Finite Group Databasebeta

← #13408173

Subgroups — #13408173

Order 1792.

1088 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#540C211yesmaximal, characteristic
2C2×#89843C211yesmaximal, characteristic, complemented
3C2×#89836C211yesmaximal, characteristic, complemented
4#11739710C211yesmaximal, characteristic, complemented
5#11729518C211yesmaximal, characteristic, complemented
6#11729495C211yesmaximal, characteristic, complemented
7#11729434C212yesmaximal, complemented
8#11729433C212yesmaximal, complemented
9#11729361C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11#11729336C211yesmaximal, characteristic, complemented
12C7×#540—24no—
13C7×#540—24no—
14C2×C7×#127C2211yescharacteristic
15C2×C7×#125C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×#121—22no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4C2212yes—
20C2×C7×#117C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C2×#6406C2212yes—
23C2×#6406C2212yes—
24C2×#6406C2212yes—
25C2×#6406C2212yes—
26#89984C2212yes—
27C2×C22⋊(C7⋊C8)C2211yescharacteristic
28C2×C22⋊(C7⋊C8)C2211yescharacteristic
29#89843—24no—
30#89836—24no—
31#89756—22no—
32C2×#6382C2212yes—
33C2×#6381C2212yes—
34#89720—22no—
35#89707C2211yescharacteristic
36#89700C2212yes—
37C2×#6377C2211yescharacteristic
38C2×C4×C7⋊C8C2211yescharacteristic
39#89324C2212yes—
40#89324C2212yes—
41#89319C2211yescharacteristic
42#89319C2211yescharacteristic
43#25682—71nomaximal, Sylow
44C22×C7×D4—22no—
45C22×C7×D4—22no—
46C22×C7×D4C2312yes—
47C23×C4×C7—22no—
48C23×C4×C7C2311yescharacteristic
49C7×#127—22no—
50C7×#127D414yes—
51C7×#125—22no—
52C7×#125D414yes—
53C7×#123—216no—
54C7×#121—216no—
55C7×#121—24no—
56C7×#121—24no—
57C4×C7×D4—24no—
58C4×C7×D4—24no—
59C7×#117—22no—
60C7×#117—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4D412yes—
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4D412yes—
67C2×C42×C7C2311yescharacteristic
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7C2312yes—
72C2×C4⋊Dic7C2312yes—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C22×C7⋊C8C2311yescharacteristic
76C22×C7⋊C8C2311yescharacteristic
77C22×C7⋊C8C2311yescharacteristic
78C22×C7⋊C8C2311yescharacteristic
79#6433—22no—
80#6433—22no—
81#6406—24no—
82#6406—24no—
83#6406—24no—
84#6406—24no—
85C7⋊#95—22no—
86C7⋊#95—22no—
87C22⋊(C7⋊C8)—22no—
88C22⋊(C7⋊C8)—22no—
89#6382—24no—
90#6381—24no—
91#6377—22no—
92C4×C7⋊C8—22no—
93C2×#540D711yescharacteristic
94C2×#499—71no—
95C2×#490—71no—
96#2629—71no—
97#2034—72no—
98#2033—72no—
99#1992—71no—
100#1957—71no—
101#1864—72no—
102#1862—72no—
103#1750—71no—
104C24×C7—22no—
105C24×C7C2×D411yescharacteristic
106C2×C7×D4—28no—
107C2×C7×D4—28no—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4—22no—
113C2×C7×D4—82no—
114C2×C7×D4—82no—
115C2×C7×D4—82no—
116C2×C7×D4—82no—
117C2×C7×D4C2×D414yes—
118C22×C4×C7—22no—
119C22×C4×C7—28no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—82no—
127C22×C4×C7—82no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D412yes—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2411yescharacteristic
132C22×Dic7—22no—
133C22×Dic7—22no—
134C7×C4⋊C4—28no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4C2×D414yes—
138C7×C22⋊C4—28no—
139C7×C22⋊C4—28no—
140C7×C22⋊C4—24no—
141C42×C7—21no—
142C42×C7—21no—
143C42×C7C2×D412yes—
144C42×C7C2×D412yes—
145C22⋊Dic7—82no—
146C4⋊Dic7—22no—
147C4⋊Dic7—22no—
148C4⋊Dic7—22no—
149C4⋊Dic7—22no—
150C4⋊Dic7—82no—
151Dic7⋊C4—44no—
152Dic7⋊C4—82no—
153C2×C7⋊C8—21no—
154C2×C7⋊C8—21no—
155C2×C7⋊C8—21no—
156C2×C7⋊C8—21no—
157C2×C7⋊C8—21no—
158C2×C7⋊C8—21no—
159C2×C7⋊C8—21no—
160C2×C7⋊C8—21no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—21no—
163C2×C7⋊C8#4112yes—
164C2×C7⋊C8#4112yes—
165#540—24no—
166#540—24no—
167C2×#127C2×D711yescharacteristic
168C2×#125—71no—
169C2×#125C2×D711yescharacteristic
170C2×#123—22no—
171C2×#121—22no—
172C2×#121C2×D712yes—
173C2×C4×D4C2×D712yes—
174C2×#117C2×D711yescharacteristic
175#499—144no—
176#490—144no—
177C2×#107—72no—
178C2×#106—72no—
179C2×#102—72no—
180C2×#102—72no—
181C2×#102—72no—
182C2×#102—72no—
183C2×C22⋊C8—71no—
184C2×C22⋊C8—71no—
185C2×#97—71no—
186C2×C4×C8—71no—
187#398—142no—
188#396—142no—
189#389—72no—
190#384—71no—
191#380—72no—
192#340—72no—
193#340—72no—
194#336—71no—
195#336—71no—
196C23×C7—21no—
197C23×C7—24no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—22no—
201C23×C7—22no—
202C23×C7—42no—
203C23×C7—42no—
204C23×C7—42no—
205C23×C7—42no—
206C23×C7—42no—
207C23×C7—42no—
208C23×C7C22×D411yescharacteristic
209C7×D4—24no—
210C7×D4—24no—
211C7×D4—42no—
212C7×D4—42no—
213C7×D4—88no—
214C7×D4—88no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—21no—
219C2×C4×C7—24no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—88no—
231C2×C4×C7#12014yes—
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C22×D411yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×C4×C7C2×#4111yescharacteristic
236C2×C4×C72−1+411yescharacteristic
237C2×C4×C72−1+411yescharacteristic
238C2×Dic7—42no—
239C2×Dic7—42no—
240C2×Dic7—42no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C7⋊C8—22no—
245C7⋊C8—22no—
246C7⋊C8—41no—
247C7⋊C8—41no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C22×D712yes—
251C23×C4—22no—
252C23×C4C22×D711yescharacteristic
253C22×C8—71no—
254C22×C8—71no—
255C22×C8—71no—
256C22×C8—71no—
257#127—22no—
258#127C22⋊D714yes—
259#125—22no—
260#125—142no—
261#125—142no—
262#125C22⋊D714yes—
263#123—216no—
264#121—216no—
265#121—24no—
266#121—24no—
267C4×D4—24no—
268C4×D4—24no—
269#117—22no—
270#117—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—72no—
274C2×C4⋊C4—142no—
275C2×C4⋊C4—142no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4C22⋊D712yes—
278C2×C4⋊C4C22×D712yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—142no—
282C2×C22⋊C4C22⋊D712yes—
283C2×C42C22×D711yescharacteristic
284C2×C42C22×D711yescharacteristic
285#107—144no—
286#106—144no—
287#102—144no—
288#102—144no—
289#102—144no—
290#102—144no—
291C22⋊C8—142no—
292C22⋊C8—142no—
293#97—142no—
294C4×C8—142no—
295#95—142no—
296#95—142no—
297C22×C7—41no—
298C22×C7—41no—
299C22×C7—41no—
300C22×C7—44no—
301C22×C7—42no—
302C22×C7—44no—
303C22×C7—42no—
304C22×C7—44no—
305C22×C7—82no—
306C22×C7—82no—
307C22×C7—82no—
308C22×C7—82no—
309C22×C7C2×#12011yescharacteristic
310C22×C7#53211yescharacteristic
311C22×C7#53611yescharacteristic
312C22×C7#57612yes—
313C22×C7#57712yes—
314C4×C7—24no—
315C4×C7—41no—
316C4×C7—42no—
317C4×C7—44no—
318C4×C7#54711yescharacteristic
319C4×C7#54711yescharacteristic
320C4×C7#54811yescharacteristic
321C4×C7#54811yescharacteristic
322Dic7—82no—
323Dic7—82no—
324C24—22no—
325C24C2×C22⋊D711yescharacteristic
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4C2×C22⋊D714yes—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4C2×C22⋊D711yescharacteristic
351C22×C4C2×C22⋊D712yes—
352C22×C4C2×C22⋊D712yes—
353C22×C4C23×D711yescharacteristic
354C2×C8—72no—
355C2×C8—72no—
356C2×C8—141no—
357C2×C8—141no—
358C2×C8—141no—
359C2×C8—141no—
360C2×C8—141no—
361C2×C8—141no—
362C2×C8—141no—
363C2×C8—141no—
364C2×C8—141no—
365C2×C8—141no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—142no—
370C4⋊C4—142no—
371C4⋊C4—142no—
372C4⋊C4—142no—
373C4⋊C4—284no—
374C4⋊C4—562no—
375C4⋊C4—562no—
376C4⋊C4C2×C22⋊D714yes—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—562no—
381C42—21no—
382C42—21no—
383C42C2×C22⋊D712yes—
384C42C2×C22⋊D712yes—
385C2×C7—81no—
386C2×C7—84no—
387C2×C7#243211yescharacteristic
388C2×C7#301212yes—
389C2×C7#317411yescharacteristic
390C2×C7#317611yescharacteristic
391C2×C7#324112yes—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—24no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23C22×C22⋊D711yescharacteristic
405D4—24no—
406D4—24no—
407D4—42no—
408D4—42no—
409D4—88no—
410D4—88no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—88no—
427C2×C4—282no—
428C2×C4—282no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4#651614yes—
434C2×C4D7×#4111yescharacteristic
435C2×C4D7×#4111yescharacteristic
436C2×C4#655411yescharacteristic
437C2×C4#655411yescharacteristic
438C2×C4C22×C22⋊D711yescharacteristic
439C2×C4C22×C22⋊D711yescharacteristic
440C8—142no—
441C8—142no—
442C8—281no—
443C8—281no—
444C7#2568211yescharacteristic, complemented, Sylow
445C22—41no—
446C22—44no—
447C22—44no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—41no—
452C22—41no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22#9051611yescharacteristic
458C22#9052311yescharacteristic
459C22#9052612yes—
460C22#9052712yes—
461C22C2×#651611yescharacteristic
462C4—24no—
463C4—41no—
464C4—42no—
465C4—44no—
466C4—562no—
467C4—562no—
468C4#9051511yescharacteristic
469C4#9051511yescharacteristic
470C4#9052211yescharacteristic
471C4#9052211yescharacteristic
472C2—81no—
473C2—84no—
474C2#1174087512yes—
475C2#1174087611yescharacteristic
476C2#1174087711yescharacteristic
477C2#1174153811yescharacteristic
478C2#1174154712yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#540C2
socleC23×C7C22×D4
hypercenterC2×#540D7
hypocenterC7#25682
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
2. C2×#89843C2semidirect
3. C2×#89836C2semidirect
4. #11739710C2semidirect
5. #11729518C2semidirect
6. #11729495C2semidirect
7. #11729434C2semidirect
8. #11729433C2semidirect
9. #11729361C2semidirect
10. #11729360C2semidirect
11. #11729336C2semidirect
444. C7#25682semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5402
C2×#898432
C2×#898362
#117397102
#117295182
#117294952
#117294342
#117294332
#117293612
#117293602
#117293362
#256827