Finite Group Databasebeta

← #13403943

Subgroups — #13403943

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2547C211yesmaximal, characteristic
2C2×#89866C211yesmaximal, characteristic, complemented
3#11739821C211yesmaximal, characteristic
4C2×D4×C7⋊C8C211yesmaximal, characteristic
5#11729541C211yesmaximal, characteristic, complemented
6#11727280C212yesmaximal, complemented
7#11727274C212yesmaximal, complemented
8#11727268C212yesmaximal, complemented
9#11727265C212yesmaximal, complemented
10C4×#6384C211yesmaximal, characteristic, complemented
11#11727221C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×#124—21no—
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6487C2212yes—
16C23×C7⋊C8C2212yes—
17C2×C4×Dic14C2211yescharacteristic
18C7×#397—22no—
19C7×#391—21no—
20C7×#380C2212yes—
21C7×#380C2212yes—
22C4×C7×C4⋊C4C2211yescharacteristic
23#89993—21no—
24#89984C2212yes—
25C2×C22⋊(C7⋊C8)C2212yes—
26#89866—24no—
27D4×C7⋊C8—24no—
28#89745—21no—
29C2×#6384C2211yescharacteristic
30C2×#6384C2211yescharacteristic
31C2×#6384C2212yes—
32C2×#6382C2212yes—
33C2×#6382C2212yes—
34#89720—22no—
35#89707C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4×C7⋊C8C2211yescharacteristic
38#89324C2212yes—
39#89324C2211yescharacteristic
40#89324C2211yescharacteristic
41#89324C2212yes—
42#89324C2212yes—
43#25341—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×D4D411yescharacteristic
46C23×C4×C7C2312yes—
47C22×Dic14C2311yescharacteristic
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8—22no—
50C7×#124—44no—
51C4×C7×D4—22no—
52C4×C7×D4—22no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C7×C4⋊C4C2312yes—
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C7×C22⋊C4—22no—
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4D412yes—
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63C2×C22⋊Dic7—22no—
64C2×C22⋊Dic7—22no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7C2311yescharacteristic
68C2×C4⋊Dic7C2312yes—
69#6487—24no—
70#6487—42no—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—22no—
73C2×C4×Dic7—21no—
74C22×C7⋊C8—24no—
75C22×C7⋊C8—24no—
76C22×C7⋊C8—42no—
77C22×C7⋊C8C2311yescharacteristic
78C22×C7⋊C8C2312yes—
79C22×C7⋊C8C2311yescharacteristic
80C4×Dic14—22no—
81C4×Dic14—22no—
82C7×#95—21no—
83C7×#95—22no—
84C7×#95—22no—
85C7⋊#95—22no—
86C22⋊(C7⋊C8)—24no—
87#6384—22no—
88#6384—24no—
89#6384—22no—
90#6382—24no—
91#6382—42no—
92C4⋊(C7⋊C8)D414yes—
93C4×C7⋊C8—22no—
94C2×#475—71no—
95C2×C8×D4—71no—
96#2547D711yescharacteristic
97#2538—71no—
98#2077—72no—
99#2035—72no—
100#1990—71no—
101#1948—71no—
102#1892—72no—
103#1807—72no—
104C4×#103—71no—
105C24×C7C2×D412yes—
106C2×C7×Q8—42no—
107C2×C7×Q8—42no—
108C2×C7×Q8—42no—
109C2×C7×D4—22no—
110C2×C7×D4—22no—
111C2×C7×D4SD1614yes—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7C2×D412yes—
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2411yescharacteristic
125C22×Dic7—21no—
126C22×Dic7—21no—
127C22×Dic7—22no—
128C2×Dic14—21no—
129C2×Dic14—21no—
130C2×Dic14—21no—
131C2×Dic14—21no—
132C2×Dic14—21no—
133C2×Dic14#4112yes—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—42no—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—44no—
144C7×C22⋊C4—44no—
145C42×C7—42no—
146C42×C7C2×D412yes—
147C42×C7#4112yes—
148C22⋊Dic7—44no—
149C22⋊Dic7—82no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—22no—
154C4⋊Dic7—42no—
155C4⋊Dic7—81no—
156Dic7⋊C4—44no—
157Dic7⋊C4—42no—
158C4×Dic7—42no—
159C2×C7⋊C8—21no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—21no—
166C2×C7⋊C8—48no—
167C2×C7⋊C8C2×D414yes—
168C2×C7⋊C8#4112yes—
169C23×C8—72no—
170C2×C4⋊Q8C2×D711yescharacteristic
171C2×#124—21no—
172C2×C22⋊Q8—72no—
173C2×C4×Q8—71no—
174C2×C4×D4C2×D711yescharacteristic
175#475—144no—
176C8×D4—144no—
177C2×#106—72no—
178C2×#106—72no—
179C2×C4⋊C8—71no—
180C2×#103—72no—
181C2×#103—71no—
182C2×#103—71no—
183C2×C22⋊C8—72no—
184C2×C4×C8—71no—
185#397—22no—
186#396—142no—
187#391—21no—
188#390—141no—
189#389—141no—
190#384—71no—
191#380—72no—
192#380C2×D712yes—
193#380C2×D712yes—
194C4×C4⋊C4C2×D711yescharacteristic
195#340—71no—
196#340—72no—
197#340—71no—
198#340—72no—
199#340—72no—
200C23×C7—22no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—22no—
206C23×C7C2×SD1614yes—
207C23×C7C22×D411yescharacteristic
208C7×Q8—84no—
209C7×D4—28no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—22no—
215C2×C4×C7—42no—
216C2×C4×C7—42no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—41no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—41no—
227C2×C4×C7—41no—
228C2×C4×C7#12112yes—
229C2×C4×C7#12112yes—
230C2×C4×C7C2×SD1612yes—
231C2×C4×C7#13512yes—
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C2×#4111yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×C4×C7C2×#4111yescharacteristic
236C2×C4×C7H(5,𝔽2)11yescharacteristic
237C2×C4×C72−1+411yescharacteristic
238C2×Dic7—22no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—41no—
246C2×Dic7—41no—
247Dic14—22no—
248Dic14—22no—
249Dic14—41no—
250Dic14—41no—
251C7⋊C8—22no—
252C7⋊C8—24no—
253C7⋊C8—41no—
254C22×Q8—21no—
255C22×Q8—71no—
256C22×D4C22⋊D711yescharacteristic
257C23×C4C22×D712yes—
258C22×C8—71no—
259C22×C8—72no—
260C22×C8—71no—
261C22×C8—144no—
262C22×C8—144no—
263C22×C8—282no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266#124—44no—
267C22⋊Q8—144no—
268C22⋊Q8—282no—
269C4×Q8—142no—
270C4×Q8—142no—
271C4×D4—22no—
272C4×D4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—71no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—141no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4C22⋊D711yescharacteristic
281C2×C4⋊C4C22×D712yes—
282C2×C4⋊C4C22×D711yescharacteristic
283C2×C4⋊C4C22×D711yescharacteristic
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—142no—
287C2×C22⋊C4—142no—
288C2×C22⋊C4C22⋊D712yes—
289C2×C42—21no—
290C2×C42—141no—
291C2×C42C22×D711yescharacteristic
292#106—144no—
293#106—282no—
294C4⋊C8—74no—
295#103—144no—
296#103—142no—
297#103—142no—
298C22⋊C8—144no—
299C4×C8—142no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—142no—
304C22×C7—24no—
305C22×C7—24no—
306C22×C7—24no—
307C22×C7—42no—
308C22×C7—42no—
309C22×C7—42no—
310C22×C7—42no—
311C22×C7C2×#12111yescharacteristic
312C22×C7#54611yescharacteristic
313C22×C7#54711yescharacteristic
314C22×C7C22×SD1611yescharacteristic
315C22×C7C2×#13511yescharacteristic
316C22×C7#57711yescharacteristic
317C22×C7#57811yescharacteristic
318C4×C7—22no—
319C4×C7—42no—
320C4×C7—81no—
321C4×C7—81no—
322C4×C7—82no—
323C4×C7#46514yes—
324C4×C7#52911yescharacteristic
325C4×C7#54211yescharacteristic
326C4×C7#54611yescharacteristic
327C4×C7#54811yescharacteristic
328Dic7—42no—
329Dic7—82no—
330Dic7—81no—
331C24C2×C22⋊D712yes—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—72no—
336C2×Q8—141no—
337C2×Q8—141no—
338C2×Q8—141no—
339C2×Q8—141no—
340C2×Q8—141no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4#114214yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—141no—
354C22×C4—141no—
355C22×C4—142no—
356C22×C4C2×C22⋊D712yes—
357C22×C4C2×C22⋊D711yescharacteristic
358C22×C4C2×C22⋊D711yescharacteristic
359C22×C4C23×D711yescharacteristic
360C2×C8—74no—
361C2×C8—72no—
362C2×C8—141no—
363C2×C8—142no—
364C2×C8—142no—
365C2×C8—141no—
366C2×C8—141no—
367C2×C8—141no—
368C2×C8—141no—
369C2×C8—288no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—42no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C4⋊C4—142no—
381C4⋊C4—142no—
382C4⋊C4—284no—
383C4⋊C4—282no—
384C4⋊C4—282no—
385C4⋊C4—561no—
386C22⋊C4—24no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—284no—
390C22⋊C4—562no—
391C42—42no—
392C42—282no—
393C42D4×D712yes—
394C42#115912yes—
395C2×C7—44no—
396C2×C7#242711yescharacteristic
397C2×C7C2×#46511yescharacteristic
398C2×C7#306311yescharacteristic
399C2×C7#329811yescharacteristic
400C2×C7#330411yescharacteristic
401C2×C7#333511yescharacteristic
402C2×C7#334211yescharacteristic
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23C2×#114214yes—
410C23C22×C22⋊D711yescharacteristic
411Q8—84no—
412Q8—142no—
413Q8—142no—
414Q8—281no—
415Q8—281no—
416D4—28no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—142no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4—281no—
439C2×C4—282no—
440C2×C4—282no—
441C2×C4—281no—
442C2×C4—282no—
443C2×C4—281no—
444C2×C4C2×#114212yes—
445C2×C4#650212yes—
446C2×C4#650212yes—
447C2×C4#651312yes—
448C2×C4C2×D4×D711yescharacteristic
449C2×C4C2×#115911yescharacteristic
450C2×C4D7×#4111yescharacteristic
451C2×C4D7×#4111yescharacteristic
452C2×C4#655311yescharacteristic
453C2×C4#655411yescharacteristic
454C8—144no—
455C8—142no—
456C8—281no—
457C7#2534111yescharacteristic, complemented, Sylow
458C22—24no—
459C22—24no—
460C22—24no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22C22×#114211yescharacteristic
466C22C2×#650211yescharacteristic
467C22C2×#651311yescharacteristic
468C22#9051511yescharacteristic
469C22#9052011yescharacteristic
470C22#9052811yescharacteristic
471C22#9052911yescharacteristic
472C4—22no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—82no—
477C4—282no—
478C4—561no—
479C4—562no—
480C4#8981314yes—
481C4#9036711yescharacteristic
482C4#9036811yescharacteristic
483C4#9037111yescharacteristic
484C4#9037311yescharacteristic
485C2—44no—
486C2C2×#8981311yescharacteristic
487C2#1174044111yescharacteristic
488C2#1174084511yescharacteristic
489C2#1174096211yescharacteristic
490C2#1174097611yescharacteristic
491C2#1174110411yescharacteristic
492C2#1174111811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2547C2
socleC23×C7C22×D4
prime radicalC22×C7×D4D4
hypercenter#2547D7
hypocenterC7#25341
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. C2×#89866C2semidirect
5. #11729541C2semidirect
6. #11727280C2semidirect
7. #11727274C2semidirect
8. #11727268C2semidirect
9. #11727265C2semidirect
10. C4×#6384C2semidirect
11. #11727221C2semidirect
457. C7#25341semidirect

Maximal subgroups

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

subgroupindex
C7×#25472
C2×#898662
#117398212
C2×D4×C7⋊C82
#117295412
#117272802
#117272742
#117272682
#117272652
C4×#63842
#117272212
#253417