Finite Group Databasebeta

← #13407447

Subgroups — #13407447

Order 1792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#541C211yesmaximal, characteristic, complemented
2C2×#89858C211yesmaximal, characteristic, complemented
3C2×#89840C211yesmaximal, characteristic, complemented
4#11739732C211yesmaximal, characteristic
5#11729539C211yesmaximal, characteristic, complemented
6#11729495C211yesmaximal, characteristic, complemented
7#11729454C212yesmaximal, complemented
8#11729453C212yesmaximal
9#11729387C212yesmaximal, complemented
10#11729386C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C7×#541—24no—
13C7×#541—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#125—22no—
17C2×C7×#125C2211yescharacteristic
18C2×C7×C22⋊Q8C2212yes—
19C2×C4×C7×Q8C2212yes—
20C2×C7×#117C2211yescharacteristic
21C22×#1144C2212yes—
22C22×#1143C2212yes—
23C2×#6493C2211yescharacteristic
24C2×#6487C2211yescharacteristic
25C2×#6438C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×C22⋊(C7⋊C8)C2212yes—
29#89858—24no—
30#89840—24no—
31#89766—22no—
32C2×#6384C2212yes—
33C2×#6383C2212yes—
34#89733—22no—
35#89711C2211yescharacteristic
36#89707C2211yescharacteristic
37#89701C2211yescharacteristic
38C2×#6377C2211yescharacteristic
39C2×C4×C7⋊C8C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#25612—71nomaximal, Sylow
44C22×C7×Q8C2312yes—
45C23×C4×C7C2311yescharacteristic
46C22×C22⋊D7—21no—
47C22×D28C2311yescharacteristic
48C22×Dic14C2311yescharacteristic
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8D414yescomplemented
51C7×#126—216no—
52C7×#125—216no—
53C7×#125—22no—
54C7×#125D414yescomplemented
55C7×C22⋊Q8—24no—
56C7×C22⋊Q8—24no—
57C4×C7×Q8—24no—
58C4×C7×Q8—24no—
59C7×#117—22no—
60C7×#117—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4D412yes—
66C2×C7×C4⋊C4C2312yes—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4D412yes—
69C2×C42×C7—22no—
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C22⋊Dic7—21no—
73C2×#1144—24no—
74C2×#1144—24no—
75C2×#1144—42no—
76C2×#1143—24no—
77C2×#1143—24no—
78C2×#1143—42no—
79#6493—42no—
80C2×(C2×D7)⋊C4—22no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—22no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86#6487—42no—
87C2×Dic7⋊C4—21no—
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2312yes—
90#6438—22no—
91#6438—22no—
92#6409—42no—
93#6409—42no—
94C7⋊#95—22no—
95C22⋊(C7⋊C8)—24no—
96#6384—24no—
97#6383—24no—
98#6377—22no—
99C4×C7⋊C8—22no—
100C2×#541D711yescharacteristic, complemented
101C2×#497—71no—
102C2×#488—71no—
103#2620—71no—
104#2001—72no—
105#2000—72no—
106#1992—71no—
107#1956—71no—
108#1880—72no—
109#1878—72no—
110#1750—71no—
111C24×C7C2×D411yescharacteristic
112C23×D7—21no—
113C2×C7×Q8—22no—
114C2×C7×Q8—22no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8C2×D414yes—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2411yescharacteristic
129C2×C22⋊D7—42no—
130C2×C22⋊D7—82no—
131C22×Dic7—21no—
132C22×Dic7—21no—
133C22×Dic7—21no—
134C2×D28—21no—
135C2×D28—21no—
136C2×D28—22no—
137C2×D28—22no—
138C2×Dic14—21no—
139C2×Dic14—21no—
140C2×Dic14—22no—
141C2×Dic14—22no—
142C7×C4⋊C4—28no—
143C7×C4⋊C4—28no—
144C7×C4⋊C4—28no—
145C7×C4⋊C4—28no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4C2×D414yes—
149C7×C22⋊C4—28no—
150C7×C22⋊C4—24no—
151C42×C7—28no—
152C42×C7—21no—
153C42×C7—21no—
154C42×C7C2×D412yes—
155C42×C7C2×D412yes—
156C22⋊Dic7—42no—
157#1144—48no—
158#1143—48no—
159(C2×D7)⋊C4—42no—
160(C2×D7)⋊C4—44no—
161(C2×D7)⋊C4—82no—
162C4⋊Dic7—22no—
163C4⋊Dic7—22no—
164C4⋊Dic7—82no—
165Dic7⋊C4—42no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8#4114yes—
172C22×Q16—72no—
173C22×SD16—72no—
174#541—24no—
175#541—24no—
176C2×C4⋊Q8C2×D711yescharacteristic
177C2×#126—22no—
178C2×#125—22no—
179C2×#125C2×D711yescharacteristic
180C2×#124—71no—
181C2×C22⋊Q8—71no—
182C2×C22⋊Q8C2×D712yes—
183C2×#121—71no—
184C2×C4×Q8C2×D712yes—
185C2×#117C2×D711yescharacteristic
186#497—144no—
187#488—144no—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#102—72no—
192C2×C22⋊C8—72no—
193C2×#97—71no—
194C2×C4×C8—71no—
195#396—142no—
196#394—142no—
197#391—71no—
198#390—71no—
199#384—71no—
200#340—72no—
201#340—72no—
202#336—72no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—21no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7C22×D411yescharacteristic
209C22×D7—41no—
210C22×D7—41no—
211C22×D7—41no—
212C22×D7—42no—
213C22×D7—42no—
214C7×Q8—24no—
215C7×Q8—24no—
216C7×Q8—42no—
217C7×Q8—42no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7#12014yes—
235C2×C4×C7C22×D411yescharacteristic
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C2×#4112yes—
238C2×C4×C7H(5,𝔽2)12yes—
239C22⋊D7—84no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—41no—
244C2×Dic7—41no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247D28—42no—
248D28—42no—
249Dic14—42no—
250Dic14—42no—
251C7⋊C8—24no—
252C7⋊C8—42no—
253C22×Q8—71no—
254C22×Q8C22×D712yes—
255C22×D4—71no—
256C22×D4—141no—
257C23×C4C22×D711yescharacteristic
258C2×Q16—144no—
259C2×Q16—144no—
260C2×Q16—282no—
261C2×SD16—144no—
262C2×SD16—144no—
263C2×SD16—282no—
264C22×C8—72no—
265C22×C8—72no—
266C4⋊Q8—22no—
267C4⋊Q8C22⋊D714yescomplemented
268#126—216no—
269#125—216no—
270#125—22no—
271#125C22⋊D714yescomplemented
272#124—142no—
273#124—142no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—282no—
277#121—282no—
278C4×Q8—24no—
279C4×Q8—24no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—71no—
287C2×C4⋊C4—71no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4C22⋊D712yes—
291C2×C4⋊C4C22×D712yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—142no—
296C2×C22⋊C4—142no—
297C2×C22⋊C4C22⋊D712yes—
298C2×C42—22no—
299C2×C42C22×D711yescharacteristic
300C2×C42C22×D711yescharacteristic
301#103—144no—
302#103—282no—
303#103—282no—
304#102—144no—
305C22⋊C8—144no—
306#97—142no—
307C4×C8—142no—
308#95—142no—
309C22×C7—42no—
310C22×C7—41no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—42no—
314C22×C7C2×#12011yescharacteristic
315C22×C7#53211yescharacteristic
316C22×C7#53511yescharacteristic
317C22×C7#57712yes—
318C22×C7#57812yes—
319C2×D7—82no—
320C2×D7—81no—
321C2×D7—81no—
322C2×D7—81no—
323C2×D7—82no—
324C4×C7—24no—
325C4×C7—41no—
326C4×C7—44no—
327C4×C7—42no—
328C4×C7—44no—
329C4×C7#54612yes—
330C4×C7#55012yes—
331Dic7—81no—
332Dic7—81no—
333Dic7—81no—
334C24—141no—
335C24C2×C22⋊D711yescharacteristic
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—142no—
342C2×Q8—142no—
343C2×Q8—141no—
344C2×Q8—141no—
345C2×Q8C2×C22⋊D714yes—
346C2×D4—142no—
347C2×D4—142no—
348C2×D4—141no—
349C2×D4—141no—
350C2×D4—282no—
351C2×D4—562no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—141no—
359C22×C4—141no—
360C22×C4—141no—
361C22×C4C2×C22⋊D712yes—
362C22×C4C2×C22⋊D711yescharacteristic
363C22×C4C2×C22⋊D712yes—
364C22×C4C23×D711yescharacteristic
365Q16—288no—
366SD16—288no—
367C2×C8—74no—
368C2×C8—142no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—142no—
372C2×C8—142no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—142no—
380C4⋊C4—142no—
381C4⋊C4—282no—
382C4⋊C4—562no—
383C4⋊C4C2×C22⋊D714yes—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—282no—
387C22⋊C4—282no—
388C22⋊C4—284no—
389C22⋊C4—562no—
390C42—28no—
391C42—21no—
392C42—21no—
393C42C2×C22⋊D712yes—
394C42C2×C22⋊D712yes—
395C2×C7—81no—
396C2×C7#243111yescharacteristic
397C2×C7#301312yes—
398C2×C7#317812yes—
399C2×C7#322211yescharacteristic
400C2×C7#322311yescharacteristic
401D7—161no—
402C23—21no—
403C23—21no—
404C23—22no—
405C23—22no—
406C23—21no—
407C23—281no—
408C23—281no—
409C23—282no—
410C23—282no—
411C23—281no—
412C23C22×C22⋊D711yescharacteristic
413Q8—24no—
414Q8—24no—
415Q8—42no—
416Q8—42no—
417Q8—282no—
418Q8—282no—
419D4—282no—
420D4—282no—
421D4—564no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4—282no—
443C2×C4—282no—
444C2×C4—281no—
445C2×C4#651614yes—
446C2×C4D7×#4112yes—
447C2×C4#655312yes—
448C2×C4C22×C22⋊D711yescharacteristic
449C2×C4C22×C22⋊D711yescharacteristic
450C8—144no—
451C8—282no—
452C7#2561211yescharacteristic, complemented, Sylow
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—561no—
459C22—562no—
460C22—562no—
461C22—561no—
462C22—561no—
463C22#9052111yescharacteristic
464C22#9052311yescharacteristic
465C22#9052812yes—
466C22#9052912yes—
467C22C2×#651611yescharacteristic
468C4—24no—
469C4—41no—
470C4—44no—
471C4—42no—
472C4—44no—
473C4—561no—
474C4—561no—
475C4—561no—
476C4#9052012yes—
477C4#9052512yes—
478C2—81no—
479C2—1121no—
480C2#1174089012yes—
481C2#1174089311yescharacteristic
482C2#1174089711yescharacteristic
483C2#1174153711yescharacteristic
484C2#1174155912yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#541C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenterC2×#541D7
hypocenterC7#25612
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C7×#541C2semidirect
2. C2×#89858C2semidirect
3. C2×#89840C2semidirect
5. #11729539C2semidirect
6. #11729495C2semidirect
7. #11729454C2semidirect
9. #11729387C2semidirect
10. #11729386C2semidirect
11. #11729336C2semidirect
50. C7×C4⋊Q8D4semidirect
54. C7×#125D4semidirect
100. C2×#541D7semidirect
267. C4⋊Q8C22⋊D7semidirect
271. #125C22⋊D7semidirect
452. C7#25612semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5412
C2×#898582
C2×#898402
#117397322
#117295392
#117294952
#117294542
#117294532
#117293872
#117293862
#117293362
#256127