Finite Group Databasebeta

← #13382613

Subgroups — #13382613

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89885C211yesmaximal, characteristic
2#11739631C211yesmaximal, characteristic
3#11739602C211yesmaximal, characteristic, complemented
4C2×#89738C211yesmaximal, characteristic
5C2×C7×#340C211yesmaximal, characteristic, complemented
6#11729719C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11726669C212yesmaximal, complemented
9#11726668C212yesmaximal, complemented
10#11726657C212yesmaximal, complemented
11#11726656C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6489C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19#89985—22no—
20#89982—22no—
21#89885—24no—
22C2×#6395C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×#6390C2211yescharacteristic
26#89760C2212yes—
27#89759—21no—
28#89759—21no—
29#89749C2212yes—
30#89748C2212yes—
31#89746C2212yes—
32#89745—21no—
33#89745—21no—
34C2×#6384C2212yes—
35#89738—24no—
36C2×#6383C2212yes—
37C7×#340C2212yes—
38C7×#340C2212yes—
39#89345C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#23634—71nomaximal, Sylow
44C23×C4×C7—21no—
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C22×C22⋊D7—21no—
48C2×#1157—22no—
49C2×#1157—22no—
50C2×#1157—22no—
51C2×#1157—22no—
52C2×#1157—22no—
53C2×#1157D414yes—
54C22×D28C2311yescharacteristic
55C22×C4×D7—21no—
56C22×Dic14C2311yescharacteristic
57C22×C7×C8—22no—
58C22×C7×C8—22no—
59C22×C7×C8—22no—
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C4⋊C4C2312yes—
70C2×C22⋊Dic7—21no—
71C2×C22⋊Dic7D412yes—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—22no—
74#6489—22no—
75#6489—22no—
76#6489—22no—
77#6489—22no—
78C2×C4⋊Dic7D411yescharacteristic
79C2×C4⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7C2312yes—
86C2×#1136—22no—
87C2×#1136—22no—
88C2×#1136—22no—
89C22×C7⋊C8C2312yes—
90C7⋊#95—22no—
91C7⋊#95—22no—
92C7⋊#95—22no—
93C7⋊#95—22no—
94#6395—22no—
95#6395—22no—
96#6390—22no—
97#6390—22no—
98#6384—24no—
99#6383—24no—
100C2×#417—71no—
101C2×#416—71no—
102#2317—71no—
103#2311—71no—
104#2074—72no—
105#2073—72no—
106#2072—72no—
107#2071—72no—
108#1736—71no—
109C2×#340D711yescharacteristic, complemented
110#1729—71no—
111C24×C7C2×D411yescharacteristic
112C23×D7—21no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—42no—
116C22×C4×C7—42no—
117C22×C4×C7—81no—
118C22×C4×C7—81no—
119C22×C4×C7—81no—
120C22×C4×C7—81no—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D412yes—
123C22×C4×C7#4112yes—
124C22×C4×C7#4112yes—
125C22×C4×C7#4112yes—
126C22×C4×C7#4112yes—
127C22×C4×C7#4112yes—
128C22×C4×C7C2411yescharacteristic
129C2×C22⋊D7—24no—
130C2×C22⋊D7—42no—
131C2×C22⋊D7—42no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C22×Dic7—21no—
135C22×Dic7C2×D412yes—
136C22×Dic7C2×D412yes—
137#1157—28no—
138#1157—28no—
139#1157—44no—
140#1157—44no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28C2×D412yes—
147C2×C4×D7—24no—
148C2×C4×D7—42no—
149C2×C4×D7—42no—
150C2×Dic14—21no—
151C2×Dic14—21no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14C2×D412yes—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—44no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—42no—
168C7×C4⋊C4—42no—
169C22⋊Dic7—24no—
170C22⋊Dic7—42no—
171(C2×D7)⋊C4—82no—
172(C2×D7)⋊C4—82no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179Dic7⋊C4—82no—
180Dic7⋊C4—82no—
181C4×Dic7—22no—
182C4×Dic7—22no—
183C4×Dic7—42no—
184#1136—44no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C22×#41—71no—
189C22×M16—71no—
190C23×C8C2×D711yescharacteristic
191C2×#117—71no—
192C2×#117—71no—
193C22×C4⋊C4C2×D711yescharacteristic
194C22×C4⋊C4C2×D711yescharacteristic
195#417—144no—
196#416—144no—
197C2×#103—71no—
198C2×#103—71no—
199C2×#103—72no—
200C2×#102—71no—
201C2×#102—71no—
202C2×#102—72no—
203#391—72no—
204#390—72no—
205#389—141no—
206#389—141no—
207#386—142no—
208#385—142no—
209#384—72no—
210#382—72no—
211#380—141no—
212#380—141no—
213#340—72no—
214#340—72no—
215#340C2×D712yes—
216#340C2×D712yes—
217#336—72no—
218#336—72no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7#12112yes—
224C23×C7#12312yes—
225C23×C7#12412yes—
226C23×C7C22×D411yescharacteristic
227C22×D7—22no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—84no—
241C2×C4×C7—84no—
242C2×C4×C7#12112yes—
243C2×C4×C7#12314yes—
244C2×C4×C7#12314yes—
245C2×C4×C7#12312yes—
246C2×C4×C7#12412yes—
247C2×C4×C7#12614yes—
248C2×C4×C7#12614yes—
249C2×C4×C7C22×D411yescharacteristic
250C2×C4×C7C2×#4111yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C22⋊D7—48no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—22no—
262C2×Dic7—22no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271D28—22no—
272D28—22no—
273D28—41no—
274D28—41no—
275C4×D7—48no—
276Dic14—22no—
277Dic14—22no—
278Dic14—41no—
279Dic14—41no—
280C7×C8—41no—
281C7×C8—41no—
282C7⋊C8—42no—
283C2×#41—74no—
284C2×#41—142no—
285C2×#41—142no—
286C2×#41—142no—
287C2×#41—142no—
288C2×#41—142no—
289C22×Q8—71no—
290C22×D4—71no—
291C22×D4—141no—
292C23×C4—21no—
293C23×C4—21no—
294C23×C4—141no—
295C23×C4C22×D711yescharacteristic
296C2×M16—142no—
297C2×M16—142no—
298C2×M16—142no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—72no—
303C22×C8C22×D711yescharacteristic
304C22×C8C22×D711yescharacteristic
305#117—142no—
306#117—142no—
307#117—142no—
308#117—142no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4C22×D712yes—
322C2×C4⋊C4C22×D712yes—
323C2×C22⋊C4—72no—
324C2×C22⋊C4—141no—
325C2×C22⋊C4—142no—
326C2×C22⋊C4—142no—
327C2×C42—72no—
328C2×C42—141no—
329#103—144no—
330#103—142no—
331#103—142no—
332#102—144no—
333#102—142no—
334#102—142no—
335#95—142no—
336#95—142no—
337#95—142no—
338#95—142no—
339C22×C7—22no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7#39714yes—
346C22×C7C2×#12111yescharacteristic
347C22×C7C2×#12311yescharacteristic
348C22×C7C2×#12411yescharacteristic
349C22×C7#57611yescharacteristic
350C22×C7#57711yescharacteristic
351C22×C7#57711yescharacteristic
352C22×C7#57811yescharacteristic
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—42no—
356C2×D7—81no—
357C2×D7—81no—
358C2×D7—81no—
359C2×D7—81no—
360C4×C7—82no—
361C4×C7—82no—
362C4×C7#39714yes—
363C4×C7C2×#12311yescharacteristic
364C4×C7C2×#12311yescharacteristic
365C4×C7C2×#12611yescharacteristic
366C4×C7C2×#12611yescharacteristic
367Dic7—42no—
368Dic7—42no—
369Dic7—42no—
370Dic7—81no—
371Dic7—81no—
372C24—141no—
373C24D4×D711yescharacteristic
374#41—148no—
375#41—148no—
376#41—284no—
377#41—284no—
378C2×Q8—72no—
379C2×Q8—141no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8—141no—
384C2×D4—72no—
385C2×D4—144no—
386C2×D4—141no—
387C2×D4—141no—
388C2×D4—141no—
389C2×D4—141no—
390C2×D4—141no—
391C2×D4—282no—
392C2×D4—282no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—141no—
404C22×C4—144no—
405C22×C4—141no—
406C22×C4—141no—
407C22×C4—282no—
408C22×C4—282no—
409C22×C4#115712yes—
410C22×C4#115712yes—
411C22×C4#115712yes—
412C22×C4D4×D712yes—
413C22×C4D4×D711yescharacteristic
414C22×C4#115912yes—
415C22×C4#116112yes—
416C22×C4C23×D711yescharacteristic
417M16—284no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—44no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—142no—
435C4⋊C4—142no—
436C4⋊C4—141no—
437C4⋊C4—141no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—562no—
441C4⋊C4—562no—
442C22⋊C4—144no—
443C22⋊C4—282no—
444C22⋊C4—562no—
445C22⋊C4—562no—
446C42—142no—
447C42—142no—
448C42—282no—
449C2×C7—24no—
450C2×C7C2×#39711yescharacteristic
451C2×C7#304511yescharacteristic
452C2×C7#304611yescharacteristic
453C2×C7#309811yescharacteristic
454C2×C7#309911yescharacteristic
455C2×C7#313611yescharacteristic
456C2×C7#313711yescharacteristic
457D7—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—142no—
463C23—281no—
464C23—281no—
465C23—281no—
466C23—281no—
467C23—281no—
468C23—281no—
469C23#644712yes—
470C23#644812yes—
471C23#645712yes—
472C23C2×D4×D711yescharacteristic
473Q8—142no—
474Q8—142no—
475Q8—281no—
476Q8—281no—
477D4—142no—
478D4—142no—
479D4—288no—
480D4—281no—
481D4—281no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—142no—
491C2×C4—142no—
492C2×C4—142no—
493C2×C4—142no—
494C2×C4—142no—
495C2×C4—142no—
496C2×C4—142no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—288no—
500C2×C4—281no—
501C2×C4—281no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4#643914yes—
507C2×C4#644614yes—
508C2×C4#644712yes—
509C2×C4#644812yes—
510C2×C4#645714yes—
511C2×C4#645712yes—
512C2×C4#646114yes—
513C2×C4C2×#115711yescharacteristic
514C2×C4C2×#115711yescharacteristic
515C2×C4C2×#115711yescharacteristic
516C2×C4C2×D4×D711yescharacteristic
517C2×C4C2×#115911yescharacteristic
518C2×C4C2×#116111yescharacteristic
519C8—41no—
520C8—41no—
521C8—282no—
522C7#2363411yescharacteristic, complemented, Sylow
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—282no—
530C22—282no—
531C22—282no—
532C22—561no—
533C22—561no—
534C22—561no—
535C22—561no—
536C22#8944714yes—
537C22C2×#644711yescharacteristic
538C22C2×#644811yescharacteristic
539C22C2×#645711yescharacteristic
540C22#9044811yescharacteristic
541C22#9045311yescharacteristic
542C22#9045311yescharacteristic
543C22#9045711yescharacteristic
544C4—82no—
545C4—82no—
546C4—282no—
547C4—282no—
548C4—282no—
549C4—561no—
550C4—561no—
551C4#8944714yes—
552C4C2×#643911yescharacteristic
553C4C2×#644611yescharacteristic
554C4C2×#645711yescharacteristic
555C4C2×#646111yescharacteristic
556C2—24no—
557C2—562no—
558C2C2×#8944711yescharacteristic
559C2#1173618211yescharacteristic
560C2#1173621211yescharacteristic
561C2#1173623011yescharacteristic
562C2#1173626011yescharacteristic
563C2#1173655711yescharacteristic
564C2#1173658611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#340C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
hypercenterC2×#340D7
hypocenterC7#23634
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #11739602C2semidirect
5. C2×C7×#340C2semidirect
6. #11729719C2semidirect
7. #11729312C2semidirect
8. #11726669C2semidirect
9. #11726668C2semidirect
10. #11726657C2semidirect
11. #11726656C2semidirect
109. C2×#340D7semidirect
522. C7#23634semidirect

Maximal subgroups

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

subgroupindex
C2×#898852
#117396312
#117396022
C2×#897382
C2×C7×#3402
#117297192
#117293122
#117266692
#117266682
#117266572
#117266562
#236347