Finite Group Databasebeta

← #13502621

Subgroups — #13502621

Order 1792.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C7×#2857C211yesmaximal, characteristic
3C2×#89862C212yesmaximal, complemented
4C2×#89822C212yesmaximal
5#11729446C212yesmaximal
6#11729284C211yesmaximal, characteristic
7#11727444C214yesmaximal, complemented
8#11727342C212yesmaximal, complemented
9C23×Dic14C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C22⋊Q8C2212yes—
12C22×C42×C7C2211yescharacteristic
13C22×#1144C2214yes—
14C22×C4⋊Dic7C2211yescharacteristic
15C22×Dic7⋊C4—21no—
16C22×C4×Dic7—21no—
17C2×C4×Dic14—21no—
18C2×C4×Dic14—21no—
19C2×C4×Dic14—28no—
20C2×C4×Dic14C2212yes—
21C2×C4×Dic14C2212yes—
22C7×#391C2212yes—
23C7×#389—21no—
24C7×#389—21no—
25C7×#384C2211yescharacteristic
26C7×#380—21no—
27C7×#380—21no—
28C2×#6409C2214yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30#89862—28no—
31#89822—28no—
32C2×#6384C2214yes—
33C2×C4⋊(C7⋊C8)C2212yes—
34#89324C2214yes—
35#89319C2212yes—
36#32906—71nomaximal, Sylow
37C22×C7×Q8C2312yes—
38C23×C4×C7D411yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7—21no—
42C23×Dic7—21no—
43C22×Dic14—21no—
44C22×Dic14—22no—
45C22×Dic14—24no—
46C22×Dic14—24no—
47C22×Dic14D414yes—
48C22×Dic14C2312yes—
49C7×C4⋊Q8—24no—
50C7×C22⋊Q8—24no—
51C7×C22⋊Q8—24no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C22⋊C4—22no—
59C2×C42×C7—24no—
60C2×C42×C7D411yescharacteristic
61C2×C42×C7D411yescharacteristic
62C2×C42×C7C2311yescharacteristic
63C2×C42×C7C2311yescharacteristic
64C2×#1144—28no—
65C2×#1144—28no—
66C2×#1144—28no—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—24no—
69C2×C4⋊Dic7C2312yes—
70C2×Dic7⋊C4—28no—
71C2×Dic7⋊C4—22no—
72C2×Dic7⋊C4—22no—
73C2×C4×Dic7—28no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C22×C7⋊C8C2314yes—
77C4×Dic14—28no—
78C4×Dic14—28no—
79C4×Dic14—24no—
80C4×Dic14—24no—
81C4×Dic14—24no—
82C4×Dic14D418yes—
83C7×#95—22no—
84C7×#95—22no—
85#6409—28no—
86C22⋊(C7⋊C8)—24no—
87#6384—28no—
88C4⋊(C7⋊C8)—24no—
89C22×C4×Q8—71no—
90C2×#462—72no—
91C2×#451—72no—
92#2857D711yescharacteristic
93#1878—74no—
94#1872—72no—
95#1803—72no—
96#1797—71no—
97C24×C7C2×D411yescharacteristic
98C2×C7×Q8—24no—
99C2×C7×Q8—22no—
100C2×C7×Q8—24no—
101C2×C7×Q8—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—41no—
111C22×C4×C7Q1614yescomplemented
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D412yes—
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7#4112yes—
119C22×C4×C7C2411yescharacteristic
120C22×Dic7—24no—
121C22×Dic7—24no—
122C22×Dic7—24no—
123C22×Dic7—22no—
124C22×Dic7—24no—
125C22×Dic7—22no—
126C22×Dic7—42no—
127C22×Dic7—42no—
128C2×Dic14—24no—
129C2×Dic14—24no—
130C2×Dic14—216no—
131C2×Dic14—24no—
132C2×Dic14—22no—
133C2×Dic14—22no—
134C2×Dic14—22no—
135C2×Dic14—216no—
136C2×Dic14C2×D414yes—
137C2×Dic14C2×D414yes—
138C2×Dic14#4114yes—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—42no—
142C7×C4⋊C4—42no—
143C7×C4⋊C4—42no—
144C7×C4⋊C4—44no—
145C7×C4⋊C4—42no—
146C7×C22⋊C4—44no—
147C42×C7—22no—
148C42×C7—22no—
149C42×C7—22no—
150C42×C7C2×D414yes—
151#1144—416no—
152C4⋊Dic7—24no—
153C4⋊Dic7—22no—
154C4⋊Dic7C2×D414yes—
155Dic7⋊C4—28no—
156Dic7⋊C4—28no—
157C4×Dic7—28no—
158C4×Dic7—28no—
159C2×C7⋊C8—24no—
160C2×C7⋊C8—24no—
161C2×C7⋊C8—24no—
162C23×Q8—71no—
163C22×Q16—74no—
164C2×C4⋊Q8C2×D711yescharacteristic
165C2×C22⋊Q8C2×D712yes—
166C2×C4×Q8—72no—
167C2×C4×Q8—72no—
168C2×C4×Q8—148no—
169C2×C4×Q8—141no—
170C2×C4×Q8—141no—
171C22×C4⋊C4—71no—
172C22×C4⋊C4—141no—
173C22×C42—141no—
174C22×C42C2×D711yescharacteristic
175#462—148no—
176#451—148no—
177C2×C4⋊C8—72no—
178C2×#103—74no—
179C2×#103—74no—
180C2×C22⋊C8—72no—
181#391C2×D712yes—
182#389—21no—
183#389—21no—
184#384C2×D711yescharacteristic
185#380—21no—
186#380—21no—
187#340—74no—
188#336—72no—
189C23×C7—21no—
190C23×C7—21no—
191C23×C7—21no—
192C23×C7—21no—
193C23×C7#12112yes—
194C23×C7C2×Q1612yes—
195C23×C7#13712yes—
196C23×C7C22×D411yescharacteristic
197C7×Q8—44no—
198C7×Q8—44no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7—21no—
205C2×C4×C7—24no—
206C2×C4×C7—24no—
207C2×C4×C7—28no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—44no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7#12012yes—
219C2×C4×C7#12012yes—
220C2×C4×C7#12112yes—
221C2×C4×C7C2×Q1612yes—
222C2×C4×C7C2×Q1612yes—
223C2×C4×C7#13712yes—
224C2×C4×C7C22×D411yescharacteristic
225C2×C4×C7C22×D412yes—
226C2×C4×C7C2×#4111yescharacteristic
227C2×C4×C7C2×#4112yes—
228C2×Dic7—216no—
229C2×Dic7—24no—
230C2×Dic7—24no—
231C2×Dic7—24no—
232C2×Dic7—24no—
233C2×Dic7—48no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236Dic14—216no—
237Dic14—24no—
238Dic14—24no—
239Dic14—24no—
240Dic14#12118yes—
241C7⋊C8—44no—
242C22×Q8—74no—
243C22×Q8—72no—
244C22×Q8—144no—
245C22×Q8—144no—
246C22×Q8—142no—
247C22×Q8—141no—
248C22×Q8C22×D712yes—
249C23×C4—141no—
250C23×C4—141no—
251C23×C4C22⋊D711yescharacteristic
252C23×C4C22⋊D711yescharacteristic
253C23×C4C22×D711yescharacteristic
254C2×Q16—148no—
255C2×Q16—148no—
256C2×Q16—148no—
257C22×C8—74no—
258C4⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C4×Q8—78no—
262C4×Q8—144no—
263C4×Q8—144no—
264C4×Q8—148no—
265C4×Q8—148no—
266C4×Q8—144no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—144no—
274C2×C4⋊C4—148no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4C22×D712yes—
279C2×C22⋊C4—22no—
280C2×C42—24no—
281C2×C42—148no—
282C2×C42—142no—
283C2×C42—142no—
284C2×C42C22⋊D711yescharacteristic
285C2×C42C22⋊D711yescharacteristic
286C2×C42C22×D711yescharacteristic
287C2×C42C22×D711yescharacteristic
288C4⋊C8—144no—
289#103—148no—
290#103—148no—
291C22⋊C8—144no—
292#95—22no—
293#95—22no—
294C22×C7—22no—
295C22×C7—22no—
296C22×C7—22no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7#46214yescomplemented
301C22×C7C2×#12011yescharacteristic
302C22×C7C2×#12111yescharacteristic
303C22×C7#52511yescharacteristic
304C22×C7C22×Q1611yescharacteristic
305C22×C7C2×#13711yescharacteristic
306C22×C7C2×#13711yescharacteristic
307C22×C7#57511yescharacteristic
308C4×C7—22no—
309C4×C7—22no—
310C4×C7—22no—
311C4×C7—41no—
312C4×C7—82no—
313C4×C7—82no—
314C4×C7#45114yescomplemented
315C4×C7C2×#12112yes—
316C4×C7#52512yes—
317Dic7—28no—
318Dic7—44no—
319C24C2×C22⋊D711yescharacteristic
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—74no—
325C2×Q8—74no—
326C2×Q8—74no—
327C2×Q8—1416no—
328C2×Q8—144no—
329C2×Q8—144no—
330C2×Q8—1416no—
331C2×Q8—144no—
332C2×Q8—142no—
333C2×Q8—142no—
334C2×Q8—142no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—144no—
345C22×C4—144no—
346C22×C4—144no—
347C22×C4—142no—
348C22×C4—144no—
349C22×C4—142no—
350C22×C4—282no—
351C22×C4—282no—
352C22×C4#114414yescomplemented
353C22×C4D4×D712yes—
354C22×C4#115912yes—
355C22×C4C2×C22⋊D711yescharacteristic
356C22×C4C2×C22⋊D711yescharacteristic
357C22×C4C2×C22⋊D711yescharacteristic
358C22×C4C2×C22⋊D711yescharacteristic
359C22×C4C2×C22⋊D711yescharacteristic
360C22×C4C23×D711yescharacteristic
361Q16—2816no—
362C2×C8—144no—
363C2×C8—144no—
364C2×C8—144no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—74no—
373C4⋊C4—148no—
374C4⋊C4—144no—
375C4⋊C4—148no—
376C4⋊C4—142no—
377C22⋊C4—44no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—148no—
382C42—148no—
383C42D4×D714yes—
384C2×C7—24no—
385C2×C7#284711yescharacteristic
386C2×C7C2×#45111yescharacteristic
387C2×C7#300711yescharacteristic
388C2×C7C2×#46211yescharacteristic
389C2×C7#303711yescharacteristic
390C2×C7#310411yescharacteristic
391C2×C7#311311yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#650112yes—
397C23#650512yes—
398C23C2×#114412yes—
399C23C22×C22⋊D711yescharacteristic
400Q8—44no—
401Q8—44no—
402Q8—78no—
403Q8—1416no—
404Q8—144no—
405Q8—144no—
406Q8—144no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—1416no—
427C2×C4—144no—
428C2×C4—144no—
429C2×C4—144no—
430C2×C4—144no—
431C2×C4—288no—
432C2×C4—282no—
433C2×C4—282no—
434C2×C4#650012yes—
435C2×C4#650012yes—
436C2×C4#650112yes—
437C2×C4C2×#114412yes—
438C2×C4C2×#114412yes—
439C2×C4#651412yes—
440C2×C4C2×D4×D711yescharacteristic
441C2×C4C2×D4×D712yes—
442C2×C4C2×#115911yescharacteristic
443C2×C4D7×#4112yes—
444C8—284no—
445C7#3290611yescharacteristic, complemented, Sylow
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22#8986214yescomplemented
453C22C2×#650011yescharacteristic
454C22C2×#650111yescharacteristic
455C22C2×#650511yescharacteristic
456C22C22×#114411yescharacteristic
457C22#9051311yescharacteristic
458C22C2×#651411yescharacteristic
459C22#9051911yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—82no—
465C4—82no—
466C4—148no—
467C4—284no—
468C4#8982214yescomplemented
469C4D7×#12112yes—
470C4#9029712yes—
471C2—24no—
472C2#1174016811yescharacteristic
473C2#1174018011yescharacteristic
474C2C2×#8982211yescharacteristic
475C2C2×#8986211yescharacteristic
476C2#1174075911yescharacteristic
477C2#1174081311yescharacteristic
478C2#1174081411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2857C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2857D7
hypocenterC7#32906
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#89862C2semidirect
7. #11727444C2semidirect
8. #11727342C2semidirect
111. C22×C4×C7Q16semidirect
300. C22×C7#462semidirect
314. C4×C7#451semidirect
352. C22×C4#1144semidirect
445. C7#32906semidirect
452. C22#89862semidirect
468. C4#89822semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C7×#28572
C2×#898622
C2×#898222
#117294462
#117292842
#117274442
#117273422
#329067