Finite Group Databasebeta

← #13370365

Subgroups — #13370365

Order 1792.

1014 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90230C211yesmaximal, characteristic
2M16×Dic14C212yesmaximal, complemented
3C4×C7×#100C211yesmaximal, characteristic
4#11729371C211yesmaximal, characteristic
5#11727817C211yesmaximal, characteristic
6#11727817C211yesmaximal, characteristic
7Dic7×#100C211yesmaximal, characteristic
8Dic7×#100C211yesmaximal, characteristic
9#11727634C212yesmaximal, complemented
10#11726729C212yesmaximal, complemented
11#11726728C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13C2×D4×Dic7C2211yescharacteristic
14C2×D4×Dic7C2211yescharacteristic
15C2×#6487C412yes—
16C22×Dic7⋊C4C412yes—
17#90230—22no—
18#90230—22no—
19#90230—22no—
20#90230—22no—
21C2×#6453C2211yescharacteristic
22C2×#6451C2211yescharacteristic
23C2×#6441C412yes—
24C2×#6441C412yes—
25C2×#6433C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C2×C7×#100C2211yescharacteristic
28C2×C7×#100C2211yescharacteristic
29C4×C7×M16C2212yes—
30C2×#6407C2211yescharacteristic
31C2×#6407C2211yescharacteristic
32#89890C2212yes—
33#89890C2212yes—
34M16×Dic7C2212yes—
35M16×Dic7C2212yes—
36#89696C2212yes—
37#89473—22no—
38#89473—22no—
39#89451—22no—
40C8×Dic14—22no—
41C7×#343C2212yes—
42#89349C2212yes—
43#89349C2212yes—
44#89349C2212yes—
45#89349C2212yes—
46#89325C2212yes—
47#22515—71nomaximal, Sylow
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7C2×C412yes—
50C23×Dic7C2×C412yes—
51C23×Dic7C2×C412yes—
52C22×Dic14C2×C412yes—
53C22×Dic14C2×C411yescharacteristic
54C2×C7×M16C2312yes—
55C2×C7×M16C2312yes—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C22⋊C4C2×C412yes—
60C2×C42×C7C2311yescharacteristic
61C2×C22⋊Dic7C2×C412yes—
62C2×C22⋊Dic7C2×C412yes—
63D4×Dic7—22no—
64D4×Dic7—22no—
65D4×Dic7—22no—
66D4×Dic7—22no—
67C2×C4⋊Dic7C2×C412yes—
68C2×C4⋊Dic7C2×C411yescharacteristic
69C2×C4⋊Dic7C2311yescharacteristic
70C2×C4⋊Dic7C2311yescharacteristic
71#6487—24no—
72#6487—24no—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4C2×C412yes—
78C2×Dic7⋊C4C2×C412yes—
79C2×Dic7⋊C4C2×C412yes—
80C2×Dic7⋊C4C2×C412yes—
81C2×Dic7⋊C4C2×C411yescharacteristic
82C2×Dic7⋊C4C2×C411yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C2×C4×Dic7C2311yescharacteristic
85C2×#1136C2312yes—
86C2×#1136C2312yes—
87#6453—22no—
88#6453—22no—
89#6451—22no—
90#6451—22no—
91#6441—24no—
92#6441—24no—
93#6441—24no—
94#6441—24no—
95#6433—22no—
96#6433—22no—
97C4×Dic14D412yes—
98C4×Dic14D412yes—
99C4×Dic14D412yes—
100C4×Dic14D412yes—
101C7×#100—22no—
102C7×#100Q814yescomplemented
103C7×#97—22no—
104C4×C7×C8—22no—
105#6407—22no—
106#6407—22no—
107#6389—22no—
108#6389—22no—
109Dic7⋊C8—22no—
110Dic7⋊C8—22no—
111Dic7⋊C8—22no—
112Dic7⋊C8—22no—
113C8×Dic7—22no—
114C8×Dic7—22no—
115C4⋊(C7⋊C8)—22no—
116C4⋊(C7⋊C8)—22no—
117C2×#554—71no—
118Q8×M16—72no—
119#1998—72no—
120#1998—72no—
121#1997—72no—
122#1800—71no—
123#1800—71no—
124#1800—71no—
125C4×#100—71no—
126C4×#100—71no—
127C4×#100D711yescharacteristic
128C24×C7C22×C412yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4C2×Q812yes—
132C2×C7×D4#4112yes—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—82no—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C411yescharacteristic
138C22×C4×C7C22×C411yescharacteristic
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—44no—
145C22×Dic7—82no—
146C22×Dic7C22×C411yescharacteristic
147C22×Dic7C22×C412yes—
148C22×Dic7C22×C412yes—
149C22×Dic7C22×C411yescharacteristic
150C22×Dic7C22×C411yescharacteristic
151C22×Dic7C22×C411yescharacteristic
152C2×Dic14—24no—
153C2×Dic14—42no—
154C2×Dic14—82no—
155C2×Dic14C22⋊C412yes—
156C2×Dic14C22⋊C412yes—
157C7×M16—22no—
158C7×M16—22no—
159C7×M16—22no—
160C7×M16C2×Q814yes—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C7×C4⋊C4—22no—
164C7×C22⋊C4—24no—
165C42×C7C2×D412yes—
166C42×C7C2×D412yes—
167C22⋊Dic7—24no—
168C22⋊Dic7—24no—
169C4⋊Dic7—22no—
170C4⋊Dic7—22no—
171C4⋊Dic7—24no—
172C4⋊Dic7C22⋊C412yes—
173C4⋊Dic7C22⋊C412yes—
174Dic7⋊C4—24no—
175Dic7⋊C4—22no—
176Dic7⋊C4—22no—
177Dic7⋊C4—22no—
178Dic7⋊C4—22no—
179Dic7⋊C4—22no—
180Dic7⋊C4—22no—
181Dic7⋊C4C22⋊C411yescharacteristic
182Dic7⋊C4C22⋊C411yescharacteristic
183Dic7⋊C4C22⋊C411yescharacteristic
184Dic7⋊C4C22⋊C411yescharacteristic
185Dic7⋊C4C22⋊C412yes—
186Dic7⋊C4C22⋊C412yes—
187C4×Dic7C2×D412yes—
188C4×Dic7C2×D412yes—
189C4×Dic7C2×D412yes—
190C4×Dic7C2×D412yes—
191#1136—22no—
192#1136—22no—
193#1136—22no—
194#1136—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197#554—142no—
198#554—142no—
199#554—142no—
200#554—142no—
201C2×C4⋊Q8—71no—
202C2×#125—71no—
203C2×#125—71no—
204C2×C22⋊Q8—72no—
205C2×C22⋊Q8—72no—
206C2×C22⋊Q8—72no—
207C2×C4×Q8—71no—
208C2×C4×D4—71no—
209C2×C4×D4—71no—
210C2×C4×D4C2×D711yescharacteristic
211C22×C4⋊C4—72no—
212#446—142no—
213#446—142no—
214#446—142no—
215C8×Q8—142no—
216#423—72no—
217#423—72no—
218#423—72no—
219C2×#100—71no—
220C2×#100—71no—
221C2×#100C2×D711yescharacteristic
222C2×#100C2×D711yescharacteristic
223C4×M16—72no—
224C4×M16—72no—
225C4×M16C2×D712yes—
226#343—72no—
227#343—72no—
228#343—72no—
229#343—72no—
230#343—72no—
231#343C2×D712yes—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7C4×Q814yes—
237C23×C7C23×C411yescharacteristic
238C7×D4—24no—
239C7×D4—24no—
240C2×C4×C7—22no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—42no—
244C2×C4×C7—82no—
245C2×C4×C7—82no—
246C2×C4×C7C2×C22⋊C411yescharacteristic
247C2×C4×C7C2×C22⋊C411yescharacteristic
248C2×C4×C7C2×C22⋊C412yes—
249C2×C4×C7C22×D412yes—
250C2×C4×C7C22×Q811yescharacteristic
251C2×C4×C7C2×#4111yescharacteristic
252C2×C4×C72−1+412yes—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—22no—
256C2×Dic7—41no—
257C2×Dic7—41no—
258C2×Dic7—41no—
259C2×Dic7—44no—
260C2×Dic7—44no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—82no—
264C2×Dic7—82no—
265C2×Dic7C2×C22⋊C411yescharacteristic
266C2×Dic7C2×C22⋊C411yescharacteristic
267C2×Dic7C2×C22⋊C411yescharacteristic
268C2×Dic7C2×C22⋊C412yes—
269C2×Dic7C2×C22⋊C412yes—
270C2×Dic7C2×C22⋊C411yescharacteristic
271Dic14—24no—
272Dic14—42no—
273Dic14—84no—
274C7×C8—24no—
275C7×C8—42no—
276C7⋊C8—42no—
277C7⋊C8—42no—
278C22×Q8—71no—
279C22×Q8—72no—
280C22×D4C22×D711yescharacteristic
281C23×C4—72no—
282C23×C4—72no—
283C23×C4C4×D712yes—
284C2×M16—72no—
285C2×M16—72no—
286C2×M16C22×D712yes—
287C2×M16C22×D712yes—
288C4⋊Q8—142no—
289C4⋊Q8—142no—
290#125—142no—
291#125—142no—
292#125—142no—
293#125—142no—
294C22⋊Q8—144no—
295C22⋊Q8—144no—
296C22⋊Q8—144no—
297C22⋊Q8—144no—
298C22⋊Q8—144no—
299C22⋊Q8—144no—
300C4×Q8—72no—
301C4×Q8—72no—
302C4×Q8—72no—
303C4×Q8—72no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—142no—
307C4×D4—142no—
308C4×D4—142no—
309C4×D4—142no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—71no—
312C2×C4⋊C4—71no—
313C2×C4⋊C4—72no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—72no—
317C2×C4⋊C4—72no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C22⋊C4—72no—
326C2×C22⋊C4—72no—
327C2×C22⋊C4C4×D712yes—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42C22×D711yescharacteristic
331C4⋊C8—142no—
332C4⋊C8—142no—
333C4⋊C8—142no—
334C4⋊C8—142no—
335C4⋊C8—142no—
336C4⋊C8—142no—
337#100—22no—
338#100—142no—
339#100—142no—
340#100Dic1414yescomplemented
341#97—22no—
342#97—142no—
343#97—142no—
344C4×C8—22no—
345C4×C8—142no—
346C4×C8—142no—
347C22×C7—21no—
348C22×C7—21no—
349C22×C7—24no—
350C22×C7—44no—
351C22×C7—44no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7C22×C22⋊C411yescharacteristic
355C22×C7C2×C4×Q811yescharacteristic
356C22×C7#51911yescharacteristic
357C4×C7—22no—
358C4×C7—21no—
359C4×C7—21no—
360C4×C7—42no—
361C4×C7#54812yes—
362C4×C7D4×Q812yes—
363Dic7—22no—
364Dic7—21no—
365Dic7—22no—
366Dic7—21no—
367Dic7—21no—
368Dic7—42no—
369Dic7—42no—
370Dic7#41312yes—
371C24C2×C4×D712yes—
372C2×Q8—72no—
373C2×Q8—72no—
374C2×Q8—144no—
375C2×Q8—282no—
376C2×Q8—562no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4C2×Dic1412yes—
380C2×D4#115712yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—71no—
384C22×C4—71no—
385C22×C4—71no—
386C22×C4—71no—
387C22×C4—72no—
388C22×C4—72no—
389C22×C4—82no—
390C22×C4—142no—
391C22×C4—142no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—284no—
395C22×C4—562no—
396C22×C4C2×C4×D712yes—
397C22×C4C2×C4×D711yescharacteristic
398C22×C4C2×C4×D711yescharacteristic
399C22×C4C23×D711yescharacteristic
400M16—22no—
401M16—22no—
402M16—22no—
403M16—142no—
404M16—142no—
405M16—142no—
406M16—142no—
407M16C2×Dic1414yes—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—142no—
411C2×C8—142no—
412C4⋊C4—22no—
413C4⋊C4—71no—
414C4⋊C4—71no—
415C4⋊C4—71no—
416C4⋊C4—71no—
417C4⋊C4—72no—
418C4⋊C4—72no—
419C4⋊C4—72no—
420C4⋊C4—72no—
421C4⋊C4—142no—
422C4⋊C4—142no—
423C4⋊C4—142no—
424C4⋊C4—142no—
425C4⋊C4—144no—
426C4⋊C4—142no—
427C4⋊C4—142no—
428C4⋊C4—144no—
429C4⋊C4—142no—
430C4⋊C4—142no—
431C22⋊C4—24no—
432C22⋊C4—144no—
433C22⋊C4—144no—
434C42—72no—
435C42—72no—
436C42—72no—
437C42—72no—
438C42D4×D712yes—
439C42D4×D712yes—
440C2×C7—21no—
441C2×C7—21no—
442C2×C7—44no—
443C2×C7Q8×C22⋊C411yescharacteristic
444C2×C7C2×#41311yescharacteristic
445C2×C7#288811yescharacteristic
446C23—22no—
447C23—22no—
448C23—42no—
449C23—42no—
450C23C4×Dic1414yes—
451C23C22×C4×D711yescharacteristic
452Q8—144no—
453Q8—282no—
454Q8—564no—
455D4—24no—
456D4—24no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—71no—
462C2×C4—72no—
463C2×C4—72no—
464C2×C4—71no—
465C2×C4—71no—
466C2×C4—71no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—142no—
470C2×C4—142no—
471C2×C4—142no—
472C2×C4—284no—
473C2×C4—281no—
474C2×C4—281no—
475C2×C4—281no—
476C2×C4—284no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—562no—
480C2×C4—562no—
481C2×C4D7×C22⋊C412yes—
482C2×C4D7×C22⋊C411yescharacteristic
483C2×C4D7×C22⋊C411yescharacteristic
484C2×C4C22×Dic1411yescharacteristic
485C2×C4C2×#115711yescharacteristic
486C2×C4C2×D4×D712yes—
487C2×C4#655412yes—
488C8—24no—
489C8—42no—
490C8—282no—
491C8—282no—
492C7#2251511yescharacteristic, complemented, Sylow
493C22—21no—
494C22—21no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—42no—
499C22—42no—
500C22C2×C4×Dic1411yescharacteristic
501C22C2×D7×C22⋊C411yescharacteristic
502C22#9020711yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—42no—
507C4—72no—
508C4—141no—
509C4—141no—
510C4—142no—
511C4—141no—
512C4—142no—
513C4—282no—
514C4—282no—
515C4D4×Dic1412yes—
516C4#9024312yes—
517C2—21no—
518C2—21no—
519C2—44no—
520C2C22⋊C4×Dic1411yescharacteristic
521C2#1173614211yescharacteristic
522C2D7×#41311yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#100C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC4×#100D7
hypocenterC7#22515
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. M16×Dic14C2semidirect
9. #11727634C2semidirect
10. #11726729C2semidirect
11. #11726728C2semidirect
102. C7×#100Q8semidirect
340. #100Dic14semidirect
492. C7#22515semidirect

Maximal subgroups

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

subgroupindex
C2×#902302
M16×Dic142
C4×C7×#1002
#117293712
#117278172
Dic7×#1002
#117276342
#117267292
#117267282
#225157