Finite Group Databasebeta

← #13358673

Subgroups — #13358673

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic7C211yesmaximal, characteristic
2C2×#89491C211yesmaximal, characteristic
3#11731308C211yesmaximal, characteristic
4C7×#1853C211yesmaximal, characteristic
5#11729309C211yesmaximal, characteristic
6#11728318C212yesmaximal, complemented
7#11728307C212yesmaximal, complemented
8#11727681C211yesmaximal, characteristic
9#11727259C211yesmaximal, characteristic
10#11726552C212yesmaximal, complemented
11#11726540C212yesmaximal, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×C4×Dic7—21no—
15C22×C4×Dic7C2211yescharacteristic
16C2×C7×#106C2212yes—
17C2×C7×C22⋊C8C2211yescharacteristic
18C7×#380C2211yescharacteristic
19#89984C2211yescharacteristic
20C2×C7⋊#95—21no—
21C2×C22⋊(C7⋊C8)C2211yescharacteristic
22C2×#6392C2212yes—
23C2×#6389C2212yes—
24C2×Dic7⋊C8C2212yes—
25#89749C2211yescharacteristic
26#89749C2211yescharacteristic
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7—21no—
29C4⋊C4×Dic7C2211yescharacteristic
30C4⋊C4×Dic7C2211yescharacteristic
31C4⋊C4×Dic7C2212yes—
32C4⋊C4×Dic7C2212yes—
33C2×#6382C2212yes—
34C2×#6381C2212yes—
35#89715—21no—
36C22⋊C4×Dic7—21no—
37#89491C418yescomplemented
38#89423—21no—
39#89420—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21363—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C22×C7×C8C2312yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C22⋊C4—21no—
57C2×C22⋊Dic7—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7—24no—
60C2×C4⋊Dic7C2311yescharacteristic
61C2×C4⋊Dic7C2311yescharacteristic
62C2×C4⋊Dic7C2312yes—
63C2×C4×Dic7—21no—
64C2×C4×Dic7—21no—
65C2×C4×Dic7—22no—
66C2×C4×Dic7—22no—
67C2×C4×Dic7—44no—
68C2×C4×Dic7C2×C414yes—
69C2×C4×Dic7D411yescharacteristic
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7Q814yes—
72C2×C4×Dic7C2311yescharacteristic
73C2×C4×Dic7C2311yescharacteristic
74C22×C7⋊C8C2312yes—
75C7×#106—24no—
76C7×C22⋊C8C2×C414yes—
77C7×#95—21no—
78C7⋊#95—21no—
79C7⋊#95—21no—
80C7⋊#95—21no—
81C7⋊#95—21no—
82C7⋊#95—21no—
83C7⋊#95—21no—
84C7⋊#95—21no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C22⋊(C7⋊C8)C2×C414yes—
88#6392—24no—
89#6389C2×C418yescomplemented
90Dic7⋊C8C2×C418yescomplemented
91#6382—24no—
92#6381—24no—
93C2×#432—71no—
94#2283—71no—
95C2×C4×C4⋊C4—71no—
96#1949—72no—
97#1949—72no—
98#1853—71no—
99#1853D711yescharacteristic
100#1852—71no—
101#1852—71no—
102#1776—72no—
103#1776—72no—
104C24×C7C2×D411yescharacteristic
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—24no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—42no—
112C22×C4×C7—42no—
113C22×C4×C7C22×C412yes—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×Q812yes—
116C22×C4×C7C2411yescharacteristic
117C22×Dic7—22no—
118C22×Dic7—22no—
119C22×Dic7—24no—
120C22×Dic7—22no—
121C22×Dic7—21no—
122C22×Dic7—21no—
123C22×Dic7—41no—
124C22×Dic7—41no—
125C22×Dic7—42no—
126C22×Dic7—42no—
127C22×Dic7C4⋊C414yes—
128C22×Dic7C4⋊C414yes—
129C22×Dic7C2×D411yescharacteristic
130C22×Dic7C2×D411yescharacteristic
131C22×Dic7C2×D411yescharacteristic
132C22×Dic7C2×D411yescharacteristic
133C2×C7×C8—22no—
134C2×C7×C8—22no—
135C2×C7×C8C22×C414yes—
136C7×C4⋊C4—24no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4#4114yes—
141C7×C22⋊C4—42no—
142C22⋊Dic7—42no—
143C4⋊Dic7—24no—
144C4⋊Dic7—22no—
145C4⋊Dic7—22no—
146C4⋊Dic7—22no—
147C4⋊Dic7#4114yes—
148C4×Dic7—42no—
149C4×Dic7—42no—
150C4×Dic7—44no—
151C4×Dic7—44no—
152C4×Dic7C4⋊C414yes—
153C4×Dic7C4⋊C414yes—
154C4×Dic7C22×C412yes—
155C4×Dic7C22×C412yes—
156C4×Dic7C2×Q812yes—
157C4×Dic7C2×Q812yes—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8C22×C414yes—
161C22×C4⋊C4—71no—
162C22×C4⋊C4C2×D711yescharacteristic
163C22×C42—71no—
164C22×C42—141no—
165#432—78no—
166C2×#107—72no—
167C2×#107—72no—
168C2×#106—72no—
169C2×#106C2×D712yes—
170C2×C4⋊C8—72no—
171C2×C22⋊C8—71no—
172C2×C22⋊C8C2×D711yescharacteristic
173C2×#97—72no—
174#383—141no—
175#382—71no—
176#382—71no—
177#381—141no—
178#380—71no—
179#380C2×D711yescharacteristic
180C4×C4⋊C4—72no—
181C4×C4⋊C4—72no—
182C4×C4⋊C4—71no—
183C4×C4⋊C4—71no—
184C4×C4⋊C4—141no—
185C4×C4⋊C4—141no—
186C4×C22⋊C4—141no—
187#376—141no—
188C2×#95—141no—
189#340—72no—
190#340—72no—
191#340—72no—
192#340C2×D712yes—
193C23×C7—21no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7—21no—
197C23×C7C2×C4⋊C412yes—
198C23×C7AGL(1,ℤ/8)12yes—
199C23×C7#13712yes—
200C23×C7C22×D411yescharacteristic
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—24no—
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—48no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7C2×C4⋊C412yes—
216C2×C4×C7C23×C411yescharacteristic
217C2×C4×C7C22×Q811yescharacteristic
218C2×C4×C7C2×#4112yes—
219C2×C4×C7C2×#4112yes—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—21no—
225C2×Dic7—21no—
226C2×Dic7—21no—
227C2×Dic7—21no—
228C2×Dic7—28no—
229C2×Dic7—24no—
230C2×Dic7—44no—
231C2×Dic7—48no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7C2×C4⋊C412yes—
241C2×Dic7C2×C4⋊C412yes—
242C2×Dic7C2×C4⋊C412yes—
243C2×Dic7C2×C4⋊C412yes—
244C7×C8—24no—
245C7⋊C8—24no—
246C23×C4—21no—
247C23×C4—71no—
248C23×C4—71no—
249C23×C4—141no—
250C23×C4C22×D711yescharacteristic
251C22×C8—72no—
252C22×C8C22×D712yes—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—72no—
256C2×C4⋊C4—71no—
257C2×C4⋊C4—71no—
258C2×C4⋊C4—144no—
259C2×C4⋊C4—141no—
260C2×C4⋊C4C22×D711yescharacteristic
261C2×C4⋊C4C22×D711yescharacteristic
262C2×C4⋊C4C22×D712yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—141no—
265C2×C42—74no—
266C2×C42—71no—
267C2×C42—71no—
268C2×C42—74no—
269C2×C42—71no—
270C2×C42—71no—
271C2×C42—141no—
272C2×C42—141no—
273C2×C42—142no—
274C2×C42—142no—
275C2×C42—284no—
276#107—144no—
277#107—144no—
278#106—24no—
279#106—144no—
280C4⋊C8—78no—
281C22⋊C8—74no—
282C22⋊C8C4×D714yes—
283#97—78no—
284#95—21no—
285#95—142no—
286#95—142no—
287#95—141no—
288#95—141no—
289#95—141no—
290#95—141no—
291#95—141no—
292#95—141no—
293#95—141no—
294C22×C7—22no—
295C22×C7—22no—
296C22×C7—22no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7#42814yes—
301C22×C7C22×C4⋊C411yescharacteristic
302C22×C7#52511yescharacteristic
303C22×C7#52511yescharacteristic
304C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C7C2×#13711yescharacteristic
306C22×C7#57511yescharacteristic
307C22×C7#57511yescharacteristic
308C4×C7—22no—
309C4×C7—44no—
310C4×C7—81no—
311C4×C7—81no—
312C4×C7C4×#4112yes—
313C4×C7#53312yes—
314Dic7—22no—
315Dic7—22no—
316Dic7—22no—
317Dic7—22no—
318Dic7—44no—
319Dic7—81no—
320Dic7—81no—
321C24D4×D711yescharacteristic
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—74no—
331C22×C4—71no—
332C22×C4—71no—
333C22×C4—71no—
334C22×C4—71no—
335C22×C4—74no—
336C22×C4—142no—
337C22×C4—142no—
338C22×C4—144no—
339C22×C4—142no—
340C22×C4—141no—
341C22×C4—141no—
342C22×C4—281no—
343C22×C4—281no—
344C22×C4—282no—
345C22×C4—282no—
346C22×C4C2×C4×D712yes—
347C22×C4D4×D711yescharacteristic
348C22×C4Q8×D712yes—
349C22×C4C23×D711yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—74no—
353C2×C8—142no—
354C2×C8—142no—
355C2×C8C2×C4×D714yes—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—74no—
361C4⋊C4—144no—
362C4⋊C4—142no—
363C4⋊C4—142no—
364C4⋊C4—142no—
365C4⋊C4#115914yes—
366C22⋊C4—42no—
367C22⋊C4—282no—
368C42—74no—
369C42—74no—
370C42—72no—
371C42—72no—
372C42—72no—
373C42—72no—
374C42—282no—
375C42—282no—
376C42—284no—
377C42—284no—
378C2×C7—24no—
379C2×C7#229611yescharacteristic
380C2×C7C2×#42811yescharacteristic
381C2×C7#291111yescharacteristic
382C2×C7#311411yescharacteristic
383C2×C7#311411yescharacteristic
384C2×C7#311711yescharacteristic
385C2×C7#311711yescharacteristic
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23D7×C4⋊C412yes—
391C23#647712yes—
392C23#647812yes—
393C23C2×D4×D711yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—72no—
409C2×C4—72no—
410C2×C4—72no—
411C2×C4—72no—
412C2×C4—148no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—144no—
422C2×C4—284no—
423C2×C4—288no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—281no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4—281no—
432C2×C4D7×C4⋊C412yes—
433C2×C4C22×C4×D711yescharacteristic
434C2×C4C2×#115912yes—
435C2×C4C2×Q8×D711yescharacteristic
436C2×C4D7×#4112yes—
437C8—24no—
438C8—144no—
439C7#2136311yescharacteristic, complemented, Sylow
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22#8963314yes—
447C22C2×D7×C4⋊C411yescharacteristic
448C22#9027711yescharacteristic
449C22#9034711yescharacteristic
450C22C2×#647711yescharacteristic
451C22C2×#647811yescharacteristic
452C22#9046411yescharacteristic
453C22#9046811yescharacteristic
454C4—22no—
455C4—44no—
456C4—81no—
457C4—81no—
458C4—142no—
459C4—142no—
460C4—142no—
461C4—142no—
462C4—284no—
463C4—561no—
464C4—561no—
465C4C4×#115912yes—
466C4#9031312yes—
467C2—24no—
468C2#1173182311yescharacteristic
469C2C2×#8963311yescharacteristic
470C2#1173665311yescharacteristic
471C2#1173747611yescharacteristic
472C2#1173758011yescharacteristic
473C2#1173875611yescharacteristic
474C2#1173886011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1853D7
hypocenterC7#21363
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
6. #11728318C2semidirect
7. #11728307C2semidirect
10. #11726552C2semidirect
11. #11726540C2semidirect
37. #89491C4semidirect
89. #6389C2×C4semidirect
90. Dic7⋊C8C2×C4semidirect
439. C7#21363semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic72
C2×#894912
#117313082
C7×#18532
#117293092
#117283182
#117283072
#117276812
#117272592
#117265522
#117265402
#213637