Finite Group Databasebeta

← #11739026

Subgroups — #11739026

Order 896.

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

normal subgroups only

This group has 460 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#90425C211yesmaximal, characteristic, complemented
2#90408C211yesmaximal, characteristic, complemented
3C7×#486C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89844C211yesmaximal, characteristic, complemented
6#89556C212yesmaximal, complemented
7#89554C212yesmaximal, complemented
8#89545C212yesmaximal, complemented
9#89542C212yesmaximal, complemented
10C4×D56C211yesmaximal, characteristic, complemented
11#89460C211yesmaximal, characteristic, complemented
12C2×D4×D7—21no—
13C2×D4×D7C2212yes—
14C7×C4⋊Q8C2211yescharacteristic, complemented
15C7×#127C2211yescharacteristic
16#6510—22no—
17#6501—21no—
18#6501C2212yes—
19#6500—22no—
20C2×#1141C2212yescomplemented
21C2×D56C2211yescharacteristic
22C2×#1131C2212yescomplemented
23#6458—21no—
24#6458C2212yescomplemented
25#6455—21no—
26#6455C2212yescomplemented
27C4×D28C2211yescharacteristic
28C4×D28C2211yescharacteristic
29C4×D28C2211yescharacteristic, complemented
30C7×#102C2212yescomplemented
31C7×#102C2212yescomplemented
32C4×C7×C8C2211yescharacteristic, complemented
33#6406C2212yescomplemented
34#6395C2211yescharacteristic
35#6395C2211yescharacteristic, complemented
36(C2×D7)⋊C8C2212yescomplemented
37#6392C2211yescharacteristic, complemented
38#6383C2212yes—
39#6382C2212yescomplemented
40C4⋊(C7⋊C8)C2211yescharacteristic, complemented
41#3325—71nomaximal, Sylow
42C23×D7—22no—
43C2×C7×Q8—21no—
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×D28—22no—
53C2×D28D411yescharacteristic
54C2×D28C2311yescharacteristic
55C2×D28C2311yescharacteristic
56C2×C4×D7—21no—
57C2×C4×D7—21no—
58C2×C4×D7C2312yes—
59C2×C7×C8C2311yescharacteristic
60C2×C7×C8C2311yescharacteristic
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2312yes—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—22no—
65#1141—42no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4D412yes—
71C4⋊Dic7D411yescharacteristic
72C4⋊Dic7C2311yescharacteristic
73C4⋊Dic7C2311yescharacteristic
74C4×Dic7—22no—
75C2×C7⋊C8C2312yes—
76D56—22no—
77#1131—42no—
78#555—71no—
79D42—71no—
80#486D711yescharacteristic, complemented
81#482—72no—
82#476—71no—
83#469—72no—
84#459—71no—
85#447—72no—
86C4×D8—71no—
87#436—71no—
88#418—72no—
89C23×C7—22no—
90C22×D7—21no—
91C22×D7—21no—
92C22×D7—44no—
93C22×D7—44no—
94C22×D7—82no—
95C22×D7C2×D412yes—
96C7×Q8—81no—
97C7×D4—22no—
98C7×D4—22no—
99C7×D4—42no—
100C7×D4—81no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C22⋊D7—44no—
106C22⋊D7—44no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7C2×D412yes—
110D28—22no—
111D28—21no—
112D28—21no—
113D28—21no—
114D28—82no—
115D28#4112yes—
116C4×D7—22no—
117C4×D7—22no—
118C4×D7—42no—
119C4×D7—42no—
120C7×C8—21no—
121C7×C8—21no—
122C7⋊C8—22no—
123C22×D4—72no—
124C22×D4—141no—
125C2×D8—72no—
126C2×D8—71no—
127C2×M16—72no—
128C4⋊Q8C2×D711yescharacteristic, complemented
129#127C2×D711yescharacteristic
130#124—142no—
131#121—72no—
132#121—72no—
133#121—141no—
134#121—141no—
135#120—142no—
136C4×D4—71no—
137C4×D4—71no—
138C4×D4—71no—
139#117—72no—
140#117—141no—
141#107—71no—
142#106—72no—
143C4⋊C8—71no—
144#102—72no—
145#102—71no—
146#102—71no—
147#102—72no—
148#102C2×D712yescomplemented
149#102C2×D712yescomplemented
150C22⋊C8—72no—
151C4×C8C2×D711yescharacteristic, complemented
152C22×C7—42no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7C22×D411yescharacteristic
156C2×D7—22no—
157C2×D7—22no—
158C2×D7—22no—
159C2×D7—22no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—41no—
164C2×D7—41no—
165C2×D7—88no—
166C4×C7—21no—
167C4×C7—42no—
168C4×C7AGL(1,ℤ/8)12yes—
169C4×C7C2×#4111yescharacteristic
170C4×C7H(5,𝔽2)11yescharacteristic
171Dic7—22no—
172Dic7—41no—
173Dic7—41no—
174C24—142no—
175C2×Q8—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—71no—
179C2×D4—71no—
180C2×D4—71no—
181C2×D4—142no—
182C2×D4—142no—
183C2×D4—144no—
184C2×D4—142no—
185C2×D4—282no—
186C2×D4—284no—
187C2×D4C22×D712yes—
188C22×C4—72no—
189C22×C4—141no—
190C22×C4—141no—
191D8—142no—
192D8—282no—
193M16—282no—
194C2×C8—72no—
195C2×C8C22×D711yescharacteristic
196C2×C8C22×D711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—71no—
199C4⋊C4—71no—
200C4⋊C4—71no—
201C4⋊C4C22×D712yes—
202C22⋊C4—72no—
203C22⋊C4—142no—
204C22⋊C4—142no—
205C22⋊C4—142no—
206C22⋊C4—141no—
207C22⋊C4—141no—
208C42—142no—
209C42C22×D711yescharacteristic
210C2×C7—82no—
211C2×C7#54611yescharacteristic
212C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C7#57611yescharacteristic
214D7—42no—
215D7—42no—
216D7—81no—
217C23—22no—
218C23—72no—
219C23—141no—
220C23—141no—
221C23—284no—
222C23—284no—
223C23—562no—
224Q8—81no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—72no—
229D4—81no—
230D4—142no—
231D4—141no—
232D4—141no—
233D4—141no—
234D4—284no—
235D4—284no—
236D4—562no—
237C2×C4—22no—
238C2×C4—72no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—141no—
242C2×C4—141no—
243C2×C4—282no—
244C2×C4—282no—
245C2×C4D4×D711yescharacteristic
246C2×C4D4×D711yescharacteristic
247C2×C4C23×D711yescharacteristic
248C8—21no—
249C8—21no—
250C8—142no—
251C7#332511yescharacteristic, complemented, Sylow
252C22—42no—
253C22—42no—
254C22—42no—
255C22—142no—
256C22—142no—
257C22—142no—
258C22—142no—
259C22—281no—
260C22—281no—
261C22—281no—
262C22—281no—
263C22—281no—
264C22—568no—
265C22C2×D4×D711yescharacteristic
266C4—21no—
267C4—42no—
268C4—142no—
269C4—281no—
270C4—281no—
271C4#647712yes—
272C4D7×#4111yescharacteristic
273C4#655311yescharacteristic
274C2—82no—
275C2—282no—
276C2—282no—
277C2—561no—
278C2#9036711yescharacteristic
279C2#9044811yescharacteristic
280C2C2×#647711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
hypercenter#486D7
hypocenterC7#3325
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90425C2semidirect
2. #90408C2semidirect
3. C7×#486C2semidirect
4. #89856C2semidirect
5. #89844C2semidirect
6. #89556C2semidirect
7. #89554C2semidirect
8. #89545C2semidirect
9. #89542C2semidirect
10. C4×D56C2semidirect
11. #89460C2semidirect
14. C7×C4⋊Q8C22semidirect
20. C2×#1141C22semidirect
22. C2×#1131C22semidirect
24. #6458C22semidirect
26. #6455C22semidirect
29. C4×D28C22semidirect
30. C7×#102C22semidirect
31. C7×#102C22semidirect
32. C4×C7×C8C22semidirect
33. #6406C22semidirect
35. #6395C22semidirect
36. (C2×D7)⋊C8C22semidirect
37. #6392C22semidirect
39. #6382C22semidirect
40. C4⋊(C7⋊C8)C22semidirect
80. #486D7semidirect
128. C4⋊Q8C2×D7semidirect
148. #102C2×D7semidirect
149. #102C2×D7semidirect
151. C4×C8C2×D7semidirect
251. C7#3325semidirect

Maximal subgroups

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

subgroupindex
#904252
#904082
C7×#4862
#898562
#898442
#895562
#895542
#895452
#895422
C4×D562
#894602
#33257