Finite Group Databasebeta

← #11736663

Subgroups — #11736663

Order 896.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#117C211yesmaximal, characteristic
2C2×#6460C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic, complemented
4#89898C211yesmaximal, characteristic
5#89892C211yesmaximal, characteristic, complemented
6#89776C211yesmaximal, characteristic, complemented
7#89736C211yesmaximal, characteristic, complemented
8#89660C212yesmaximal, complemented
9#89656C212yesmaximal, complemented
10#89639C212yesmaximal, complemented
11#89634C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×M16C2211yescharacteristic
15C7×#117C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4—21no—
18#6489C2211yescharacteristic
19C2×C4⋊Dic7—21no—
20C2×#1136C2211yescharacteristic
21#6460—21no—
22#6460—21no—
23#6460C2212yes—
24#6460C2212yes—
25#6455C2211yescharacteristic
26#6455C2211yescharacteristic
27D7×C4⋊C4C2211yescharacteristic
28D7×C4⋊C4C2211yescharacteristic
29D7×C22⋊C4—21no—
30#6440—21no—
31#6435—21no—
32C42×D7—21no—
33C7×#107C2212yescomplemented
34C7×#106C2212yescomplemented
35(C2×D7)⋊C8C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6390C2212yescomplemented
38#6390C2212yescomplemented
39#6384C2212yescomplemented
40#6384C2212yescomplemented
41#6382C2212yescomplemented
42#6381C2212yescomplemented
43#3095—71nomaximal, Sylow
44C23×D7D411yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C22×Dic7—21no—
48C22×Dic7D411yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—22no—
51C2×C4×D7—21no—
52C2×C4×D7—21no—
53C2×C4×D7D411yescharacteristic
54C2×C4×D7D411yescharacteristic
55C2×C4×D7C2311yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14—22no—
58C2×Dic14—22no—
59C2×Dic14—21no—
60C2×Dic14C2312yes—
61C7×M16—22no—
62C2×C7×C8C2312yes—
63C7×C4⋊C4—21no—
64C7×C4⋊C4C2311yescharacteristic
65C7×C4⋊C4C2312yes—
66C7×C4⋊C4C2311yescharacteristic
67C7×C22⋊C4—21no—
68C42×C7—21no—
69C22⋊Dic7—21no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—22no—
74C4⋊Dic7—22no—
75C4⋊Dic7—22no—
76C4⋊Dic7C2311yescharacteristic
77C4⋊Dic7C2311yescharacteristic
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81C4×Dic7—21no—
82C4×Dic7—21no—
83#1136—22no—
84C2×C7⋊C8C2312yes—
85C2×C22⋊Q8—71no—
86C2×#117—71no—
87#485—72no—
88#485—72no—
89#484—72no—
90#483—72no—
91#428—71no—
92#428D711yescharacteristic, complemented
93#419—71no—
94#419—71no—
95#407—71no—
96C23×C7C2×D411yescharacteristic
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—41no—
100C22×D7C2×D411yescharacteristic
101C22×D7C2×D411yescharacteristic
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—42no—
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7#4112yes—
108C2×C4×C7#4112yes—
109C2×C4×C7C2411yescharacteristic
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—42no—
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117C4×D7—22no—
118C4×D7—22no—
119C4×D7—21no—
120C4×D7—21no—
121C4×D7—21no—
122C4×D7—21no—
123C4×D7—42no—
124C4×D7—42no—
125Dic14—22no—
126Dic14—22no—
127Dic14—42no—
128C7×C8—22no—
129C7⋊C8—22no—
130C22×Q8—71no—
131C23×C4—71no—
132C2×M16—71no—
133C2×M16C2×D711yescharacteristic
134C22⋊Q8—72no—
135C22⋊Q8—72no—
136C22⋊Q8—141no—
137C22⋊Q8—141no—
138#117—71no—
139#117—71no—
140#117—71no—
141#117—141no—
142#117—141no—
143#117C2×D711yescharacteristic
144C2×C4⋊C4—71no—
145C2×C4⋊C4—71no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4C2×D711yescharacteristic
148C2×C22⋊C4—141no—
149C2×C22⋊C4—141no—
150C2×C42—141no—
151#107—72no—
152#107C2×D712yescomplemented
153#106—72no—
154#106C2×D712yescomplemented
155#103—72no—
156#103—72no—
157#103—72no—
158#103—72no—
159C22⋊C8—72no—
160C22⋊C8—72no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7#12312yes—
164C22×C7C22×D411yescharacteristic
165C2×D7—21no—
166C2×D7—21no—
167C2×D7—21no—
168C2×D7—21no—
169C2×D7—21no—
170C2×D7—44no—
171C2×D7#13712yes—
172C4×C7—41no—
173C4×C7—41no—
174C4×C7—42no—
175C4×C7#12312yes—
176C4×C7C2×#4111yescharacteristic
177C4×C7C2×#4111yescharacteristic
178Dic7—21no—
179Dic7—21no—
180Dic7—42no—
181Dic7—41no—
182Dic7—41no—
183C24—71no—
184C2×Q8—72no—
185C2×Q8—141no—
186C2×Q8—142no—
187C2×Q8—142no—
188C22×C4—21no—
189C22×C4—71no—
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—141no—
195C22×C4—141no—
196C22×C4—142no—
197C22×C4—142no—
198C22×C4—141no—
199C22×C4C22×D711yescharacteristic
200M16—22no—
201M16—142no—
202C2×C8—72no—
203C2×C8C22×D712yes—
204C4⋊C4—21no—
205C4⋊C4—71no—
206C4⋊C4—71no—
207C4⋊C4—141no—
208C4⋊C4—141no—
209C4⋊C4—142no—
210C4⋊C4—142no—
211C4⋊C4C22×D711yescharacteristic
212C4⋊C4C22×D711yescharacteristic
213C4⋊C4C22×D712yes—
214C22⋊C4—21no—
215C22⋊C4—141no—
216C22⋊C4—141no—
217C22⋊C4—142no—
218C22⋊C4—142no—
219C22⋊C4—141no—
220C42—21no—
221C42—141no—
222C42—141no—
223C42—141no—
224C2×C7—22no—
225C2×C7C2×#12311yescharacteristic
226C2×C7C2×#13711yescharacteristic
227C2×C7#57511yescharacteristic
228D7—22no—
229D7—41no—
230C23—71no—
231C23—71no—
232C23—142no—
233C23—142no—
234C23—281no—
235C23D4×D711yescharacteristic
236Q8—142no—
237Q8—142no—
238Q8—282no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—71no—
244C2×C4—71no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—142no—
255C2×C4—282no—
256C2×C4—282no—
257C2×C4—282no—
258C2×C4#115712yes—
259C2×C4D4×D711yescharacteristic
260C2×C4#116112yes—
261C2×C4C23×D711yescharacteristic
262C8—22no—
263C8—142no—
264C7#309511yescharacteristic, complemented, Sylow
265C22—21no—
266C22—21no—
267C22—72no—
268C22—141no—
269C22—141no—
270C22—141no—
271C22—141no—
272C22—141no—
273C22—284no—
274C22#645712yes—
275C22C2×D4×D711yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—141no—
280C4—141no—
281C4—282no—
282C4—281no—
283C4—281no—
284C4#645712yes—
285C4C2×#115711yescharacteristic
286C4C2×#116111yescharacteristic
287C2—22no—
288C2—142no—
289C2—281no—
290C2C2×#645711yescharacteristic
291C2#9046411yescharacteristic
292C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC23×D7D4
hypercenter#428D7
hypocenterC7#3095
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. C7×#428C2semidirect
5. #89892C2semidirect
6. #89776C2semidirect
7. #89736C2semidirect
8. #89660C2semidirect
9. #89656C2semidirect
10. #89639C2semidirect
11. #89634C2semidirect
33. C7×#107C22semidirect
34. C7×#106C22semidirect
37. #6390C22semidirect
38. #6390C22semidirect
39. #6384C22semidirect
40. #6384C22semidirect
41. #6382C22semidirect
42. #6381C22semidirect
92. #428D7semidirect
152. #107C2×D7semidirect
154. #106C2×D7semidirect
264. C7#3095semidirect

Maximal subgroups

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

subgroupindex
D7×#1172
C2×#64602
C7×#4282
#898982
#898922
#897762
#897362
#896602
#896562
#896392
#896342
#30957