
203C4×Dic7—41no—
204#1136—41no—
205#1136—41no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—41no—
208C2×C7⋊C8—41no—
209C2×C7⋊C8—41no—
210C2×C7⋊C8—41no—
211Q82Dic711yescharacteristic, complemented
212#557—21no—
213#557Dic711yescharacteristic, complemented
214#554—22no—
215#554—21no—
216#554—21no—
217#554Dic711yescharacteristic, complemented
218D4×Q8—21no—
219D4×Q8Dic711yescharacteristic, complemented
220#533—22no—
221#533—21no—
222#533—21no—
223#533C2×D711yescharacteristic
224C4×#41—21no—
225C4×#41C2×D711yescharacteristic
226C2×C4×Q8—21no—
227C2×C4×Q8C2×D711yescharacteristic
228#431—71no—
229#423—71no—
230C2×#104—141no—
231C2×#103—141no—
232C2×C4×C8—71no—
233C4×C4⋊C4—71no—
234#340—141no—
235#339—141no—
236#328—142no—
237Q8⋊C8—141no—
238Q8⋊C8—141no—
239Q8⋊C8—141no—
240Q8⋊C8—142no—
241D4⋊C8—141no—
242C23×C7—21no—
243C23×C7#12011yescharacteristic
244C7×Q8—21no—
245C7×Q8—21no—
246C7×Q8—21no—
247C7×Q8—21no—
248C7×Q8—42no—
249C7×Q8—42no—
250C7×Q8—44no—
251C7×Q8—41no—
252C7×Q8—41no—
253C7×Q8—82no—
254C7×Q8#10312yescomplemented
255C7×Q8#10412yescomplemented
256C7×D4—22no—
257C7×D4—22no—
258C7×D4—21no—
259C7×D4#10312yescomplemented
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—44no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—81no—
279C2×C4×C7—81no—
280C2×C4×C7—82no—
281C2×C4×C7C2×C22⋊C411yescharacteristic
282C2×C4×C7C2×C22⋊C411yescharacteristic
283C2×C4×C7C2×C22⋊C411yescharacteristic
284C2×C4×C7#12011yescharacteristic
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7C2×SD1611yescharacteristic
288C2×C4×C7C2×Q1611yescharacteristic
289C2×C4×C7#13711yescharacteristic
290C2×C4×C7#13711yescharacteristic
291C2×Dic7—42no—
292C2×Dic7—81no—
293C2×Dic7—81no—
294C7⋊C8—42no—
295C7⋊C8—81no—
296C7⋊C8—81no—
297C2×#41C22⋊D711yescharacteristic
298C22×Q8—21no—
299C22×Q8C22⋊D711yescharacteristic
300C2×M16—141no—
301C22×C8—141no—
302C4⋊Q8—22no—
303C4⋊Q8—21no—
304C4⋊Q8—22no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8C2×Dic711yescharacteristic
309C4⋊Q8C2×Dic711yescharacteristic
310#125—22no—
311#125—22no—
312#125—22no—
313#125—21no—
314#125—21no—
315#125—21no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—22no—
326C22⋊Q8C22⋊D712yes—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8C2×Dic711yescharacteristic
338C4×Q8C2×Dic711yescharacteristic
339C4×Q8C2×Dic711yescharacteristic
340C4×Q8C22⋊D711yescharacteristic
341C4×Q8C22⋊D711yescharacteristic
342C4×Q8C22⋊D711yescharacteristic
343C4×Q8C22⋊D712yes—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4C2×Dic711yescharacteristic
351C4×D4C22⋊D711yescharacteristic
352#117—22no—
353#117—21no—
354#117—21no—
355#117C22⋊D711yescharacteristic
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4C22⋊D711yescharacteristic
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—141no—
365C2×C42C22×D711yescharacteristic
366C4⋊C8—141no—
367C4⋊C8—141no—
368C4⋊C8—141no—
369#104—282no—
370#103—144no—
371C22⋊C8—141no—
372C4×C8—141no—
373C4×C8—141no—
374C4×C8—141no—
375#95—141no—
376C22×C7—21no—
377C22×C7—21no—
378C22×C7—22no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7#37911yescharacteristic
382C22×C7#44811yescharacteristic
383C22×C7#45111yescharacteristic
384C4×C7—21no—
385C4×C7—21no—
386C4×C7—22no—
387C4×C7—21no—
388C4×C7—22no—
389C4×C7—41no—
390C4×C7—42no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—42no—
395C4×C7—41no—
396C4×C7C2×#10311yescharacteristic
397C4×C7#41911yescharacteristic
398C4×C7C2×#10411yescharacteristic
399C4×C7#42111yescharacteristic
400C4×C7#44811yescharacteristic
401C4×C7#45111yescharacteristic
402Dic7—82no—
403#41—21no—
404#41—21no—
405#41#114312yes—
406#41#114412yes—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8C22⋊Dic711yescharacteristic
416C2×Q8C22⋊Dic711yescharacteristic
417C2×Q8C22⋊Dic711yescharacteristic
418C2×Q8C22⋊Dic712yes—
419C2×D4—21no—
420C2×D4C22⋊Dic711yescharacteristic
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—281no—
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432M16—281no—
433M16—281no—
434C2×C8—142no—
435C2×C8—281no—
436C2×C8—281no—
437C2×C8—281no—
438C2×C8—281no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—142no—
460C4⋊C4—281no—
461C4⋊C4C22⋊Dic711yescharacteristic
462C4⋊C4C22⋊Dic711yescharacteristic
463C4⋊C4C22⋊Dic711yescharacteristic
464C4⋊C4C22⋊Dic711yescharacteristic
465C4⋊C4C22⋊Dic712yes—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—281no—
483C42—281no—
484C42C22×Dic711yescharacteristic
485C42C2×C22⋊D711yescharacteristic
486C42C2×C22⋊D711yescharacteristic
487C42C2×C22⋊D711yescharacteristic
488C2×C7—21no—
489C2×C7—21no—
490C2×C7—42no—
491C2×C7#178711yescharacteristic
492C2×C7#178911yescharacteristic
493C2×C7#179411yescharacteristic
494C23—21no—
495C23#651611yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—42no—
501Q8—42no—
502Q8—44no—
503Q8—41no—
504Q8—41no—
505Q8—82no—
506Q8#640912yescomplemented
507Q8#641112yescomplemented
508D4—22no—
509D4—22no—
510D4—21no—
511D4#640912yescomplemented
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—282no—
534C2×C4—561no—
535C2×C4—561no—
536C2×C4C2×#114311yescharacteristic
537C2×C4#650511yescharacteristic
538C2×C4#650511yescharacteristic
539C2×C4C2×#114411yescharacteristic
540C2×C4C2×C22⋊Dic711yescharacteristic
541C2×C4C2×C22⋊Dic711yescharacteristic
542C2×C4C2×C22⋊Dic711yescharacteristic
543C2×C4#651611yescharacteristic
544C2×C4#651611yescharacteristic
545C2×C4#651611yescharacteristic
546C8—282no—
547C8—561no—
548C8—561no—
549C7#843911yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22#8999911yescharacteristic
556C22#9000011yescharacteristic
557C22#9001911yescharacteristic
558C4—22no—
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—562no—
571C4C2×#640911yescharacteristic
572C4#8999711yescharacteristic
573C4#8999911yescharacteristic
574C4#9000011yescharacteristic
575C4C2×#641111yescharacteristic
576C4#9000811yescharacteristic
577C2—21no—
578C2—21no—
579C2—42no—
580C2#1172988611yescharacteristic
581C2#1172989211yescharacteristic
582C2#1172990011yescharacteristic
subgroupG/H
centerC2×C4#6516
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×Q8×#41C2
socleC22×C7#379
prime radicalC2×C7×#41D4
cosocleC2×C42×C7C23
hypercenterQ8×#41D7
hypocenterC7#8439
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
2. #11729589C2semidirect
3. #11729580C2semidirect
6. C4×#6409C2semidirect
7. #11729376C2semidirect
8. C7×Q82C4semidirect
10. C7×#557C4semidirect
14. C7×#554C4semidirect
16. C7×D4×Q8C4semidirect
211. Q82Dic7semidirect
213. #557Dic7semidirect
217. #554Dic7semidirect
219. D4×Q8Dic7semidirect
254. C7×Q8#103semidirect
255. C7×Q8#104semidirect
259. C7×D4#103semidirect
506. Q8#6409semidirect
507. Q8#6411semidirect
511. D4#6409semidirect
549. C7#8439semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×#412
#117295892
#117295802
#117294002
#117293922
C4×#64092
#117293762
#84397