Finite Group Databasebeta

← #13245609

Subgroups — #13245609

Order 1792.

1020 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#543C211yesmaximal, characteristic
2#11729446C212yesmaximal, complemented
3C2×#89333C212yesmaximal, complemented
4#11729300C211yesmaximal, characteristic
5C2×#89320C211yesmaximal, characteristic
6C7×#543—24no—
7C7×#543—22no—
8C7×#543C414yescomplemented
9C2×C7×C22⋊Q8—22no—
10C2×C7×C22⋊Q8—21no—
11C2×C7×C22⋊Q8—22no—
12C2×C7×C22⋊Q8C2212yes—
13C22×C7×C22⋊C4—21no—
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic7—21no—
16C22×C4⋊Dic7C2211yescharacteristic
17C2×#6409—22no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2212yes—
21#89333—24no—
22#89324—22no—
23#89320—22no—
24#8405—71nomaximal, Sylow
25C25×C7D411yescharacteristic
26C22×C7×Q8—21no—
27C22×C7×Q8D412yes—
28C23×C4×C7—21no—
29C23×C4×C7—21no—
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C23×Dic7—21no—
33C23×Dic7—21no—
34C7×C22⋊Q8—28no—
35C7×C22⋊Q8—28no—
36C7×C22⋊Q8—24no—
37C7×C22⋊Q8—28no—
38C7×C22⋊Q8—24no—
39C7×C22⋊Q8—22no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8C2×C414yes—
43C7×C22⋊Q8D414yes—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4D412yes—
48C2×C7×C22⋊C4—22no—
49C2×C7×C22⋊C4—24no—
50C2×C7×C22⋊C4—24no—
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4C2×C412yes—
55C2×C7×C22⋊C4D412yes—
56C2×C22⋊Dic7—42no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7—42no—
59C2×C4⋊Dic7—22no—
60C2×C4⋊Dic7—42no—
61C22×C7⋊C8—22no—
62#6409—44no—
63C7⋊#95—22no—
64C7⋊#95—42no—
65C22⋊(C7⋊C8)—22no—
66C22⋊(C7⋊C8)—22no—
67C2×#543D711yescharacteristic
68#1872—72no—
69C2×#328—72no—
70C2×#323—71no—
71#1439—71no—
72C24×C7—21no—
73C24×C7—21no—
74C24×C7—21no—
75C24×C7—21no—
76C24×C7—41no—
77C24×C7—41no—
78C24×C7—41no—
79C24×C7—41no—
80C24×C7C22⋊C412yes—
81C24×C7SD1612yes—
82C24×C7Q1612yes—
83C24×C7C2×D411yescharacteristic
84C2×C7×Q8—24no—
85C2×C7×Q8—22no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×Q8—81no—
89C2×C7×Q8—81no—
90C2×C7×Q8C22⋊C414yescomplemented
91C22×C4×C7—22no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—24no—
95C22×C4×C7—24no—
96C22×C4×C7—21no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—41no—
101C22×C4×C7—81no—
102C22×C4×C7—81no—
103C22×C4×C7—81no—
104C22×C4×C7—81no—
105C22×C4×C7C22⋊C412yes—
106C22×C4×C7C22×C411yescharacteristic
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2×D411yescharacteristic
110C22×Dic7—22no—
111C22×Dic7—22no—
112C22×Dic7—41no—
113C22×Dic7—41no—
114C22×Dic7—41no—
115C22×Dic7—41no—
116C22×Dic7—41no—
117C22×Dic7—41no—
118C22×Dic7—41no—
119C22×Dic7—41no—
120C22×Dic7—41no—
121C22×Dic7—41no—
122C22×Dic7—41no—
123C22×Dic7—41no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4—28no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4C22⋊C414yescomplemented
130C7×C22⋊C4—28no—
131C7×C22⋊C4—24no—
132C7×C22⋊C4—24no—
133C7×C22⋊C4—24no—
134C7×C22⋊C4—24no—
135C22⋊Dic7—48no—
136C4⋊Dic7—24no—
137C4⋊Dic7—42no—
138C2×C7⋊C8—42no—
139C2×C7⋊C8—42no—
140C2×C7⋊C8—42no—
141#543—24no—
142#543—22no—
143#543Dic714yescomplemented
144C2×C22⋊Q8—22no—
145C2×C22⋊Q8—21no—
146C2×C22⋊Q8—22no—
147C2×C22⋊Q8C2×D712yes—
148C22×C4⋊C4—71no—
149C22×C22⋊C4—21no—
150C22×C22⋊C4—141no—
151C22×C22⋊C4C2×D711yescharacteristic
152C2×#103—142no—
153C2×C22⋊C8—72no—
154C2×#95—71no—
155C2×#95—141no—
156#340—142no—
157#328—144no—
158#323—142no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—21no—
163C23×C7—22no—
164C23×C7—22no—
165C23×C7—22no—
166C23×C7—22no—
167C23×C7—22no—
168C23×C7—22no—
169C23×C7—42no—
170C23×C7—42no—
171C23×C7—42no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7—42no—
175C23×C7—41no—
176C23×C7—41no—
177C23×C7—41no—
178C23×C7—41no—
179C23×C7—81no—
180C23×C7—81no—
181C23×C7—81no—
182C23×C7—81no—
183C23×C7—81no—
184C23×C7—81no—
185C23×C7#10314yescomplemented
186C23×C7C2×C22⋊C411yescharacteristic
187C23×C7#12011yescharacteristic
188C23×C7#12011yescharacteristic
189C23×C7C2×SD1611yescharacteristic
190C23×C7C2×Q1611yescharacteristic
191C23×C7#13711yescharacteristic
192C23×C7#13711yescharacteristic
193C7×Q8—44no—
194C7×Q8—82no—
195C7×Q8—84no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—24no—
199C2×C4×C7—22no—
200C2×C4×C7—24no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—42no—
204C2×C4×C7—41no—
205C2×C4×C7—42no—
206C2×C4×C7—41no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—81no—
212C2×C4×C7—84no—
213C2×C4×C7—84no—
214C2×C4×C7C2×C22⋊C412yes—
215C2×C4×C7#12012yes—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7—81no—
223C2×Dic7—81no—
224C2×Dic7—81no—
225C2×Dic7—81no—
226C2×Dic7—81no—
227C2×Dic7—81no—
228C2×Dic7—81no—
229C2×Dic7—81no—
230C7⋊C8—82no—
231C25C22⋊D711yescharacteristic
232C22×Q8—21no—
233C22×Q8C22⋊D712yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—141no—
237C23×C4—141no—
238C23×C4C22⋊D711yescharacteristic
239C23×C4C22×D711yescharacteristic
240C22×C8—142no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—28no—
249C22⋊Q8C2×Dic714yes—
250C22⋊Q8C22⋊D714yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—142no—
255C2×C4⋊C4—282no—
256C2×C4⋊C4C22⋊D712yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—282no—
264C2×C22⋊C4—282no—
265C2×C22⋊C4—282no—
266C2×C22⋊C4C2×Dic712yes—
267C2×C22⋊C4C22⋊D712yes—
268#103—284no—
269C22⋊C8—142no—
270C22⋊C8—142no—
271#95—142no—
272#95—282no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—21no—
283C22×C7—21no—
284C22×C7—21no—
285C22×C7—24no—
286C22×C7—44no—
287C22×C7—42no—
288C22×C7—42no—
289C22×C7—42no—
290C22×C7—42no—
291C22×C7—42no—
292C22×C7—42no—
293C22×C7—82no—
294C22×C7—82no—
295C22×C7—82no—
296C22×C7#37911yescharacteristic
297C22×C7C2×#10311yescharacteristic
298C22×C7#41911yescharacteristic
299C22×C7#44811yescharacteristic
300C22×C7#44811yescharacteristic
301C22×C7#45111yescharacteristic
302C22×C7#45111yescharacteristic
303C22×C7#45311yescharacteristic
304C22×C7#45511yescharacteristic
305C22×C7B(4,𝔽2)11yescharacteristic
306C22×C7#45811yescharacteristic
307C4×C7—22no—
308C4×C7—22no—
309C4×C7—42no—
310C4×C7—44no—
311C4×C7—44no—
312Dic7—82no—
313Dic7—82no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24#114312yes—
323C24#114412yes—
324C24C22⋊Dic712yes—
325C24C2×C22⋊D711yescharacteristic
326C2×Q8—24no—
327C2×Q8—22no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—81no—
331C2×Q8—81no—
332C2×Q8C22⋊Dic714yescomplemented
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—281no—
350C22×C4—281no—
351C22×C4—281no—
352C22×C4—281no—
353C22×C4—281no—
354C22×C4—281no—
355C22×C4—281no—
356C22×C4—281no—
357C22×C4—281no—
358C22×C4—281no—
359C22×C4—281no—
360C22×C4—281no—
361C22×C4C22⋊Dic712yes—
362C22×C4C22×Dic711yescharacteristic
363C22×C4C2×C22⋊D712yes—
364C22×C4C2×C22⋊D712yes—
365C22×C4C2×C22⋊D711yescharacteristic
366C2×C8—282no—
367C2×C8—282no—
368C2×C8—282no—
369C4⋊C4—24no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—144no—
375C4⋊C4—282no—
376C4⋊C4C22⋊Dic714yescomplemented
377C22⋊C4—28no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—288no—
383C2×C7—21no—
384C2×C7—21no—
385C2×C7—21no—
386C2×C7—21no—
387C2×C7—44no—
388C2×C7#159611yescharacteristic
389C2×C7#160211yescharacteristic
390C2×C7#160811yescharacteristic
391C2×C7#161411yescharacteristic
392C2×C7#178711yescharacteristic
393C2×C7#179011yescharacteristic
394C2×C7#179211yescharacteristic
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23#640914yescomplemented
422C23C2×#114311yescharacteristic
423C23#650511yescharacteristic
424C23#650511yescharacteristic
425C23C2×#114411yescharacteristic
426C23C2×C22⋊Dic711yescharacteristic
427C23#651611yescharacteristic
428C23#651611yescharacteristic
429Q8—44no—
430Q8—82no—
431Q8—84no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—81no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—282no—
451C2×C4—282no—
452C2×C4—282no—
453C2×C4—282no—
454C2×C4—282no—
455C2×C4—282no—
456C2×C4—561no—
457C2×C4—561no—
458C2×C4—561no—
459C2×C4—561no—
460C2×C4—561no—
461C2×C4—561no—
462C2×C4—561no—
463C2×C4—561no—
464C2×C4C2×C22⋊Dic712yes—
465C2×C4#651612yes—
466C8—562no—
467C7#840511yescharacteristic, complemented, Sylow
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22C2×#640911yescharacteristic
492C22#8999711yescharacteristic
493C22#8999911yescharacteristic
494C22#8999911yescharacteristic
495C22#9000011yescharacteristic
496C22#9000011yescharacteristic
497C22#9001311yescharacteristic
498C22#9001411yescharacteristic
499C22#9001511yescharacteristic
500C22#9001611yescharacteristic
501C22#9001911yescharacteristic
502C4—22no—
503C4—22no—
504C4—42no—
505C4—44no—
506C4—44no—
507C4—562no—
508C4—562no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—44no—
514C2#1172964411yescharacteristic
515C2#1172964511yescharacteristic
516C2#1172964611yescharacteristic
517C2#1172964711yescharacteristic
518C2#1172988611yescharacteristic
519C2#1172989411yescharacteristic
520C2#1172989611yescharacteristic
subgroupG/H
centerC23#6516
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#543C2
socleC23×C7#120
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenterC2×#543D7
hypocenterC7#8405
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #11729446C2semidirect
3. C2×#89333C2semidirect
8. C7×#543C4semidirect
90. C2×C7×Q8C22⋊C4semidirect
129. C7×C4⋊C4C22⋊C4semidirect
143. #543Dic7semidirect
185. C23×C7#103semidirect
332. C2×Q8C22⋊Dic7semidirect
376. C4⋊C4C22⋊Dic7semidirect
421. C23#6409semidirect
467. C7#8405semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5432
#117294462
C2×#893332
#117293002
C2×#893202
#84057