Finite Group Databasebeta

← #11737844

Subgroups — #11737844

Order 896.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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#90408C211yesmaximal, characteristic
2#90275C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic, complemented
4#89946C212yesmaximal, complemented
5#89941C212yesmaximal, complemented
6#89842C211yesmaximal, characteristic, complemented
7C4×#1143C211yesmaximal, characteristic, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89610C211yesmaximal, characteristic, complemented
10#89556C212yesmaximal, complemented
11#89542C212yesmaximal, complemented
12C2×D4×D7—21no—
13C2×D4×D7C2212yes—
14C2×C7×SD16C2212yescomplemented
15C7×#127C2211yescharacteristic
16C4×C7×Q8C2211yescharacteristic, complemented
17#6509C2212yescomplemented
18C2×#1143C2211yescharacteristic
19#6501—21no—
20#6501C2212yes—
21#6500—22no—
22C2×#1131C2212yescomplemented
23#6461—22no—
24#6455C2212yescomplemented
25#6438—21no—
26C4×D28C2211yescharacteristic
27C4×D28C2211yescharacteristic
28#6432C2211yescharacteristic, complemented
29C4×Dic14—21no—
30C7×C4⋊C8C2211yescharacteristic, complemented
31C7×#102C2212yescomplemented
32#6409C2211yescharacteristic, complemented
33#6406C2212yescomplemented
34#6406C2212yescomplemented
35#6395C2212yes—
36(C2×D7)⋊C8C2212yescomplemented
37#6392C2212yescomplemented
38#6383C2211yescharacteristic
39#6382C2211yescharacteristic, complemented
40C4×C7⋊C8C2211yescharacteristic, complemented
41#3324—71nomaximal, Sylow
42C23×D7—22no—
43C2×C7×Q8C2311yescharacteristic
44C2×C7×D4—21no—
45C2×C7×D4—21no—
46C2×C7×D4C2312yes—
47C2×C22⋊D7—22no—
48C2×C22⋊D7—22no—
49D4×D7—24no—
50D4×D7—42no—
51D4×D7—44no—
52C2×D28D411yescharacteristic
53C2×D28C2311yescharacteristic
54C2×C4×D7—21no—
55C2×C4×D7C2312yes—
56C2×Dic14—21no—
57C7×SD16—42no—
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2311yescharacteristic
61C42×C7—21no—
62C42×C7C2311yescharacteristic
63C22⋊Dic7—22no—
64#1143—22no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4D412yes—
69C4⋊Dic7—21no—
70C4⋊Dic7D411yescharacteristic
71C4⋊Dic7C2312yes—
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—22no—
74C4×Dic7—22no—
75C2×C7⋊C8C2311yescharacteristic
76C2×C7⋊C8C2311yescharacteristic
77#1131—42no—
78#556—71no—
79D42—71no—
80#486—71no—
81#482—72no—
82#474—71no—
83#468—72no—
84#460D711yescharacteristic, complemented
85#450—72no—
86C4×SD16—71no—
87#436—71no—
88#418—72no—
89C23×C7—22no—
90C22×D7—21no—
91C22×D7—44no—
92C22×D7—44no—
93C22×D7—82no—
94C22×D7C2×D412yes—
95C7×Q8—21no—
96C7×Q8—21no—
97C7×D4—22no—
98C7×D4—22no—
99C7×D4—42no—
100C7×D4—81no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2411yescharacteristic
106C22⋊D7—44no—
107C22⋊D7—44no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7C2×D412yes—
111D28—22no—
112D28—21no—
113D28#4112yes—
114C4×D7—22no—
115C4×D7—22no—
116C4×D7—42no—
117Dic14—42no—
118C7×C8—22no—
119C7⋊C8—21no—
120C7⋊C8—21no—
121C22×D4—72no—
122C22×D4—141no—
123C2×SD16—71no—
124C2×SD16C2×D712yescomplemented
125C2×M16—72no—
126C4⋊Q8—71no—
127#127C2×D711yescharacteristic
128#126—142no—
129#124—141no—
130C22⋊Q8—72no—
131#121—72no—
132#121—141no—
133#120—142no—
134C4×Q8—141no—
135C4×Q8C2×D711yescharacteristic, complemented
136C4×D4—71no—
137C4×D4—71no—
138#117—72no—
139#107—72no—
140#106—71no—
141C4⋊C8C2×D711yescharacteristic, complemented
142#103—71no—
143#102—72no—
144#102—72no—
145#102—72no—
146#102—71no—
147#102C2×D712yescomplemented
148C22⋊C8—72no—
149C4×C8—71no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7C22×D411yescharacteristic
154C2×D7—22no—
155C2×D7—22no—
156C2×D7—22no—
157C2×D7—22no—
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—88no—
161C4×C7—21no—
162C4×C7—41no—
163C4×C7—41no—
164C4×C7AGL(1,ℤ/8)12yes—
165C4×C7C2×#4111yescharacteristic
166C4×C7H(5,𝔽2)11yescharacteristic
167Dic7—22no—
168Dic7—42no—
169Dic7—41no—
170C24—142no—
171C2×Q8—141no—
172C2×Q8C22×D711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—71no—
176C2×D4—71no—
177C2×D4—142no—
178C2×D4—144no—
179C2×D4—142no—
180C2×D4—282no—
181C2×D4—284no—
182C2×D4C22×D712yes—
183C22×C4—72no—
184C22×C4—141no—
185SD16—42no—
186SD16—142no—
187M16—282no—
188C2×C8—71no—
189C2×C8—71no—
190C2×C8C22×D712yes—
191C4⋊C4—21no—
192C4⋊C4—71no—
193C4⋊C4—72no—
194C4⋊C4—71no—
195C4⋊C4—142no—
196C4⋊C4—141no—
197C4⋊C4C22×D711yescharacteristic
198C22⋊C4—72no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C22⋊C4—141no—
203C42—21no—
204C42—142no—
205C42C22×D711yescharacteristic
206C2×C7—82no—
207C2×C7#54611yescharacteristic
208C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
209C2×C7#57711yescharacteristic
210D7—42no—
211D7—42no—
212C23—22no—
213C23—72no—
214C23—141no—
215C23—284no—
216C23—284no—
217C23—562no—
218Q8—21no—
219Q8—21no—
220Q8—282no—
221D4—22no—
222D4—22no—
223D4—42no—
224D4—72no—
225D4—81no—
226D4—142no—
227D4—141no—
228D4—284no—
229D4—284no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—72no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—141no—
237C2×C4—282no—
238C2×C4D4×D711yescharacteristic
239C2×C4D4×D711yescharacteristic
240C2×C4C23×D711yescharacteristic
241C8—22no—
242C8—141no—
243C8—141no—
244C7#332411yescharacteristic, complemented, Sylow
245C22—42no—
246C22—42no—
247C22—42no—
248C22—142no—
249C22—142no—
250C22—142no—
251C22—142no—
252C22—281no—
253C22—281no—
254C22—568no—
255C22C2×D4×D711yescharacteristic
256C4—21no—
257C4—41no—
258C4—41no—
259C4—142no—
260C4—281no—
261C4—282no—
262C4#647712yes—
263C4#654811yescharacteristic
264C4D7×#4111yescharacteristic
265C2—82no—
266C2—282no—
267C2—282no—
268C2#9030311yescharacteristic
269C2C2×#647711yescharacteristic
270C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
prime radical#90408C2
hypercenter#460D7
hypocenterC7#3324
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #90275C2semidirect
3. C7×#460C2semidirect
4. #89946C2semidirect
5. #89941C2semidirect
6. #89842C2semidirect
7. C4×#1143C2semidirect
8. #89618C2semidirect
9. #89610C2semidirect
10. #89556C2semidirect
11. #89542C2semidirect
14. C2×C7×SD16C22semidirect
16. C4×C7×Q8C22semidirect
17. #6509C22semidirect
22. C2×#1131C22semidirect
24. #6455C22semidirect
28. #6432C22semidirect
30. C7×C4⋊C8C22semidirect
31. C7×#102C22semidirect
32. #6409C22semidirect
33. #6406C22semidirect
34. #6406C22semidirect
36. (C2×D7)⋊C8C22semidirect
37. #6392C22semidirect
39. #6382C22semidirect
40. C4×C7⋊C8C22semidirect
84. #460D7semidirect
124. C2×SD16C2×D7semidirect
135. C4×Q8C2×D7semidirect
141. C4⋊C8C2×D7semidirect
147. #102C2×D7semidirect
244. C7#3324semidirect

Maximal subgroups

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

subgroupindex
#904082
#902752
C7×#4602
#899462
#899412
#898422
C4×#11432
#896182
#896102
#895562
#895422
#33247