Finite Group Databasebeta

← #13387733

Subgroups — #13387733

Order 1792.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90242C211yesmaximal, characteristic
2C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
3#11739945C211yesmaximal, characteristic, complemented
4C2×#89792C211yesmaximal, characteristic, complemented
5#11729808C212yesmaximal, complemented
6#11729358C211yesmaximal, characteristic, complemented
7#11728333C212yesmaximal, complemented
8#11728022C212yesmaximal, complemented
9#11727961C212yesmaximal, complemented
10#11727541C211yesmaximal, characteristic, complemented
11#11727531C211yesmaximal, characteristic, complemented
12C22×#1157—21no—
13C22×C7×SD16C2211yescharacteristic
14C2×C4×C7×Q8C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6509C2212yes—
17C2×#6501C2212yes—
18C2×#6487—21no—
19#90242—24no—
20#90242—24no—
21C2×D7×C4⋊C4C2212yes—
22C2×#6446—22no—
23C2×C4×D28C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C4×C7×SD16—24no—
26C2×C7×#106C2211yescharacteristic
27C2×C7×#103C2211yescharacteristic
28C2×C7×#102C2211yescharacteristic
29C2×C4×C7×C8C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6406C2212yes—
32C2×(C2×D7)⋊C8C2212yes—
33C2×(C2×D7)⋊C8C2212yes—
34#89792—24no—
35#89766—22no—
36#89759C2212yes—
37C2×#6382C2212yes—
38#89711—21no—
39#89707C2211yescharacteristic
40#89700—21no—
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#24120—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C22×C22⋊D7—22no—
50C2×#1157—28no—
51C2×#1157—44no—
52C2×#1157—44no—
53C22×D28D411yescharacteristic
54C22×C4×D7—22no—
55C22×C4×D7C2312yes—
56C22×Dic14—21no—
57C2×C7×SD16—22no—
58C2×C7×SD16—22no—
59C2×C7×SD16—22no—
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2311yescharacteristic
62C4×C7×Q8—22no—
63C4×C7×Q8—22no—
64C4×C7×D4—22no—
65C4×C7×D4D414yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72C2×C22⋊Dic7—22no—
73#6509—24no—
74#6509—42no—
75#6501—24no—
76#6501—24no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4D412yes—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7D411yescharacteristic
84C2×C4⋊Dic7C2312yes—
85C2×C4⋊Dic7C2312yes—
86#6487—28no—
87C2×Dic7⋊C4—22no—
88C2×Dic7⋊C4—22no—
89C22×C7⋊C8C2312yes—
90D7×C4⋊C4—24no—
91D7×C4⋊C4—24no—
92#6446—216no—
93C4×D28—22no—
94C4×D28—22no—
95#6432—22no—
96#6432D414yes—
97C7×#106—22no—
98C7×#103—22no—
99C7×#102—22no—
100C4×C7×C8—22no—
101#6409—24no—
102#6406—42no—
103C7⋊#95—22no—
104(C2×D7)⋊C8—24no—
105(C2×D7)⋊C8—42no—
106#6382—24no—
107C4⋊(C7⋊C8)—22no—
108C2×#547—71no—
109C2×#475—71no—
110C2×C4×SD16D711yescharacteristic, complemented
111#2538—71no—
112#2034—72no—
113#2000—72no—
114#1990—71no—
115#1954—71no—
116#1872—72no—
117#1853—72no—
118#1802—71no—
119C24×C7—21no—
120C23×D7C2×D412yes—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8#4112yescomplemented
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4C2×D412yes—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—24no—
136C22×C4×C7—21no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C2×C22⋊D7—28no—
143C2×C22⋊D7—44no—
144C2×C22⋊D7—44no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7C2×D412yes—
148#1157—432no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28SD1614yescomplemented
152C2×C4×D7—28no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—44no—
160C2×C4×D7—44no—
161C2×Dic14—24no—
162C2×Dic14—42no—
163C2×Dic14—42no—
164C7×SD16—28no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8#4112yescomplemented
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4C2×D412yes—
176C7×C22⋊C4—24no—
177C42×C7—42no—
178C42×C7C2×D412yes—
179C42×C7C2×D412yes—
180C22⋊Dic7—28no—
181(C2×D7)⋊C4—24no—
182(C2×D7)⋊C4—44no—
183(C2×D7)⋊C4—82no—
184C4⋊Dic7—24no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—42no—
190C4⋊Dic7—81no—
191C4⋊Dic7C2×D414yes—
192Dic7⋊C4—28no—
193Dic7⋊C4—44no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C22×#41—141no—
198C22×SD16C2×D711yescharacteristic
199#547—144no—
200#547—144no—
201C2×C4⋊Q8—71no—
202C2×#123—142no—
203C2×C22⋊Q8—72no—
204C2×C22⋊Q8—141no—
205C2×#121—72no—
206C2×C4×Q8C2×D711yescharacteristic
207C2×C4×D4—71no—
208C2×C4×D4C2×D711yescharacteristic
209C22×C4⋊C4—72no—
210#475—144no—
211C4×SD16—24no—
212C2×#106—72no—
213C2×#106C2×D711yescharacteristic
214C2×C4⋊C8—71no—
215C2×#103—72no—
216C2×#103C2×D711yescharacteristic
217C2×#102—72no—
218C2×#102C2×D711yescharacteristic
219C2×C22⋊C8—72no—
220C2×C22⋊C8—72no—
221C2×C4×C8C2×D711yescharacteristic
222#396—142no—
223#390—141no—
224#389—141no—
225#384—71no—
226#380—72no—
227#340—72no—
228#340—72no—
229#340—72no—
230C23×C7—22no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7C22×D411yescharacteristic
238C22×D7—22no—
239C22×D7—22no—
240C22×D7—22no—
241C22×D7—22no—
242C22×D7—22no—
243C22×D7—22no—
244C22×D7C2×SD1614yes—
245C7×Q8—22no—
246C7×Q8—22no—
247C7×Q8—41no—
248C7×Q8—41no—
249C7×D4—22no—
250C7×D4—22no—
251C7×D4—41no—
252C7×D4—41no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—48no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7#12012yes—
266C2×C4×C7#12012yes—
267C2×C4×C7C2×SD1612yes—
268C2×C4×C7AGL(1,ℤ/8)12yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C72−1+411yescharacteristic
275C22⋊D7—416no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285D28—28no—
286C4×D7—416no—
287C4×D7—42no—
288C4×D7—42no—
289C4×D7—42no—
290C4×D7—42no—
291Dic14—48no—
292C7×C8—22no—
293C7×C8—41no—
294C7⋊C8—42no—
295C2×#41—148no—
296C2×#41—284no—
297C2×#41—284no—
298C22×Q8—141no—
299C22×Q8C22×D711yescharacteristic
300C22×D4—71no—
301C22×D4—142no—
302C22×D4C22×D711yescharacteristic
303C23×C4—21no—
304C23×C4—72no—
305C23×C4—142no—
306C2×SD16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C22×C8—72no—
310C22×C8C22×D711yescharacteristic
311C22×C8C22×D711yescharacteristic
312C4⋊Q8—74no—
313C4⋊Q8—142no—
314#123—1416no—
315C22⋊Q8—148no—
316C22⋊Q8—144no—
317C22⋊Q8—282no—
318#121—144no—
319#121—144no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—142no—
324C4×D4—142no—
325C4×D4C22⋊D714yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—72no—
330C2×C4⋊C4—144no—
331C2×C4⋊C4—142no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—144no—
334C2×C4⋊C4—142no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—72no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C42—21no—
345C2×C42C22×D711yescharacteristic
346#106—22no—
347#106—144no—
348C4⋊C8—142no—
349#103—22no—
350#103—144no—
351#102—22no—
352#102—282no—
353C22⋊C8—144no—
354C22⋊C8—282no—
355C4×C8—22no—
356#95—142no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7C2×#12011yescharacteristic
365C22×C7#54611yescharacteristic
366C22×C7#54711yescharacteristic
367C22×C7C22×SD1611yescharacteristic
368C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C7#57711yescharacteristic
370C22×C7#57811yescharacteristic
371C2×D7—24no—
372C2×D7—24no—
373C2×D7—24no—
374C2×D7—42no—
375C2×D7—42no—
376C2×D7—42no—
377C2×D7—42no—
378C4×C7—22no—
379C4×C7—42no—
380C4×C7—42no—
381C4×C7—81no—
382C4×C7#45014yes—
383C4×C7#52511yescharacteristic
384C4×C7#53711yescharacteristic
385C4×C7#54811yescharacteristic
386C4×C7#55011yescharacteristic
387Dic7—42no—
388Dic7—44no—
389Dic7—82no—
390C24—21no—
391C24—72no—
392#41—2832no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—144no—
399C2×Q8—282no—
400C2×Q8—282no—
401C2×Q8#115712yescomplemented
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—74no—
408C2×D4—148no—
409C2×D4—142no—
410C2×D4—142no—
411C2×D4—284no—
412C2×D4—284no—
413C2×D4C2×C22⋊D712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—72no—
421C22×C4—148no—
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×C4—284no—
431C22×C4—284no—
432C22×C4D4×D711yescharacteristic
433C22×C4D4×D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435SD16—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—142no—
442C2×C8—142no—
443C2×C8—142no—
444C2×C8#115712yescomplemented
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—74no—
450C4⋊C4—148no—
451C4⋊C4—142no—
452C4⋊C4—144no—
453C4⋊C4—142no—
454C4⋊C4—142no—
455C4⋊C4—142no—
456C4⋊C4—284no—
457C4⋊C4—282no—
458C4⋊C4—561no—
459C4⋊C4C2×C22⋊D712yes—
460C22⋊C4—24no—
461C22⋊C4—148no—
462C22⋊C4—144no—
463C22⋊C4—284no—
464C22⋊C4—562no—
465C42—42no—
466C42D4×D712yes—
467C42C2×C22⋊D712yes—
468C2×C7—82no—
469C2×C7#240711yescharacteristic
470C2×C7C2×#45011yescharacteristic
471C2×C7#301311yescharacteristic
472C2×C7#329811yescharacteristic
473C2×C7#333211yescharacteristic
474C2×C7#333511yescharacteristic
475C2×C7#335311yescharacteristic
476D7—44no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—74no—
485C23—142no—
486C23—142no—
487C23—142no—
488C23—142no—
489C23—142no—
490C23—142no—
491C23C2×D4×D711yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496Q8—288no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—148no—
502D4—2816no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—48no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—142no—
516C2×C4—144no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—2816no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4D7×SD1612yes—
530C2×C4#647712yes—
531C2×C4#650012yes—
532C2×C4#650012yes—
533C2×C4C2×#115711yescharacteristic
534C2×C4C2×#115711yescharacteristic
535C2×C4C2×D4×D711yescharacteristic
536C2×C4#655311yescharacteristic
537C2×C4#655411yescharacteristic
538C2×C4C22×C22⋊D711yescharacteristic
539C8—22no—
540C8—41no—
541C8—282no—
542C7#2412011yescharacteristic, complemented, Sylow
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—144no—
551C22—144no—
552C22—144no—
553C22—282no—
554C22—282no—
555C22—282no—
556C22—282no—
557C22#9024111yescharacteristic
558C22#9024211yescharacteristic
559C22C2×D7×SD1611yescharacteristic
560C22C2×#647711yescharacteristic
561C22#9046111yescharacteristic
562C22#9046211yescharacteristic
563C22C2×#650011yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—284no—
569C4—282no—
570C4—562no—
571C4#8994114yes—
572C4#9024311yescharacteristic
573C4#9024411yescharacteristic
574C4#9048211yescharacteristic
575C4#9052411yescharacteristic
576C2—82no—
577C2—284no—
578C2#1173712311yescharacteristic
579C2#1173713211yescharacteristic
580C2#1173713311yescharacteristic
581C2#1173713711yescharacteristic
582C2#1173984711yescharacteristic
583C2C2×#8994111yescharacteristic
584C2#1174135811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radicalC2×#90242C2
hypercenterC2×C4×SD16D7
hypocenterC7#24120
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C4×C7×SD16C2semidirect
3. #11739945C2semidirect
4. C2×#89792C2semidirect
5. #11729808C2semidirect
6. #11729358C2semidirect
7. #11728333C2semidirect
8. #11728022C2semidirect
9. #11727961C2semidirect
10. #11727541C2semidirect
11. #11727531C2semidirect
110. C2×C4×SD16D7semidirect
126. C2×C7×Q8#41semidirect
151. C2×D28SD16semidirect
170. C2×C7×C8#41semidirect
401. C2×Q8#1157semidirect
444. C2×C8#1157semidirect
542. C7#24120semidirect

Maximal subgroups

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

subgroupindex
C2×#902422
C2×C4×C7×SD162
#117399452
C2×#897922
#117298082
#117293582
#117283332
#117280222
#117279612
#117275412
#117275312
#241207