Finite Group Databasebeta

← #13253971

Subgroups — #13253971

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731443C211yesmaximal, characteristic
2#11727662C211yesmaximal, characteristic
3Dic7×#95C211yesmaximal, characteristic
4C7×#1294C211yesmaximal, characteristic
5#11725481C211yesmaximal, characteristic
6#11725430C211yesmaximal, characteristic
7#11725418C211yesmaximal, characteristic
8C22×C4×Dic7—21no—
9C22×C4×Dic7C2211yescharacteristic
10C2×#6453C411yescharacteristic, complemented
11C2×C7×C22⋊C8C2211yescharacteristic
12C7×#380C2211yescharacteristic
13C2×C7×#95C2211yescharacteristic
14#89984C2211yescharacteristic
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2211yescharacteristic
17C2×C22⋊(C7⋊C8)C2211yescharacteristic
18C2×Dic7⋊C8—21no—
19#89757C411yescharacteristic, complemented
20#89749C411yescharacteristic, complemented
21#89745C411yescharacteristic, complemented
22#89722—21no—
23#89718—21no—
24#89427—21no—
25#89426—21no—
26#89345—21no—
27#9910—71nomaximal, Sylow
28C23×C4×C7—21no—
29C23×C4×C7C2311yescharacteristic
30C23×Dic7—21no—
31C23×Dic7D411yescharacteristic
32C23×Dic7D411yescharacteristic
33C22×C7×C8—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C22⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7—24no—
45C2×C4×Dic7—42no—
46C2×C4×Dic7—42no—
47C2×C4×Dic7—42no—
48C2×C4×Dic7C2×C411yescharacteristic
49C2×C4×Dic7C2×C411yescharacteristic
50C2×C4×Dic7D411yescharacteristic
51C2×C4×Dic7D411yescharacteristic
52C22×C7⋊C8—21no—
53#6453—42no—
54C7×C22⋊C8—21no—
55C7×C22⋊C8—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95Q812yes—
59C7⋊#95—24no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62C7⋊#95—21no—
63C7⋊#95—21no—
64C7⋊#95—21no—
65C7⋊#95—21no—
66C7⋊#95Q812yes—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69Dic7⋊C8—42no—
70#2851—71no—
71#1797—71no—
72C4×#95—71no—
73#1294—71no—
74#1294—71no—
75#1294—71no—
76#1294D711yescharacteristic
77C24×C7C2×D411yescharacteristic
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—22no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—41no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7—41no—
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×Q811yescharacteristic
92C22×C4×C7C2×Q811yescharacteristic
93C22×C4×C7#4111yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C22×Dic7—21no—
96C22×Dic7—21no—
97C22×Dic7—22no—
98C22×Dic7—22no—
99C22×Dic7—22no—
100C22×Dic7—22no—
101C22×Dic7—22no—
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—22no—
105C22×Dic7—21no—
106C22×Dic7—21no—
107C22×Dic7—41no—
108C22×Dic7—41no—
109C22×Dic7—41no—
110C22×Dic7—41no—
111C22×Dic7—41no—
112C22×Dic7C22⋊C411yescharacteristic
113C22×Dic7C22⋊C411yescharacteristic
114C22×Dic7C22⋊C411yescharacteristic
115C22×Dic7C22⋊C411yescharacteristic
116C2×C7×C8—41no—
117C2×C7×C8—41no—
118C2×C7×C8—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C22⋊Dic7—41no—
124C22⋊Dic7—41no—
125C4⋊Dic7—42no—
126C4⋊Dic7—42no—
127C4⋊Dic7—41no—
128C4⋊Dic7—41no—
129Dic7⋊C4—42no—
130Dic7⋊C4—42no—
131C4×Dic7—22no—
132C4×Dic7—22no—
133C4×Dic7—41no—
134C4×Dic7—41no—
135C4×Dic7—48no—
136C2×C7⋊C8—41no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139C2×#125—71no—
140C22×C42—71no—
141C22×C42—141no—
142C2×C4⋊C8—141no—
143C2×C22⋊C8—71no—
144C2×C22⋊C8C2×D711yescharacteristic
145#389—71no—
146#389—71no—
147#387—141no—
148#387—141no—
149#385—141no—
150#385—141no—
151#382—71no—
152#380—71no—
153#380C2×D711yescharacteristic
154C2×#95—71no—
155C2×#95—141no—
156C2×#95C2×D711yescharacteristic
157#336—141no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7C2×C22⋊C411yescharacteristic
163C23×C7C22⋊Q811yescharacteristic
164C23×C7C22⋊Q811yescharacteristic
165C23×C7#13511yescharacteristic
166C23×C7#13511yescharacteristic
167C23×C7AGL(1,ℤ/8)11yescharacteristic
168C23×C7#13711yescharacteristic
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—24no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7C4×Q811yescharacteristic
191C2×C4×C7C4×Q811yescharacteristic
192C2×C4×C7#12411yescharacteristic
193C2×C4×C7C4⋊Q811yescharacteristic
194C2×Dic7—24no—
195C2×Dic7—24no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—24no—
203C2×Dic7—21no—
204C2×Dic7—21no—
205C2×Dic7—21no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7—42no—
210C2×Dic7—42no—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7—41no—
215C2×Dic7—41no—
216C2×Dic7—41no—
217C2×Dic7—41no—
218C2×Dic7—41no—
219C2×Dic7—41no—
220C2×Dic7#9912yes—
221C2×Dic7#10412yes—
222C2×Dic7#10412yes—
223C7×C8—81no—
224C7⋊C8—81no—
225C23×C4—21no—
226C23×C4—71no—
227C23×C4—71no—
228C23×C4—141no—
229C23×C4C22×D711yescharacteristic
230C22×C8—21no—
231C22×C8—141no—
232#125—282no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—141no—
236C2×C4⋊C4—141no—
237C2×C4⋊C4—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4C4×D711yescharacteristic
240C2×C4⋊C4C4×D711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—141no—
243C2×C42—71no—
244C2×C42—71no—
245C2×C42—71no—
246C2×C42—71no—
247C2×C42—144no—
248C2×C42—282no—
249C2×C42—282no—
250C2×C42—282no—
251C4⋊C8—282no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—141no—
255C22⋊C8—141no—
256#95—21no—
257#95—21no—
258#95—72no—
259#95—141no—
260#95—141no—
261#95—141no—
262#95—141no—
263#95—144no—
264#95—141no—
265#95—141no—
266#95Dic1412yes—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7#39111yescharacteristic
280C22×C7C2×#9911yescharacteristic
281C22×C7#41011yescharacteristic
282C22×C7#41611yescharacteristic
283C22×C7#41711yescharacteristic
284C22×C7C2×#10411yescharacteristic
285C22×C7C2×#10411yescharacteristic
286C22×C7#47811yescharacteristic
287C22×C7#47811yescharacteristic
288C22×C7#47911yescharacteristic
289C22×C7#47911yescharacteristic
290C4×C7—21no—
291C4×C7—21no—
292C4×C7—21no—
293C4×C7—21no—
294C4×C7—44no—
295C4×C7—81no—
296C4×C7—81no—
297Dic7—22no—
298Dic7—22no—
299Dic7—22no—
300Dic7—44no—
301Dic7—41no—
302Dic7—81no—
303Dic7—81no—
304C24D4×D711yescharacteristic
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—71no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—141no—
324C22×C4—141no—
325C22×C4—142no—
326C22×C4—142no—
327C22×C4—142no—
328C22×C4—141no—
329C22×C4—141no—
330C22×C4—141no—
331C22×C4—141no—
332C22×C4—281no—
333C22×C4—281no—
334C22×C4—281no—
335C22×C4—281no—
336C22×C4—281no—
337C22×C4C2×Dic1411yescharacteristic
338C22×C4C2×C4×D711yescharacteristic
339C22×C4#115711yescharacteristic
340C22×C4D4×D711yescharacteristic
341C22×C4#115911yescharacteristic
342C22×C4Q8×D711yescharacteristic
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—281no—
347C2×C8—281no—
348C2×C8—281no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—281no—
352C4⋊C4—281no—
353C4⋊C4—282no—
354C4⋊C4—282no—
355C4⋊C4—282no—
356C4⋊C4—282no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—281no—
360C22⋊C4—281no—
361C42—142no—
362C42—142no—
363C42—281no—
364C42—281no—
365C42—288no—
366C2×C7—21no—
367C2×C7—21no—
368C2×C7—21no—
369C2×C7—21no—
370C2×C7#150011yescharacteristic
371C2×C7#150111yescharacteristic
372C2×C7#150411yescharacteristic
373C2×C7#150911yescharacteristic
374C2×C7#198911yescharacteristic
375C2×C7#199411yescharacteristic
376C2×C7#199611yescharacteristic
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23#644111yescharacteristic
382C23D7×C22⋊C411yescharacteristic
383C23#645911yescharacteristic
384C23#647411yescharacteristic
385C23#647811yescharacteristic
386C23#647911yescharacteristic
387C23#648211yescharacteristic
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—72no—
410C2×C4—72no—
411C2×C4—72no—
412C2×C4—144no—
413C2×C4—144no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—144no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—282no—
427C2×C4—282no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—281no—
433C2×C4—281no—
434C2×C4—281no—
435C2×C4—281no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4C4×Dic1411yescharacteristic
439C2×C4#644811yescharacteristic
440C2×C4#645011yescharacteristic
441C2×C4#645111yescharacteristic
442C8—81no—
443C8—561no—
444C7#991011yescharacteristic, complemented, Sylow
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22#8941811yescharacteristic
458C22#8951211yescharacteristic
459C22D7×#9911yescharacteristic
460C22#8953511yescharacteristic
461C22#8954311yescharacteristic
462C22#8956811yescharacteristic
463C22#8957611yescharacteristic
464C22D7×#10411yescharacteristic
465C22D7×#10411yescharacteristic
466C22#8962911yescharacteristic
467C22#8966311yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—44no—
473C4—81no—
474C4—81no—
475C4—142no—
476C4—142no—
477C4—142no—
478C4—284no—
479C4—281no—
480C4—561no—
481C4—561no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2#1172603311yescharacteristic
487C2#1172605611yescharacteristic
488C2#1172606211yescharacteristic
489C2#1172608711yescharacteristic
490C2#1172648511yescharacteristic
491C2#1172655511yescharacteristic
492C2#1172670911yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1294C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1294D7
hypocenterC7#9910
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
10. C2×#6453C4semidirect
19. #89757C4semidirect
20. #89749C4semidirect
21. #89745C4semidirect
444. C7#9910semidirect

Maximal subgroups

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

subgroupindex
#117314432
#117276622
Dic7×#952
C7×#12942
#117254812
#117254302
#117254182
#99107