Finite Group Databasebeta

← #13413074

Subgroups — #13413074

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×C4⋊C4C211yesmaximal, characteristic
2#11731170C211yesmaximal, characteristic, complemented
3C4×#6406C211yesmaximal, characteristic
4#11729360C212yesmaximal
5#11729359C212yesmaximal
6#11729358C211yesmaximal, characteristic
7#11727377C212yesmaximal, complemented
8#11727367C212yesmaximal, complemented
9#11727328C211yesmaximal, characteristic, complemented
10#11727321C211yesmaximal, characteristic, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C4×C7×D4C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C22×C7×C4⋊C4—21no—
15C7×#389C2212yes—
16C7×#385—22no—
17C7×#384C2211yescharacteristic
18C7×#380—21no—
19C4×C7×C4⋊C4C2211yescharacteristic
20C2×#6406C2212yes—
21C2×#6406C2211yescharacteristic
22C2×#6406C2211yescharacteristic
23#89749—21no—
24#89745—21no—
25C2×#6381C2212yes—
26#89708C2212yes—
27#89707C2211yescharacteristic
28C4×C4⋊Dic7C2211yescharacteristic
29#89700C2212yes—
30C2×C4⋊(C7⋊C8)C2211yescharacteristic
31C2×C4⋊(C7⋊C8)C2212yes—
32C2×C4×C7⋊C8C2211yescharacteristic
33#89432—22no—
34#89324C2211yescharacteristic
35#89324C2212yes—
36#89324C2212yes—
37#89324C2212yes—
38#89324C2211yescharacteristic
39#89324C2212yes—
40#89319C2212yes—
41#26111—71nomaximal, Sylow
42C22×C7×D4C2311yescharacteristic
43C23×C4×C7—21no—
44C23×C4×C7—22no—
45C4×C7×D4—24no—
46C4×C7×D4—22no—
47C4×C7×D4—42no—
48C4×C7×D4Q814yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—44no—
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—22no—
59C2×C42×C7C2312yes—
60C2×C42×C7C2311yescharacteristic
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4⋊Dic7C2312yes—
65C2×C4⋊Dic7C2311yescharacteristic
66C2×Dic7⋊C4—22no—
67C2×C4×Dic7—21no—
68C22×C7⋊C8C2311yescharacteristic
69C22×C7⋊C8C2311yescharacteristic
70C22×C7⋊C8C2312yes—
71C7×#95—22no—
72C7×#95D412yes—
73#6406—22no—
74#6406—24no—
75#6406—22no—
76C7⋊#95—21no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80#6381—42no—
81C4⋊(C7⋊C8)—22no—
82C4⋊(C7⋊C8)—42no—
83C4×C7⋊C8—22no—
84#2571—71no—
85D4×C4⋊C4D711yescharacteristic
86#2101—72no—
87#2087—72no—
88#1947—71no—
89#1939—71no—
90#1862—72no—
91#1834—71no—
92#1804—72no—
93#1802—71no—
94C4×#102—71no—
95C24×C7—21no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4C2×Q812yes—
100C2×C7×D4#4112yes—
101C2×C7×D4#4112yes—
102C22×C4×C7—21no—
103C22×C4×C7—24no—
104C22×C4×C7—22no—
105C22×C4×C7—44no—
106C22×C4×C7—44no—
107C22×C4×C7—42no—
108C22×C4×C7—42no—
109C22×C4×C7—82no—
110C22×C4×C7C2×D412yes—
111C22×C4×C7C2×D412yes—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2411yescharacteristic
115C22×Dic7—21no—
116C22×Dic7—21no—
117C22×Dic7—22no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—44no—
124C7×C4⋊C4—42no—
125C7×C4⋊C4—81no—
126C7×C4⋊C4C2×Q812yes—
127C7×C22⋊C4—24no—
128C7×C22⋊C4—44no—
129C42×C7—22no—
130C42×C7—22no—
131C42×C7C2×Q812yes—
132C42×C7#4112yes—
133C4⋊Dic7—21no—
134C4⋊Dic7—22no—
135C4⋊Dic7—21no—
136C4⋊Dic7—21no—
137C4⋊Dic7—22no—
138C4⋊Dic7—21no—
139C4⋊Dic7—42no—
140C4⋊Dic7—81no—
141Dic7⋊C4—82no—
142C4×Dic7—42no—
143C2×C7⋊C8—21no—
144C2×C7⋊C8—21no—
145C2×C7⋊C8—22no—
146C2×C7⋊C8—21no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—22no—
151C2×C7⋊C8—21no—
152C2×C4×D4C2×D712yes—
153C2×C4×D4C2×D711yescharacteristic
154C22×C4⋊C4—21no—
155C2×#107—72no—
156C2×C4⋊C8—71no—
157C2×C4⋊C8—72no—
158C2×#102—72no—
159C2×#102—71no—
160C2×#102—71no—
161C2×C4×C8—71no—
162#400—142no—
163#389—72no—
164#389—141no—
165#389C2×D712yes—
166#385—22no—
167#384—71no—
168#384C2×D711yescharacteristic
169#382—72no—
170#382—141no—
171#380—21no—
172C4×C4⋊C4—71no—
173C4×C4⋊C4C2×D711yescharacteristic
174#340—72no—
175#340—72no—
176#340—72no—
177#340—71no—
178#340—72no—
179#340—71no—
180#336—72no—
181C23×C7—22no—
182C23×C7—22no—
183C23×C7—22no—
184C23×C7—41no—
185C23×C7—41no—
186C23×C7—41no—
187C23×C7—41no—
188C23×C7C22×D411yescharacteristic
189C7×D4—22no—
190C7×D4—22no—
191C7×D4—22no—
192C7×D4#12514yes—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—22no—
201C2×C4×C7—22no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—48no—
205C2×C4×C7—42no—
206C2×C4×C7—42no—
207C2×C4×C7—42no—
208C2×C4×C7—88no—
209C2×C4×C7C22⋊Q812yes—
210C2×C4×C7C22⋊Q812yes—
211C2×C4×C7#13512yes—
212C2×C4×C7AGL(1,ℤ/8)12yes—
213C2×C4×C7C22×Q811yescharacteristic
214C2×C4×C7C2×#4111yescharacteristic
215C2×C4×C7C2×#4111yescharacteristic
216C2×C4×C7C2×#4111yescharacteristic
217C2×C4×C7H(5,𝔽2)11yescharacteristic
218C2×C4×C72−1+411yescharacteristic
219C2×Dic7—41no—
220C2×Dic7—42no—
221C2×Dic7—41no—
222C2×Dic7—42no—
223C2×Dic7—41no—
224C2×Dic7—41no—
225C2×Dic7—41no—
226C2×Dic7—42no—
227C2×Dic7—41no—
228C7⋊C8—41no—
229C7⋊C8—41no—
230C7⋊C8—42no—
231C22×D4C22×D711yescharacteristic
232C23×C4—21no—
233C23×C4—22no—
234C22×C8—71no—
235C22×C8—72no—
236C22×C8—71no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4—42no—
240C4×D4Dic1414yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4—72no—
247C2×C4⋊C4—71no—
248C2×C4⋊C4—141no—
249C2×C4⋊C4—142no—
250C2×C4⋊C4—141no—
251C2×C4⋊C4C22⋊D711yescharacteristic
252C2×C4⋊C4C22⋊D711yescharacteristic
253C2×C4⋊C4C22×D712yes—
254C2×C4⋊C4C22×D711yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42—141no—
258C2×C42C22×D712yes—
259C2×C42C22×D711yescharacteristic
260#107—282no—
261C4⋊C8—142no—
262C4⋊C8—282no—
263#102—142no—
264#102—144no—
265#102—142no—
266C4×C8—142no—
267#95—22no—
268#95—141no—
269#95—142no—
270#95—142no—
271#95—142no—
272#95C22⋊D712yes—
273C22×C7—24no—
274C22×C7—42no—
275C22×C7—42no—
276C22×C7—42no—
277C22×C7—42no—
278C22×C7—42no—
279C22×C7—42no—
280C22×C7C2×C22⋊Q811yescharacteristic
281C22×C7#54611yescharacteristic
282C22×C7#54811yescharacteristic
283C22×C7C2×#13511yescharacteristic
284C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C7#57611yescharacteristic
286C22×C7#57811yescharacteristic
287C4×C7—22no—
288C4×C7—42no—
289C4×C7—42no—
290C4×C7—42no—
291C4×C7—82no—
292C4×C7#47814yes—
293C4×C7C2×#12511yescharacteristic
294C4×C7#54411yescharacteristic
295C4×C7#55211yescharacteristic
296C4×C7#55311yescharacteristic
297Dic7—82no—
298Dic7—81no—
299Dic7—81no—
300C24—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C2×Dic1412yes—
305C2×D4#115912yes—
306C2×D4#116112yes—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—141no—
316C22×C4—141no—
317C22×C4—142no—
318C22×C4C2×C22⋊D712yes—
319C22×C4C2×C22⋊D712yes—
320C22×C4C2×C22⋊D711yescharacteristic
321C22×C4C2×C22⋊D711yescharacteristic
322C22×C4C23×D711yescharacteristic
323C2×C8—141no—
324C2×C8—141no—
325C2×C8—142no—
326C2×C8—141no—
327C2×C8—141no—
328C2×C8—142no—
329C2×C8—142no—
330C2×C8—141no—
331C2×C8—141no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C4⋊C4—141no—
341C4⋊C4—141no—
342C4⋊C4—141no—
343C4⋊C4—142no—
344C4⋊C4—141no—
345C4⋊C4—142no—
346C4⋊C4—282no—
347C4⋊C4—562no—
348C4⋊C4—561no—
349C4⋊C4C2×Dic1412yes—
350C22⋊C4—24no—
351C22⋊C4—44no—
352C42—22no—
353C42—22no—
354C42—282no—
355C42C2×Dic1412yes—
356C42#115712yes—
357C2×C7—44no—
358C2×C7#244211yescharacteristic
359C2×C7C2×#47811yescharacteristic
360C2×C7#308311yescharacteristic
361C2×C7#330211yescharacteristic
362C2×C7#332511yescharacteristic
363C2×C7#336111yescharacteristic
364C2×C7#336711yescharacteristic
365C23—22no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23C22×C22⋊D711yescharacteristic
373D4—22no—
374D4—22no—
375D4—22no—
376D4#645314yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—88no—
393C2×C4—281no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—282no—
398C2×C4—282no—
399C2×C4—281no—
400C2×C4—282no—
401C2×C4—281no—
402C2×C4#648712yes—
403C2×C4#648712yes—
404C2×C4#649512yes—
405C2×C4#651312yes—
406C2×C4C22×Dic1411yescharacteristic
407C2×C4C2×#115711yescharacteristic
408C2×C4C2×#115911yescharacteristic
409C2×C4#654811yescharacteristic
410C2×C4C2×#116111yescharacteristic
411C2×C4#655111yescharacteristic
412C8—281no—
413C8—281no—
414C8—282no—
415C7#2611111yescharacteristic, complemented, Sylow
416C22—24no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22C2×#648711yescharacteristic
424C22C2×#649511yescharacteristic
425C22#9049711yescharacteristic
426C22#9050811yescharacteristic
427C22C2×#651311yescharacteristic
428C22#9052611yescharacteristic
429C22#9052911yescharacteristic
430C4—22no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—561no—
436C4—561no—
437C4—562no—
438C4#8978114yes—
439C4C2×#645311yescharacteristic
440C4#9019111yescharacteristic
441C4#9020411yescharacteristic
442C4#9020511yescharacteristic
443C2—44no—
444C2#1173961711yescharacteristic
445C2C2×#8978111yescharacteristic
446C2#1174000411yescharacteristic
447C2#1174093211yescharacteristic
448C2#1174095511yescharacteristic
449C2#1174100811yescharacteristic
450C2#1174108811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×D4×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterD4×C4⋊C4D7
hypocenterC7#26111
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11731170C2semidirect
7. #11727377C2semidirect
8. #11727367C2semidirect
9. #11727328C2semidirect
10. #11727321C2semidirect
11. #11727238C2semidirect
415. C7#26111semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C4⋊C42
#117311702
C4×#64062
#117293602
#117293592
#117293582
#117273772
#117273672
#117273282
#117273212
#117272382
#261117