Finite Group Databasebeta

← #11736798

Subgroups — #11736798

Order 896.

490 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 125 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#90170C211yesmaximal, characteristic, complemented
2C2×#6437C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89620C212yesmaximal, complemented
5#89615C212yesmaximal, complemented
6#89504C212yesmaximal, complemented
7#89503C212yesmaximal, complemented
8#89485C212yesmaximal, complemented
9#89484C212yesmaximal, complemented
10C22×D28—21no—
11C22×D28—21no—
12C22×D28C2212yes—
13C2×C42×C7C2211yescharacteristic
14#6493C2211yescharacteristic
15#6492—21no—
16C4×C22⋊D7—21no—
17#6487C2211yescharacteristic
18C2×D56C2214yescomplemented
19C2×#1132C2214yescomplemented
20#6439—21no—
21#6438C2211yescharacteristic, complemented
22#6437—21no—
23#6437—21no—
24#6437C2212yes—
25#6437C2212yes—
26C4×D28C2211yescharacteristic, complemented
27#6435—21no—
28#6433C2211yescharacteristic, complemented
29C4×Dic14C2211yescharacteristic, complemented
30C7×C4⋊C8C2212yescomplemented
31C7×C22⋊C8C2212yes—
32C7×#97C2212yescomplemented
33#6395C2214yescomplemented
34#6395C2214yescomplemented
35#3217—71nomaximal, Sylow
36C23×D7—22no—
37C22×C4×C7D411yescharacteristic
38C22×C4×C7D411yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—21no—
41C2×D28—22no—
42C2×D28—22no—
43C2×D28—22no—
44C2×D28—22no—
45C2×D28—22no—
46C2×D28—44no—
47C2×D28—44no—
48C2×D28—44no—
49C2×D28C2311yescharacteristic
50C2×D28C2314yes—
51C2×C4×D7—21no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2314yes—
54C42×C7D411yescharacteristic
55C42×C7D411yescharacteristic
56C42×C7C2311yescharacteristic
57C42×C7C2311yescharacteristic
58C22⋊Dic7—21no—
59(C2×D7)⋊C4—21no—
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—21no—
63C4⋊Dic7C2311yescharacteristic
64C4⋊Dic7C2311yescharacteristic
65Dic7⋊C4—21no—
66Dic7⋊C4—21no—
67Dic7⋊C4—21no—
68Dic7⋊C4—21no—
69C4×Dic7—21no—
70D56—44no—
71#1132—44no—
72C2×#127—71no—
73#529—71no—
74#496—72no—
75#490—72no—
76#460—72no—
77#459—72no—
78#450—72no—
79#447—72no—
80#432D711yescharacteristic, complemented
81C23×C7C2×D411yescharacteristic
82C22×D7—21no—
83C22×D7—24no—
84C22×D7—44no—
85C22×D7—44no—
86C22×D7—44no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2×D411yescharacteristic
95C2×C4×C7C2411yescharacteristic
96C22⋊D7—42no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100D28—24no—
101D28—22no—
102D28—24no—
103D28—44no—
104D28—44no—
105D28—44no—
106D28—44no—
107D28—44no—
108C4×D7—42no—
109Dic14—22no—
110C7×C8—24no—
111C22×D4—72no—
112C22×D4—141no—
113C22×D4—141no—
114C2×SD16—74no—
115C2×D8—74no—
116#127—72no—
117#127—72no—
118#127—141no—
119#127—141no—
120#126—141no—
121#125—71no—
122#124—71no—
123#123—141no—
124C22⋊Q8—71no—
125#121—71no—
126C4×Q8—71no—
127C4×D4—71no—
128C4×D4—141no—
129#117—141no—
130C2×C42C2×D711yescharacteristic
131C4⋊C8C2×D712yescomplemented
132#102—74no—
133#102—74no—
134C22⋊C8C2×D712yes—
135#97C2×D712yescomplemented
136C22×C7—21no—
137C22×C7—21no—
138C22×C7AGL(1,ℤ/8)12yes—
139C22×C7C22×D411yescharacteristic
140C2×D7—44no—
141C2×D7—41no—
142C2×D7—41no—
143C2×D7—44no—
144C2×D7—44no—
145C2×D7—41no—
146C2×D7—88no—
147C4×C7—21no—
148C4×C7—21no—
149C4×C7—21no—
150C4×C7—21no—
151C4×C7AGL(1,ℤ/8)12yes—
152C4×C7H(5,𝔽2)12yes—
153Dic7—41no—
154Dic7—41no—
155Dic7—41no—
156C24—142no—
157C2×Q8—71no—
158C2×D4—74no—
159C2×D4—71no—
160C2×D4—141no—
161C2×D4—142no—
162C2×D4—142no—
163C2×D4—142no—
164C2×D4—142no—
165C2×D4—142no—
166C2×D4—284no—
167C2×D4—284no—
168C2×D4—284no—
169C22×C4—141no—
170C22×C4D2811yescharacteristic
171C22×C4D2811yescharacteristic
172C22×C4C22×D711yescharacteristic
173SD16—284no—
174D8—284no—
175C2×C8C22×D714yes—
176C4⋊C4—71no—
177C4⋊C4—71no—
178C4⋊C4—141no—
179C4⋊C4—141no—
180C4⋊C4—141no—
181C4⋊C4—141no—
182C22⋊C4—141no—
183C22⋊C4—141no—
184C22⋊C4—141no—
185C22⋊C4—141no—
186C22⋊C4—141no—
187C42—141no—
188C42D2811yescharacteristic
189C42D2811yescharacteristic
190C42C22×D711yescharacteristic
191C42C22×D711yescharacteristic
192C2×C7—22no—
193C2×C7#53511yescharacteristic
194C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
196D7—81no—
197D7—84no—
198C23—141no—
199C23—144no—
200C23—284no—
201C23—284no—
202C23—284no—
203C23C2×D2811yescharacteristic
204Q8—142no—
205D4—142no—
206D4—144no—
207D4—144no—
208D4—284no—
209D4—284no—
210D4—284no—
211D4—284no—
212D4—282no—
213D4—284no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—141no—
220C2×C4—282no—
221C2×C4C2×D2811yescharacteristic
222C2×C4C2×D2811yescharacteristic
223C2×C4C2×D2811yescharacteristic
224C2×C4C2×D2811yescharacteristic
225C2×C4C2×D2811yescharacteristic
226C2×C4C23×D711yescharacteristic
227C8—24no—
228C7#321711yescharacteristic, complemented, Sylow
229C22—21no—
230C22—21no—
231C22—281no—
232C22—284no—
233C22—284no—
234C22—281no—
235C22—284no—
236C22—281no—
237C22—568no—
238C22#647112yes—
239C22C22×D2811yescharacteristic
240C4—21no—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—281no—
245C4—281no—
246C4—281no—
247C4#647112yes—
248C4#655312yes—
249C2—22no—
250C2—564no—
251C2—561no—
252C2#9021611yescharacteristic
253C2C2×#647111yescharacteristic
254C2C2×#647111yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
hypercenter#432D7
hypocenterC7#3217
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. C2×#6437C2semidirect
3. C7×#432C2semidirect
4. #89620C2semidirect
5. #89615C2semidirect
6. #89504C2semidirect
7. #89503C2semidirect
8. #89485C2semidirect
9. #89484C2semidirect
18. C2×D56C22semidirect
19. C2×#1132C22semidirect
21. #6438C22semidirect
26. C4×D28C22semidirect
28. #6433C22semidirect
29. C4×Dic14C22semidirect
30. C7×C4⋊C8C22semidirect
32. C7×#97C22semidirect
33. #6395C22semidirect
34. #6395C22semidirect
80. #432D7semidirect
131. C4⋊C8C2×D7semidirect
135. #97C2×D7semidirect
228. C7#3217semidirect

Maximal subgroups

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

subgroupindex
#901702
C2×#64372
C7×#4322
#896202
#896152
#895042
#895032
#894852
#894842
#32177