Finite Group Databasebeta

← #13453619

Subgroups — #13453619

Order 1792.

978 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
1C4×#6503C211yesmaximal, characteristic
2#11731801C211yesmaximal, characteristic, complemented
3C7×#1802C211yesmaximal, characteristic, complemented
4#11729358C211yesmaximal, characteristic, complemented
5#11728233C211yesmaximal, characteristic, complemented
6Dic7×C4⋊C8C211yesmaximal, characteristic, complemented
7#11727917C212yesmaximal, complemented
8#11727911C212yesmaximal
9#11727338C211yesmaximal, characteristic, complemented
10#11726652C212yesmaximal, complemented
11#11726601C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6503C2211yescharacteristic
14C2×C4×C22⋊D7—21no—
15C2×C4×C22⋊D7—21no—
16C2×#6451—21no—
17C2×#6451—21no—
18C2×#6451C2212yes—
19C2×C4×D28C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#102C2212yes—
22C7×#384C2211yescharacteristic
23#89993C2211yescharacteristic
24C2×#6406C2212yes—
25C2×#6395C2212yes—
26C2×C8×Dic7C2212yes—
27#89760C2211yescharacteristic
28#89748—21no—
29#89748—21no—
30#89748C2211yescharacteristic
31#89748C2212yes—
32C2×#6383C2212yes—
33#89729—21no—
34#89729—21no—
35#89707C2211yescharacteristic
36C42×Dic7C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C7×#340C2212yes—
39#89346C2212yes—
40#89345C2212yes—
41#89324C2212yes—
42#89324C2212yes—
43#29139—71nomaximal, Sylow
44C22×C7×D4C2311yescharacteristic
45C23×C4×C7—21no—
46C22×C22⋊D7—21no—
47C22×C22⋊D7—21no—
48C22×D28C2311yescharacteristic
49C22×C4×D7—21no—
50C22×Dic14—22no—
51C22×C7×C8C2312yes—
52C4×C7×D4—22no—
53C4×C7×D4—22no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C2×C22⋊Dic7—21no—
61C2×C22⋊Dic7—21no—
62#6503—22no—
63#6503Q814yescomplemented
64C4×C22⋊D7—44no—
65C4×C22⋊D7—44no—
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2312yes—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—21no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D411yescharacteristic
82C2×C4×Dic7C2311yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85#6451—42no—
86#6451—42no—
87#6451—44no—
88#6451—44no—
89C4×D28—22no—
90C4×D28—22no—
91C7×C4⋊C8—22no—
92C7×#102—42no—
93#6406—42no—
94#6395—42no—
95C8×Dic7—24no—
96#6383—42no—
97C4⋊(C7⋊C8)—22no—
98#2868—71no—
99C4×#127—71no—
100#2023—72no—
101#2023—72no—
102#1957—72no—
103#1834—72no—
104#1802—71no—
105#1802—71no—
106#1802—71no—
107#1802D711yescharacteristic, complemented
108C4×C4⋊C8—71no—
109C24×C7—21no—
110C23×D7—21no—
111C2×C7×D4—21no—
112C2×C7×D4—21no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C2×C7×D4—21no—
116C2×C7×D4C2×Q812yes—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—42no—
120C22×C4×C7—42no—
121C22×C4×C7—42no—
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2411yescharacteristic
125C2×C22⋊D7—24no—
126C2×C22⋊D7—24no—
127C2×C22⋊D7—42no—
128C2×C22⋊D7—42no—
129C2×C22⋊D7—42no—
130C2×C22⋊D7—42no—
131C22×Dic7—22no—
132C22×Dic7—21no—
133C22×Dic7C2×D411yescharacteristic
134C22×Dic7C2×D411yescharacteristic
135C22×Dic7C2×D411yescharacteristic
136C22×Dic7C2×D411yescharacteristic
137C2×D28—21no—
138C2×D28—21no—
139C2×D28—21no—
140C2×D28—21no—
141C2×D28—21no—
142C2×D28C2×Q812yes—
143C2×C4×D7—42no—
144C2×C4×D7—42no—
145C2×C4×D7—42no—
146C2×Dic14—44no—
147C2×Dic14—44no—
148C2×Dic14—44no—
149C2×C7×C8—22no—
150C2×C7×C8—22no—
151C2×C7×C8—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—42no—
157C7×C4⋊C4—42no—
158C7×C22⋊C4—42no—
159C42×C7#4112yes—
160C42×C7#4112yes—
161C22⋊Dic7—42no—
162C22⋊Dic7—42no—
163(C2×D7)⋊C4—42no—
164(C2×D7)⋊C4—42no—
165(C2×D7)⋊C4—42no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—22no—
169C4⋊Dic7—22no—
170C4⋊Dic7—42no—
171C4⋊Dic7—42no—
172Dic7⋊C4—44no—
173Dic7⋊C4—44no—
174Dic7⋊C4—42no—
175Dic7⋊C4—42no—
176Dic7⋊C4—44no—
177C4×Dic7—21no—
178C4×Dic7—21no—
179C4×Dic7—22no—
180C4×Dic7—22no—
181C4×Dic7D814yes—
182C4×Dic7SD1614yes—
183C4×Dic7C2×Q812yes—
184C4×Dic7#4112yes—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188C2×C4⋊Q8—72no—
189C2×C4⋊Q8—141no—
190C2×C4⋊Q8—141no—
191C2×#127—71no—
192C2×C4×D4—71no—
193C2×C4×D4—141no—
194C2×C4×D4—141no—
195C2×C4×D4C2×D711yescharacteristic
196C2×C4⋊C8—71no—
197C2×C4⋊C8C2×D711yescharacteristic
198C2×#102—72no—
199C2×#102—72no—
200C2×#102—72no—
201C2×#102C2×D712yes—
202C2×C4×C8—72no—
203#390—71no—
204#390—71no—
205#390—141no—
206#390—141no—
207#384—71no—
208#384—72no—
209#384—71no—
210#384—141no—
211#384—141no—
212#384C2×D711yescharacteristic
213C43—71no—
214#340—72no—
215#340—72no—
216#340—72no—
217#340C2×D712yes—
218#336—72no—
219C23×C7—22no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7C22×D411yescharacteristic
227C22×D7—22no—
228C22×D7—41no—
229C22×D7—41no—
230C22×D7—41no—
231C22×D7—41no—
232C22×D7—41no—
233C22×D7—41no—
234C7×D4—22no—
235C7×D4—22no—
236C7×D4—41no—
237C7×D4—41no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—41no—
241C2×C4×C7—42no—
242C2×C4×C7—41no—
243C2×C4×C7—42no—
244C2×C4×C7—41no—
245C2×C4×C7—42no—
246C2×C4×C7—84no—
247C2×C4×C7#12412yes—
248C2×C4×C7#12412yes—
249C2×C4×C7C2×D812yes—
250C2×C4×C7C2×SD1612yes—
251C2×C4×C7C22×Q811yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C2×C4×C7H(5,𝔽2)11yescharacteristic
256C2×C4×C7H(5,𝔽2)11yescharacteristic
257C22⋊D7—48no—
258C22⋊D7—48no—
259C2×Dic7—21no—
260C2×Dic7—21no—
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—41no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—41no—
267C2×Dic7—41no—
268C2×Dic7—42no—
269C2×Dic7C22⋊Q812yes—
270C2×Dic7C22⋊Q812yes—
271C2×Dic7C2×D812yes—
272C2×Dic7C2×D812yes—
273C2×Dic7C2×SD1612yes—
274C2×Dic7C2×SD1612yes—
275C2×Dic7AGL(1,ℤ/8)12yes—
276C2×Dic7#13712yes—
277D28—22no—
278D28—22no—
279D28—41no—
280D28—41no—
281C4×D7—84no—
282Dic14—88no—
283C7×C8—42no—
284C7⋊C8—42no—
285C22×Q8—142no—
286C22×D4—71no—
287C22×D4—141no—
288C22×D4—141no—
289C22×D4C22×D711yescharacteristic
290C23×C4—21no—
291C23×C4—141no—
292C22×C8—72no—
293C22×C8C22×D712yes—
294C4⋊Q8—284no—
295C4⋊Q8—282no—
296C4⋊Q8—282no—
297C4⋊Q8—284no—
298#127—74no—
299#127—142no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—142no—
303C4×D4—142no—
304C4×D4—284no—
305C4×D4—284no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—72no—
310C2×C4⋊C4—141no—
311C2×C4⋊C4—141no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—142no—
314C2×C4⋊C4—142no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4C22×D711yescharacteristic
318C2×C4⋊C4C22×D712yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—141no—
321C2×C22⋊C4—141no—
322C2×C22⋊C4—141no—
323C2×C22⋊C4—141no—
324C2×C22⋊C4—141no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42C22×D711yescharacteristic
332C4⋊C8—22no—
333C4⋊C8—142no—
334#102—42no—
335#102—282no—
336#102—282no—
337#102—282no—
338C4×C8—144no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—81no—
343C22×C7—81no—
344C22×C7—81no—
345C22×C7—81no—
346C22×C7C2×C22⋊Q811yescharacteristic
347C22×C7C2×#12411yescharacteristic
348C22×C7#53511yescharacteristic
349C22×C7C22×D811yescharacteristic
350C22×C7C22×SD1611yescharacteristic
351C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C7C2×#13711yescharacteristic
353C2×D7—42no—
354C2×D7—42no—
355C2×D7—42no—
356C2×D7—81no—
357C2×D7—81no—
358C2×D7—81no—
359C2×D7—81no—
360C4×C7—22no—
361C4×C7—81no—
362C4×C7—82no—
363C4×C7#48614yes—
364C4×C7#55411yescharacteristic
365C4×C7#55411yescharacteristic
366C4×C7#55511yescharacteristic
367C4×C7#55511yescharacteristic
368Dic7—22no—
369Dic7—22no—
370Dic7—81no—
371Dic7—82no—
372Dic7#47414yes—
373Dic7#47614yes—
374C24—21no—
375C24—141no—
376C2×Q8—284no—
377C2×Q8—284no—
378C2×Q8—284no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—72no—
385C2×D4—144no—
386C2×D4—144no—
387C2×D4—141no—
388C2×D4—141no—
389C2×D4—141no—
390C2×D4—141no—
391C2×D4—141no—
392C2×D4—282no—
393C2×D4—282no—
394C2×D4—282no—
395C2×D4—282no—
396C2×D4Q8×D712yes—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—71no—
405C22×C4—71no—
406C22×C4—141no—
407C22×C4—142no—
408C22×C4—282no—
409C22×C4—282no—
410C22×C4—282no—
411C22×C4D4×D711yescharacteristic
412C22×C4D4×D711yescharacteristic
413C22×C4C23×D711yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—142no—
418C2×C8—142no—
419C2×C8—142no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—141no—
427C4⋊C4—141no—
428C4⋊C4—142no—
429C4⋊C4—142no—
430C4⋊C4—282no—
431C4⋊C4—282no—
432C4⋊C4—282no—
433C4⋊C4—284no—
434C4⋊C4—284no—
435C4⋊C4—284no—
436C4⋊C4—282no—
437C22⋊C4—42no—
438C22⋊C4—282no—
439C22⋊C4—282no—
440C22⋊C4—282no—
441C22⋊C4—282no—
442C22⋊C4—282no—
443C42—74no—
444C42—74no—
445C42—72no—
446C42—72no—
447C42—142no—
448C42—142no—
449C42—141no—
450C42—141no—
451C42#115712yes—
452C42#115912yes—
453C2×C7—82no—
454C2×C7#256311yescharacteristic
455C2×C7C2×#47411yescharacteristic
456C2×C7C2×#47611yescharacteristic
457C2×C7C2×#48611yescharacteristic
458C2×C7#314111yescharacteristic
459C2×C7#320511yescharacteristic
460C2×C7#320611yescharacteristic
461D7—82no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—142no—
470C23—281no—
471C23—281no—
472C23—281no—
473C23—281no—
474C23—281no—
475C23—281no—
476C23C2×D4×D711yescharacteristic
477Q8—568no—
478D4—22no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—142no—
483D4—142no—
484D4—288no—
485D4—288no—
486D4—281no—
487D4—281no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—72no—
497C2×C4—72no—
498C2×C4—72no—
499C2×C4—72no—
500C2×C4—72no—
501C2×C4—72no—
502C2×C4—72no—
503C2×C4—72no—
504C2×C4—84no—
505C2×C4—141no—
506C2×C4—141no—
507C2×C4—141no—
508C2×C4—141no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—282no—
512C2×C4—281no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—564no—
516C2×C4#644812yes—
517C2×C4#644812yes—
518C2×C4D7×D812yes—
519C2×C4D7×SD1612yes—
520C2×C4C2×#115711yescharacteristic
521C2×C4C2×#115911yescharacteristic
522C2×C4#654811yescharacteristic
523C2×C4C2×Q8×D711yescharacteristic
524C2×C4D7×#4111yescharacteristic
525C2×C4#655311yescharacteristic
526C8—42no—
527C8—282no—
528C7#2913911yescharacteristic, complemented, Sylow
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—282no—
537C22—282no—
538C22—282no—
539C22—561no—
540C22—561no—
541C22—561no—
542C22—561no—
543C22C2×#644811yescharacteristic
544C22#9029811yescharacteristic
545C22D7×C22⋊Q811yescharacteristic
546C22C2×D7×D811yescharacteristic
547C22C2×D7×SD1611yescharacteristic
548C22D7×AGL(1,ℤ/8)11yescharacteristic
549C22D7×#13711yescharacteristic
550C4—22no—
551C4—74no—
552C4—74no—
553C4—81no—
554C4—82no—
555C4—142no—
556C4—142no—
557C4—561no—
558C4—562no—
559C4#8953214yes—
560C4#9020211yescharacteristic
561C4#9022311yescharacteristic
562C4#9030711yescharacteristic
563C4#9032611yescharacteristic
564C2—82no—
565C2—562no—
566C2#1173138711yescharacteristic
567C2C2×#8953211yescharacteristic
568C2#1173632311yescharacteristic
569C2#1173778111yescharacteristic
570C2#1173787711yescharacteristic
571C2D7×#47411yescharacteristic
572C2D7×#47611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1802C2
socleC23×C7C22×D4
prime radicalC2×#6503C22
hypercenter#1802D7
hypocenterC7#29139
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #11731801C2semidirect
3. C7×#1802C2semidirect
4. #11729358C2semidirect
5. #11728233C2semidirect
6. Dic7×C4⋊C8C2semidirect
7. #11727917C2semidirect
9. #11727338C2semidirect
10. #11726652C2semidirect
11. #11726601C2semidirect
63. #6503Q8semidirect
107. #1802D7semidirect
528. C7#29139semidirect

Maximal subgroups

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

subgroupindex
C4×#65032
#117318012
C7×#18022
#117293582
#117282332
Dic7×C4⋊C82
#117279172
#117279112
#117273382
#117266522
#117266012
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