Finite Group Databasebeta

← #13502622

Subgroups — #13502622

Order 1792.

1436 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic7C211yesmaximal, characteristic
2C2×C7×#451C211yesmaximal, characteristic
3C2×#90000C211yesmaximal, characteristic
4C2×#89954C212yesmaximal, complemented
5#11740256C211yesmaximal, characteristic
6#11729819C212yesmaximal, complemented
7#11729446C211yesmaximal, characteristic
8#11727993C212yesmaximal, complemented
9#11727975C212yesmaximal, complemented
10#11727684C211yesmaximal, characteristic
11#11727662C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×C7×Q16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×Q8×Dic7—28no—
16C2×Q8×Dic7—21no—
17C2×Q8×Dic7—21no—
18C2×Q8×Dic7C2212yes—
19C2×Q8×Dic7C2212yes—
20C2×#6507C2211yescharacteristic
21C22×#1144C2212yes—
22C22×C4⋊Dic7—21no—
23C22×C4⋊Dic7C2211yescharacteristic
24C2×#6487C2211yescharacteristic
25C22×C4×Dic7—21no—
26C22×C4×Dic7C2211yescharacteristic
27C7×#451—24no—
28C2×C7×#103C2212yes—
29C2×C7×C22⋊C8C2211yescharacteristic
30#90002C2211yescharacteristic
31#90000—24no—
32C2×#6409C2212yes—
33C2×#6409C2212yes—
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35#89954—28no—
36C2×#6390C2212yes—
37C2×Dic7⋊C8C2212yes—
38#89748C2211yescharacteristic
39#89746C2211yescharacteristic
40#89744—21no—
41#89744—21no—
42#89714—21no—
43#89714—21no—
44#89346C2212yes—
45#89345C2212yes—
46#89324C2212yes—
47#32906—71nomaximal, Sylow
48C22×C7×Q8—24no—
49C22×C7×Q8—24no—
50C22×C7×Q8—21no—
51C22×C7×Q8—22no—
52C22×C7×Q8D414yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×Q8C2312yes—
55C23×C4×C7—21no—
56C23×C4×C7C2311yescharacteristic
57C23×Dic7—21no—
58C23×Dic7D411yescharacteristic
59C23×Dic7D411yescharacteristic
60C22×Dic14C2311yescharacteristic
61C2×C7×Q16—24no—
62C2×C7×Q16—24no—
63C2×C7×Q16—24no—
64C22×C7×C8C2312yes—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C22⋊Dic7—21no—
71Q8×Dic7—28no—
72Q8×Dic7—28no—
73Q8×Dic7—24no—
74Q8×Dic7—24no—
75Q8×Dic7—24no—
76Q8×Dic7D418yes—
77#6507—24no—
78C2×#1144—24no—
79C2×#1144—24no—
80C2×#1144—24no—
81C2×C4⋊Dic7—28no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—24no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2312yes—
88#6487—22no—
89#6487—22no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—28no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7—22no—
98C2×C4×Dic7—24no—
99C2×C4×Dic7D411yescharacteristic
100C2×C4×Dic7D411yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2311yescharacteristic
103C22×C7⋊C8C2312yes—
104C7×#103—24no—
105C7×C22⋊C8—22no—
106#6409—24no—
107#6409—24no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C22⋊(C7⋊C8)—22no—
113#6390—24no—
114Dic7⋊C8—24no—
115C22×C4×Q8—71no—
116C2×#462—72no—
117C2×#451—71no—
118C2×#451D711yescharacteristic
119#2857—71no—
120#1878—72no—
121#1878—72no—
122#1872—71no—
123#1872—71no—
124#1803—72no—
125#1797—71no—
126C24×C7C2×D411yescharacteristic
127C2×C7×Q8—24no—
128C2×C7×Q8—216no—
129C2×C7×Q8—24no—
130C2×C7×Q8—216no—
131C2×C7×Q8—24no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—21no—
137C2×C7×Q8—21no—
138C2×C7×Q8—22no—
139C2×C7×Q8C2×D414yes—
140C2×C7×Q8C2×D414yes—
141C2×C7×Q8#4114yes—
142C22×C4×C7—24no—
143C22×C4×C7—24no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—24no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—41no—
163C22×Dic7—42no—
164C22×Dic7—42no—
165C22×Dic7Q1614yescomplemented
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C2×Dic14—21no—
171C2×Dic14—22no—
172C2×Dic14—21no—
173C2×Dic14—22no—
174C7×Q16—48no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—42no—
181C7×C22⋊C4—42no—
182C22⋊Dic7—42no—
183#1144—48no—
184C4⋊Dic7—28no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7—28no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7C2×D414yes—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196C4×Dic7—28no—
197C4×Dic7—22no—
198C4×Dic7—28no—
199C4×Dic7—22no—
200C4×Dic7—22no—
201C4×Dic7C2×D414yes—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C23×Q8C2×D711yescharacteristic
206C22×Q16—72no—
207C22×Q16C2×D712yes—
208C2×C4⋊Q8—71no—
209C2×C22⋊Q8—71no—
210C2×C22⋊Q8C2×D711yescharacteristic
211C2×C4×Q8—72no—
212C2×C4×Q8—72no—
213C2×C4×Q8—141no—
214C2×C4×Q8—141no—
215C2×C4×Q8—148no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—141no—
218C22×C42—71no—
219C22×C42—141no—
220#462—148no—
221#451—24no—
222#451—144no—
223C2×C4⋊C8—72no—
224C2×#103—72no—
225C2×#103—72no—
226C2×#103—72no—
227C2×#103C2×D712yes—
228C2×C22⋊C8—71no—
229C2×C22⋊C8C2×D711yescharacteristic
230#391—71no—
231#391—71no—
232#389—141no—
233#389—141no—
234#384—71no—
235#380—141no—
236#380—141no—
237#340—72no—
238#340—72no—
239#336—72no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7#12112yes—
245C23×C7C2×Q1612yes—
246C23×C7#13712yes—
247C23×C7C22×D411yescharacteristic
248C7×Q8—216no—
249C7×Q8—24no—
250C7×Q8—24no—
251C7×Q8—24no—
252C7×Q8—42no—
253C7×Q8—42no—
254C7×Q8#12118yes—
255C2×C4×C7—216no—
256C2×C4×C7—24no—
257C2×C4×C7—21no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—21no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7#12112yes—
270C2×C4×C7C22×D412yes—
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C2×#4112yes—
273C2×C4×C7C2×#4111yescharacteristic
274C2×Dic7—21no—
275C2×Dic7—21no—
276C2×Dic7—21no—
277C2×Dic7—21no—
278C2×Dic7—28no—
279C2×Dic7—21no—
280C2×Dic7—24no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—44no—
284C2×Dic7—48no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7#12012yes—
294C2×Dic7#12012yes—
295C2×Dic7C2×Q1612yes—
296C2×Dic7C2×Q1612yes—
297C2×Dic7#13712yes—
298Dic14—42no—
299Dic14—42no—
300C7×C8—42no—
301C7⋊C8—42no—
302C22×Q8—24no—
303C22×Q8—24no—
304C22×Q8—21no—
305C22×Q8—22no—
306C22×Q8—71no—
307C22×Q8C22⋊D714yes—
308C22×Q8C22×D711yescharacteristic
309C22×Q8C22×D712yes—
310C23×C4—21no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—141no—
314C23×C4C22×D711yescharacteristic
315C2×Q16—24no—
316C2×Q16—24no—
317C2×Q16—24no—
318C2×Q16—144no—
319C2×Q16—144no—
320C2×Q16—144no—
321C22×C8—72no—
322C22×C8C22×D712yes—
323C4⋊Q8—144no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—142no—
327C22⋊Q8—142no—
328C4×Q8—78no—
329C4×Q8—144no—
330C4×Q8—144no—
331C4×Q8—148no—
332C4×Q8—148no—
333C4×Q8—144no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—72no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—148no—
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×C22⋊C4—21no—
349C2×C22⋊C4—141no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—144no—
355C2×C42—142no—
356C2×C42—142no—
357C2×C42—148no—
358C4⋊C8—144no—
359#103—24no—
360#103—144no—
361#103—144no—
362#103—144no—
363C22⋊C8—22no—
364C22⋊C8—142no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7#46214yescomplemented
376C22×C7C2×#12011yescharacteristic
377C22×C7C2×#12111yescharacteristic
378C22×C7#52511yescharacteristic
379C22×C7C22×Q1611yescharacteristic
380C22×C7C2×#13711yescharacteristic
381C22×C7C2×#13711yescharacteristic
382C22×C7#57511yescharacteristic
383C4×C7—28no—
384C4×C7—22no—
385C4×C7—81no—
386C4×C7—81no—
387C4×C7C2×#12112yes—
388C4×C7#52512yes—
389Dic7—22no—
390Dic7—22no—
391Dic7—41no—
392Dic7—44no—
393Dic7—81no—
394Dic7—81no—
395Dic7#45114yescomplemented
396C24D4×D711yescharacteristic
397C2×Q8—24no—
398C2×Q8—216no—
399C2×Q8—24no—
400C2×Q8—216no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—142no—
410C2×Q8—141no—
411C2×Q8—142no—
412C2×Q8—141no—
413C2×Q8D4×D714yes—
414C2×Q8#115914yes—
415C2×Q8C2×C22⋊D714yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—71no—
425C22×C4—74no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—144no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—281no—
438C22×C4—282no—
439C22×C4—282no—
440C22×C4D4×D711yescharacteristic
441C22×C4#115912yes—
442C22×C4C2×C22⋊D712yes—
443C22×C4C23×D711yescharacteristic
444Q16—48no—
445Q16—288no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—142no—
450C2×C8—142no—
451C2×C8—142no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—74no—
456C4⋊C4—148no—
457C4⋊C4—148no—
458C4⋊C4—144no—
459C4⋊C4—142no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—282no—
463C4⋊C4—282no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4—282no—
467C22⋊C4—42no—
468C22⋊C4—282no—
469C42—74no—
470C42—142no—
471C42—142no—
472C42—148no—
473C42—148no—
474C42—142no—
475C2×C7—24no—
476C2×C7#284711yescharacteristic
477C2×C7C2×#45111yescharacteristic
478C2×C7#300711yescharacteristic
479C2×C7C2×#46211yescharacteristic
480C2×C7#303711yescharacteristic
481C2×C7#310411yescharacteristic
482C2×C7#311311yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23D7×Q1612yes—
488C23#648112yes—
489C23#650212yes—
490C23C2×D4×D711yescharacteristic
491Q8—216no—
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497Q8—282no—
498Q8—282no—
499Q8#650218yes—
500C2×C4—216no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—72no—
515C2×C4—72no—
516C2×C4—72no—
517C2×C4—72no—
518C2×C4—72no—
519C2×C4—148no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—144no—
527C2×C4—141no—
528C2×C4—284no—
529C2×C4—288no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4#650212yes—
539C2×C4C2×D4×D712yes—
540C2×C4C2×#115912yes—
541C2×C4C2×#115911yescharacteristic
542C2×C4C22×C22⋊D711yescharacteristic
543C8—42no—
544C8—282no—
545C7#3290611yescharacteristic, complemented, Sylow
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22#8995414yescomplemented
553C22#9027711yescharacteristic
554C22D7×#12011yescharacteristic
555C22C2×D7×Q1611yescharacteristic
556C22C2×#648111yescharacteristic
557C22D7×#13711yescharacteristic
558C22#9046811yescharacteristic
559C22C2×#650211yescharacteristic
560C4—28no—
561C4—22no—
562C4—74no—
563C4—81no—
564C4—81no—
565C4—142no—
566C4—142no—
567C4—281no—
568C4—284no—
569C4—561no—
570C4—561no—
571C4#9027712yes—
572C4C2×#650212yes—
573C2—24no—
574C2#1173761711yescharacteristic
575C2#1173761911yescharacteristic
576C2D7×#45111yescharacteristic
577C2#1173762911yescharacteristic
578C2#1174004611yescharacteristic
579C2C2×#8995411yescharacteristic
580C2#1174141111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#451C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#451D7
hypocenterC7#32906
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
4. C2×#89954C2semidirect
6. #11729819C2semidirect
8. #11727993C2semidirect
9. #11727975C2semidirect
165. C22×Dic7Q16semidirect
375. C22×C7#462semidirect
395. Dic7#451semidirect
545. C7#32906semidirect
552. C22#89954semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic72
C2×C7×#4512
C2×#900002
C2×#899542
#117402562
#117298192
#117294462
#117279932
#117279752
#117276842
#117276622
#329067