Finite Group Databasebeta

← #13271357

Subgroups — #13271357

Order 1792.

624 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
1C7×#3011C211yesmaximal, characteristic
2#11741557C211yesmaximal, characteristic
3#11729898C211yesmaximal, characteristic
4#11729627C211yesmaximal, characteristic, complemented
5#11729624C211yesmaximal, characteristic, complemented
6#11725499C211yesmaximal, characteristic, complemented
7#11725498C211yesmaximal, characteristic, complemented
8C2×C7×2−1+4C2211yescharacteristic
9C2×C7×#137C411yescharacteristic, complemented
10C22×C7×Q16C411yescharacteristic, complemented
11C7×#536C2211yescharacteristic
12#90510—21no—
13C7×#452—21no—
14C7×#452—21no—
15C7×#451—21no—
16C7×#451—21no—
17C7×#419C411yescharacteristic, complemented
18C7×#416C411yescharacteristic, complemented
19C7×#408C2211yescharacteristic
20#90012C2211yescharacteristic
21#90009—22no—
22#90009C2211yescharacteristic
23#90009C2211yescharacteristic
24#90009C2211yescharacteristic
25#90008—21no—
26C2×#6410—21no—
27C2×#6410—22no—
28#89997—21no—
29#89343—22no—
30#89334—22no—
31#13096—71nomaximal, Sylow
32C7×2−1+4—42no—
33C7×2−1+4—42no—
34C2×C7×#41—21no—
35C2×C7×#41—21no—
36C2×C7×#41—21no—
37C2×C7×#41C2×C411yescharacteristic
38C2×C7×#41D411yescharacteristic
39C2×C7×#41D411yescharacteristic
40C2×C7×#41C2311yescharacteristic
41C22×C7×Q8—21no—
42C22×C7×Q8C2×C411yescharacteristic
43C22×C7×Q8C2×C411yescharacteristic
44C22×C7×Q8D411yescharacteristic
45C22×C7×Q8D411yescharacteristic
46C7×#137—42no—
47C2×C7×Q16—21no—
48C2×C7×Q16—21no—
49C2×C7×Q16—21no—
50C2×C7×Q16—42no—
51C2×C7×SD16—21no—
52C2×C7×M16C2×C411yescharacteristic
53C22×C7×C8C2×C411yescharacteristic
54C7×C4⋊Q8—21no—
55C7×#124—21no—
56C7×#123—21no—
57C7×#123—21no—
58C7×C22⋊Q8—21no—
59C7×C22⋊Q8—21no—
60C7×#117C2×C411yescharacteristic
61#6511—42no—
62#6511—42no—
63#6489—21no—
64C2×#1136—24no—
65C2×#1136—21no—
66C2×#1136—21no—
67C2×#1136—21no—
68C7×#103—21no—
69C7×#103—21no—
70C7×#103—21no—
71C7×#102—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74#6411—42no—
75#6410—24no—
76#6410—21no—
77#6410—21no—
78#6410D412yes—
79#6409—42no—
80#6408—21no—
81#6408—21no—
82#6407—21no—
83#6407—21no—
84#6407D412yes—
85#3011D711yescharacteristic
86#2889—71no—
87#2135—71no—
88#2123—71no—
89#1796—71no—
90#1395—71no—
91#1392—71no—
92C7×#41—22no—
93C7×#41—22no—
94C7×#41—22no—
95C7×#41—22no—
96C7×#41—41no—
97C7×#41—41no—
98C7×#41—41no—
99C7×#41—41no—
100C7×#41—44no—
101C7×#41—44no—
102C7×#41—44no—
103C2×C7×Q8—21no—
104C2×C7×Q8—21no—
105C2×C7×Q8—21no—
106C2×C7×Q8—22no—
107C2×C7×Q8—22no—
108C2×C7×Q8—21no—
109C2×C7×Q8—21no—
110C2×C7×Q8—41no—
111C2×C7×Q8—41no—
112C2×C7×Q8—41no—
113C2×C7×Q8—41no—
114C2×C7×Q8—44no—
115C2×C7×Q8—41no—
116C2×C7×Q8—41no—
117C2×C7×Q8C22⋊C411yescharacteristic
118C2×C7×Q8C22⋊C411yescharacteristic
119C2×C7×Q8C22⋊C411yescharacteristic
120C2×C7×Q8C2×D411yescharacteristic
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4C22⋊C411yescharacteristic
125C2×C7×D4C2×D411yescharacteristic
126C2×C7×D4#4111yescharacteristic
127C2×C7×D4#4111yescharacteristic
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7C22×C411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C7×Q16—42no—
139C7×Q16—42no—
140C7×Q16—42no—
141C7×SD16—42no—
142C7×M16—41no—
143C2×C7×C8—21no—
144C2×C7×C8—21no—
145C2×C7×C8—41no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—41no—
148C7×C4⋊C4—41no—
149C7×C22⋊C4—21no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—41no—
152C42×C7—21no—
153C22⋊Dic7—41no—
154C4⋊Dic7—41no—
155C4×Dic7—41no—
156#1136—22no—
157#1136—22no—
158#1136—44no—
159#1136—41no—
160#1136—41no—
161#1136—41no—
162#1136—41no—
163C2×C7⋊C8—44no—
164C2×C7⋊C8—41no—
165C2×C7⋊C8—41no—
166C2×C7⋊C8—41no—
167C2×2−1+4C2×D711yescharacteristic
168C2×#137Dic711yescharacteristic, complemented
169C22×Q16Dic711yescharacteristic, complemented
170#568—141no—
171#536C2×D711yescharacteristic
172#452—21no—
173#452—21no—
174#451—21no—
175#451—21no—
176#421—141no—
177#419—141no—
178#419Dic711yescharacteristic, complemented
179#416Dic711yescharacteristic, complemented
180#413—71no—
181#413—71no—
182#413—71no—
183#413—142no—
184C2×#101—142no—
185C2×#101—141no—
186#410—71no—
187#408C2×D711yescharacteristic
188#356—142no—
189#352—142no—
190C23×C7—21no—
191C23×C7C4×D411yescharacteristic
192C23×C7#12111yescharacteristic
193C23×C7#12311yescharacteristic
194C7×Q8—21no—
195C7×Q8—21no—
196C7×Q8—21no—
197C7×Q8—21no—
198C7×Q8—21no—
199C7×Q8—41no—
200C7×Q8—42no—
201C7×Q8—42no—
202C7×Q8—42no—
203C7×Q8—41no—
204C7×Q8—41no—
205C7×Q8—81no—
206C7×Q8#9912yes—
207C7×D4—24no—
208C7×D4—21no—
209C7×D4—41no—
210C7×D4—41no—
211C7×D4—41no—
212C7×D4—42no—
213C7×D4—42no—
214C7×D4#9912yes—
215C2×C4×C7—24no—
216C2×C4×C7—21no—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—21no—
220C2×C4×C7—21no—
221C2×C4×C7—21no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—81no—
233C2×C4×C7C2×C22⋊C411yescharacteristic
234C2×C4×C7C4×D411yescharacteristic
235C2×C4×C7#12011yescharacteristic
236C2×C4×C7#12111yescharacteristic
237C2×Dic7—81no—
238C7×C8—41no—
239C7×C8—41no—
240C7⋊C8—42no—
241C7⋊C8—42no—
242C7⋊C8—81no—
2432−1+4—42no—
2442−1+4—42no—
245C2×#41—21no—
246C2×#41—21no—
247C2×#41—21no—
248C2×#41C2×Dic711yescharacteristic
249C2×#41C22⋊D711yescharacteristic
250C2×#41C22⋊D711yescharacteristic
251C2×#41C22×D711yescharacteristic
252C22×Q8—21no—
253C22×Q8C2×Dic711yescharacteristic
254C22×Q8C2×Dic711yescharacteristic
255C22×Q8C22⋊D711yescharacteristic
256C22×Q8C22⋊D711yescharacteristic
257#137—42no—
258C2×Q16—21no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×Q16—42no—
262C2×SD16—21no—
263#131—282no—
264#131—282no—
265C2×M16—144no—
266C2×M16—141no—
267C2×M16—141no—
268C2×M16—141no—
269C2×M16C2×Dic711yescharacteristic
270C22×C8C2×Dic711yescharacteristic
271C4⋊Q8—21no—
272#124—21no—
273#123—21no—
274#123—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277#117—141no—
278#117C2×Dic711yescharacteristic
279#104—282no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—282no—
284#102—21no—
285#101—72no—
286#101—144no—
287#101—141no—
288#101—141no—
289#100—72no—
290#100—141no—
291#100—141no—
292#99—141no—
293#99—141no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22×C7—21no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—22no—
300C22×C7—41no—
301C22×C7—41no—
302C22×C7—41no—
303C22×C7—81no—
304C22×C7#38611yescharacteristic
305C4×C7—21no—
306C4×C7—21no—
307C4×C7—22no—
308C4×C7—21no—
309C4×C7—41no—
310C4×C7—41no—
311C4×C7—81no—
312C4×C7—81no—
313C4×C7C2×#9911yescharacteristic
314C4×C7C2×#9911yescharacteristic
315Dic7—161no—
316#41—22no—
317#41—22no—
318#41—22no—
319#41—22no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—44no—
325#41—44no—
326#41—44no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—44no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8C22⋊Dic711yescharacteristic
342C2×Q8C22⋊Dic711yescharacteristic
343C2×Q8C22⋊Dic711yescharacteristic
344C2×Q8D4×D711yescharacteristic
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4C22⋊Dic711yescharacteristic
349C2×D4D4×D711yescharacteristic
350C2×D4#115911yescharacteristic
351C2×D4#115911yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4C22×Dic711yescharacteristic
360C22×C4C2×C22⋊D711yescharacteristic
361C22×C4C2×C22⋊D711yescharacteristic
362Q16—42no—
363Q16—42no—
364Q16—42no—
365SD16—42no—
366M16—41no—
367M16—142no—
368M16—142no—
369M16—284no—
370M16—281no—
371M16—281no—
372M16—281no—
373M16—281no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—41no—
377C2×C8—284no—
378C2×C8—281no—
379C2×C8—281no—
380C2×C8—281no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—281no—
385C22⋊C4—21no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—281no—
389C42—21no—
390C42—281no—
391C2×C7—21no—
392C2×C7—41no—
393C2×C7—41no—
394C2×C7—41no—
395C2×C7—42no—
396C2×C7#189811yescharacteristic
397C23—21no—
398C23D4×Dic711yescharacteristic
399C23#649811yescharacteristic
400C23#650211yescharacteristic
401Q8—21no—
402Q8—21no—
403Q8—21no—
404Q8—21no—
405Q8—21no—
406Q8—41no—
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—41no—
411Q8—41no—
412Q8—81no—
413Q8#640812yes—
414D4—24no—
415D4—21no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—42no—
420D4—42no—
421D4#640812yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—561no—
441C2×C4D4×Dic711yescharacteristic
442C2×C4#650011yescharacteristic
443C2×C4#650211yescharacteristic
444C2×C4C2×C22⋊Dic711yescharacteristic
445C8—41no—
446C8—41no—
447C8—282no—
448C8—282no—
449C8—561no—
450C7#1309611yescharacteristic, complemented, Sylow
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22#8999211yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—81no—
467C4—81no—
468C4—1121no—
469C4C2×#640811yescharacteristic
470C4C2×#640811yescharacteristic
471C2—21no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2—42no—
476C2#1172942411yescharacteristic
subgroupG/H
centerC2#11729424
commutatorC22×C4×C7C22×C4
FrattiniC2×#41C22×D7
FittingC7×#3011C2
socleC2×C7#1898
prime radicalC2×C7×2−1+4C22
cosocleC2×C7×#41C23
hypercenter#3011D7
hypocenterC7#13096
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ #3011

Complements

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

NH ≅ G/Ntype
4. #11729627C2semidirect
5. #11729624C2semidirect
6. #11725499C2semidirect
7. #11725498C2semidirect
9. C2×C7×#137C4semidirect
10. C22×C7×Q16C4semidirect
17. C7×#419C4semidirect
18. C7×#416C4semidirect
168. C2×#137Dic7semidirect
169. C22×Q16Dic7semidirect
178. #419Dic7semidirect
179. #416Dic7semidirect
450. C7#13096semidirect

Maximal subgroups

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

subgroupindex
C7×#30112
#117415572
#117298982
#117296272
#117296242
#117254992
#117254982
#130967