Finite Group Databasebeta

← #11736376

Subgroups — #11736376

Order 896.

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

normal subgroups only

This group has 492 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
1#90496C211yesmaximal, characteristic
2C22×#1142C211yesmaximal, characteristic, complemented
3C2×#6470C211yesmaximal, characteristic, complemented
4#90217C211yesmaximal, characteristic, complemented
5C7×#418C211yesmaximal, characteristic, complemented
6#89892C211yesmaximal, characteristic, complemented
7#89770C211yesmaximal, characteristic, complemented
8#89552C214yesmaximal, complemented
9#89550C214yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C2×#1157C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6516—22no—
16#6503—21no—
17#6501C2214yes—
18#6499—21no—
19C2×#1142—24no—
20C2×#1142C2212yes—
21C2×#1142C2212yes—
22#6489C2211yescharacteristic
23C22×C7⋊C8C2211yescharacteristic
24#6470—24no—
25C2×#1131C2212yescomplemented
26C2×C8×D7C2212yescomplemented
27#6460C2214yescomplemented
28#6449—22no—
29#6438—21no—
30#6432—21no—
31C7×#102C2214yes—
32#6390C2214yescomplemented
33#6382C2212yescomplemented
34#6381C2212yescomplemented
35#3057—71nomaximal, Sylow
36C24×C7—21no—
37C2×C7×D4—22no—
38C2×C7×D4—21no—
39C2×C7×D4—22no—
40C2×C7×D4C2312yes—
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—24no—
43C2×C22⋊D7D412yes—
44C22×Dic7—21no—
45#1157—22no—
46#1157—22no—
47C2×D28D411yescharacteristic
48C2×C4×D7C2312yes—
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—21no—
52C2×Dic14D411yescharacteristic
53C2×Dic14C2312yes—
54C7×M16—22no—
55C2×C7×C8C2312yes—
56C7×C4⋊C4C2312yes—
57C7×C22⋊C4—21no—
58C42×C7—21no—
59C22⋊Dic7—21no—
60C22⋊Dic7—24no—
61#1142—24no—
62#1142—24no—
63(C2×D7)⋊C4—24no—
64C4⋊Dic7—24no—
65C4⋊Dic7C2312yes—
66C4×Dic7—21no—
67C2×C7⋊C8D414yescomplemented
68C2×C7⋊C8C2311yescharacteristic
69C2×C7⋊C8C2311yescharacteristic
70#1131—24no—
71C8×D7—24no—
72C22×SD16—71no—
73C2×#131—71no—
74#536—71no—
75#535—71no—
76#468—74no—
77#465—74no—
78#427—71no—
79#419—71no—
80#418D711yescharacteristic, complemented
81C23×C7—22no—
82C23×C7—42no—
83C23×C7—82no—
84C23×C7C2×D411yescharacteristic
85C22×D7C2×D412yes—
86C7×D4—22no—
87C7×D4—22no—
88C7×D4—42no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D412yes—
92C2×C4×C7#4112yes—
93C2×C4×C7C2411yescharacteristic
94C22⋊D7—24no—
95C22⋊D7—84no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—42no—
99C2×Dic7C2×D412yes—
100D28—22no—
101C4×D7—22no—
102C4×D7—22no—
103Dic14—22no—
104Dic14—22no—
105Dic14—22no—
106Dic14—42no—
107C7×C8—22no—
108C7⋊C8C2×D412yes—
109C7⋊C8C2×D412yes—
110C2×#41—71no—
111C22×Q8—71no—
112C22×D4C2×D711yescharacteristic
113C2×SD16—72no—
114C2×SD16—72no—
115C2×SD16—144no—
116#131—144no—
117C2×M16—72no—
118C2×M16C2×D711yescharacteristic
119C22×C8—71no—
120C22×C8—72no—
121C4⋊Q8—141no—
122#127—141no—
123#124—141no—
124#124—141no—
125#123—142no—
126C22⋊Q8—74no—
127#121—74no—
128#120—142no—
129#117—71no—
130#117C2×D711yescharacteristic
131#107—72no—
132#106—72no—
133#103—74no—
134#102C2×D714yes—
135C22×C7—22no—
136C22×C7—42no—
137C22×C7—44no—
138C22×C7—84no—
139C22×C7#12112yes—
140C22×C7C22×D411yescharacteristic
141C2×D7—22no—
142C2×D7—24no—
143C4×C7—42no—
144C4×C7#12112yes—
145C4×C7C22×D411yescharacteristic
146C4×C7C2×#4111yescharacteristic
147Dic7—22no—
148Dic7—42no—
149Dic7—42no—
150C24—21no—
151#41—142no—
152#41—142no—
153C2×Q8—71no—
154C2×Q8—72no—
155C2×Q8—141no—
156C2×Q8—142no—
157C2×Q8—142no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—72no—
162C2×D4—71no—
163C2×D4—144no—
164C2×D4C22×D712yes—
165C22×C4—72no—
166C22×C4—141no—
167C22×C4C22×D711yescharacteristic
168SD16—144no—
169SD16—144no—
170M16—22no—
171M16—144no—
172C2×C8—74no—
173C2×C8—71no—
174C2×C8—71no—
175C2×C8—144no—
176C2×C8C22×D712yes—
177C4⋊C4—72no—
178C4⋊C4—144no—
179C4⋊C4C22×D712yes—
180C22⋊C4—21no—
181C22⋊C4—144no—
182C22⋊C4—144no—
183C22⋊C4—141no—
184C42—21no—
185C42—141no—
186C2×C7—22no—
187C2×C7—82no—
188C2×C7C2×#12111yescharacteristic
189C2×C7#57712yes—
190D7—42no—
191C23—22no—
192C23—42no—
193C23—72no—
194C23—82no—
195C23D4×D711yescharacteristic
196Q8—142no—
197Q8—142no—
198Q8—142no—
199Q8—282no—
200D4—22no—
201D4—22no—
202D4—42no—
203D4—144no—
204D4—142no—
205D4—564no—
206C2×C4—22no—
207C2×C4—72no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—282no—
213C2×C4#115712yes—
214C2×C4D4×D711yescharacteristic
215C2×C4D4×D712yes—
216C2×C4C23×D711yescharacteristic
217C8—22no—
218C8—72no—
219C8—72no—
220C7#305711yescharacteristic, complemented, Sylow
221C22—22no—
222C22—42no—
223C22—44no—
224C22—84no—
225C22—142no—
226C22—144no—
227C22#644712yes—
228C22C2×D4×D711yescharacteristic
229C4—42no—
230C4—142no—
231C4—282no—
232C4—282no—
233C4#644712yes—
234C4C2×#115711yescharacteristic
235C4C2×D4×D711yescharacteristic
236C2—22no—
237C2—82no—
238C2—282no—
239C2C2×#644711yescharacteristic
240C2#9046612yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#418C2
socleC22×C7C22×D4
prime radical#90496C2
hypercenter#418D7
hypocenterC7#3057
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C22×#1142C2semidirect
3. C2×#6470C2semidirect
4. #90217C2semidirect
5. C7×#418C2semidirect
6. #89892C2semidirect
7. #89770C2semidirect
8. #89552C2semidirect
9. #89550C2semidirect
25. C2×#1131C22semidirect
26. C2×C8×D7C22semidirect
27. #6460C22semidirect
32. #6390C22semidirect
33. #6382C22semidirect
34. #6381C22semidirect
67. C2×C7⋊C8D4semidirect
80. #418D7semidirect
220. C7#3057semidirect

Maximal subgroups

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

subgroupindex
#904962
C22×#11422
C2×#64702
#902172
C7×#4182
#898922
#897702
#895522
#895502
#30577