Finite Group Databasebeta

← #13337285

Subgroups — #13337285

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3152C211yesmaximal, characteristic, complemented
2#11740755C211yesmaximal, characteristic
3#11740660C211yesmaximal, characteristic, complemented
4#11735967C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6C2×#89342C211yesmaximal, characteristic, complemented
7#11729060C212yesmaximal, complemented
8#11729059C212yesmaximal, complemented
9#11729022C212yesmaximal, complemented
10#11729021C212yesmaximal, complemented
11#11727604C211yesmaximal, characteristic, complemented
12C2×C7×#137—21no—
13C2×C7×#137—21no—
14C7×#532C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×#6470—21no—
17#90168C2211yescharacteristic
18C7×#497—22no—
19C7×#494C2212yes—
20C7×#492C2212yes—
21C4×C7×M16C2211yescharacteristic
22#89901—21no—
23#89860—21no—
24#89859C2212yes—
25#89859C2212yes—
26#89859C2212yes—
27#89850—21no—
28#89844C2212yes—
29#89810—22no—
30#89787—21no—
31#89779—21no—
32#89739—22no—
33#89697C2211yescharacteristic
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36#89478—22no—
37#89460C2212yes—
38C8×D28C2212yes—
39#89342C2212yes—
40#89342C2212yes—
41#89342C2211yescharacteristic
42#89342C2211yescharacteristic
43#89339C2211yescharacteristic
44#89339C2211yescharacteristic
45#89253C2212yes—
46#89252C2212yes—
47#19231—71nomaximal, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C2×#1157—21no—
51C7×#137—44no—
52C7×#137—44no—
53C2×C7×Q16—22no—
54C2×C7×Q16—22no—
55C2×C7×SD16—22no—
56C2×C7×SD16—22no—
57C2×C7×M16D411yescharacteristic
58C2×C7×M16D411yescharacteristic
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8C2312yes—
63C7×C4⋊Q8C2311yescharacteristic
64C7×#127C2311yescharacteristic
65C7×#124—21no—
66C7×#121—21no—
67C7×#121—21no—
68C4×C7×D4—21no—
69C4×C7×D4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C42×C7C2311yescharacteristic
73C2×#1141—22no—
74#6493—21no—
75#6493D412yes—
76C22×C7⋊C8—21no—
77C22×C7⋊C8—21no—
78#6470—44no—
79C2×#1131—22no—
80C2×C8×D7—22no—
81#6438—21no—
82#6437D411yescharacteristic
83C4×D28—21no—
84C4×D28C2312yes—
85#6432D411yescharacteristic
86C7×#97D412yes—
87C4×C7×C8C2312yes—
88#6406—22no—
89C22⋊(C7⋊C8)—21no—
90(C2×D7)⋊C8—22no—
91(C2×D7)⋊C8—22no—
92#6383—22no—
93#6383—22no—
94#6383—22no—
95#6381—22no—
96#6381—22no—
97#6381—22no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)—21no—
100C4⋊(C7⋊C8)C2311yescharacteristic
101C4⋊(C7⋊C8)C2311yescharacteristic
102C4⋊(C7⋊C8)C2312yes—
103C4⋊(C7⋊C8)C2311yescharacteristic
104C4⋊(C7⋊C8)C2311yescharacteristic
105C4×C7⋊C8—21no—
106#3245—71no—
107#3226—71no—
108#3152D711yescharacteristic, complemented
109#2935—71no—
110#1689—72no—
111#1685—72no—
112#1669—72no—
113#1665—72no—
114#1565—71no—
115#1540—71no—
116C2×#332—71no—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—41no—
122C2×C7×Q8—42no—
123C2×C7×Q8—42no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—41no—
128C22×C4×C7—41no—
129C22×C4×C7—41no—
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7#4111yescharacteristic
133C2×C22⋊D7—22no—
134#1157—42no—
135#1157—42no—
136C2×D28—21no—
137C2×D28—21no—
138C2×D28C2×D412yes—
139C2×C4×D7—22no—
140C2×Dic14—21no—
141C7×Q16—44no—
142C7×Q16—44no—
143C7×SD16—44no—
144C7×SD16—44no—
145C7×M16—22no—
146C7×M16D814yescomplemented
147C2×C7×C8C2×D412yes—
148C2×C7×C8C2×D412yes—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—41no—
154C7×C4⋊C4—41no—
155C7×C22⋊C4—41no—
156C7×C22⋊C4—41no—
157C42×C7C2×D411yescharacteristic
158C42×C7C2×D411yescharacteristic
159C42×C7#4111yescharacteristic
160C42×C7C2411yescharacteristic
161#1141—82no—
162(C2×D7)⋊C4—22no—
163(C2×D7)⋊C4—22no—
164C4⋊Dic7—21no—
165C4⋊Dic7C2×D412yes—
166C2×C7⋊C8—21no—
167C2×C7⋊C8—21no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—42no—
172C2×C7⋊C8—81no—
173#1131—44no—
174C8×D7—44no—
175C2×#137—21no—
176C2×#137—21no—
177C2×#131—141no—
178#532—71no—
179#532C2×D711yescharacteristic
180C2×C4⋊Q8C2×D711yescharacteristic
181#500—141no—
182#497—22no—
183#494C2×D712yes—
184#492C2×D712yes—
185#486—141no—
186#480—142no—
187#474—72no—
188#474—72no—
189#474—72no—
190#474—141no—
191#466—142no—
192#459—72no—
193#459—141no—
194#436—72no—
195#435—142no—
196C8×D4—72no—
197#431—71no—
198C2×C4⋊C8—71no—
199C2×C4⋊C8—71no—
200#408—141no—
201C4×M16C2×D711yescharacteristic
202#335—72no—
203#334—72no—
204#332—72no—
205#332—72no—
206#332—71no—
207#332—71no—
208#331—71no—
209#331—71no—
210C23×C7—41no—
211C23×C7#12111yescharacteristic
212C22×D7—22no—
213C7×Q8—42no—
214C7×Q8—41no—
215C7×Q8—42no—
216C7×Q8—41no—
217C7×Q8—82no—
218C7×D4—41no—
219C7×D4—41no—
220C7×D4—41no—
221C7×D4—41no—
222C7×D4—81no—
223C7×D4—82no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—42no—
229C2×C4×C7—82no—
230C2×C4×C7—82no—
231C2×C4×C7#12111yescharacteristic
232C2×C4×C7#12111yescharacteristic
233C2×C4×C7#12111yescharacteristic
234C2×C4×C7C2×D812yes—
235C2×C4×C7AGL(1,ℤ/8)12yes—
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C2×#4111yescharacteristic
239C22⋊D7—44no—
240C2×Dic7—22no—
241D28—22no—
242D28—22no—
243D28—42no—
244D28—81no—
245C4×D7—42no—
246C4×D7—42no—
247Dic14—42no—
248C7×C8—22no—
249C7×C8C2×D814yes—
250C7⋊C8—41no—
251C7⋊C8—41no—
252C7⋊C8—41no—
253C7⋊C8—42no—
254C7⋊C8—41no—
255C2×#41—21no—
256C2×#41—141no—
257C22×Q8—21no—
258#137—44no—
259#137—44no—
260C2×Q16—22no—
261C2×Q16—22no—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×D8—142no—
265#131—284no—
266C2×M16—142no—
267C2×M16C22⋊D711yescharacteristic
268C2×M16C22⋊D711yescharacteristic
269C22×C8—141no—
270C22×C8—141no—
271C22×C8—142no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—71no—
276C4⋊Q8C22×D712yes—
277C4⋊Q8C22×D711yescharacteristic
278#127—71no—
279#127C22×D711yescharacteristic
280#124—21no—
281#124—141no—
282#121—21no—
283#121—21no—
284#121—72no—
285#121—141no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—72no—
289C4×D4—141no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42C22×D711yescharacteristic
293#107—142no—
294#107—142no—
295#107—142no—
296C4⋊C8—71no—
297C4⋊C8—71no—
298C4⋊C8—71no—
299C4⋊C8—71no—
300C4⋊C8—72no—
301C4⋊C8—141no—
302C4⋊C8—141no—
303#102—142no—
304#102—142no—
305#102—142no—
306#102—142no—
307C22⋊C8—141no—
308C22⋊C8—142no—
309C22⋊C8—142no—
310#97C22⋊D712yes—
311C4×C8—141no—
312C4×C8C22×D712yes—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—81no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7#45912yes—
319C22×C7C2×#12111yescharacteristic
320C2×D7—42no—
321C2×D7—42no—
322C2×D7—42no—
323C4×C7—21no—
324C4×C7—21no—
325C4×C7—21no—
326C4×C7—81no—
327C4×C7—82no—
328C4×C7#45912yes—
329C4×C7C22×D811yescharacteristic
330C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
331C4×C7#57711yescharacteristic
332C4×C7#57811yescharacteristic
333Dic7—42no—
334#41—42no—
335#41—42no—
336#41—282no—
337#41—282no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—141no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—72no—
349C2×D4—142no—
350C2×D4—141no—
351C2×D4—141no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—142no—
355C22×C4D4×D711yescharacteristic
356C22×C4#115911yescharacteristic
357C22×C4C2×C22⋊D711yescharacteristic
358Q16—44no—
359Q16—44no—
360SD16—44no—
361SD16—44no—
362D8—562no—
363M16—22no—
364M16—284no—
365M16#114114yescomplemented
366C2×C8—141no—
367C2×C8—141no—
368C2×C8—142no—
369C2×C8—141no—
370C2×C8—141no—
371C2×C8—282no—
372C2×C8—284no—
373C2×C8—561no—
374C2×C8C2×C22⋊D712yes—
375C2×C8C2×C22⋊D712yes—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—72no—
383C4⋊C4—141no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—142no—
387C22⋊C4—142no—
388C42D4×D711yescharacteristic
389C42#115911yescharacteristic
390C42C2×C22⋊D711yescharacteristic
391C42C23×D711yescharacteristic
392C2×C7—22no—
393C2×C7—161no—
394C2×C7C2×#45911yescharacteristic
395C2×C7#306311yescharacteristic
396C2×C7#306911yescharacteristic
397D7—82no—
398C23—41no—
399C23—142no—
400C23#650111yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—42no—
404Q8—41no—
405Q8—82no—
406Q8—282no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—81no—
412D4—82no—
413D4—142no—
414D4—142no—
415D4—284no—
416D4—282no—
417D4—561no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—142no—
426C2×C4—282no—
427C2×C4—282no—
428C2×C4C2×#114112yes—
429C2×C4#649512yes—
430C2×C4#650111yescharacteristic
431C2×C4#650111yescharacteristic
432C2×C4#650111yescharacteristic
433C2×C4C2×D4×D711yescharacteristic
434C2×C4C2×#115911yescharacteristic
435C2×C4C22×C22⋊D711yescharacteristic
436C8—22no—
437C8—281no—
438C8—282no—
439C8—281no—
440C8—281no—
441C8—281no—
442C8C2×#114114yes—
443C7#1923111yescharacteristic, complemented, Sylow
444C22—21no—
445C22—21no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—282no—
450C22—282no—
451C22—282no—
452C22#8984412yes—
453C22C2×#650111yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—82no—
459C4—282no—
460C4#8984412yes—
461C4#9046911yescharacteristic
462C4#9047011yescharacteristic
463C4C22×#114111yescharacteristic
464C4C2×#649511yescharacteristic
465C2—22no—
466C2—161no—
467C2—562no—
468C2C2×#8984411yescharacteristic
469C2#1174143111yescharacteristic
470C2#1174144311yescharacteristic
subgroupG/H
centerC22C2×#6501
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3152C2
socleC22×C7C2×#121
prime radical#11740755C2
hypercenter#3152D7
hypocenterC7#19231
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C7×#3152C2semidirect
3. #11740660C2semidirect
4. #11735967C2semidirect
5. #11729558C2semidirect
6. C2×#89342C2semidirect
7. #11729060C2semidirect
8. #11729059C2semidirect
9. #11729022C2semidirect
10. #11729021C2semidirect
11. #11727604C2semidirect
108. #3152D7semidirect
146. C7×M16D8semidirect
365. M16#1141semidirect
443. C7#19231semidirect

Maximal subgroups

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

subgroupindex
C7×#31522
#117407552
#117406602
#117359672
#117295582
C2×#893422
#117290602
#117290592
#117290222
#117290212
#117276042
#192317