Finite Group Databasebeta

← #13408221

Subgroups — #13408221

Order 1792.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#541C211yesmaximal, characteristic
2C2×#89853C211yesmaximal, characteristic, complemented
3C2×#89837C211yesmaximal, characteristic, complemented
4#11739709C211yesmaximal, characteristic
5#11729540C211yesmaximal, characteristic, complemented
6#11729494C211yesmaximal, characteristic, complemented
7#11729449C212yesmaximal
8#11729448C212yesmaximal
9#11729383C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic
12C7×#541—24no—
13C7×#541—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—21no—
16C2×C7×#126—21no—
17C2×C7×#125—21no—
18C2×C7×#125—21no—
19C2×C7×#125C2211yescharacteristic
20C2×C7×C22⋊Q8C2211yescharacteristic
21C2×C7×C22⋊Q8C2211yescharacteristic
22C2×C4×C7×Q8C2211yescharacteristic
23C2×C4×C7×Q8C2211yescharacteristic
24C2×C7×#117C2211yescharacteristic
25C2×#6432C2211yescharacteristic
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30#89984C2212yes—
31C2×C22⋊(C7⋊C8)C2212yes—
32#89853—24no—
33#89837—24no—
34#89758—21no—
35#89751—21no—
36C2×#6382C2212yes—
37C2×#6381C2212yes—
38#89721—21no—
39#89719—21no—
40#89708C2211yescharacteristic
41#89700C2212yes—
42C2×#6377C2211yescharacteristic
43C2×C4×C7⋊C8C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#25686—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×Q8C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C22×Dic14—21no—
52C7×C4⋊Q8—22no—
53C7×C4⋊Q8D414yes—
54C7×#126—28no—
55C7×#126—28no—
56C7×#125—28no—
57C7×#125—28no—
58C7×#125—22no—
59C7×#125D414yes—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—22no—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C7×#117—22no—
69C7×#117—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4D411yescharacteristic
85C2×C7×C22⋊C4D411yescharacteristic
86C2×C42×C7—21no—
87C2×C42×C7—21no—
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—22no—
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2312yes—
97#6432—22no—
98#6432—22no—
99#6409—24no—
100#6409—24no—
101#6409—24no—
102#6409—24no—
103C7⋊#95—22no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107C22⋊(C7⋊C8)—24no—
108#6382—24no—
109#6381—24no—
110#6377—22no—
111C4×C7⋊C8—22no—
112C2×#541D711yescharacteristic
113C2×#501—71no—
114C2×#487—71no—
115#2628—71no—
116#2036—72no—
117#2035—72no—
118#1991—71no—
119#1960—71no—
120#1865—72no—
121#1863—72no—
122#1750—71no—
123C24×C7C2×D411yescharacteristic
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8C2×D412yes—
131C2×C7×Q8C2×D412yes—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2411yescharacteristic
146C22×Dic7—22no—
147C22×Dic7—22no—
148C2×Dic14—42no—
149C2×Dic14—82no—
150C7×C4⋊C4—24no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4C2×D412yes—
163C7×C4⋊C4C2×D412yes—
164C7×C22⋊C4—24no—
165C7×C22⋊C4—24no—
166C7×C22⋊C4—22no—
167C7×C22⋊C4—22no—
168C42×C7—24no—
169C42×C7—24no—
170C42×C7—21no—
171C42×C7—21no—
172C42×C7C2×D412yes—
173C42×C7C2×D412yes—
174C22⋊Dic7—82no—
175C4⋊Dic7—22no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—42no—
180Dic7⋊C4—82no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8#4114yes—
187#541—24no—
188#541—24no—
189C2×C4⋊Q8—71no—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#126—21no—
192C2×#126—21no—
193C2×#125—21no—
194C2×#125—21no—
195C2×#125C2×D711yescharacteristic
196C2×C22⋊Q8C2×D711yescharacteristic
197C2×C22⋊Q8C2×D711yescharacteristic
198C2×C4×Q8C2×D711yescharacteristic
199C2×C4×Q8C2×D711yescharacteristic
200C2×#117C2×D711yescharacteristic
201#501—144no—
202#487—144no—
203C2×#107—72no—
204C2×#106—72no—
205C2×#103—72no—
206C2×#103—72no—
207C2×#103—72no—
208C2×#103—72no—
209C2×C22⋊C8—72no—
210C2×#97—71no—
211C2×C4×C8—71no—
212#400—141no—
213#399—141no—
214#395—141no—
215#393—141no—
216#389—72no—
217#382—71no—
218#380—72no—
219#340—72no—
220#340—72no—
221#336—72no—
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7C22×D411yescharacteristic
228C7×Q8—24no—
229C7×Q8—24no—
230C7×Q8—42no—
231C7×Q8—42no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7#12012yes—
254C2×C4×C7#12012yes—
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C2×#4112yes—
258C2×C4×C72−1+412yes—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265Dic14—84no—
266C7⋊C8—24no—
267C7⋊C8—42no—
268C22×Q8—141no—
269C22×Q8C22×D711yescharacteristic
270C22×Q8C22×D711yescharacteristic
271C23×C4C22×D711yescharacteristic
272C22×C8—72no—
273C22×C8—72no—
274C4⋊Q8—22no—
275C4⋊Q8—142no—
276C4⋊Q8—142no—
277C4⋊Q8C22⋊D714yes—
278#126—28no—
279#126—28no—
280#125—28no—
281#125—28no—
282#125—22no—
283#125C22⋊D714yes—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292#117—22no—
293#117—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—72no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—142no—
306C2×C4⋊C4C22⋊D711yescharacteristic
307C2×C4⋊C4C22⋊D711yescharacteristic
308C2×C4⋊C4C22×D711yescharacteristic
309C2×C4⋊C4C22×D711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—142no—
313C2×C22⋊C4C22⋊D711yescharacteristic
314C2×C22⋊C4C22⋊D711yescharacteristic
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C22×D711yescharacteristic
318C2×C42C22×D711yescharacteristic
319#107—144no—
320#106—144no—
321#103—144no—
322#103—144no—
323#103—144no—
324#103—144no—
325C22⋊C8—144no—
326#97—142no—
327C4×C8—142no—
328#95—142no—
329#95—141no—
330#95—141no—
331#95—142no—
332C22×C7—42no—
333C22×C7—41no—
334C22×C7—42no—
335C22×C7—41no—
336C22×C7—41no—
337C22×C7C2×#12011yescharacteristic
338C22×C7#53211yescharacteristic
339C22×C7#53611yescharacteristic
340C22×C7#57712yes—
341C22×C7#57812yes—
342C4×C7—22no—
343C4×C7—22no—
344C4×C7—41no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—41no—
348C4×C7—42no—
349C4×C7—41no—
350C4×C7—42no—
351C4×C7#54712yes—
352C4×C7#54812yes—
353Dic7—82no—
354Dic7—82no—
355C24C2×C22⋊D711yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—282no—
363C2×Q8—562no—
364C2×Q8C2×C22⋊D712yes—
365C2×Q8C2×C22⋊D712yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4C2×C22⋊D711yescharacteristic
377C22×C4C2×C22⋊D711yescharacteristic
378C22×C4C2×C22⋊D711yescharacteristic
379C22×C4C2×C22⋊D711yescharacteristic
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C23×D711yescharacteristic
382C2×C8—74no—
383C2×C8—142no—
384C2×C8—142no—
385C2×C8—142no—
386C2×C8—142no—
387C2×C8—142no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—142no—
401C4⋊C4—142no—
402C4⋊C4—142no—
403C4⋊C4—142no—
404C4⋊C4—282no—
405C4⋊C4—562no—
406C4⋊C4C2×C22⋊D712yes—
407C4⋊C4C2×C22⋊D712yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—562no—
413C42—24no—
414C42—24no—
415C42—21no—
416C42—21no—
417C42C2×C22⋊D712yes—
418C42C2×C22⋊D712yes—
419C2×C7—81no—
420C2×C7#243211yescharacteristic
421C2×C7#301011yescharacteristic
422C2×C7#301111yescharacteristic
423C2×C7#317812yes—
424C2×C7#324212yes—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—22no—
429C23—21no—
430C23C22×C22⋊D711yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—42no—
435Q8—564no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—282no—
458C2×C4—282no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4#651612yes—
464C2×C4#651612yes—
465C2×C4D7×#4112yes—
466C2×C4#655412yes—
467C2×C4C22×C22⋊D711yescharacteristic
468C2×C4C22×C22⋊D711yescharacteristic
469C8—144no—
470C8—282no—
471C7#2568611yescharacteristic, complemented, Sylow
472C22—41no—
473C22—41no—
474C22—41no—
475C22—42no—
476C22—42no—
477C22#9051611yescharacteristic
478C22#9052311yescharacteristic
479C22#9052812yes—
480C22#9052912yes—
481C22C2×#651611yescharacteristic
482C4—22no—
483C4—22no—
484C4—41no—
485C4—42no—
486C4—42no—
487C4—41no—
488C4—42no—
489C4—41no—
490C4—42no—
491C4—562no—
492C4—562no—
493C4#9051512yes—
494C4#9052212yes—
495C2—81no—
496C2#1174088712yes—
497C2#1174089012yes—
498C2#1174153811yescharacteristic
499C2#1174155811yescharacteristic
500C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#541C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#541D7
hypocenterC7#25686
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
2. C2×#89853C2semidirect
3. C2×#89837C2semidirect
5. #11729540C2semidirect
6. #11729494C2semidirect
9. #11729383C2semidirect
10. #11729382C2semidirect
471. C7#25686semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5412
C2×#898532
C2×#898372
#117397092
#117295402
#117294942
#117294492
#117294482
#117293832
#117293822
#117293362
#256867