Finite Group Databasebeta

← #13403311

Subgroups — #13403311

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2568C211yesmaximal, characteristic, complemented
2C2×#89855C211yesmaximal, characteristic
3#11739945C211yesmaximal, characteristic
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729539C211yesmaximal, characteristic, complemented
6#11727436C212yesmaximal, complemented
7#11727434C212yesmaximal
8#11727430C212yesmaximal
9#11727428C212yesmaximal, complemented
10C4×#6383C211yesmaximal, characteristic
11#11727221C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×#6509—21no—
15C22×#1143C2212yes—
16C2×C4×D28C2211yescharacteristic
17C7×#400—22no—
18C7×#391—21no—
19C7×#391C2212yes—
20C7×#391C2212yes—
21C7×#382—21no—
22C4×C7×C4⋊C4C2211yescharacteristic
23C2×#6409C2212yes—
24C2×#6409C2212yes—
25C2×#6409C2212yes—
26#89855—24no—
27Q8×C7⋊C8—24no—
28#89766—22no—
29#89759—21no—
30C2×#6383C2211yescharacteristic
31C2×#6383C2211yescharacteristic
32C2×#6382C2212yes—
33#89711C2212yes—
34#89707C2211yescharacteristic
35#89700C2212yes—
36C2×C4⋊(C7⋊C8)C2212yes—
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C2×C4×C7⋊C8C2212yes—
39C2×C4×C7⋊C8C2211yescharacteristic
40#89324C2211yescharacteristic
41#89324C2212yes—
42#89324C2211yescharacteristic
43#25293—71nomaximal, Sylow
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8C2312yes—
46C22×D28C2311yescharacteristic
47C22×C4×D7—21no—
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8—22no—
50C4×C7×Q8—22no—
51C4×C7×Q8—22no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C4⋊C4D412yes—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C42×C7—21no—
59C2×C42×C7C2312yes—
60C2×C42×C7C2311yescharacteristic
61#6509—44no—
62C2×#1143—24no—
63C2×#1143—24no—
64C2×#1143—42no—
65C2×(C2×D7)⋊C4—21no—
66C2×(C2×D7)⋊C4—22no—
67C2×(C2×D7)⋊C4—22no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2312yes—
72C2×Dic7⋊C4—22no—
73C22×C7⋊C8C2311yescharacteristic
74C22×C7⋊C8C2312yes—
75C22×C7⋊C8C2311yescharacteristic
76C4×D28—22no—
77C4×D28—22no—
78C7×#95—21no—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95—22no—
82C7×#95—22no—
83#6409—24no—
84#6409—24no—
85#6409—42no—
86C7⋊#95—22no—
87#6383—22no—
88#6383—22no—
89#6382—42no—
90C4⋊(C7⋊C8)—24no—
91C4⋊(C7⋊C8)D414yes—
92C4×C7⋊C8—22no—
93C4×C7⋊C8—24no—
94C2×#461—71no—
95C2×C8×Q8—71no—
96#2568D711yescharacteristic, complemented
97#2538—71no—
98#2066—72no—
99#2055—72no—
100#1968—72no—
101#1956—71no—
102#1948—71no—
103#1933—72no—
104C4×#102—71no—
105C23×D7—21no—
106C2×C7×Q8—22no—
107C2×C7×Q8—22no—
108C2×C7×Q8—22no—
109C2×C7×Q8—22no—
110C2×C7×Q8—22no—
111C2×C7×Q8—22no—
112C2×C7×Q8—22no—
113C2×C7×Q8—22no—
114C2×C7×Q8SD1614yescomplemented
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—22no—
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2411yescharacteristic
123C22×Dic7—21no—
124C22×Dic7—22no—
125C2×D28—21no—
126C2×D28—21no—
127C2×D28—21no—
128C2×D28—21no—
129C2×D28—21no—
130C2×D28#4112yes—
131C2×C4×D7—42no—
132C2×C4×D7—42no—
133C2×C4×D7—42no—
134C7×C4⋊C4—24no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—42no—
141C7×C4⋊C4—42no—
142C42×C7—22no—
143C42×C7—22no—
144C42×C7—42no—
145C42×C7C2×D412yes—
146C42×C7#4112yes—
147#1143—48no—
148(C2×D7)⋊C4—42no—
149(C2×D7)⋊C4—44no—
150(C2×D7)⋊C4—82no—
151C4⋊Dic7—21no—
152C4⋊Dic7—22no—
153C4⋊Dic7—21no—
154C4⋊Dic7—22no—
155C4⋊Dic7—42no—
156C4⋊Dic7—81no—
157Dic7⋊C4—44no—
158C2×C7⋊C8—21no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—21no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8C2×D414yes—
166C2×C7⋊C8#4112yes—
167C22×SD16—72no—
168C2×C4⋊Q8C2×D711yescharacteristic
169C2×C22⋊Q8—141no—
170C2×C4×Q8C2×D711yescharacteristic
171C2×C4×D4—71no—
172#461—144no—
173C8×Q8—144no—
174C2×#106—72no—
175C2×C4⋊C8—71no—
176C2×C4⋊C8—72no—
177C2×#103—72no—
178C2×#103—72no—
179C2×#103—72no—
180C2×#102—71no—
181C2×#102—71no—
182C2×C4×C8—72no—
183C2×C4×C8—71no—
184#400—22no—
185#396—142no—
186#391—21no—
187#391C2×D712yes—
188#391C2×D712yes—
189#390—72no—
190#389—72no—
191#384—71no—
192#382—21no—
193#380—141no—
194C4×C4⋊C4C2×D711yescharacteristic
195#340—72no—
196#340—71no—
197#340—71no—
198C23×C7C22×D411yescharacteristic
199C22×D7—22no—
200C22×D7—41no—
201C22×D7—41no—
202C22×D7—41no—
203C22×D7—41no—
204C22×D7—41no—
205C22×D7—41no—
206C7×Q8—28no—
207C7×Q8—42no—
208C7×Q8—42no—
209C7×Q8—42no—
210C7×Q8—42no—
211C2×C4×C7—22no—
212C2×C4×C7—22no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7#12112yes—
228C2×C4×C7#12112yes—
229C2×C4×C7C2×SD1614yes—
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—41no—
239C2×Dic7—42no—
240C2×Dic7—41no—
241C2×Dic7—42no—
242C2×Dic7—41no—
243C2×Dic7—42no—
244D28—22no—
245D28—22no—
246D28—41no—
247D28—41no—
248C4×D7—84no—
249C7⋊C8—24no—
250C7⋊C8—22no—
251C7⋊C8—41no—
252C22×Q8C22⋊D711yescharacteristic
253C22×Q8C22×D712yes—
254C22×D4—71no—
255C23×C4—141no—
256C2×SD16—144no—
257C2×SD16—144no—
258C2×SD16—282no—
259C22×C8—72no—
260C22×C8—71no—
261C22×C8—71no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C22⋊Q8—284no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×D4—142no—
268C4×D4—142no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—142no—
274C2×C4⋊C4—141no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4C22⋊D711yescharacteristic
277C2×C4⋊C4C22⋊D712yes—
278C2×C4⋊C4C22×D711yescharacteristic
279C2×C4⋊C4C22×D711yescharacteristic
280C2×C22⋊C4—141no—
281C2×C22⋊C4—142no—
282C2×C22⋊C4—142no—
283C2×C42—21no—
284C2×C42C22×D712yes—
285C2×C42C22×D711yescharacteristic
286#106—282no—
287C4⋊C8—74no—
288C4⋊C8—144no—
289#103—144no—
290#103—144no—
291#103—282no—
292#102—142no—
293#102—142no—
294C4×C8—144no—
295C4×C8—142no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—142no—
302C22×C7C2×#12111yescharacteristic
303C22×C7#54811yescharacteristic
304C22×C7#55011yescharacteristic
305C22×C7C22×SD1611yescharacteristic
306C22×C7C2×#13511yescharacteristic
307C22×C7#57711yescharacteristic
308C22×C7#57811yescharacteristic
309C2×D7—42no—
310C2×D7—42no—
311C2×D7—42no—
312C2×D7—81no—
313C2×D7—81no—
314C2×D7—81no—
315C2×D7—81no—
316C4×C7—24no—
317C4×C7—22no—
318C4×C7—42no—
319C4×C7—81no—
320C4×C7—81no—
321C4×C7—82no—
322C4×C7#46514yes—
323C4×C7#52911yescharacteristic
324C4×C7#54211yescharacteristic
325C4×C7#54611yescharacteristic
326C4×C7#54811yescharacteristic
327Dic7—82no—
328Dic7—81no—
329C24—141no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8#114314yescomplemented
339C2×D4—72no—
340C2×D4—141no—
341C2×D4—141no—
342C2×D4—141no—
343C2×D4—141no—
344C2×D4—141no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—141no—
349C22×C4—142no—
350C22×C4—282no—
351C22×C4—282no—
352C22×C4—282no—
353C22×C4C2×C22⋊D712yes—
354C22×C4C2×C22⋊D711yescharacteristic
355C22×C4C2×C22⋊D712yes—
356C22×C4C2×C22⋊D711yescharacteristic
357C22×C4C23×D711yescharacteristic
358SD16—288no—
359C2×C8—74no—
360C2×C8—72no—
361C2×C8—141no—
362C2×C8—142no—
363C2×C8—142no—
364C2×C8—141no—
365C2×C8—141no—
366C2×C8—141no—
367C2×C8—141no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—141no—
377C4⋊C4—141no—
378C4⋊C4—142no—
379C4⋊C4—142no—
380C4⋊C4—284no—
381C4⋊C4—282no—
382C4⋊C4—561no—
383C22⋊C4—282no—
384C22⋊C4—284no—
385C22⋊C4—562no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42D4×D712yes—
390C42#115912yes—
391C2×C7#242611yescharacteristic
392C2×C7C2×#46511yescharacteristic
393C2×C7#306311yescharacteristic
394C2×C7#335911yescharacteristic
395C2×C7#336011yescharacteristic
396C2×C7#339211yescharacteristic
397C2×C7#339311yescharacteristic
398D7—82no—
399C23—142no—
400C23—281no—
401C23—281no—
402C23—281no—
403C23—281no—
404C23—281no—
405C23—281no—
406C23C22×C22⋊D711yescharacteristic
407Q8—28no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412D4—142no—
413D4—142no—
414D4—281no—
415D4—281no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—281no—
433C2×C4—282no—
434C2×C4—281no—
435C2×C4—281no—
436C2×C4—282no—
437C2×C4—282no—
438C2×C4—564no—
439C2×C4#650212yes—
440C2×C4#650212yes—
441C2×C4C2×#114314yes—
442C2×C4C2×#114312yes—
443C2×C4#651312yes—
444C2×C4C2×D4×D711yescharacteristic
445C2×C4C2×#115911yescharacteristic
446C2×C4D7×#4111yescharacteristic
447C2×C4D7×#4111yescharacteristic
448C2×C4#655311yescharacteristic
449C2×C4#655411yescharacteristic
450C8—142no—
451C8—144no—
452C8—281no—
453C7#2529311yescharacteristic, complemented, Sylow
454C22—282no—
455C22—282no—
456C22—282no—
457C22—561no—
458C22—561no—
459C22—561no—
460C22—561no—
461C22C2×#650211yescharacteristic
462C22C22×#114311yescharacteristic
463C22C2×#651311yescharacteristic
464C22#9052211yescharacteristic
465C22#9052511yescharacteristic
466C22#9052811yescharacteristic
467C22#9052911yescharacteristic
468C4—24no—
469C4—22no—
470C4—42no—
471C4—82no—
472C4—81no—
473C4—81no—
474C4—561no—
475C4—562no—
476C4#8982014yes—
477C4#9036711yescharacteristic
478C4#9036811yescharacteristic
479C4#9037111yescharacteristic
480C4#9037311yescharacteristic
481C2—562no—
482C2C2×#8982011yescharacteristic
483C2#1174044011yescharacteristic
484C2#1174084511yescharacteristic
485C2#1174111211yescharacteristic
486C2#1174111511yescharacteristic
487C2#1174113011yescharacteristic
488C2#1174113311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2568C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2568D7
hypocenterC7#25293
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

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

NH ≅ G/Ntype
1. C7×#2568C2semidirect
4. C2×Q8×C7⋊C8C2semidirect
5. #11729539C2semidirect
6. #11727436C2semidirect
9. #11727428C2semidirect
11. #11727221C2semidirect
96. #2568D7semidirect
114. C2×C7×Q8SD16semidirect
338. C2×Q8#1143semidirect
453. C7#25293semidirect

Maximal subgroups

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

subgroupindex
C7×#25682
C2×#898552
#117399452
C2×Q8×C7⋊C82
#117295392
#117274362
#117274342
#117274302
#117274282
C4×#63832
#117272212
#252937