Finite Group Databasebeta

← #13319865

Subgroups — #13319865

Order 1792.

1450 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#123C211yesmaximal, characteristic
2C2×#89979C212yesmaximal, complemented
3#11729456C218yesmaximal, complemented
4#11729306C212yesmaximal, complemented
5#11729293C212yesmaximal, complemented
6C23×C7×D4C411yescharacteristic
7C24×C4×C7C2211yescharacteristic
8C2×C7×#123—22no—
9C2×C7×#123—22no—
10C2×C7×#123—22no—
11C2×C7×#123—22no—
12C2×C7×#123C414yes—
13C2×C7×#123C2214yes—
14C22×C7×C4⋊C4C411yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C411yescharacteristic
17C22×C7×C22⋊C4C411yescharacteristic
18C22×C7×C22⋊C4C2211yescharacteristic
19C22×#1136—22no—
20#89979—22no—
21#89979—22no—
22#89979C2214yes—
23#89979C2214yescomplemented
24C2×C22⋊(C7⋊C8)—22no—
25C2×C22⋊(C7⋊C8)C2214yescomplemented
26#89323C2218yescomplemented
27#89320C2218yescomplemented
28#17392—71nomaximal, Sylow
29C25×C7C2×C411yescharacteristic
30C25×C7C2×C411yescharacteristic
31C22×C7×D4—22no—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4—44no—
35C22×C7×D4C2×C414yes—
36C23×C4×C7—44no—
37C23×C4×C7C2×C411yescharacteristic
38C23×C4×C7C2×C411yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2×C411yescharacteristic
41C23×C4×C7C2×C412yes—
42C23×C4×C7D412yes—
43C23×C4×C7D412yes—
44C23×C4×C7D412yes—
45C23×C4×C7D412yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2311yescharacteristic
48C7×#123—24no—
49C7×#123—24no—
50C7×#123—24no—
51C7×#123—24no—
52C7×#123—24no—
53C7×#123—24no—
54C7×#123—24no—
55C7×#123—24no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4C2×C414yes—
62C2×C7×C4⋊C4C2×C412yes—
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4C2×C414yes—
73C2×C7×C22⋊C4C2×C414yes—
74C2×C7×C22⋊C4C2×C412yes—
75C2×C7×C22⋊C4C2312yes—
76C2×#1136—24no—
77C2×#1136—24no—
78C2×#1136—48no—
79C22×C7⋊C8—24no—
80C22⋊(C7⋊C8)—24no—
81C22⋊(C7⋊C8)—24no—
82C22⋊(C7⋊C8)—24no—
83C22⋊(C7⋊C8)C2318yes—
84C22×#123D711yescharacteristic
85C2×#407—72no—
86#1468—78no—
87#1463—72no—
88#1462—72no—
89C24×C7—24no—
90C24×C7—21no—
91C24×C7—21no—
92C24×C7—42no—
93C24×C7—42no—
94C24×C7—42no—
95C24×C7—42no—
96C24×C7—42no—
97C24×C7C22⋊C412yes—
98C24×C7C22⋊C411yescharacteristic
99C24×C7C22⋊C411yescharacteristic
100C24×C7C22×C411yescharacteristic
101C24×C7C22×C412yes—
102C24×C7C22×C412yes—
103C2×C7×D4—24no—
104C2×C7×D4—24no—
105C2×C7×D4—48no—
106C2×C7×D4—44no—
107C2×C7×D4—44no—
108C2×C7×D4—48no—
109C22×C4×C7—28no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—42no—
117C22×C4×C7—44no—
118C22×C4×C7—44no—
119C22×C4×C7—44no—
120C22×C4×C7—44no—
121C22×C4×C7—44no—
122C22×C4×C7—42no—
123C22×C4×C7—42no—
124C22×C4×C7—42no—
125C22×C4×C7—42no—
126C22×C4×C7C22⋊C412yes—
127C22×C4×C7C22⋊C412yes—
128C22×C4×C7C22⋊C412yes—
129C22×C4×C7C22⋊C411yescharacteristic
130C22×C4×C7C22⋊C411yescharacteristic
131C22×C4×C7C22⋊C412yes—
132C22×C4×C7C22⋊C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C22×C411yescharacteristic
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2411yescharacteristic
145C7×C4⋊C4—24no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—24no—
149C7×C22⋊C4—24no—
150C7×C22⋊C4—24no—
151C7×C22⋊C4—24no—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—24no—
155#1136—48no—
156#1136—48no—
157C2×C7⋊C8—28no—
158C2×C7⋊C8—48no—
159C23×D4Dic711yescharacteristic
160C24×C4C2×D711yescharacteristic
161C22×M16—142no—
162C2×#123—22no—
163C2×#123—22no—
164C2×#123—22no—
165C2×#123—22no—
166C2×#123Dic714yes—
167C2×#123C2×D714yes—
168C22×C4⋊C4Dic711yescharacteristic
169C22×C4⋊C4C2×D711yescharacteristic
170C22×C22⋊C4Dic711yescharacteristic
171C22×C22⋊C4Dic711yescharacteristic
172C22×C22⋊C4C2×D711yescharacteristic
173#407—74no—
174#407—74no—
175#407—142no—
176#407—142no—
177C2×C22⋊C8—74no—
178C2×C22⋊C8—142no—
179#324—78no—
180#323—78no—
181C23×C7—22no—
182C23×C7—22no—
183C23×C7—22no—
184C23×C7—24no—
185C23×C7—24no—
186C23×C7—22no—
187C23×C7—24no—
188C23×C7—24no—
189C23×C7—22no—
190C23×C7—24no—
191C23×C7—41no—
192C23×C7—42no—
193C23×C7—41no—
194C23×C7—41no—
195C23×C7—41no—
196C23×C7—42no—
197C23×C7—416no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—88no—
207C23×C7#9914yes—
208C23×C7C2×C22⋊C412yes—
209C23×C7C2×C22⋊C412yes—
210C23×C7C2×C22⋊C411yescharacteristic
211C23×C7C2×C22⋊C411yescharacteristic
212C23×C7C23×C411yescharacteristic
213C7×D4—48no—
214C7×D4—48no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—24no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—24no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—41no—
235C2×C4×C7—42no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7C2×C22⋊C412yes—
244C2×C4×C7C2×C22⋊C412yes—
245C2×C4×C7C2×C22⋊C412yes—
246C2×C4×C7C22×D412yes—
247C7⋊C8—48no—
248C25C2×Dic711yescharacteristic
249C25C2×Dic711yescharacteristic
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—44no—
254C22×D4C2×Dic714yes—
255C23×C4—44no—
256C23×C4C2×Dic711yescharacteristic
257C23×C4C2×Dic711yescharacteristic
258C23×C4C2×Dic711yescharacteristic
259C23×C4C2×Dic711yescharacteristic
260C23×C4C2×Dic712yes—
261C23×C4C22⋊D712yes—
262C23×C4C22⋊D712yes—
263C23×C4C22⋊D712yes—
264C23×C4C22⋊D712yes—
265C23×C4C22×D711yescharacteristic
266C23×C4C22×D712yes—
267C2×M16—144no—
268C2×M16—144no—
269C2×M16—288no—
270C22×C8—144no—
271#123—24no—
272#123—24no—
273#123—24no—
274#123—24no—
275#123—24no—
276#123—24no—
277#123—24no—
278#123—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4C2×Dic714yes—
285C2×C4⋊C4C2×Dic712yes—
286C2×C4⋊C4C22×D712yes—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4C2×Dic714yes—
296C2×C22⋊C4C2×Dic714yes—
297C2×C22⋊C4C2×Dic712yes—
298C2×C22⋊C4C22×D712yes—
299C22⋊C8—78no—
300C22⋊C8—144no—
301C22⋊C8—144no—
302C22⋊C8—144no—
303C22×C7—28no—
304C22×C7—21no—
305C22×C7—21no—
306C22×C7—22no—
307C22×C7—21no—
308C22×C7—21no—
309C22×C7—21no—
310C22×C7—21no—
311C22×C7—22no—
312C22×C7—22no—
313C22×C7—22no—
314C22×C7—22no—
315C22×C7—44no—
316C22×C7—48no—
317C22×C7—48no—
318C22×C7—44no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—42no—
322C22×C7—84no—
323C22×C7—84no—
324C22×C7—84no—
325C22×C7C2×#9912yes—
326C22×C7C2×#9914yes—
327C22×C7#41312yes—
328C22×C7C22×C22⋊C411yescharacteristic
329C22×C7#56812yes—
330C4×C7—22no—
331C4×C7—22no—
332C4×C7—22no—
333C4×C7—22no—
334C4×C7—42no—
335C4×C7—42no—
336C4×C7—42no—
337C24—24no—
338C24—21no—
339C24—21no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C22⋊Dic712yes—
346C24C22⋊Dic711yescharacteristic
347C24C22⋊Dic711yescharacteristic
348C24C22×Dic711yescharacteristic
349C24C22×Dic712yes—
350C24C22×Dic712yes—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—48no—
357C22×C4—28no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C22⋊Dic712yes—
375C22×C4C22⋊Dic712yes—
376C22×C4C22⋊Dic712yes—
377C22×C4C22⋊Dic711yescharacteristic
378C22×C4C22⋊Dic711yescharacteristic
379C22×C4C22⋊Dic712yes—
380C22×C4C22⋊Dic712yes—
381C22×C4C22×Dic712yes—
382C22×C4C22×Dic712yes—
383C22×C4C22×Dic712yes—
384C22×C4C22×Dic712yes—
385C22×C4C22×Dic711yescharacteristic
386C22×C4C2×C22⋊D712yes—
387C22×C4C2×C22⋊D712yes—
388C22×C4C2×C22⋊D712yes—
389C22×C4C2×C22⋊D712yes—
390C22×C4C2×C22⋊D712yes—
391C22×C4C2×C22⋊D712yes—
392C22×C4C23×D711yescharacteristic
393M16—288no—
394M16—288no—
395C2×C8—148no—
396C2×C8—288no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C2×C7—22no—
408C2×C7—22no—
409C2×C7—22no—
410C2×C7—24no—
411C2×C7—42no—
412C2×C7—82no—
413C2×C7#287911yescharacteristic
414C2×C7C22×#9911yescharacteristic
415C2×C7C2×#41311yescharacteristic
416C23—22no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—24no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—416no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—88no—
442C23#640814yes—
443C23C2×C22⋊Dic712yes—
444C23C2×C22⋊Dic712yes—
445C23C2×C22⋊Dic711yescharacteristic
446C23C2×C22⋊Dic711yescharacteristic
447C23C23×Dic711yescharacteristic
448D4—48no—
449D4—48no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4C2×C22⋊Dic712yes—
479C2×C4C2×C22⋊Dic712yes—
480C2×C4C2×C22⋊Dic712yes—
481C2×C4C22×C22⋊D712yes—
482C8—288no—
483C7#1739211yescharacteristic, complemented, Sylow
484C22—28no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—48no—
498C22—48no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22C2×#640812yes—
507C22C2×#640814yes—
508C22#9000912yes—
509C22#9051012yes—
510C22C22×C22⋊Dic711yescharacteristic
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C2—22no—
519C2—22no—
520C2—22no—
521C2—24no—
522C2—42no—
523C2—82no—
524C2C22×#640811yescharacteristic
525C2#1174151911yescharacteristic
526C2C2×#9000911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#123C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7×D4C4
cosocleC22×C4×C7C24
hypercenterC22×#123D7
hypocenterC7#17392
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C2×#89979C2semidirect
3. #11729456C2semidirect
4. #11729306C2semidirect
5. #11729293C2semidirect
23. #89979C22semidirect
25. C2×C22⋊(C7⋊C8)C22semidirect
26. #89323C22semidirect
27. #89320C22semidirect
483. C7#17392semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1232
C2×#899792
#117294562
#117293062
#117292932
#173927