Finite Group Databasebeta

← #13411925

Subgroups — #13411925

Order 1792.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731769C211yesmaximal, characteristic
2#11731160C211yesmaximal, characteristic, complemented
3C7×#1948C211yesmaximal, characteristic, complemented
4#11728339C212yesmaximal
5#11728312C212yesmaximal, complemented
6#11728233C211yesmaximal, characteristic
7#11728222C211yesmaximal, characteristic, complemented
8#11727328C211yesmaximal, characteristic, complemented
9C4×#6383C211yesmaximal, characteristic
10#11726652C212yesmaximal
11#11726553C212yesmaximal, complemented
12C2×#6455—21no—
13C2×#6451C2212yes—
14C2×C4×D28C2211yescharacteristic
15C2×#6432—21no—
16C2×C7×#106C2212yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C7×#384C2211yescharacteristic
19C4×C7×C4⋊C4C2211yescharacteristic
20C2×#6395C2212yes—
21C2×Dic7⋊C8C2212yes—
22#89762—22no—
23#89760C2212yes—
24#89759—21no—
25#89757C2212yes—
26C4⋊C4×Dic7C2212yes—
27C2×#6383C2211yescharacteristic
28C2×#6383C2211yescharacteristic
29#89707C2211yescharacteristic
30C4×C4⋊Dic7C2211yescharacteristic
31#89700—21no—
32C2×C4×C7⋊C8C2211yescharacteristic
33#89428—22no—
34C7×#340C2212yes—
35#89346C2212yes—
36#89345C2212yes—
37#89324C2212yes—
38#89324C2211yescharacteristic
39#89324C2211yescharacteristic
40#89324C2212yes—
41#26024—71nomaximal, Sylow
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C22×Dic14—22no—
45C22×C7×C8C2312yes—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2311yescharacteristic
50C2×C7×C4⋊C4C2311yescharacteristic
51C2×C42×C7—21no—
52C2×C42×C7C2311yescharacteristic
53C2×(C2×D7)⋊C4—21no—
54C2×(C2×D7)⋊C4—22no—
55C2×(C2×D7)⋊C4—22no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—22no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7D411yescharacteristic
60C2×C4⋊Dic7D411yescharacteristic
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×Dic7⋊C4—22no—
64C2×C4×Dic7—22no—
65C2×C4×Dic7C2312yes—
66C22×C7⋊C8C2311yescharacteristic
67C22×C7⋊C8C2311yescharacteristic
68#6455—44no—
69#6451—24no—
70#6451—42no—
71C4×D28—22no—
72C4×D28Q814yescomplemented
73#6432—44no—
74C7×#106—24no—
75C7×C4⋊C8—22no—
76C7×#95—21no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95D412yes—
80#6395—42no—
81Dic7⋊C8—42no—
82#6383—22no—
83#6383—22no—
84C4×C7⋊C8—22no—
85#2551—71no—
86#2354—71no—
87#2065—72no—
88#2023—72no—
89#1987—72no—
90#1948D711yescharacteristic, complemented
91#1939—71no—
92#1920—72no—
93#1834—71no—
94#1802—71no—
95C4×#102—71no—
96C23×D7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2411yescharacteristic
103C22×Dic7—21no—
104C22×Dic7—22no—
105C22×Dic7C2×D412yes—
106C22×Dic7C2×D412yes—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—21no—
111C2×D28—21no—
112C2×D28C2×Q812yes—
113C2×C4×D7—24no—
114C2×C4×D7—42no—
115C2×C4×D7—42no—
116C2×Dic14—44no—
117C2×Dic14—44no—
118C2×Dic14—82no—
119C2×C7×C8—22no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—42no—
127C7×C4⋊C4—81no—
128C7×C4⋊C4#4112yes—
129C7×C4⋊C4#4112yes—
130C42×C7—42no—
131C42×C7C2×Q812yes—
132C42×C7#4112yes—
133(C2×D7)⋊C4—24no—
134(C2×D7)⋊C4—44no—
135(C2×D7)⋊C4—82no—
136C4⋊Dic7—22no—
137C4⋊Dic7—21no—
138C4⋊Dic7—22no—
139C4⋊Dic7—22no—
140C4⋊Dic7—44no—
141C4⋊Dic7—42no—
142C4⋊Dic7—81no—
143C4⋊Dic7SD1614yes—
144C4⋊Dic7C2×Q812yes—
145Dic7⋊C4—44no—
146C4×Dic7—22no—
147C4×Dic7—22no—
148C4×Dic7—44no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—21no—
151C2×C7⋊C8—21no—
152C2×C7⋊C8—21no—
153C2×C7⋊C8—21no—
154C2×C7⋊C8—21no—
155C2×C4⋊Q8—72no—
156C2×C4⋊Q8—141no—
157C2×C4×D4—71no—
158C2×#117—141no—
159C2×#106C2×D712yes—
160C2×C4⋊C8—72no—
161C2×C4⋊C8C2×D711yescharacteristic
162C2×#102—71no—
163C2×#102—71no—
164C2×#102—72no—
165C2×C4×C8—71no—
166#395—142no—
167#390—72no—
168#389—72no—
169#389—141no—
170#388—142no—
171#384—71no—
172#384C2×D711yescharacteristic
173#380—141no—
174C4×C4⋊C4—71no—
175C4×C4⋊C4—72no—
176C4×C4⋊C4C2×D711yescharacteristic
177#340—72no—
178#340—72no—
179#340—72no—
180#340—71no—
181#340—71no—
182#340C2×D712yes—
183#336—72no—
184C23×C7C22×D411yescharacteristic
185C22×D7—22no—
186C22×D7—41no—
187C22×D7—41no—
188C22×D7—41no—
189C22×D7—41no—
190C22×D7—41no—
191C22×D7—41no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7C22⋊Q812yes—
204C2×C4×C7C22⋊Q812yes—
205C2×C4×C7C2×SD1612yes—
206C2×C4×C7AGL(1,ℤ/8)12yes—
207C2×C4×C7C22×Q811yescharacteristic
208C2×C4×C7C2×#4111yescharacteristic
209C2×C4×C7C2×#4111yescharacteristic
210C2×C4×C7C2×#4111yescharacteristic
211C2×C4×C7H(5,𝔽2)11yescharacteristic
212C2×C4×C72−1+411yescharacteristic
213C2×Dic7—22no—
214C2×Dic7—22no—
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—22no—
218C2×Dic7—22no—
219C2×Dic7—41no—
220C2×Dic7—41no—
221C2×Dic7—42no—
222C2×Dic7—42no—
223C2×Dic7—42no—
224C2×Dic7C2×SD1614yes—
225D28—22no—
226D28—22no—
227D28—41no—
228D28—41no—
229C4×D7—48no—
230Dic14—88no—
231C7×C8—42no—
232C7⋊C8—41no—
233C7⋊C8—41no—
234C22×Q8—142no—
235C22×D4—71no—
236C23×C4—141no—
237C22×C8—71no—
238C22×C8—71no—
239C22×C8C22×D712yes—
240C4⋊Q8—144no—
241C4⋊Q8—284no—
242C4⋊Q8—282no—
243C4×D4—74no—
244C4×D4—142no—
245#117—284no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—71no—
249C2×C4⋊C4—71no—
250C2×C4⋊C4—71no—
251C2×C4⋊C4—72no—
252C2×C4⋊C4—142no—
253C2×C4⋊C4—142no—
254C2×C4⋊C4—141no—
255C2×C4⋊C4—141no—
256C2×C4⋊C4C22×D712yes—
257C2×C4⋊C4C22×D711yescharacteristic
258C2×C4⋊C4C22×D711yescharacteristic
259C2×C22⋊C4—141no—
260C2×C22⋊C4—142no—
261C2×C22⋊C4—142no—
262C2×C42—21no—
263C2×C42—72no—
264C2×C42—142no—
265C2×C42C22×D711yescharacteristic
266#106—24no—
267C4⋊C8—22no—
268C4⋊C8—282no—
269#102—142no—
270#102—142no—
271#102—282no—
272C4×C8—142no—
273#95—21no—
274#95—72no—
275#95—142no—
276#95—142no—
277C22×C7C2×C22⋊Q811yescharacteristic
278C22×C7#54611yescharacteristic
279C22×C7#54811yescharacteristic
280C22×C7C22×SD1611yescharacteristic
281C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C7#57611yescharacteristic
283C22×C7#57811yescharacteristic
284C2×D7—42no—
285C2×D7—42no—
286C2×D7—42no—
287C2×D7—81no—
288C2×D7—81no—
289C2×D7—81no—
290C2×D7—81no—
291C4×C7—22no—
292C4×C7—42no—
293C4×C7—42no—
294C4×C7—82no—
295C4×C7#47614yes—
296C4×C7#53311yescharacteristic
297C4×C7#54411yescharacteristic
298C4×C7#55511yescharacteristic
299C4×C7#55611yescharacteristic
300Dic7—24no—
301Dic7—42no—
302Dic7—42no—
303Dic7—82no—
304C24—141no—
305C2×Q8—284no—
306C2×Q8—284no—
307C2×Q8—562no—
308C2×D4—72no—
309C2×D4—141no—
310C2×D4—141no—
311C2×D4—141no—
312C2×D4—141no—
313C2×D4—141no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—72no—
318C22×C4—72no—
319C22×C4—141no—
320C22×C4—144no—
321C22×C4—142no—
322C22×C4—282no—
323C22×C4—282no—
324C22×C4D4×D711yescharacteristic
325C22×C4D4×D711yescharacteristic
326C22×C4C23×D711yescharacteristic
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—141no—
331C2×C8—141no—
332C2×C8—141no—
333C2×C8—141no—
334C2×C8—141no—
335C2×C8—141no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—74no—
342C4⋊C4—72no—
343C4⋊C4—81no—
344C4⋊C4—142no—
345C4⋊C4—141no—
346C4⋊C4—142no—
347C4⋊C4—142no—
348C4⋊C4—284no—
349C4⋊C4—284no—
350C4⋊C4—282no—
351C4⋊C4—561no—
352C4⋊C4#115912yes—
353C4⋊C4#116112yes—
354C22⋊C4—144no—
355C22⋊C4—284no—
356C22⋊C4—562no—
357C42—42no—
358C42—142no—
359C42—142no—
360C42—284no—
361C42#115712yes—
362C42Q8×D712yes—
363C2×C7#243911yescharacteristic
364C2×C7C2×#47611yescharacteristic
365C2×C7#308311yescharacteristic
366C2×C7#329811yescharacteristic
367C2×C7#332511yescharacteristic
368C2×C7#336211yescharacteristic
369C2×C7#336711yescharacteristic
370D7—82no—
371C23—142no—
372C23—281no—
373C23—281no—
374C23—281no—
375C23—281no—
376C23—281no—
377C23—281no—
378C23C2×D4×D711yescharacteristic
379Q8—568no—
380D4—142no—
381D4—142no—
382D4—281no—
383D4—281no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—74no—
396C2×C4—142no—
397C2×C4—142no—
398C2×C4—142no—
399C2×C4—142no—
400C2×C4—142no—
401C2×C4—142no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—288no—
405C2×C4—282no—
406C2×C4—282no—
407C2×C4—282no—
408C2×C4#645912yes—
409C2×C4#645912yes—
410C2×C4D7×SD1612yes—
411C2×C4#647712yes—
412C2×C4C2×#115711yescharacteristic
413C2×C4C2×#115911yescharacteristic
414C2×C4C2×Q8×D711yescharacteristic
415C2×C4C2×#116111yescharacteristic
416C2×C4#655311yescharacteristic
417C2×C4#655411yescharacteristic
418C8—42no—
419C8—281no—
420C8—281no—
421C7#2602411yescharacteristic, complemented, Sylow
422C22—282no—
423C22—282no—
424C22—282no—
425C22—561no—
426C22—561no—
427C22—561no—
428C22—561no—
429C22C2×#645911yescharacteristic
430C22#9028911yescharacteristic
431C22#9032111yescharacteristic
432C22C2×D7×SD1611yescharacteristic
433C22C2×#647711yescharacteristic
434C22#9046511yescharacteristic
435C22#9047011yescharacteristic
436C4—22no—
437C4—42no—
438C4—42no—
439C4—82no—
440C4—144no—
441C4—282no—
442C4—282no—
443C4—562no—
444C4#8964214yes—
445C4#9025411yescharacteristic
446C4#9027511yescharacteristic
447C4#9038011yescharacteristic
448C4#9041911yescharacteristic
449C2—562no—
450C2#1173366811yescharacteristic
451C2C2×#8964211yescharacteristic
452C2#1173668811yescharacteristic
453C2#1173781411yescharacteristic
454C2#1173786111yescharacteristic
455C2#1173846711yescharacteristic
456C2#1173852411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1948D7
hypocenterC7#26024
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #11731160C2semidirect
3. C7×#1948C2semidirect
5. #11728312C2semidirect
7. #11728222C2semidirect
8. #11727328C2semidirect
11. #11726553C2semidirect
72. C4×D28Q8semidirect
90. #1948D7semidirect
421. C7#26024semidirect

Maximal subgroups

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

subgroupindex
#117317692
#117311602
C7×#19482
#117283392
#117283122
#117282332
#117282222
#117273282
C4×#63832
#117266522
#117265532
#260247