Finite Group Databasebeta

← #13285141

Subgroups — #13285141

Order 1792.

1662 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6452C211yesmaximal, characteristic
2C2×C42×Dic7C211yesmaximal, characteristic
3C2×#89422C211yesmaximal, characteristic
4C7×#1436C211yesmaximal, characteristic
5#11729318C211yesmaximal, characteristic
6#11729290C211yesmaximal, characteristic
7#11727161C212yesmaximal
8#11727160C212yesmaximal
9#11727159C211yesmaximal, characteristic
10Dic7⋊#95C212yesmaximal
11Dic7⋊#95C212yesmaximal
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C22×Dic7⋊C4C2212yes—
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C22×Dic7⋊C4C2211yescharacteristic
19C22×Dic7⋊C4C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C22×C4×Dic7C2212yes—
23C22×C4×Dic7C2211yescharacteristic
24C2×#6452—24no—
25C2×#6452—24no—
26C2×#6452C418yescomplemented
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2211yescharacteristic
38C42×Dic7—24no—
39#89422—24no—
40#14709—71nomaximal, Sylow
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C23×Dic7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4C2×C414yes—
55C2×C42×C7—22no—
56C2×C42×C7—24no—
57C2×C4⋊Dic7—22no—
58C2×C4⋊Dic7—22no—
59C2×C4⋊Dic7C2×C414yes—
60C2×Dic7⋊C4—22no—
61C2×Dic7⋊C4—22no—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4—24no—
71C2×Dic7⋊C4C2×C414yes—
72C2×Dic7⋊C4C2×C414yes—
73C2×Dic7⋊C4C2×C414yes—
74C2×Dic7⋊C4C2×C414yes—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—24no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—24no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7C2×C414yes—
83C2×C4×Dic7D414yes—
84C2×C4×Dic7D414yes—
85C2×C4×Dic7Q818yes—
86#6452—232no—
87C7×#95—22no—
88C7×#95—24no—
89C7×#95—24no—
90C7⋊#95—22no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—24no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C22×#125—71no—
99C2×#382—71no—
100C2×C43—71no—
101#1436—71no—
102#1436—71no—
103#1436—72no—
104#1436—72no—
105#1436—71no—
106#1436D711yescharacteristic
107C4⋊#95—72no—
108C4⋊#95—72no—
109C24×C7C2411yescharacteristic
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7C22×C412yes—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×Q814yes—
137C22×C4×C7#4114yes—
138C22×C4×C7#4114yes—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7C22×C412yes—
161C22×Dic7C22×C412yes—
162C22×Dic7C22×C412yes—
163C22×Dic7C22×C412yes—
164C22×Dic7C2×D412yes—
165C22×Dic7C2×D412yes—
166C22×Dic7C2×D412yes—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×Q814yes—
169C22×Dic7C2×Q814yes—
170C22×Dic7#4114yes—
171C22×Dic7#4114yes—
172C22×Dic7#4114yes—
173C7×C4⋊C4—28no—
174C42×C7—28no—
175C4⋊Dic7—28no—
176Dic7⋊C4—28no—
177Dic7⋊C4—28no—
178Dic7⋊C4—28no—
179Dic7⋊C4—28no—
180Dic7⋊C4—48no—
181C4×Dic7—216no—
182C4×Dic7—28no—
183C4×Dic7—216no—
184C4×Dic7C22⋊C4116yes—
185C2×#125—78no—
186C2×#125—144no—
187C2×#125—144no—
188C22×C4⋊C4—71no—
189C22×C4⋊C4—71no—
190C22×C4⋊C4—71no—
191C22×C4⋊C4—71no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4—72no—
194C22×C4⋊C4C2×D711yescharacteristic
195C22×C42—71no—
196C22×C42—72no—
197C22×C42—72no—
198C22×C42—71no—
199C22×C42C2×D711yescharacteristic
200#382—144no—
201C2×#95—72no—
202C2×#95—72no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—71no—
209C2×#95C2×D712yes—
210C2×#95C2×D712yes—
211C2×#95C2×D711yescharacteristic
212C43—144no—
213C23×C7C23×C411yescharacteristic
214C23×C7C22×D411yescharacteristic
215C23×C7C22×D411yescharacteristic
216C23×C7C22×Q812yes—
217C23×C7C2×#4112yes—
218C23×C7C2×#4112yes—
219C23×C7C2×#4112yes—
220C23×C7C2×#4112yes—
221C23×C7C2×#4112yes—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—24no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—24no—
237C2×C4×C7—42no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7C2×C22⋊C414yes—
250C2×C4×C7#12518yescomplemented
251C2×Dic7—22no—
252C2×Dic7—22no—
253C2×Dic7—24no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—24no—
266C2×Dic7—42no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7C2×C22⋊C414yes—
279C2×Dic7C2×C22⋊C414yes—
280C2×Dic7C4×Q818yes—
281C2×Dic7C22⋊Q818yes—
282C2×Dic7#12414yes—
283C2×Dic7#12518yescomplemented
284C2×Dic7C4⋊Q814yes—
285C23×C4—71no—
286C23×C4—71no—
287C23×C4—72no—
288C23×C4—71no—
289C23×C4—71no—
290C23×C4—72no—
291C23×C4C22×D712yes—
292C23×C4C22×D711yescharacteristic
293C23×C4C22×D711yescharacteristic
294C23×C4C22×D711yescharacteristic
295C23×C4C22×D712yes—
296#125—1432no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—74no—
300C2×C4⋊C4—74no—
301C2×C4⋊C4—74no—
302C2×C4⋊C4—74no—
303C2×C4⋊C4—74no—
304C2×C4⋊C4—142no—
305C2×C4⋊C4—142no—
306C2×C4⋊C4—142no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4—142no—
309C2×C4⋊C4—142no—
310C2×C4⋊C4—142no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—144no—
316C2×C4⋊C4—144no—
317C2×C4⋊C4C4×D714yes—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—74no—
321C2×C42—74no—
322C2×C42—74no—
323C2×C42—78no—
324C2×C42—144no—
325C2×C42—144no—
326C2×C42—144no—
327C2×C42—144no—
328C2×C42—144no—
329C2×C42—144no—
330C2×C42—142no—
331#95—24no—
332#95—22no—
333#95—24no—
334#95—144no—
335#95—144no—
336#95—144no—
337#95—144no—
338#95—144no—
339#95—144no—
340#95—142no—
341#95—144no—
342C22×C7C22×C22⋊C411yescharacteristic
343C22×C7C2×C4×Q812yes—
344C22×C7C4×#4111yescharacteristic
345C22×C7C4×#4111yescharacteristic
346C22×C7C4×#4112yes—
347C22×C7C2×C22⋊Q812yes—
348C22×C7#52512yes—
349C22×C7#52511yescharacteristic
350C22×C7#52511yescharacteristic
351C22×C7C2×#12411yescharacteristic
352C22×C7C2×#12512yes—
353C22×C7C2×#12512yes—
354C22×C7#52912yes—
355C22×C7#52912yes—
356C22×C7#52912yes—
357C22×C7#52912yes—
358C22×C7C2×C4⋊Q811yescharacteristic
359C22×C7#53212yes—
360C22×C7#53212yes—
361C22×C7#53312yes—
362C22×C7#53312yes—
363C4×C7—28no—
364C4×C7—24no—
365C4×C7—42no—
366C4×C7—42no—
367C4×C7—82no—
368Dic7—28no—
369Dic7—24no—
370Dic7—42no—
371Dic7—42no—
372Dic7—82no—
373Dic7#39118yes—
374C24C23×D711yescharacteristic
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—72no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—74no—
400C22×C4—74no—
401C22×C4—72no—
402C22×C4—72no—
403C22×C4—72no—
404C22×C4—74no—
405C22×C4—74no—
406C22×C4—74no—
407C22×C4—72no—
408C22×C4—72no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4C2×C4×D712yes—
431C22×C4C2×C4×D712yes—
432C22×C4C2×C4×D712yes—
433C22×C4#115714yes—
434C22×C4D4×D712yes—
435C22×C4D4×D712yes—
436C22×C4#115914yes—
437C22×C4Q8×D714yes—
438C4⋊C4—28no—
439C4⋊C4—148no—
440C4⋊C4—148no—
441C4⋊C4—148no—
442C4⋊C4—148no—
443C4⋊C4—148no—
444C4⋊C4—288no—
445C42—28no—
446C42—716no—
447C42—1416no—
448C42—1416no—
449C42—148no—
450C2×C7C2×#39111yescharacteristic
451C2×C7#229711yescharacteristic
452C2×C7#229711yescharacteristic
453C2×C7C4×#12512yes—
454C2×C7C4×#12512yes—
455C2×C7#285112yes—
456C2×C7#285112yes—
457C2×C7#285912yes—
458C2×C7#286112yes—
459C23C22×C4×D711yescharacteristic
460C23C2×#115712yes—
461C23C2×D4×D711yescharacteristic
462C23C2×D4×D711yescharacteristic
463C23C2×#115912yes—
464C23C2×Q8×D712yes—
465C23D7×#4112yes—
466C23D7×#4112yes—
467C23D7×#4112yes—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—74no—
496C2×C4—74no—
497C2×C4—78no—
498C2×C4—78no—
499C2×C4—74no—
500C2×C4—78no—
501C2×C4—74no—
502C2×C4—144no—
503C2×C4—142no—
504C2×C4—142no—
505C2×C4—144no—
506C2×C4—144no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—144no—
510C2×C4—144no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—144no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—281no—
520C2×C4—282no—
521C2×C4—281no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4D7×C22⋊C414yes—
530C2×C4#645218yescomplemented
531C7#1470911yescharacteristic, complemented, Sylow
532C22C4×#115711yescharacteristic
533C22C2×D7×C22⋊C411yescharacteristic
534C22C2×#645212yes—
535C22#9021311yescharacteristic
536C22#9021912yes—
537C22C4×#115912yes—
538C22#9022912yes—
539C22#9023812yes—
540C22#9023912yes—
541C22C4×Q8×D712yes—
542C22#9026912yes—
543C22#9027711yescharacteristic
544C22#9028712yes—
545C22#9031312yes—
546C22D7×C22⋊Q812yes—
547C22#9034012yes—
548C22#9034711yescharacteristic
549C22D7×#12411yescharacteristic
550C22D7×#12512yes—
551C22#9039712yes—
552C22D7×C4⋊Q811yescharacteristic
553C4—28no—
554C4—24no—
555C4—42no—
556C4—42no—
557C4—78no—
558C4—82no—
559C4—144no—
560C4—148no—
561C4—282no—
562C4—282no—
563C4—562no—
564C2#1173100811yescharacteristic
565C2C4×#645212yes—
566C2#1173203112yes—
567C2#1173210311yescharacteristic
568C2#1173238812yes—
569C2#1173289012yes—
570C2#1173326612yes—
571C2#1173363512yes—
572C2D7×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1436C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1436D7
hypocenterC7#14709
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
26. C2×#6452C4semidirect
250. C2×C4×C7#125semidirect
283. C2×Dic7#125semidirect
530. C2×C4#6452semidirect
531. C7#14709semidirect

Maximal subgroups

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

subgroupindex
C22×#64522
C2×C42×Dic72
C2×#894222
C7×#14362
#117293182
#117292902
#117271612
#117271602
#117271592
Dic7⋊#952
#147097