Finite Group Databasebeta

← #11739760

Subgroups — #11739760

Order 896.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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×C4×C7×D4C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3#89990C212yesmaximal
4#89791C212yesmaximal, complemented
5#89790C214yesmaximal, complemented
6#89779C212yesmaximal, complemented
7#89742C212yesmaximal, complemented
8#89698C211yesmaximal, characteristic, complemented
9C22×C7×D4C2211yescharacteristic
10C23×C4×C7—21no—
11C4×C7×D4—21no—
12C4×C7×D4—21no—
13C4×C7×D4C2212yes—
14C4×C7×D4C2212yes—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4—21no—
17C2×C42×C7C2211yescharacteristic
18C2×#1142C2214yes—
19#6489—21no—
20#6487—21no—
21#6487C2212yes—
22#6433—21no—
23#6432C2211yescharacteristic, complemented
24#6432C2211yescharacteristic, complemented
25C4×Dic14—21no—
26C4×Dic14C2212yescomplemented
27#6406C2214yes—
28C22⋊(C7⋊C8)C2212yes—
29#6384C2214yescomplemented
30#6381C2214yescomplemented
31C4⋊(C7⋊C8)C2212yescomplemented
32#6377C2212yescomplemented
33#3110—71nomaximal, Sylow
34C24×C7—21no—
35C2×C7×D4—24no—
36C2×C7×D4—21no—
37C2×C7×D4C2312yes—
38C22×C4×C7—21no—
39C22×C4×C7—22no—
40C22×C4×C7—22no—
41C22×C4×C7—44no—
42C22×C4×C7D411yescharacteristic
43C22×C4×C7D411yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C2×Dic14—21no—
46C2×Dic14C2312yes—
47C7×C4⋊C4—21no—
48C7×C4⋊C4C2312yes—
49C7×C22⋊C4—22no—
50C7×C22⋊C4—22no—
51C42×C7D411yescharacteristic
52C42×C7D411yescharacteristic
53C42×C7C2311yescharacteristic
54C42×C7C2311yescharacteristic
55C22⋊Dic7—22no—
56#1142—28no—
57C4⋊Dic7—21no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7C2312yes—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63C4×Dic7—22no—
64C2×C7⋊C8C2314yes—
65#533—71no—
66C2×C4×D4D711yescharacteristic
67#484—72no—
68#474—72no—
69#461—72no—
70#450—72no—
71#440—74no—
72#432—71no—
73C23×C7—24no—
74C23×C7—22no—
75C23×C7—42no—
76C23×C7—42no—
77C23×C7C2×D411yescharacteristic
78C7×D4—24no—
79C7×D4—22no—
80C7×D4#4114yes—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—44no—
86C2×C4×C7—42no—
87C2×C4×C7—44no—
88C2×C4×C7C2×D411yescharacteristic
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2411yescharacteristic
94C2×Dic7—22no—
95C2×Dic7—22no—
96Dic14—24no—
97Dic14—42no—
98C7⋊C8—24no—
99C22×D4C2×D711yescharacteristic
100C23×C4—21no—
101C2×SD16—74no—
102C4⋊Q8—71no—
103C4⋊Q8—71no—
104#125—141no—
105C22⋊Q8—72no—
106C22⋊Q8—141no—
107C4×Q8—72no—
108C4×Q8—141no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4C2×D712yes—
112C4×D4C2×D712yes—
113#117—141no—
114C2×C4⋊C4C2×D711yescharacteristic
115C2×C22⋊C4—21no—
116C2×C42C2×D711yescharacteristic
117#107—74no—
118C4⋊C8—72no—
119#103—74no—
120#102—74no—
121C22⋊C8—72no—
122#97—72no—
123C22×C7—21no—
124C22×C7—21no—
125C22×C7—24no—
126C22×C7—48no—
127C22×C7—42no—
128C22×C7—42no—
129C22×C7#13712yes—
130C22×C7C22×D411yescharacteristic
131C4×C7—21no—
132C4×C7—21no—
133C4×C7—21no—
134C4×C7—21no—
135C4×C7—42no—
136C4×C7AGL(1,ℤ/8)12yes—
137C4×C7C2×#4112yes—
138Dic7—42no—
139Dic7—42no—
140C24—21no—
141C2×Q8—72no—
142C2×Q8—141no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4C22×D712yes—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—44no—
150C22×C4C22⋊D711yescharacteristic
151C22×C4C22⋊D711yescharacteristic
152C22×C4C22×D711yescharacteristic
153SD16—148no—
154C2×C8—74no—
155C4⋊C4—21no—
156C4⋊C4—72no—
157C4⋊C4—142no—
158C4⋊C4—142no—
159C4⋊C4—141no—
160C4⋊C4—141no—
161C4⋊C4—141no—
162C4⋊C4C22×D712yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—142no—
166C42—142no—
167C42C22⋊D711yescharacteristic
168C42C22⋊D711yescharacteristic
169C42C22×D711yescharacteristic
170C42C22×D711yescharacteristic
171C2×C7—22no—
172C2×C7—44no—
173C2×C7#52511yescharacteristic
174C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
175C2×C7C2×#13711yescharacteristic
176C23—24no—
177C23—22no—
178C23—42no—
179C23—42no—
180C23C2×C22⋊D711yescharacteristic
181Q8—144no—
182Q8—282no—
183D4—24no—
184D4—22no—
185D4#115714yes—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4C2×C22⋊D711yescharacteristic
196C2×C4C2×C22⋊D711yescharacteristic
197C2×C4C2×C22⋊D711yescharacteristic
198C2×C4C2×C22⋊D711yescharacteristic
199C2×C4C2×C22⋊D711yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—144no—
202C7#311011yescharacteristic, complemented, Sylow
203C22—21no—
204C22—21no—
205C22—24no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22#651412yes—
210C22C22×C22⋊D711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—282no—
217C4—282no—
218C4#649512yes—
219C4C2×#115712yes—
220C2—22no—
221C2—44no—
222C2#9048211yescharacteristic
223C2C2×#649511yescharacteristic
224C2C2×#651411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C4×C7×D4C2
socleC22×C7C22×D4
prime radicalC22×C7×D4C22
hypercenterC2×C4×D4D7
hypocenterC7#3110
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
4. #89791C2semidirect
5. #89790C2semidirect
6. #89779C2semidirect
7. #89742C2semidirect
8. #89698C2semidirect
23. #6432C22semidirect
24. #6432C22semidirect
26. C4×Dic14C22semidirect
29. #6384C22semidirect
30. #6381C22semidirect
31. C4⋊(C7⋊C8)C22semidirect
32. #6377C22semidirect
202. C7#3110semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
#901612
#899902
#897912
#897902
#897792
#897422
#896982
#31107