Finite Group Databasebeta

← #13319685

Subgroups — #13319685

Order 1792.

1214 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2#11736064C212yesmaximal
3C7×#1463C211yesmaximal, characteristic
4#11729306C211yesmaximal, characteristic
5#11727725C212yesmaximal
6#11725881C214yesmaximal, complemented
7#11725879C214yesmaximal, complemented
8C23×Dic14C411yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C22×Dic7⋊C4C411yescharacteristic
11C22×Dic7⋊C4C2211yescharacteristic
12C22×C4×Dic7C411yescharacteristic
13C22×C4×Dic7C411yescharacteristic
14C22×C4×Dic7C2211yescharacteristic
15C2×#6450—22no—
16C2×#6450—22no—
17C2×#6450—22no—
18C2×#6450—22no—
19C2×#6450C412yes—
20C2×#6450C412yes—
21C2×#6450C2212yes—
22C2×#6450C2212yes—
23C7×#407C2212yes—
24#89979C2212yes—
25#89890—22no—
26M16×Dic7—22no—
27#89491—22no—
28#89491C2214yes—
29#89490—22no—
30#89490C2214yes—
31C7×#324C2214yes—
32#89323C2214yes—
33#89264C2218yes—
34#17365—71nomaximal, Sylow
35C23×C4×C7C2×C411yescharacteristic
36C23×C4×C7C2×C411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×Dic7C2×C411yescharacteristic
39C23×Dic7C2×C411yescharacteristic
40C23×Dic7C2×C411yescharacteristic
41C23×Dic7C2×C411yescharacteristic
42C22×Dic14—22no—
43C22×Dic14—22no—
44C22×Dic14—28no—
45C22×Dic14C2×C412yes—
46C22×Dic14C2×C412yes—
47C2×C7×M16—22no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4C2×C412yes—
51C2×C7×C4⋊C4C2312yes—
52C2×Dic7⋊C4—22no—
53C2×Dic7⋊C4—22no—
54C2×Dic7⋊C4—22no—
55C2×Dic7⋊C4—22no—
56C2×Dic7⋊C4C2×C412yes—
57C2×Dic7⋊C4C2×C412yes—
58C2×Dic7⋊C4C2×C412yes—
59C2×Dic7⋊C4C2312yes—
60C2×C4×Dic7—21no—
61C2×C4×Dic7—21no—
62C2×C4×Dic7—21no—
63C2×C4×Dic7—21no—
64C2×C4×Dic7—21no—
65C2×C4×Dic7—21no—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7C2×C412yes—
69C2×C4×Dic7C2×C412yes—
70C2×C4×Dic7C2×C412yes—
71C2×C4×Dic7C2×C412yes—
72C2×C4×Dic7C2×C411yescharacteristic
73C2×C4×Dic7C2×C411yescharacteristic
74C2×C4×Dic7D412yes—
75C2×C4×Dic7D412yes—
76C2×C4×Dic7D412yes—
77C2×C4×Dic7D412yes—
78C2×C4×Dic7C2311yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×#1136—22no—
81#6450—24no—
82#6450—24no—
83#6450—24no—
84#6450—24no—
85#6450—24no—
86#6450—24no—
87#6450—24no—
88#6450—24no—
89C7×C22⋊C8—22no—
90C7×C22⋊C8C2314yes—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2314yes—
93#6389—24no—
94Dic7⋊C8—24no—
95Dic7⋊C8—24no—
96C8×Dic7—24no—
97C22×C4×Q8—71no—
98#2920—72no—
99#1467—74no—
100#1463—72no—
101#1463—71no—
102#1463D711yescharacteristic
103#1461—74no—
104C24×C7C22×C411yescharacteristic
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—24no—
108C22×C4×C7—21no—
109C22×C4×C7—21no—
110C22×C4×C7—42no—
111C22×C4×C7—42no—
112C22×C4×C7C22×C412yes—
113C22×C4×C7C22×C412yes—
114C22×C4×C7C22×C411yescharacteristic
115C22×C4×C7C2×D412yes—
116C22×C4×C7C2×D412yes—
117C22×C4×C7C2411yescharacteristic
118C22×Dic7—21no—
119C22×Dic7—21no—
120C22×Dic7—24no—
121C22×Dic7—24no—
122C22×Dic7—24no—
123C22×Dic7—21no—
124C22×Dic7—21no—
125C22×Dic7—42no—
126C22×Dic7C22⋊C411yescharacteristic
127C22×Dic7C22⋊C411yescharacteristic
128C22×Dic7C22⋊C411yescharacteristic
129C22×Dic7C22⋊C412yes—
130C22×Dic7C22⋊C412yes—
131C22×Dic7C22⋊C411yescharacteristic
132C22×Dic7C22×C411yescharacteristic
133C22×Dic7C22×C411yescharacteristic
134C22×Dic7C22×C411yescharacteristic
135C22×Dic7C22×C412yes—
136C22×Dic7C22×C412yes—
137C22×Dic7C22×C411yescharacteristic
138C2×Dic14—216no—
139C2×Dic14—216no—
140C2×Dic14—24no—
141C2×Dic14—24no—
142C2×Dic14—24no—
143C2×Dic14—24no—
144C2×Dic14—24no—
145C2×Dic14M1618yescomplemented
146C7×M16—44no—
147C2×C7×C8—24no—
148C7×C4⋊C4—24no—
149C7×C4⋊C4—24no—
150Dic7⋊C4—24no—
151Dic7⋊C4—24no—
152Dic7⋊C4—24no—
153Dic7⋊C4—24no—
154C4×Dic7—22no—
155C4×Dic7—22no—
156C4×Dic7—22no—
157C4×Dic7—22no—
158C4×Dic7—22no—
159C4×Dic7—22no—
160C4×Dic7—22no—
161C4×Dic7—22no—
162C4×Dic7C22⋊C412yes—
163C4×Dic7C22⋊C412yes—
164C4×Dic7C22⋊C412yes—
165C4×Dic7C22⋊C412yes—
166C4×Dic7C2×D412yes—
167C4×Dic7C2×D412yes—
168C4×Dic7C2×D412yes—
169C4×Dic7C2×D412yes—
170#1136—44no—
171C2×C7⋊C8—24no—
172C23×Q8—71no—
173C2×C4×Q8—72no—
174C2×C4×Q8—72no—
175C2×C4×Q8—72no—
176C2×C4×Q8—72no—
177C2×C4×Q8—142no—
178C2×C4×Q8—142no—
179C2×C4×Q8—142no—
180C2×C4×Q8—142no—
181C22×C4⋊C4—71no—
182C22×C4⋊C4—71no—
183C22×C4⋊C4C2×D711yescharacteristic
184C22×C42—71no—
185C22×C42—71no—
186C22×C42—71no—
187#432—74no—
188#432—142no—
189#431—74no—
190#431—142no—
191#423—142no—
192#407—72no—
193#407C2×D712yes—
194C4×M16—142no—
195#324—74no—
196#324—78no—
197#324C2×D714yes—
198C23×C7—22no—
199C23×C7—22no—
200C23×C7—22no—
201C23×C7C2×C22⋊C411yescharacteristic
202C23×C7C2×C22⋊C411yescharacteristic
203C23×C7C23×C411yescharacteristic
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—24no—
209C2×C4×C7—22no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7C2×C22⋊C412yes—
217C2×C4×C7C2×M1614yes—
218C2×C4×C7C22×D412yes—
219C2×Dic7—24no—
220C2×Dic7—24no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—24no—
226C2×Dic7—24no—
227C2×Dic7—22no—
228C2×Dic7—22no—
229C2×Dic7—22no—
230C2×Dic7—22no—
231C2×Dic7—22no—
232C2×Dic7—42no—
233C2×Dic7—42no—
234C2×Dic7—42no—
235C2×Dic7#10114yes—
236C2×Dic7C2×C22⋊C411yescharacteristic
237C2×Dic7C2×C22⋊C411yescharacteristic
238C2×Dic7C2×C22⋊C412yes—
239C2×Dic7C2×C22⋊C412yes—
240C2×Dic7C2×C22⋊C411yescharacteristic
241C2×Dic7C2×C22⋊C411yescharacteristic
242C2×Dic7C2×M1614yes—
243C2×Dic7C2×M1614yes—
244Dic14—216no—
245Dic14—216no—
246C7×C8—44no—
247C7⋊C8—44no—
248C22×Q8—72no—
249C22×Q8—72no—
250C22×Q8—142no—
251C22×Q8—142no—
252C22×Q8—148no—
253C23×C4—71no—
254C23×C4—71no—
255C23×C4—71no—
256C23×C4—71no—
257C23×C4C4×D711yescharacteristic
258C23×C4C4×D711yescharacteristic
259C23×C4C22×D711yescharacteristic
260C2×M16—22no—
261C2×M16—142no—
262C4×Q8—144no—
263C4×Q8—144no—
264C4×Q8—144no—
265C4×Q8—144no—
266C4×Q8—144no—
267C4×Q8—144no—
268C4×Q8—144no—
269C4×Q8—144no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—72no—
274C2×C4⋊C4—72no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—142no—
279C2×C4⋊C4—142no—
280C2×C4⋊C4C4×D712yes—
281C2×C4⋊C4C22×D712yes—
282C2×C42—71no—
283C2×C42—71no—
284C2×C42—72no—
285C2×C42—72no—
286C2×C42—72no—
287C2×C42—72no—
288C2×C42—71no—
289C2×C42—71no—
290C2×C42—72no—
291C2×C42—72no—
292C2×C42—72no—
293C2×C42—72no—
294C2×C42—141no—
295C2×C42—141no—
296C2×C42—141no—
297C2×C42—141no—
298C2×C42—141no—
299C2×C42—141no—
300C2×C42—141no—
301C2×C42—141no—
302C4⋊C8—144no—
303C4⋊C8—144no—
304C22⋊C8—22no—
305C22⋊C8—74no—
306C22⋊C8—142no—
307C22⋊C8C22×D714yes—
308#97—144no—
309C4×C8—144no—
310C22×C7—28no—
311C22×C7—21no—
312C22×C7—21no—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—21no—
316C22×C7—21no—
317C22×C7#41012yes—
318C22×C7C2×#10112yes—
319C22×C7C22×C22⋊C411yescharacteristic
320C22×C7C22×M1612yes—
321C4×C7—24no—
322C4×C7—22no—
323C4×C7—22no—
324C4×C7—42no—
325Dic7—22no—
326Dic7—24no—
327Dic7—21no—
328Dic7—21no—
329Dic7—42no—
330Dic7#40714yes—
331Dic7C2×#10112yes—
332Dic7C2×#10112yes—
333C24C2×C4×D711yescharacteristic
334C2×Q8—78no—
335C2×Q8—144no—
336C2×Q8—144no—
337C2×Q8—144no—
338C2×Q8—144no—
339C2×Q8—1416no—
340C2×Q8—1416no—
341C2×Q8—144no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4—71no—
354C22×C4—72no—
355C22×C4—72no—
356C22×C4—71no—
357C22×C4—71no—
358C22×C4—71no—
359C22×C4—72no—
360C22×C4—72no—
361C22×C4—144no—
362C22×C4—144no—
363C22×C4—144no—
364C22×C4—141no—
365C22×C4—141no—
366C22×C4—141no—
367C22×C4—141no—
368C22×C4—282no—
369C22×C4C2×C4×D712yes—
370C22×C4C2×C4×D712yes—
371C22×C4C2×C4×D711yescharacteristic
372C22×C4D4×D712yes—
373C22×C4D4×D712yes—
374C22×C4C23×D711yescharacteristic
375M16—44no—
376M16—284no—
377C2×C8—24no—
378C2×C8—144no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—144no—
382C4⋊C4—144no—
383C4⋊C4—144no—
384C4⋊C4—144no—
385C42—72no—
386C42—72no—
387C42—72no—
388C42—72no—
389C42—72no—
390C42—72no—
391C42—72no—
392C42—72no—
393C42—142no—
394C42—142no—
395C42—142no—
396C42—142no—
397C42—142no—
398C42—142no—
399C42—142no—
400C42—142no—
401C2×C7—22no—
402C2×C7—22no—
403C2×C7—22no—
404C2×C7C2×#40711yescharacteristic
405C2×C7C2×#41011yescharacteristic
406C2×C7C22×#10111yescharacteristic
407C23—22no—
408C23—22no—
409C23—22no—
410C23D7×C22⋊C411yescharacteristic
411C23D7×C22⋊C411yescharacteristic
412C23C22×C4×D711yescharacteristic
413Q8—1416no—
414Q8—1416no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—71no—
428C2×C4—71no—
429C2×C4—72no—
430C2×C4—74no—
431C2×C4—74no—
432C2×C4—71no—
433C2×C4—71no—
434C2×C4—72no—
435C2×C4—74no—
436C2×C4—142no—
437C2×C4—142no—
438C2×C4—142no—
439C2×C4—142no—
440C2×C4—144no—
441C2×C4—144no—
442C2×C4—144no—
443C2×C4—144no—
444C2×C4—142no—
445C2×C4—142no—
446C2×C4—142no—
447C2×C4—142no—
448C2×C4—142no—
449C2×C4—282no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4D7×C22⋊C412yes—
453C2×C4D7×M1614yes—
454C2×C4C2×D4×D712yes—
455C8—44no—
456C8—284no—
457C7#1736511yescharacteristic, complemented, Sylow
458C22—28no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22#8951212yes—
466C22D7×#10112yes—
467C22C2×D7×C22⋊C411yescharacteristic
468C22C2×D7×M1612yes—
469C4—24no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—74no—
474C4—72no—
475C4—72no—
476C4—142no—
477C4—144no—
478C4—141no—
479C4—141no—
480C4—282no—
481C2—22no—
482C2—22no—
483C2—22no—
484C2D7×#40711yescharacteristic
485C2C2×#8951211yescharacteristic
486C2C2×D7×#10111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1463C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1463D7
hypocenterC7#17365
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
6. #11725881C2semidirect
7. #11725879C2semidirect
145. C2×Dic14M16semidirect
457. C7#17365semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
#117360642
C7×#14632
#117293062
#117277252
#117258812
#117258792
#173657