Finite Group Databasebeta

← #11739319

Subgroups — #11739319

Order 896.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#90419C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic
3C7×#497C211yesmaximal, characteristic, complemented
4#89958C212yesmaximal, complemented
5#89957C212yesmaximal, complemented
6#89941C212yesmaximal
7#89938C212yesmaximal, complemented
8#89858C211yesmaximal, characteristic, complemented
9#89827C211yesmaximal, characteristic, complemented
10#89486C211yesmaximal, characteristic, complemented
11#89477C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2211yescharacteristic
13C2×D4×D7C2211yescharacteristic
14C2×C7×Q16C2212yescomplemented
15C2×C7×SD16C2212yes—
16C7×C4⋊Q8C2211yescharacteristic, complemented
17C7×#124C2211yescharacteristic
18#6510C2211yescharacteristic
19#6509C2212yes—
20Q8×Dic7C2212yescomplemented
21C2×#1144C2212yescomplemented
22C2×#1143C2212yes—
23#6499C2211yescharacteristic
24#6460—21no—
25#6455—21no—
26#6453—21no—
27#6450—21no—
28#6448—21no—
29#6448—21no—
30D7×C22⋊C4—21no—
31D7×C22⋊C4—21no—
32#6438C2211yescharacteristic
33C42×D7C2211yescharacteristic
34#6432C2211yescharacteristic, complemented
35C7×#97C2211yescharacteristic, complemented
36#6409C2212yescomplemented
37#6406C2212yes—
38#6395C2212yes—
39(C2×D7)⋊C8C2212yes—
40#6390C2212yescomplemented
41Dic7⋊C8C2212yescomplemented
42#6377C2211yescharacteristic, complemented
43#3215—71nomaximal, Sylow
44C23×D7—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×C7×Q8C2312yes—
47C2×C7×D4C2311yescharacteristic
48C2×C22⋊D7—21no—
49Q8×D7—42no—
50D4×D7—42no—
51C2×D28C2311yescharacteristic
52C2×C4×D7—21no—
53C2×C4×D7D411yescharacteristic
54C2×C4×D7D411yescharacteristic
55C2×C4×D7C2311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14—21no—
58C2×Dic14C2311yescharacteristic
59C7×Q16—42no—
60C7×SD16—42no—
61C2×C7×C8C2312yes—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—21no—
64C7×C22⋊C4—21no—
65C7×C22⋊C4—21no—
66C42×C7C2311yescharacteristic
67C22⋊Dic7—21no—
68C22⋊Dic7—21no—
69#1144—42no—
70#1143—42no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—21no—
73(C2×D7)⋊C4—21no—
74(C2×D7)⋊C4—21no—
75(C2×D7)⋊C4—21no—
76C4⋊Dic7—21no—
77C4⋊Dic7—22no—
78C4⋊Dic7C2312yes—
79Dic7⋊C4—22no—
80C4×Dic7—22no—
81C4×Dic7D411yescharacteristic
82C4×Dic7D411yescharacteristic
83C4×Dic7C2311yescharacteristic
84C2×C7⋊C8C2312yes—
85#533—71no—
86C2×#124—71no—
87#497—71no—
88#497D711yescharacteristic, complemented
89#489—71no—
90#489—71no—
91#464—72no—
92#464—72no—
93#451—72no—
94#450—72no—
95#432—71no—
96C23×C7—21no—
97C22×D7—21no—
98C22×D7—42no—
99C22×D7—42no—
100C22×D7—42no—
101C22×D7C2×D411yescharacteristic
102C7×Q8—22no—
103C7×Q8—22no—
104C7×Q8—22no—
105C7×D4—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C22⋊D7—42no—
112C2×Dic7—22no—
113C2×Dic7—21no—
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116C2×Dic7C2×D411yescharacteristic
117D28—22no—
118C4×D7—22no—
119C4×D7—22no—
120C4×D7—41no—
121C4×D7—42no—
122Dic14—22no—
123Dic14—42no—
124Dic14—42no—
125C7×C8—22no—
126C7⋊C8—22no—
127C22×Q8—71no—
128C22×D4—71no—
129C2×Q16—72no—
130C2×Q16C2×D712yescomplemented
131C2×SD16—72no—
132C2×SD16C2×D712yes—
133C4⋊Q8—71no—
134C4⋊Q8C2×D711yescharacteristic, complemented
135#125—141no—
136#124—71no—
137#124—71no—
138#124—71no—
139#124—141no—
140#124—141no—
141#124C2×D711yescharacteristic
142C22⋊Q8—72no—
143C22⋊Q8—141no—
144C4×Q8—72no—
145C4×Q8—141no—
146#117—141no—
147C2×C22⋊C4—141no—
148C2×C22⋊C4—141no—
149C2×C42—71no—
150C4⋊C8—72no—
151#103—72no—
152#103—72no—
153#102—72no—
154#102—72no—
155C22⋊C8—72no—
156#97—71no—
157#97C2×D711yescharacteristic, complemented
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7C22×D411yescharacteristic
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7—84no—
168C2×D7#13712yes—
169C4×C7—21no—
170C4×C7—21no—
171C4×C7—41no—
172C4×C7—42no—
173C4×C7H(5,𝔽2)12yes—
174Dic7—21no—
175Dic7—21no—
176Dic7—21no—
177Dic7—41no—
178Dic7—42no—
179C24—141no—
180C2×Q8—71no—
181C2×Q8—141no—
182C2×Q8—141no—
183C2×Q8—282no—
184C2×Q8C22×D711yescharacteristic
185C2×Q8C22×D712yes—
186C2×D4—71no—
187C2×D4—141no—
188C2×D4—282no—
189C2×D4C22×D711yescharacteristic
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—141no—
194Q16—42no—
195Q16—282no—
196SD16—42no—
197SD16—282no—
198C2×C8—72no—
199C2×C8C22×D712yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—72no—
203C4⋊C4—142no—
204C4⋊C4—142no—
205C4⋊C4—141no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—142no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C22⋊C4—141no—
214C22⋊C4—141no—
215C42—71no—
216C42—71no—
217C42—71no—
218C42—142no—
219C42C22×D711yescharacteristic
220C2×C7—81no—
221C2×C7#53511yescharacteristic
222C2×C7C2×#13711yescharacteristic
223C2×C7#57511yescharacteristic
224D7—22no—
225D7—81no—
226C23—21no—
227C23—71no—
228C23—141no—
229C23—282no—
230C23—282no—
231C23—282no—
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—142no—
236Q8—282no—
237Q8—282no—
238D4—22no—
239D4—142no—
240D4—282no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—71no—
244C2×C4—71no—
245C2×C4—71no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—281no—
251C2×C4—282no—
252C2×C4D4×D711yescharacteristic
253C2×C4D4×D711yescharacteristic
254C2×C4C23×D711yescharacteristic
255C8—22no—
256C8—142no—
257C7#321511yescharacteristic, complemented, Sylow
258C22—41no—
259C22—41no—
260C22—41no—
261C22—72no—
262C22—141no—
263C22—141no—
264C22—281no—
265C22—281no—
266C22—281no—
267C22—564no—
268C22C2×D4×D711yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—141no—
274C4—141no—
275C4—141no—
276C4—281no—
277C4—282no—
278C4#654812yes—
279C2—81no—
280C2—142no—
281C2—561no—
282C2#9040611yescharacteristic
283C2D7×#13711yescharacteristic
284C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#497D7
hypocenterC7#3215
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#497C2semidirect
4. #89958C2semidirect
5. #89957C2semidirect
7. #89938C2semidirect
8. #89858C2semidirect
9. #89827C2semidirect
10. #89486C2semidirect
11. #89477C2semidirect
14. C2×C7×Q16C22semidirect
16. C7×C4⋊Q8C22semidirect
20. Q8×Dic7C22semidirect
21. C2×#1144C22semidirect
34. #6432C22semidirect
35. C7×#97C22semidirect
36. #6409C22semidirect
40. #6390C22semidirect
41. Dic7⋊C8C22semidirect
42. #6377C22semidirect
88. #497D7semidirect
130. C2×Q16C2×D7semidirect
134. C4⋊Q8C2×D7semidirect
157. #97C2×D7semidirect
257. C7#3215semidirect

Maximal subgroups

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

subgroupindex
#904192
D7×#1242
C7×#4972
#899582
#899572
#899412
#899382
#898582
#898272
#894862
#894772
#32157