Finite Group Databasebeta

← #13375618

Subgroups — #13375618

Order 1792.

822 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic7⋊C4C211yesmaximal, characteristic
2C2×#89563C211yesmaximal, characteristic
3#11731097C211yesmaximal, characteristic
4C4×C7×#103C211yesmaximal, characteristic
5#11729383C211yesmaximal, characteristic
6#11727992C211yesmaximal, characteristic
7Dic7×#103C211yesmaximal, characteristic
8Dic7×#103C211yesmaximal, characteristic
9C8×Dic7⋊C4C211yesmaximal, characteristic
10#11727523C211yesmaximal, characteristic
11#11727214C211yesmaximal, characteristic
12#11726612C211yesmaximal, characteristic
13#11726572C211yesmaximal, characteristic
14#11726557C211yesmaximal, characteristic
15#11726550C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×C4×Dic14—21no—
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89757C2211yescharacteristic
33#89748—21no—
34#89748C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745—21no—
38#89745C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41#89705—21no—
42C4×Dic7⋊C4—21no—
43C4×Dic7⋊C4C2211yescharacteristic
44#89700—21no—
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89563C418yescomplemented
48#89427—21no—
49#89418—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22964—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×Dic14—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C4×C7×Q8—22no—
66C4×C7×Q8—41no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74Q8×Dic7—22no—
75Q8×Dic7—22no—
76Q8×Dic7—41no—
77Q8×Dic7C2×C414yes—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4D411yescharacteristic
92C2×Dic7⋊C4D411yescharacteristic
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7C2311yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6451—22no—
102#6451C2×C414yes—
103C4×Dic14—44no—
104C7×#103C2×C414yes—
105C7×#103Q814yescomplemented
106C4×C7×C8—41no—
107C7×#95—21no—
108#6409—22no—
109#6409C2×C414yes—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95D411yescharacteristic
117C7⋊#95D411yescharacteristic
118#6391—41no—
119#6391C2×C414yes—
120Dic7⋊C8C2×C414yes—
121C8×Dic7—22no—
122C8×Dic7—41no—
123#6382—41no—
124#6382C2×C414yes—
125C2×#475—71no—
126#2365—71no—
127Q8×C4⋊C4—71no—
128#1990—71no—
129#1948—71no—
130#1941—71no—
131#1941—71no—
132#1920—71no—
133#1920—71no—
134#1865—71no—
135#1865—71no—
136C8×C4⋊C4—71no—
137C4×#103—71no—
138C4×#103—71no—
139C4×#103D711yescharacteristic
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8C22×C412yes—
144C2×C7×Q8C2×Q812yes—
145C2×C7×Q8#4112yes—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C2×Dic14—24no—
162C2×Dic14—42no—
163C2×Dic14—42no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8C22×C412yes—
169C2×C7×C8C2×Q812yes—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4C22×C412yes—
176C7×C4⋊C4C2×Q812yes—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—42no—
180C42×C7—42no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—24no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—42no—
189C4⋊Dic7—42no—
190C4⋊Dic7C22×C412yes—
191C4⋊Dic7C22×C412yes—
192Dic7⋊C4—24no—
193Dic7⋊C4—22no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198Dic7⋊C4—81no—
199Dic7⋊C4SD1614yes—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—24no—
203C4×Dic7—42no—
204C4×Dic7—42no—
205C4×Dic7—42no—
206C4×Dic7C22×C412yes—
207C4×Dic7#4112yes—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8C22×C412yes—
214C2×C4⋊Q8—71no—
215C2×C4×Q8—71no—
216C2×C4×Q8—71no—
217C2×C4×Q8—141no—
218C2×C4×Q8C2×D711yescharacteristic
219#475—78no—
220C2×#106—71no—
221C2×#106—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×#103—71no—
226C2×#103—71no—
227C2×#103C2×D711yescharacteristic
228C2×#103C2×D711yescharacteristic
229C2×C4×C8—71no—
230C2×C4×C8—71no—
231C2×C4×C8C2×D711yescharacteristic
232#391—141no—
233#389—71no—
234#389—71no—
235#389—71no—
236#389—71no—
237#389—141no—
238#389—141no—
239#387—141no—
240#387—141no—
241#384—71no—
242#384—141no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—141no—
247C4×C4⋊C4C2×D711yescharacteristic
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340C2×D711yescharacteristic
255#340C2×D711yescharacteristic
256#336—71no—
257#336—71no—
258C23×C7C22×D411yescharacteristic
259C7×Q8—22no—
260C7×Q8—22no—
261C7×Q8—22no—
262C7×Q8C4×Q814yescomplemented
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7C2×SD1612yes—
281C2×C4×C7C23×C411yescharacteristic
282C2×C4×C7C22×Q811yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7H(5,𝔽2)11yescharacteristic
286C2×C4×C72−1+411yescharacteristic
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7C4×D412yes—
307C2×Dic7C4×D412yes—
308C2×Dic7C2×SD1612yes—
309C2×Dic7C2×SD1612yes—
310C2×Dic7#13512yes—
311Dic14—48no—
312C7×C8—22no—
313C7×C8—22no—
314C7⋊C8—22no—
315C7⋊C8—41no—
316C22×Q8—141no—
317C22×Q8C22×D711yescharacteristic
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322C4⋊Q8—74no—
323C4⋊Q8—142no—
324C4×Q8—22no—
325C4×Q8—41no—
326C4×Q8—74no—
327C4×Q8—142no—
328C4×Q8—142no—
329C4×Q8—284no—
330C4×Q8—281no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—141no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42C22×D711yescharacteristic
359#106—74no—
360#106—74no—
361#106—281no—
362#106—281no—
363C4⋊C8—74no—
364#103—74no—
365#103—142no—
366#103Dic1414yescomplemented
367#103C4×D714yes—
368C4×C8—41no—
369C4×C8—142no—
370C4×C8—281no—
371#95—21no—
372#95—71no—
373#95—71no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380C22×C7C2×C4×D411yescharacteristic
381C22×C7#54611yescharacteristic
382C22×C7D4×Q811yescharacteristic
383C22×C7C22×SD1611yescharacteristic
384C22×C7C2×#13511yescharacteristic
385C22×C7#57711yescharacteristic
386C22×C7#57811yescharacteristic
387C4×C7—24no—
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—41no—
392C4×C7—81no—
393C4×C7C2×C4×Q811yescharacteristic
394C4×C7#52011yescharacteristic
395C4×C7#55211yescharacteristic
396C4×C7#55411yescharacteristic
397Dic7—22no—
398Dic7—22no—
399Dic7—42no—
400Dic7—42no—
401Dic7—41no—
402Dic7—81no—
403Dic7—81no—
404Dic7C4×SD1614yes—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—144no—
409C2×Q8—282no—
410C2×Q8—282no—
411C2×Q8C2×Dic1412yes—
412C2×Q8C2×C4×D712yes—
413C2×Q8#115712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4D4×D711yescharacteristic
427C22×C4D4×D711yescharacteristic
428C22×C4C23×D711yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—72no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8C2×Dic1412yes—
440C2×C8C2×C4×D712yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—42no—
446C4⋊C4—74no—
447C4⋊C4—72no—
448C4⋊C4—72no—
449C4⋊C4—142no—
450C4⋊C4—144no—
451C4⋊C4—144no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4—561no—
465C4⋊C4C2×Dic1412yes—
466C4⋊C4C2×C4×D712yes—
467C42—21no—
468C42—21no—
469C42—42no—
470C42—42no—
471C42—72no—
472C42—72no—
473C42—144no—
474C42—141no—
475C42—141no—
476C42—282no—
477C42—282no—
478C42—282no—
479C2×C7#236011yescharacteristic
480C2×C7C2×C4×SD1611yescharacteristic
481C2×C7#294711yescharacteristic
482C2×C7#329811yescharacteristic
483C2×C7#330411yescharacteristic
484C2×C7Q8×SD1611yescharacteristic
485C2×C7#338311yescharacteristic
486C23C2×D4×D711yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—22no—
490Q8—288no—
491Q8C4×Dic1414yescomplemented
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—72no—
510C2×C4—72no—
511C2×C4—72no—
512C2×C4—72no—
513C2×C4—72no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4D7×SD1612yes—
534C2×C4C22×Dic1411yescharacteristic
535C2×C4C22×C4×D711yescharacteristic
536C2×C4C2×#115711yescharacteristic
537C2×C4#654811yescharacteristic
538C2×C4D7×#4111yescharacteristic
539C2×C4#655411yescharacteristic
540C8—22no—
541C8—22no—
542C8—142no—
543C8—281no—
544C7#2296411yescharacteristic, complemented, Sylow
545C22#9018111yescharacteristic
546C22D4×Dic1411yescharacteristic
547C22C4×D4×D711yescharacteristic
548C22C2×D7×SD1611yescharacteristic
549C22#9045711yescharacteristic
550C22D7×#13511yescharacteristic
551C22#9046111yescharacteristic
552C4—22no—
553C4—24no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—74no—
558C4—81no—
559C4—142no—
560C4—142no—
561C4—282no—
562C4—282no—
563C4—281no—
564C4—561no—
565C4—561no—
566C4C2×C4×Dic1411yescharacteristic
567C4#9022211yescharacteristic
568C4#9023011yescharacteristic
569C4#9023711yescharacteristic
570C2#1173091111yescharacteristic
571C2#1173625011yescharacteristic
572C2#1173629611yescharacteristic
573C2SD16×Dic1411yescharacteristic
574C2C4×D7×SD1611yescharacteristic
575C2#1173715211yescharacteristic
576C2#1173716411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#103C2
socle, prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC4×#103D7
hypocenterC7#22964
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
47. #89563C4semidirect
105. C7×#103Q8semidirect
262. C7×Q8C4×Q8semidirect
366. #103Dic14semidirect
491. Q8C4×Dic14semidirect
544. C7#22964semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic7⋊C42
C2×#895632
#117310972
C4×C7×#1032
#117293832
#117279922
Dic7×#1032
C8×Dic7⋊C42
#117275232
#117272142
#117266122
#117265722
#117265572
#117265502
#229647