Finite Group Databasebeta

← #13375640

Subgroups — #13375640

Order 1792.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89663C211yesmaximal, characteristic
2#11733648C211yesmaximal, characteristic
3#11733434C211yesmaximal, characteristic, complemented
4C7×#1942C211yesmaximal, characteristic, complemented
5#11728392C211yesmaximal, characteristic
6#11728370C211yesmaximal, characteristic, complemented
7#11728364C211yesmaximal, characteristic, complemented
8C8×Dic7⋊C4C211yesmaximal, characteristic, complemented
9C4×#6395C211yesmaximal, characteristic
10#11727380C211yesmaximal, characteristic
11#11727356C211yesmaximal, characteristic, complemented
12#11726612C211yesmaximal, characteristic, complemented
13#11726558C211yesmaximal, characteristic, complemented
14#11726534C211yesmaximal, characteristic
15#11726534C211yesmaximal, characteristic
16C2×#6456C2211yescharacteristic
17C2×#6456C2211yescharacteristic
18C2×#6453C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×C8×Dic7C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89761—21no—
31#89759—21no—
32#89757C2211yescharacteristic
33#89750—21no—
34#89749C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37#89745—21no—
38#89745C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C4×Dic7⋊C4C2211yescharacteristic
45#89663C418yescomplemented
46#89440—21no—
47D7×#95—21no—
48#89425—21no—
49#89421—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#22966—71nomaximal, Sylow
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4D411yescharacteristic
85C2×Dic7⋊C4D411yescharacteristic
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93#6456—22no—
94#6456—22no—
95#6456—41no—
96#6456C2×C414yes—
97#6453—22no—
98#6453C2×C414yescomplemented
99C4×D28—22no—
100C4×D28—41no—
101C7×#107—41no—
102C7×#107C2×C414yescomplemented
103C4×C7×C8—41no—
104C7×#95—21no—
105C7×#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115#6395C2×C414yes—
116#6395Q814yes—
117Dic7⋊C8C2×C414yescomplemented
118C8×Dic7—22no—
119C8×Dic7—41no—
120#6383—22no—
121#6383C2×C414yes—
122#6382—41no—
123#6382C2×C414yescomplemented
124C2×#478—71no—
125#2370—71no—
126#2349—71no—
127#1991—71no—
128#1949—71no—
129#1942D711yescharacteristic, complemented
130#1941—71no—
131#1921—71no—
132#1920—71no—
133#1864—71no—
134#1862—71no—
135C8×C4⋊C4—71no—
136C4×#102—71no—
137C4×#102—71no—
138C4×#102—71no—
139C23×D7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28C22×C412yes—
158C2×D28C2×Q812yes—
159C2×D28#4112yes—
160C2×C4×D7—24no—
161C2×C4×D7—42no—
162C2×C4×D7—42no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—81no—
166C2×C4×D7—81no—
167C2×C4×D7—81no—
168C2×C4×D7—81no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8C22×C412yes—
174C2×C7×C8C2×Q812yes—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—81no—
182C7×C4⋊C4—81no—
183C7×C4⋊C4C22×C412yes—
184C7×C4⋊C4C22×C412yes—
185C42×C7—21no—
186C42×C7—21no—
187(C2×D7)⋊C4—24no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—42no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—24no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—81no—
196C4⋊Dic7C22×C412yes—
197C4⋊Dic7C2×Q812yes—
198Dic7⋊C4—22no—
199Dic7⋊C4—24no—
200Dic7⋊C4—22no—
201Dic7⋊C4—42no—
202C4×Dic7—21no—
203C4×Dic7—21no—
204C4×Dic7—42no—
205C4×Dic7—81no—
206C4×Dic7—81no—
207C4×Dic7C22×C412yes—
208C4×Dic7#4112yes—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C412yes—
215C2×#125—71no—
216C2×C4×D4—71no—
217C2×C4×D4—71no—
218C2×C4×D4—71no—
219#478—78no—
220C2×#107C2×D711yescharacteristic
221C2×#107C2×D711yescharacteristic
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×C4×C8—71no—
230C2×C4×C8—71no—
231C2×C4×C8C2×D711yescharacteristic
232#389—71no—
233#389—71no—
234#389—141no—
235#389C2×D711yescharacteristic
236#389C2×D711yescharacteristic
237#387—141no—
238#387—141no—
239#385—141no—
240#385—141no—
241#382—71no—
242#380—141no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—141no—
248C2×#95—141no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340C2×D711yescharacteristic
256#340C2×D711yescharacteristic
257#336—71no—
258#336—71no—
259C23×C7C22×D411yescharacteristic
260C22×D7—22no—
261C22×D7—22no—
262C22×D7—22no—
263C22×D7—41no—
264C22×D7—41no—
265C22×D7—41no—
266C22×D7—41no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
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×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7#13512yes—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×Q811yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7H(5,𝔽2)11yescharacteristic
290C2×C4×C72−1+411yescharacteristic
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—22no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7C4×D412yes—
310C2×Dic7C4×D412yes—
311C2×Dic7#13512yes—
312D28—22no—
313D28—22no—
314D28—22no—
315D28C4×Q814yes—
316C4×D7—84no—
317C4×D7—84no—
318C4×D7—84no—
319C7×C8—22no—
320C7×C8—22no—
321C7⋊C8—22no—
322C7⋊C8—41no—
323C22×D4—71no—
324C23×C4—141no—
325C23×C4—141no—
326C23×C4—141no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8C22×D711yescharacteristic
330C22×C8C22×D711yescharacteristic
331#125—74no—
332#125—142no—
333C4×D4—74no—
334C4×D4—142no—
335C4×D4—142no—
336C4×D4—142no—
337C4×D4—281no—
338C4×D4—281no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—141no—
360C2×C42—141no—
361C2×C42—141no—
362C2×C42C22×D711yescharacteristic
363#107—41no—
364#107C4×D714yescomplemented
365#106—74no—
366#106—281no—
367C4⋊C8—74no—
368#102—74no—
369#102—74no—
370#102—74no—
371#102—142no—
372C4×C8—41no—
373C4×C8—142no—
374C4×C8—281no—
375#95—21no—
376#95—21no—
377#95—71no—
378#95—71no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386C22×C7C2×C4×D411yescharacteristic
387C22×C7#54611yescharacteristic
388C22×C7D4×Q811yescharacteristic
389C22×C7C2×#13511yescharacteristic
390C22×C7C2×#13511yescharacteristic
391C22×C7#57611yescharacteristic
392C22×C7#57711yescharacteristic
393C2×D7—24no—
394C2×D7—42no—
395C2×D7—42no—
396C2×D7—42no—
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—42no—
400C4×C7—22no—
401C4×C7—42no—
402C4×C7—42no—
403C4×C7—41no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7C2×C4×Q811yescharacteristic
407C4×C7#52011yescharacteristic
408C4×C7#55211yescharacteristic
409C4×C7#55411yescharacteristic
410Dic7—22no—
411Dic7—22no—
412Dic7—22no—
413Dic7—42no—
414Dic7—42no—
415Dic7—41no—
416Dic7—81no—
417C24—141no—
418C2×D4—72no—
419C2×D4—72no—
420C2×D4—72no—
421C2×D4—141no—
422C2×D4—141no—
423C2×D4—141no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—144no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—282no—
439C22×C4—282no—
440C22×C4—561no—
441C22×C4—561no—
442C22×C4—561no—
443C22×C4—561no—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—72no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8C2×C4×D712yes—
458C2×C8Q8×D712yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—72no—
466C4⋊C4—72no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—142no—
470C4⋊C4—144no—
471C4⋊C4—144no—
472C4⋊C4—142no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—282no—
478C4⋊C4—561no—
479C4⋊C4C2×C4×D712yes—
480C4⋊C4C2×C4×D712yes—
481C22⋊C4—144no—
482C22⋊C4—282no—
483C22⋊C4—282no—
484C42—21no—
485C42—21no—
486C42—72no—
487C42—72no—
488C42—141no—
489C42—141no—
490C42—282no—
491C42—561no—
492C42—561no—
493C2×C7#236011yescharacteristic
494C2×C7C4×#13511yescharacteristic
495C2×C7#294611yescharacteristic
496C2×C7#329611yescharacteristic
497C2×C7#329711yescharacteristic
498C2×C7#338011yescharacteristic
499C2×C7#338111yescharacteristic
500D7—44no—
501C23—142no—
502C23—142no—
503C23—142no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23C2×D4×D711yescharacteristic
509D4—74no—
510D4—142no—
511D4—142no—
512D4—142no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—142no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—142no—
543C2×C4—141no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—564no—
552C2×C4—564no—
553C2×C4—564no—
554C2×C4#648212yes—
555C2×C4C22×C4×D711yescharacteristic
556C2×C4#654811yescharacteristic
557C2×C4C2×Q8×D711yescharacteristic
558C2×C4#655111yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4D7×#4111yescharacteristic
561C8—22no—
562C8—22no—
563C8—142no—
564C8—281no—
565C7#2296611yescharacteristic, complemented, Sylow
566C22—144no—
567C22—282no—
568C22—282no—
569C22—282no—
570C22—282no—
571C22—282no—
572C22—282no—
573C22C4×D4×D711yescharacteristic
574C22#9030111yescharacteristic
575C22#9041811yescharacteristic
576C22#9044811yescharacteristic
577C22C2×#648211yescharacteristic
578C22D7×#13511yescharacteristic
579C22#9046111yescharacteristic
580C4—22no—
581C4—41no—
582C4—42no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—142no—
587C4—142no—
588C4—142no—
589C4—282no—
590C4—282no—
591C4—281no—
592C4—561no—
593C4#9019511yescharacteristic
594C4C4×Q8×D711yescharacteristic
595C4#9032711yescharacteristic
596C4#9038411yescharacteristic
597C2—284no—
598C2#1173342811yescharacteristic
599C2#1173703211yescharacteristic
600C2C4×#648211yescharacteristic
601C2#1173817011yescharacteristic
602C2#1173822011yescharacteristic
603C2#1173935811yescharacteristic
604C2#1173939811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1942C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1942D7
hypocenterC7#22966
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
3. #11733434C2semidirect
4. C7×#1942C2semidirect
6. #11728370C2semidirect
7. #11728364C2semidirect
8. C8×Dic7⋊C4C2semidirect
11. #11727356C2semidirect
12. #11726612C2semidirect
13. #11726558C2semidirect
45. #89663C4semidirect
98. #6453C2×C4semidirect
102. C7×#107C2×C4semidirect
117. Dic7⋊C8C2×C4semidirect
123. #6382C2×C4semidirect
129. #1942D7semidirect
364. #107C4×D7semidirect
565. C7#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#896632
#117336482
#117334342
C7×#19422
#117283922
#117283702
#117283642
C8×Dic7⋊C42
C4×#63952
#117273802
#117273562
#117266122
#117265582
#117265342
#229667