Finite Group Databasebeta

← #13332571

Subgroups — #13332571

Order 1792.

1192 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic, complemented
3#11739886C211yesmaximal, characteristic, complemented
4#11739753C211yesmaximal, characteristic
5#11729670C214yesmaximal, complemented
6#11729668C214yesmaximal, complemented
7#11729527C211yesmaximal, characteristic, complemented
8#11729375C211yesmaximal, characteristic, complemented
9#11729352C211yesmaximal, characteristic, complemented
10C23×C7×Q8—21no—
11C7×D4×Q8—22no—
12C7×D4×Q8—22no—
13C7×D4×Q8—22no—
14C7×D4×Q8C2214yes—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—22no—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×#6505—21no—
23#90168C2211yescharacteristic
24#89989—21no—
25#89860C2212yescomplemented
26#89859—22no—
27#89850C2212yescomplemented
28#89803C2212yescomplemented
29#89800—22no—
30#89799C2212yescomplemented
31#89787C2212yes—
32C4×#1141—22no—
33#89779C2212yes—
34#89739—21no—
35#89739—22no—
36#89738—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yescomplemented
41Q8⋊(C7⋊C8)C2214yescomplemented
42D4⋊(C7⋊C8)C2214yes—
43#18776—71nomaximal, Sylow
44C22×C7×Q8—22no—
45C22×C7×Q8—24no—
46C22×C7×Q8—24no—
47C22×C7×Q8—21no—
48C22×C7×Q8—44no—
49C22×C7×Q8—44no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4D411yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C2×#1157—21no—
55C7×C4⋊Q8—24no—
56C7×C4⋊Q8—24no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D412yes—
60C7×C4⋊Q8C2312yes—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8—28no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—28no—
65C4×C7×Q8—21no—
66C4×C7×Q8—21no—
67C4×C7×Q8D412yes—
68C4×C7×Q8C2312yes—
69C4×C7×D4—24no—
70C4×C7×D4—24no—
71C4×C7×D4—21no—
72C4×C7×D4—21no—
73C4×C7×D4D412yes—
74C4×C7×D4C2312yes—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4D412yes—
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×#1144—22no—
86#6505—44no—
87C2×#1143—22no—
88C2×#1141—22no—
89#6493—21no—
90#6493—21no—
91C2×#1136—21no—
92#6438—21no—
93#6437C2311yescharacteristic
94C4×D28—21no—
95C4×D28—21no—
96#6432C2311yescharacteristic
97#6406—22no—
98C22⋊(C7⋊C8)—21no—
99C22⋊(C7⋊C8)—21no—
100#6384—22no—
101#6383—24no—
102#6383—22no—
103#6383—22no—
104#6381—24no—
105#6381—22no—
106C4⋊(C7⋊C8)—21no—
107C4⋊(C7⋊C8)—21no—
108C4⋊(C7⋊C8)C2312yes—
109C4⋊(C7⋊C8)C2312yes—
110C4×C7⋊C8—21no—
111C4×C7⋊C8C2312yes—
112C2×D4×Q8D711yescharacteristic, complemented
113#3226—71no—
114#3100—71no—
115#3037—71no—
116#1649—74no—
117#1627—74no—
118#1546—71no—
119#1486—71no—
120#1479—71no—
121C24×C7—21no—
122C2×C7×Q8—28no—
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—48no—
128C2×C7×Q8—48no—
129C2×C7×Q8—416no—
130C2×C7×Q8—48no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×Q8D814yescomplemented
134C2×C7×Q8C2×D412yes—
135C2×C7×D4—24no—
136C2×C7×D4—21no—
137C2×C7×D4C2×D412yes—
138C22×C4×C7—21no—
139C22×C4×C7—28no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—48no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C2×C22⋊D7—41no—
152#1157—42no—
153#1157—42no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×C4×D7—41no—
158C2×Dic14—21no—
159C7×C4⋊C4—28no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4C2×D412yes—
168C7×C4⋊C4C2×D412yes—
169C7×C4⋊C4C2×D414yes—
170C7×C22⋊C4—28no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—22no—
173C42×C7—22no—
174C42×C7—22no—
175C42×C7C2×D411yescharacteristic
176C42×C7C2×D411yescharacteristic
177C42×C7C2×D411yescharacteristic
178C42×C7C2411yescharacteristic
179#1144—44no—
180#1143—44no—
181#1141—44no—
182(C2×D7)⋊C4—41no—
183(C2×D7)⋊C4—41no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186#1136—42no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—22no—
189C2×C7⋊C8—22no—
190C23×Q8—21no—
191C2×#137—141no—
192D4×Q8—22no—
193D4×Q8—22no—
194D4×Q8—22no—
195D4×Q8C2×D714yes—
196#532—71no—
197C2×C4⋊Q8—21no—
198C2×C4⋊Q8C2×D711yescharacteristic
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—22no—
201C2×C4×Q8C2×D711yescharacteristic
202C2×C4×D4—21no—
203C2×C4×D4C2×D711yescharacteristic
204#500—72no—
205#486—72no—
206#480—141no—
207#480—142no—
208#474—72no—
209#474—142no—
210#464—142no—
211#462—72no—
212#460—72no—
213#459—72no—
214#447—141no—
215C4×D8—142no—
216#431—71no—
217#431—71no—
218#423—71no—
219#417—141no—
220#332—74no—
221Q8⋊C8—74no—
222D4⋊C8—74no—
223C23×C7—24no—
224C23×C7—22no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7#12011yescharacteristic
228C22×D7—41no—
229C7×Q8—22no—
230C7×Q8—24no—
231C7×Q8—416no—
232C7×Q8—42no—
233C7×Q8—42no—
234C7×Q8—44no—
235C7×Q8C2×D814yes—
236C7×D4—24no—
237C7×D4—22no—
238C7×D4#13714yescomplemented
239C2×C4×C7—22no—
240C2×C4×C7—24no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—24no—
244C2×C4×C7—22no—
245C2×C4×C7—41no—
246C2×C4×C7—416no—
247C2×C4×C7—42no—
248C2×C4×C7—44no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—44no—
253C2×C4×C7#12011yescharacteristic
254C2×C4×C7#12011yescharacteristic
255C2×C4×C7#12011yescharacteristic
256C2×C4×C7C2×D812yes—
257C2×C4×C7AGL(1,ℤ/8)12yes—
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C22×D411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C22⋊D7—82no—
262C2×Dic7—41no—
263D28—42no—
264D28—41no—
265D28—41no—
266D28—42no—
267C4×D7—82no—
268Dic14—41no—
269Dic14—41no—
270C7⋊C8—42no—
271C7⋊C8—42no—
272C7⋊C8—42no—
273C2×#41—141no—
274C22×Q8—22no—
275C22×Q8—24no—
276C22×Q8—24no—
277C22×Q8—21no—
278C22×Q8—44no—
279C22×Q8—44no—
280C22×Q8C22⋊D711yescharacteristic
281C22×D4C22⋊D711yescharacteristic
282C23×C4—22no—
283C23×C4—21no—
284#137—284no—
285C2×Q16—142no—
286C2×SD16—142no—
287C2×D8—142no—
288C2×M16—141no—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—71no—
294C4⋊Q8C22⋊D712yes—
295C4⋊Q8C22×D712yes—
296#127—71no—
297#124—141no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—28no—
302#121—141no—
303#121—141no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8C22⋊D712yes—
307C4×Q8C22×D712yes—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—141no—
313C4×D4—141no—
314C4×D4C22⋊D712yes—
315C4×D4C22×D712yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C22⋊D711yescharacteristic
320C2×C4⋊C4C22⋊D711yescharacteristic
321C2×C4⋊C4C22⋊D712yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C42—21no—
325C2×C42C22×D711yescharacteristic
326#107—142no—
327#107—144no—
328C4⋊C8—72no—
329C4⋊C8—72no—
330C4⋊C8—141no—
331C4⋊C8—141no—
332#103—142no—
333#102—142no—
334#102—142no—
335#102—142no—
336#102—144no—
337C22⋊C8—141no—
338C22⋊C8—141no—
339C4×C8—72no—
340C4×C8—141no—
341C22×C7—21no—
342C22×C7—21no—
343C22×C7—24no—
344C22×C7—48no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7#44712yes—
348C22×C7C2×#12011yescharacteristic
349C2×D7—81no—
350C2×D7—81no—
351C2×D7—81no—
352C4×C7—24no—
353C4×C7—21no—
354C4×C7—21no—
355C4×C7—21no—
356C4×C7—44no—
357C4×C7—42no—
358C4×C7—42no—
359C4×C7#44712yes—
360C4×C7C22×D811yescharacteristic
361C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
362C4×C7C2×#13712yes—
363Dic7—81no—
364C24—21no—
365#41—282no—
366#41—282no—
367C2×Q8—28no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—141no—
379C2×Q8#114114yescomplemented
380C2×Q8C2×C22⋊D712yes—
381C2×D4—24no—
382C2×D4—21no—
383C2×D4—141no—
384C2×D4—141no—
385C2×D4—141no—
386C2×D4—281no—
387C2×D4C2×C22⋊D712yes—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—281no—
399C22×C4C2×C22⋊D711yescharacteristic
400C22×C4C2×C22⋊D711yescharacteristic
401C22×C4C2×C22⋊D711yescharacteristic
402Q16—284no—
403SD16—284no—
404D8—284no—
405M16—282no—
406C2×C8—142no—
407C2×C8—142no—
408C2×C8—142no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—141no—
418C4⋊C4—141no—
419C4⋊C4C2×C22⋊D712yes—
420C4⋊C4C2×C22⋊D712yes—
421C4⋊C4C2×C22⋊D714yes—
422C22⋊C4—28no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—281no—
426C22⋊C4—281no—
427C42—22no—
428C42—22no—
429C42C2×C22⋊D711yescharacteristic
430C42C2×C22⋊D711yescharacteristic
431C42C2×C22⋊D711yescharacteristic
432C42C23×D711yescharacteristic
433C2×C7—22no—
434C2×C7—44no—
435C2×C7C2×#44711yescharacteristic
436C2×C7#300411yescharacteristic
437C2×C7#302011yescharacteristic
438D7—161no—
439C23—24no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23—281no—
444C23#651611yescharacteristic
445Q8—22no—
446Q8—24no—
447Q8—416no—
448Q8—42no—
449Q8—42no—
450Q8—44no—
451Q8—281no—
452Q8—281no—
453Q8C2×#114114yes—
454D4—24no—
455D4—22no—
456D4—282no—
457D4—281no—
458D4—281no—
459D4—282no—
460D4—562no—
461D4#650514yescomplemented
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—41no—
469C2×C4—416no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—281no—
477C2×C4—562no—
478C2×C4C2×#114112yes—
479C2×C4#649512yes—
480C2×C4#651611yescharacteristic
481C2×C4#651611yescharacteristic
482C2×C4#651611yescharacteristic
483C2×C4C22×C22⋊D711yescharacteristic
484C2×C4C22×C22⋊D711yescharacteristic
485C2×C4C22×C22⋊D711yescharacteristic
486C8—282no—
487C8—282no—
488C8—282no—
489C7#1877611yescharacteristic, complemented, Sylow
490C22—21no—
491C22—21no—
492C22—24no—
493C22—48no—
494C22—42no—
495C22—42no—
496C22—561no—
497C22—561no—
498C22—561no—
499C22#8998912yes—
500C22C2×#651611yescharacteristic
501C4—24no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—561no—
509C4#8998912yes—
510C4C22×#114111yescharacteristic
511C4C2×#649511yescharacteristic
512C4C2×#650512yes—
513C2—22no—
514C2—44no—
515C2—1121no—
516C2C2×#8998911yescharacteristic
517C2#1174150411yescharacteristic
518C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×D4×Q8C2
socleC22×C7C2×#120
prime radical#11739753C2
hypercenterC2×D4×Q8D7
hypocenterC7#18776
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×D4×Q8C2semidirect
2. #11740660C2semidirect
3. #11739886C2semidirect
5. #11729670C2semidirect
6. #11729668C2semidirect
7. #11729527C2semidirect
8. #11729375C2semidirect
9. #11729352C2semidirect
25. #89860C22semidirect
27. #89850C22semidirect
28. #89803C22semidirect
30. #89799C22semidirect
40. #89342C22semidirect
41. Q8⋊(C7⋊C8)C22semidirect
112. C2×D4×Q8D7semidirect
133. C2×C7×Q8D8semidirect
238. C7×D4#137semidirect
379. C2×Q8#1141semidirect
461. D4#6505semidirect
489. C7#18776semidirect

Maximal subgroups

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

subgroupindex
C2×C7×D4×Q82
#117406602
#117398862
#117397532
#117296702
#117296682
#117295272
#117293752
#117293522
#187767