Finite Group Databasebeta

← #11737982

Subgroups — #11737982

Order 896.

494 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×SD16C211yesmaximal, characteristic, complemented
2C2×#6474C211yesmaximal, characteristic
3#90366C211yesmaximal, characteristic, complemented
4#90236C211yesmaximal, characteristic, complemented
5C7×#463C211yesmaximal, characteristic, complemented
6#89934C211yesmaximal, characteristic, complemented
7#89925C211yesmaximal, characteristic, complemented
8#89829C211yesmaximal, characteristic, complemented
9#89791C211yesmaximal, characteristic, complemented
10#89613C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic, complemented
13#89574C211yesmaximal, characteristic, complemented
14#89559C211yesmaximal, characteristic, complemented
15#89544C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×D4×D7C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C2×C7×D8C2211yescharacteristic
21C7×#124C2211yescharacteristic, complemented
22C4×C7×D4C2211yescharacteristic
23#6507C2211yescharacteristic, complemented
24C2×#1143C2211yescharacteristic
25#6499C2211yescharacteristic, complemented
26D4×Dic7C2211yescharacteristic
27C2×#1142C2211yescharacteristic
28C2×#1142C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30C4×C22⋊D7—21no—
31C2×(C2×D7)⋊C4—21no—
32D7×SD16—24no—
33#6474—24no—
34C2×#1132C2211yescharacteristic
35C2×#1132C2211yescharacteristic
36C2×#1131C2211yescharacteristic
37C2×C8×D7C2211yescharacteristic
38#6456C2211yescharacteristic
39#6455C2211yescharacteristic
40#6446—21no—
41#6446—21no—
42(C22⋊D7)⋊C4—21no—
43D7×C22⋊C4—21no—
44#6441—21no—
45#6441—21no—
46C4×D28C2211yescharacteristic
47#6435C2211yescharacteristic
48#6432C2211yescharacteristic, complemented
49C7×C4⋊C8C2211yescharacteristic, complemented
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52#6409C2211yescharacteristic, complemented
53#6406C2211yescharacteristic, complemented
54#6390C2211yescharacteristic, complemented
55#6390C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58#6384C2211yescharacteristic, complemented
59#6384C2211yescharacteristic, complemented
60C4⋊(C7⋊C8)C2211yescharacteristic, complemented
61#3042—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71Q8×D7—22no—
72Q8×D7—22no—
73#1159—22no—
74#1159—22no—
75D4×D7—22no—
76D4×D7D414yes—
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7C2311yescharacteristic
82C2×Dic14—21no—
83C2×Dic14C2311yescharacteristic
84C2×Dic14C2311yescharacteristic
85C7×SD16—22no—
86C7×D8—22no—
87C2×C7×C8C2311yescharacteristic
88C2×C7×C8C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C42×C7C2311yescharacteristic
94C22⋊Dic7—21no—
95C22⋊Dic7—21no—
96C22⋊Dic7—21no—
97#1143—22no—
98#1142—22no—
99#1142—22no—
100#1141—22no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4D411yescharacteristic
105(C2×D7)⋊C4D411yescharacteristic
106C4⋊Dic7—21no—
107C4⋊Dic7—21no—
108C4⋊Dic7C2311yescharacteristic
109Dic7⋊C4—21no—
110Dic7⋊C4—21no—
111Dic7⋊C4D411yescharacteristic
112Dic7⋊C4D411yescharacteristic
113C4×Dic7—21no—
114C4×Dic7C2311yescharacteristic
115C2×C7⋊C8C2311yescharacteristic
116C2×C7⋊C8C2311yescharacteristic
117#1132—22no—
118#1132—22no—
119#1131—22no—
120C8×D7—22no—
121C2×AGL(1,ℤ/8)—71no—
122C22×SD16—71no—
123#536—71no—
124#518—71no—
125#463—71no—
126#463—71no—
127#463—71no—
128#463D711yescharacteristic, complemented
129#461—71no—
130#461—71no—
131#461—71no—
132#461—71no—
133#424—71no—
134#419—71no—
135#416—71no—
136C23×C7—21no—
137C23×C7—21no—
138C22×D7—21no—
139C22×D7—24no—
140C22×D7—81no—
141C22×D7—81no—
142C22×D7C2×D411yescharacteristic
143C7×Q8—21no—
144C7×Q8—21no—
145C7×D4—21no—
146C7×D4—21no—
147C7×D4—21no—
148C7×D4C2×D412yes—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—81no—
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2411yescharacteristic
155C22⋊D7—24no—
156C22⋊D7—42no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—42no—
161C2×Dic7—81no—
162C2×Dic7C2×D411yescharacteristic
163C2×Dic7C2×D411yescharacteristic
164C2×Dic7C2×D411yescharacteristic
165D28—21no—
166D28C2×D412yes—
167C4×D7—42no—
168C4×D7—81no—
169C4×D7—81no—
170C4×D7C2×D412yes—
171C4×D7#4112yes—
172Dic14—21no—
173Dic14—21no—
174Dic14—21no—
175Dic14—21no—
176Dic14—42no—
177C7×C8—21no—
178C7×C8—21no—
179C7⋊C8—21no—
180C7⋊C8—21no—
181C2×#41—71no—
182C22×Q8—71no—
183C22×D4—71no—
184AGL(1,ℤ/8)—144no—
185C2×SD16—71no—
186C2×SD16—71no—
187C2×SD16—71no—
188C2×SD16—71no—
189C2×SD16—71no—
190C2×SD16—144no—
191C2×SD16C2×D711yescharacteristic
192C2×D8—71no—
193C2×D8C2×D711yescharacteristic
194C2×M16—71no—
195C22×C8—71no—
196C4⋊Q8—71no—
197C4⋊Q8—71no—
198#124—71no—
199#124C2×D711yescharacteristic, complemented
200#123—141no—
201#123—141no—
202C22⋊Q8—141no—
203C22⋊Q8—141no—
204C4×D4—71no—
205C4×D4—71no—
206C4×D4—71no—
207C4×D4—141no—
208C4×D4—141no—
209C4×D4C2×D711yescharacteristic
210#117—71no—
211#117—71no—
212C2×C22⋊C4—141no—
213C2×C22⋊C4—141no—
214C4⋊C8—71no—
215C4⋊C8—71no—
216C4⋊C8—71no—
217C4⋊C8C2×D711yescharacteristic, complemented
218#103—71no—
219#103—71no—
220#103—71no—
221#103—71no—
222#103—71no—
223#103C2×D711yescharacteristic, complemented
224#102—71no—
225#102C2×D711yescharacteristic, complemented
226C22×C7—22no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7C22×D411yescharacteristic
233C2×D7—21no—
234C2×D7—21no—
235C2×D7—22no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—84no—
239C2×D7#12112yes—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7C22×D411yescharacteristic
245C4×C7C2×#4111yescharacteristic
246Dic7—21no—
247Dic7—21no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7#12112yes—
252C24—141no—
253#41—142no—
254#41—142no—
255C2×Q8—71no—
256C2×Q8—71no—
257C2×Q8—141no—
258C2×Q8—142no—
259C2×Q8—142no—
260C2×Q8C22×D711yescharacteristic
261C2×D4—74no—
262C2×D4—71no—
263C2×D4—142no—
264C2×D4—141no—
265C2×D4—141no—
266C2×D4C22×D711yescharacteristic
267C2×D4C22×D711yescharacteristic
268C22×C4—21no—
269C22×C4—71no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—141no—
275SD16—22no—
276SD16—142no—
277SD16—142no—
278SD16—142no—
279SD16—142no—
280SD16—142no—
281D8—22no—
282D8—142no—
283M16—142no—
284C2×C8—71no—
285C2×C8—71no—
286C2×C8—142no—
287C2×C8C22×D711yescharacteristic
288C2×C8C22×D711yescharacteristic
289C4⋊C4—71no—
290C4⋊C4—71no—
291C4⋊C4—71no—
292C4⋊C4—141no—
293C4⋊C4—141no—
294C4⋊C4—141no—
295C4⋊C4—141no—
296C4⋊C4C22×D711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—71no—
301C22⋊C4—71no—
302C22⋊C4—141no—
303C22⋊C4—141no—
304C22⋊C4—141no—
305C22⋊C4—141no—
306C22⋊C4—141no—
307C22⋊C4—141no—
308C42—71no—
309C42—141no—
310C42C22×D711yescharacteristic
311C2×C7—42no—
312C2×C7—81no—
313C2×C7C2×#12111yescharacteristic
314C2×C7#57711yescharacteristic
315C2×C7#57711yescharacteristic
316D7—22no—
317D7—42no—
318C23—21no—
319C23—21no—
320C23—71no—
321C23—144no—
322C23—141no—
323C23—561no—
324C23—561no—
325Q8—21no—
326Q8—21no—
327Q8—141no—
328Q8—141no—
329Q8—141no—
330Q8—141no—
331Q8—282no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—72no—
336D4—144no—
337D4—141no—
338D4—282no—
339D4D4×D712yes—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—71no—
343C2×C4—72no—
344C2×C4—72no—
345C2×C4—71no—
346C2×C4—71no—
347C2×C4—81no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—282no—
352C2×C4—282no—
353C2×C4—561no—
354C2×C4—561no—
355C2×C4—561no—
356C2×C4D4×D711yescharacteristic
357C2×C4D4×D711yescharacteristic
358C2×C4C23×D711yescharacteristic
359C8—21no—
360C8—21no—
361C8—141no—
362C8—141no—
363C7#304211yescharacteristic, complemented, Sylow
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—72no—
371C22—141no—
372C22—141no—
373C22—142no—
374C22—281no—
375C22—281no—
376C22—564no—
377C22C2×D4×D711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—72no—
383C4—141no—
384C4—141no—
385C4—281no—
386C4—281no—
387C4—281no—
388C4C2×D4×D711yescharacteristic
389C4D7×#4111yescharacteristic
390C2—42no—
391C2—81no—
392C2—142no—
393C2—282no—
394C2D7×#12111yescharacteristic
395C2#9046111yescharacteristic
396C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
prime radicalC2×#6474C2
hypercenter#463D7
hypocenterC7#3042
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×D7×SD16C2semidirect
3. #90366C2semidirect
4. #90236C2semidirect
5. C7×#463C2semidirect
6. #89934C2semidirect
7. #89925C2semidirect
8. #89829C2semidirect
9. #89791C2semidirect
10. #89613C2semidirect
11. #89611C2semidirect
12. #89590C2semidirect
13. #89574C2semidirect
14. #89559C2semidirect
15. #89544C2semidirect
21. C7×#124C22semidirect
23. #6507C22semidirect
25. #6499C22semidirect
48. #6432C22semidirect
49. C7×C4⋊C8C22semidirect
50. C7×#103C22semidirect
51. C7×#102C22semidirect
52. #6409C22semidirect
53. #6406C22semidirect
54. #6390C22semidirect
55. #6390C22semidirect
56. Dic7⋊C8C22semidirect
57. Dic7⋊C8C22semidirect
58. #6384C22semidirect
59. #6384C22semidirect
60. C4⋊(C7⋊C8)C22semidirect
128. #463D7semidirect
199. #124C2×D7semidirect
217. C4⋊C8C2×D7semidirect
223. #103C2×D7semidirect
225. #102C2×D7semidirect
363. C7#3042semidirect

Maximal subgroups

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

subgroupindex
C2×D7×SD162
C2×#64742
#903662
#902362
C7×#4632
#899342
#899252
#898292
#897912
#896132
#896112
#895902
#895742
#895592
#895442
#30427