Finite Group Databasebeta

← #11736796

Subgroups — #11736796

Order 896.

400 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#6438C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89617C214yesmaximal, complemented
5#89511C212yesmaximal, complemented
6#89503C212yesmaximal
7#89487C212yesmaximal, complemented
8#89486C212yesmaximal, complemented
9C22×D28C2211yescharacteristic
10C22×Dic14C2211yescharacteristic
11C2×C42×C7C2211yescharacteristic
12C2×(C2×D7)⋊C4—21no—
13C2×(C2×D7)⋊C4—21no—
14#6489—21no—
15#6487—21no—
16#6487C2212yes—
17C2×Dic28C2214yescomplemented
18C2×#1132C2214yes—
19#6438—21no—
20#6438—21no—
21#6438C2212yes—
22#6438C2212yes—
23#6433—21no—
24#6432C2211yescharacteristic, complemented
25#6432C2211yescharacteristic, complemented
26C4×Dic14—21no—
27C4×Dic14C2212yescomplemented
28C7×C4⋊C8C2212yescomplemented
29C7×C22⋊C8C2212yes—
30C7×#97C2212yescomplemented
31#6395C2214yes—
32#6390C2214yescomplemented
33#3215—71nomaximal, Sylow
34C23×D7—21no—
35C22×C4×C7D411yescharacteristic
36C22×C4×C7D411yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×Dic7—21no—
39C2×D28—21no—
40C2×D28—42no—
41C2×D28C2312yes—
42C2×Dic14—21no—
43C2×Dic14—21no—
44C2×Dic14—42no—
45C2×Dic14C2312yes—
46C2×Dic14C2312yes—
47C2×C7×C8C2314yes—
48C42×C7D411yescharacteristic
49C42×C7D411yescharacteristic
50C42×C7C2311yescharacteristic
51C42×C7C2311yescharacteristic
52C22⋊Dic7—22no—
53(C2×D7)⋊C4—22no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57C4⋊Dic7—21no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63C4×Dic7—22no—
64Dic28—44no—
65#1132—44no—
66#533—71no—
67C2×#124—71no—
68#497—72no—
69#489—72no—
70#464—74no—
71#451—72no—
72#450—72no—
73#432D711yescharacteristic, complemented
74C23×C7C2×D411yescharacteristic
75C22×D7—22no—
76C22×D7—42no—
77C22×D7—42no—
78C22×D7—42no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—41no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2411yescharacteristic
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—42no—
92D28—24no—
93D28—42no—
94Dic14—24no—
95Dic14—24no—
96Dic14—42no—
97Dic14—42no—
98C7×C8—24no—
99C22×Q8—71no—
100C22×D4—71no—
101C2×Q16—74no—
102C2×SD16—74no—
103C4⋊Q8—71no—
104C4⋊Q8—71no—
105#125—141no—
106#124—72no—
107#124—72no—
108#124—141no—
109#124—141no—
110C22⋊Q8—72no—
111C22⋊Q8—141no—
112C4×Q8—72no—
113C4×Q8—141no—
114#117—141no—
115C2×C22⋊C4—141no—
116C2×C22⋊C4—141no—
117C2×C42C2×D711yescharacteristic
118C4⋊C8C2×D712yescomplemented
119#103—74no—
120#102—74no—
121C22⋊C8C2×D712yes—
122#97C2×D712yescomplemented
123C22×C7—21no—
124C22×C7—21no—
125C22×C7#13712yes—
126C22×C7C22×D411yescharacteristic
127C2×D7—42no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—84no—
131C4×C7—21no—
132C4×C7—21no—
133C4×C7—21no—
134C4×C7—21no—
135C4×C7—21no—
136C4×C7H(5,𝔽2)12yes—
137Dic7—42no—
138Dic7—42no—
139Dic7—42no—
140C24—141no—
141C2×Q8—72no—
142C2×Q8—72no—
143C2×Q8—141no—
144C2×Q8—141no—
145C2×Q8—282no—
146C2×D4—72no—
147C2×D4—141no—
148C2×D4—282no—
149C22×C4—141no—
150C22×C4D2811yescharacteristic
151C22×C4D2811yescharacteristic
152C22×C4C22×D711yescharacteristic
153Q16—284no—
154SD16—284no—
155C2×C8C22×D714yes—
156C4⋊C4—72no—
157C4⋊C4—142no—
158C4⋊C4—142no—
159C4⋊C4—141no—
160C4⋊C4—141no—
161C4⋊C4—141no—
162C22⋊C4—142no—
163C22⋊C4—142no—
164C22⋊C4—142no—
165C22⋊C4—142no—
166C22⋊C4—142no—
167C42—142no—
168C42D2811yescharacteristic
169C42D2811yescharacteristic
170C42C22×D711yescharacteristic
171C42C22×D711yescharacteristic
172C2×C7—22no—
173C2×C7#53511yescharacteristic
174C2×C7C2×#13711yescharacteristic
175C2×C7#57511yescharacteristic
176D7—82no—
177C23—142no—
178C23—282no—
179C23—282no—
180C23—282no—
181C23C2×D2811yescharacteristic
182Q8—144no—
183Q8—144no—
184Q8—282no—
185Q8—282no—
186D4—144no—
187D4—282no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—282no—
195C2×C4C2×D2811yescharacteristic
196C2×C4C2×D2811yescharacteristic
197C2×C4C2×D2811yescharacteristic
198C2×C4C2×D2811yescharacteristic
199C2×C4C2×D2811yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—24no—
202C7#321511yescharacteristic, complemented, Sylow
203C22—21no—
204C22—21no—
205C22—282no—
206C22—282no—
207C22—282no—
208C22—564no—
209C22#647212yes—
210C22C22×D2811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—282no—
217C4—282no—
218C4—282no—
219C4#655312yes—
220C2—22no—
221C2—562no—
222C2#9021611yescharacteristic
223C2C2×#647211yescharacteristic
224C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
prime radicalC22×D28C22
hypercenter#432D7
hypocenterC7#3215
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
2. #90161C2semidirect
3. C7×#432C2semidirect
4. #89617C2semidirect
5. #89511C2semidirect
7. #89487C2semidirect
8. #89486C2semidirect
17. C2×Dic28C22semidirect
24. #6432C22semidirect
25. #6432C22semidirect
27. C4×Dic14C22semidirect
28. C7×C4⋊C8C22semidirect
30. C7×#97C22semidirect
32. #6390C22semidirect
73. #432D7semidirect
118. C4⋊C8C2×D7semidirect
122. #97C2×D7semidirect
202. C7#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#64382
#901612
C7×#4322
#896172
#895112
#895032
#894872
#894862
#32157