Finite Group Databasebeta

← #13245404

Subgroups — #13245404

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2#11729892C212yesmaximal, complemented
3#11729442C212yesmaximal, complemented
4C4×#6408C211yesmaximal, characteristic, complemented
5#11729356C211yesmaximal, characteristic, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C22×C7×#41C2211yescharacteristic
8C7×#547—24no—
9C7×#547—24no—
10C2×C7×C4⋊Q8C411yescharacteristic, complemented
11C2×C7×#123—21no—
12C2×C7×#123—21no—
13C2×C7×C22⋊Q8C411yescharacteristic, complemented
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×C7×#121—21no—
16C2×C4×C7×D4C411yescharacteristic, complemented
17C2×C4×C7×D4C2211yescharacteristic
18C22×C7×C4⋊C4—21no—
19C2×#6411—24no—
20C2×#6408—21no—
21C4×C22⋊Dic7C2212yes—
22#89979C2212yes—
23#89333—24no—
24#89323—21no—
25#89321—21no—
26#89320—21no—
27#8364—71nomaximal, Sylow
28C2×C7×#41—22no—
29C2×C7×#41—22no—
30C2×C7×#41—22no—
31C2×C7×#41—44no—
32C2×C7×#41—44no—
33C2×C7×#41D414yes—
34C22×C7×Q8C2×C411yescharacteristic
35C22×C7×Q8C2×C411yescharacteristic
36C22×C7×D4—21no—
37C22×C7×D4—21no—
38C22×C7×D4C2×C411yescharacteristic
39C22×C7×D4D411yescharacteristic
40C23×C4×C7—21no—
41C23×C4×C7—21no—
42C23×C4×C7—21no—
43C23×C4×C7C2×C411yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C7×C4⋊Q8—22no—
46C7×C4⋊Q8—22no—
47C7×#123—28no—
48C7×#123—28no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8D414yes—
53C7×#121—28no—
54C4×C7×D4—22no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4—28no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4C2×C411yescharacteristic
64C2×C7×C4⋊C4D411yescharacteristic
65C2×C7×C22⋊C4—21no—
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4D411yescharacteristic
69C2×C7×C22⋊C4D411yescharacteristic
70C2×C42×C7C2×C411yescharacteristic
71C2×C22⋊Dic7—22no—
72C2×C4×Dic7—22no—
73C2×#1136—22no—
74#6411—48no—
75#6408—24no—
76C7⋊#95—22no—
77C22⋊(C7⋊C8)—22no—
78C22⋊(C7⋊C8)—22no—
79C2×#547D711yescharacteristic
80#1789—72no—
81C4×#99—71no—
82#1508—72no—
83#1465—71no—
84C24×C7—21no—
85C24×C7C22⋊C411yescharacteristic
86C24×C7C2×D411yescharacteristic
87C7×#41—28no—
88C7×#41—44no—
89C7×#41—48no—
90C7×#41—48no—
91C7×#41—44no—
92C7×#41—44no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—42no—
96C2×C7×Q8—42no—
97C2×C7×Q8C22⋊C412yes—
98C2×C7×Q8C22⋊C412yes—
99C2×C7×D4—24no—
100C2×C7×D4—24no—
101C2×C7×D4—22no—
102C2×C7×D4—24no—
103C2×C7×D4—22no—
104C2×C7×D4—21no—
105C2×C7×D4—41no—
106C2×C7×D4—44no—
107C2×C7×D4—44no—
108C2×C7×D4C22⋊C412yes—
109C22×C4×C7—24no—
110C22×C4×C7—24no—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—24no—
117C22×C4×C7—42no—
118C22×C4×C7—42no—
119C22×C4×C7—42no—
120C22×C4×C7—44no—
121C22×C4×C7—44no—
122C22×C4×C7—42no—
123C22×C4×C7C22⋊C411yescharacteristic
124C22×C4×C7C22⋊C411yescharacteristic
125C22×C4×C7C22⋊C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×C4×C7C2×D411yescharacteristic
130C22×Dic7—42no—
131C7×C4⋊C4—24no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—24no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—24no—
137C7×C4⋊C4C22⋊C412yes—
138C7×C22⋊C4—24no—
139C7×C22⋊C4—24no—
140C7×C22⋊C4—24no—
141C7×C22⋊C4—22no—
142C7×C22⋊C4—22no—
143C42×C7—21no—
144C42×C7—21no—
145C22⋊Dic7—24no—
146C22⋊Dic7—42no—
147C4×Dic7—44no—
148#1136—44no—
149C2×C7⋊C8—42no—
150C22×#41Dic711yescharacteristic, complemented
151C22×#41C2×D711yescharacteristic
152#547—24no—
153#547—24no—
154C2×C4⋊Q8Dic711yescharacteristic, complemented
155C2×#123—21no—
156C2×#123—21no—
157C2×C22⋊Q8Dic711yescharacteristic, complemented
158C2×C22⋊Q8C2×D711yescharacteristic
159C2×#121—21no—
160C2×C4×D4Dic711yescharacteristic, complemented
161C2×C4×D4C2×D711yescharacteristic
162C22×C4⋊C4—21no—
163C2×#104—144no—
164C2×#99—141no—
165#407—72no—
166C4×C22⋊C4—72no—
167#342—141no—
168#328—144no—
169#324—141no—
170#323—141no—
171C23×C7—24no—
172C23×C7—22no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7—21no—
176C23×C7—21no—
177C23×C7—21no—
178C23×C7—41no—
179C23×C7—41no—
180C23×C7—42no—
181C23×C7—41no—
182C23×C7—41no—
183C23×C7—41no—
184C23×C7—41no—
185C23×C7C2×C22⋊C411yescharacteristic
186C23×C7#12012yes—
187C7×Q8—24no—
188C7×Q8—42no—
189C7×Q8—42no—
190C7×Q8—42no—
191C7×D4—24no—
192C7×D4—22no—
193C7×D4—44no—
194C7×D4—48no—
195C7×D4—48no—
196C7×D4#9914yescomplemented
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—24no—
200C2×C4×C7—22no—
201C2×C4×C7—21no—
202C2×C4×C7—22no—
203C2×C4×C7—21no—
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—41no—
207C2×C4×C7—42no—
208C2×C4×C7—48no—
209C2×C4×C7—48no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—48no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—42no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7#10412yescomplemented
220C2×C4×C7C2×C22⋊C411yescharacteristic
221C2×C4×C7C2×C22⋊C411yescharacteristic
222C2×C4×C7#12012yes—
223C2×Dic7—44no—
224C2×Dic7—82no—
225C2×Dic7—82no—
226C7⋊C8—82no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C2×#41—44no—
231C2×#41—44no—
232C2×#41C22⋊D714yes—
233C22×Q8C2×Dic711yescharacteristic
234C22×Q8C2×Dic711yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic711yescharacteristic
238C22×D4C22⋊D711yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4C2×Dic711yescharacteristic
243C23×C4C22×D711yescharacteristic
244C2×M16—142no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247#123—28no—
248#123—28no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8C22⋊D714yes—
253#121—28no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Dic711yescharacteristic
264C2×C4⋊C4C22⋊D711yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—142no—
269C2×C22⋊C4C22⋊D711yescharacteristic
270C2×C22⋊C4C22⋊D711yescharacteristic
271C2×C42—142no—
272C2×C42C2×Dic711yescharacteristic
273#104—288no—
274#99—144no—
275C22⋊C8—142no—
276C22⋊C8—142no—
277#95—142no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—22no—
281C22×C7—22no—
282C22×C7—24no—
283C22×C7—22no—
284C22×C7—44no—
285C22×C7—44no—
286C22×C7—42no—
287C22×C7—42no—
288C22×C7—41no—
289C22×C7—41no—
290C22×C7—41no—
291C22×C7—81no—
292C22×C7—81no—
293C22×C7—81no—
294C22×C7—81no—
295C22×C7#37911yescharacteristic
296C22×C7C2×#10411yescharacteristic
297C22×C7#42111yescharacteristic
298C4×C7—22no—
299C4×C7—21no—
300C4×C7—22no—
301C4×C7—41no—
302C4×C7—41no—
303C4×C7—42no—
304C4×C7—42no—
305C4×C7—42no—
306C4×C7C2×#9912yes—
307Dic7—84no—
308C24—21no—
309C24C22⋊Dic711yescharacteristic
310C24C2×C22⋊D711yescharacteristic
311#41—28no—
312#41—44no—
313#41—48no—
314#41—48no—
315#41—44no—
316#41—44no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8C22⋊Dic712yes—
322C2×Q8C22⋊Dic712yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4C22⋊Dic712yes—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—282no—
348C22×C4C22⋊Dic711yescharacteristic
349C22×C4C22⋊Dic711yescharacteristic
350C22×C4C22⋊Dic711yescharacteristic
351C22×C4C22×Dic711yescharacteristic
352C22×C4C2×C22⋊D712yes—
353C22×C4C2×C22⋊D712yes—
354C22×C4C2×C22⋊D711yescharacteristic
355M16—284no—
356C2×C8—282no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4C22⋊Dic712yes—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—144no—
370C22⋊C4—282no—
371C42—21no—
372C42—21no—
373C42—284no—
374C2×C7—21no—
375C2×C7—21no—
376C2×C7—42no—
377C2×C7—44no—
378C2×C7—81no—
379C2×C7#178111yescharacteristic
380C2×C7#178911yescharacteristic
381C2×C7#179611yescharacteristic
382C23—24no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×C22⋊Dic711yescharacteristic
397C23#651612yes—
398Q8—24no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402D4—24no—
403D4—22no—
404D4—44no—
405D4—48no—
406D4—48no—
407D4#640814yescomplemented
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—284no—
431C2×C4—562no—
432C2×C4—562no—
433C2×C4#641112yescomplemented
434C2×C4C2×C22⋊Dic711yescharacteristic
435C2×C4C2×C22⋊Dic711yescharacteristic
436C2×C4#651612yes—
437C8—562no—
438C7#836411yescharacteristic, complemented, Sylow
439C22—21no—
440C22—21no—
441C22—22no—
442C22—24no—
443C22—22no—
444C22—22no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C2×#641111yescharacteristic
457C22#9000811yescharacteristic
458C22#9001911yescharacteristic
459C4—22no—
460C4—21no—
461C4—22no—
462C4—41no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—564no—
468C4C2×#640812yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—44no—
473C2—81no—
474C2#1172988411yescharacteristic
475C2#1172989211yescharacteristic
476C2#1172989811yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC2×C7×#547C2
socleC22×C7#379
cosocleC23×C4×C7C23
hypercenterC2×#547D7
hypocenterC7#8364
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. #11729892C2semidirect
3. #11729442C2semidirect
4. C4×#6408C2semidirect
5. #11729356C2semidirect
6. C22×C7×#41C4semidirect
10. C2×C7×C4⋊Q8C4semidirect
13. C2×C7×C22⋊Q8C4semidirect
16. C2×C4×C7×D4C4semidirect
150. C22×#41Dic7semidirect
154. C2×C4⋊Q8Dic7semidirect
157. C2×C22⋊Q8Dic7semidirect
160. C2×C4×D4Dic7semidirect
196. C7×D4#99semidirect
219. C2×C4×C7#104semidirect
407. D4#6408semidirect
433. C2×C4#6411semidirect
438. C7#8364semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
#117298922
#117294422
C4×#64082
#117293562
#83647