Finite Group Databasebeta

← #11736390

Subgroups — #11736390

Order 896.

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

normal subgroups only

This group has 490 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×#6472C211yesmaximal, characteristic, complemented
4#90207C211yesmaximal, characteristic, complemented
5C7×#418C211yesmaximal, characteristic, complemented
6#89891C211yesmaximal, characteristic, complemented
7#89774C211yesmaximal, characteristic, complemented
8#89539C214yesmaximal, complemented
9#89534C214yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C2×#1157C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×M16C2211yescharacteristic
14C7×#117C2211yescharacteristic
15#6516—22no—
16#6503C2212yes—
17#6501—22no—
18#6499C2212yes—
19C2×#1142—24no—
20C2×#1142C2214yes—
21#6489C2211yescharacteristic
22C22×C7⋊C8C2211yescharacteristic
23#6472—24no—
24C2×Dic28C2212yescomplemented
25C2×#1132C2212yescomplemented
26#6450C2214yescomplemented
27#6440—22no—
28C4×Dic14—22no—
29C7×#102C2214yes—
30Dic7⋊C8C2214yescomplemented
31#6384C2212yescomplemented
32#6383C2212yescomplemented
33#3043—71nomaximal, Sylow
34C24×C7—21no—
35C2×C7×D4—22no—
36C2×C7×D4—21no—
37C2×C7×D4—22no—
38C2×C7×D4C2312yes—
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—21no—
41C2×C22⋊D7—24no—
42C22×Dic7—21no—
43#1157—22no—
44#1157—22no—
45C2×D28C2311yescharacteristic
46C2×C4×D7—21no—
47C2×Dic14—21no—
48C2×Dic14—22no—
49C2×Dic14D414yescomplemented
50C2×Dic14C2311yescharacteristic
51C2×Dic14C2312yes—
52C7×M16—22no—
53C2×C7×C8C2312yes—
54C7×C4⋊C4C2312yes—
55C7×C22⋊C4—21no—
56C42×C7—21no—
57C22⋊Dic7—24no—
58C22⋊Dic7D412yes—
59#1142—28no—
60C4⋊Dic7D412yes—
61Dic7⋊C4—24no—
62C4×Dic7—24no—
63C4×Dic7C2312yes—
64C2×C7⋊C8—22no—
65C2×C7⋊C8C2312yes—
66Dic28—24no—
67#1132—24no—
68C2×#137—71no—
69C22×SD16—71no—
70#535—71no—
71#519—71no—
72#464—74no—
73#460—74no—
74#424—71no—
75#418D711yescharacteristic, complemented
76#416—71no—
77C23×C7—22no—
78C23×C7—42no—
79C23×C7—82no—
80C23×C7C2×D411yescharacteristic
81C22×D7—21no—
82C7×D4—22no—
83C7×D4—22no—
84C7×D4—42no—
85C2×C4×C7—22no—
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2×D412yes—
88C2×C4×C7#4112yes—
89C2×C4×C7C2411yescharacteristic
90C22⋊D7—42no—
91C22⋊D7—84no—
92C2×Dic7—24no—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7C2×D414yes—
96D28—21no—
97D28—21no—
98C4×D7—42no—
99Dic14—24no—
100Dic14—22no—
101Dic14—21no—
102Dic14—21no—
103Dic14C2×D414yes—
104C7×C8—22no—
105C7⋊C8—22no—
106C2×#41—71no—
107C22×Q8—71no—
108C22×D4C2×D711yescharacteristic
109#137—144no—
110C2×Q16—72no—
111C2×SD16—74no—
112C2×SD16—72no—
113C2×SD16—144no—
114C2×M16C2×D711yescharacteristic
115C22×C8—71no—
116#127—72no—
117#124—72no—
118#121—142no—
119#120—142no—
120C4×Q8—74no—
121C4×Q8—142no—
122#117—71no—
123#117—142no—
124#117C2×D711yescharacteristic
125C4⋊C8—74no—
126#103—72no—
127#102—72no—
128#102C2×D714yes—
129C22×C7—22no—
130C22×C7—42no—
131C22×C7—44no—
132C22×C7—84no—
133C22×C7#12112yes—
134C22×C7C22×D411yescharacteristic
135C2×D7—41no—
136C2×D7—42no—
137C4×C7—42no—
138C4×C7#12112yes—
139C4×C7C22×D411yescharacteristic
140C4×C7C2×#4111yescharacteristic
141Dic7—24no—
142Dic7—24no—
143Dic7—41no—
144C24—21no—
145#41—142no—
146#41—142no—
147C2×Q8—74no—
148C2×Q8—72no—
149C2×Q8—71no—
150C2×Q8—142no—
151C2×Q8—141no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—71no—
156C2×D4—144no—
157C2×D4—141no—
158C2×D4C22×D712yes—
159C22×C4—141no—
160C22×C4—141no—
161C22×C4C22×D711yescharacteristic
162Q16—144no—
163SD16—144no—
164SD16—148no—
165M16—22no—
166C2×C8—72no—
167C2×C8—142no—
168C2×C8C22×D712yes—
169C4⋊C4—72no—
170C4⋊C4—144no—
171C4⋊C4C22×D712yes—
172C22⋊C4—21no—
173C22⋊C4—72no—
174C22⋊C4—144no—
175C42—21no—
176C42—72no—
177C42—144no—
178C2×C7—22no—
179C2×C7—82no—
180C2×C7C2×#12111yescharacteristic
181C2×C7#57712yes—
182D7—81no—
183C23—22no—
184C23—42no—
185C23—82no—
186C23—141no—
187C23D4×D711yescharacteristic
188Q8—74no—
189Q8—144no—
190Q8—142no—
191Q8—141no—
192Q8—141no—
193D4—22no—
194D4—22no—
195D4—42no—
196D4—141no—
197D4—141no—
198D4—282no—
199D4—564no—
200C2×C4—22no—
201C2×C4—74no—
202C2×C4—144no—
203C2×C4—142no—
204C2×C4—141no—
205C2×C4—282no—
206C2×C4#115712yes—
207C2×C4D4×D711yescharacteristic
208C2×C4D4×D712yes—
209C2×C4C23×D711yescharacteristic
210C8—22no—
211C8—142no—
212C7#304311yescharacteristic, complemented, Sylow
213C22—22no—
214C22—42no—
215C22—44no—
216C22—84no—
217C22—281no—
218C22—282no—
219C22#644712yes—
220C22C2×D4×D711yescharacteristic
221C4—42no—
222C4—144no—
223C4—144no—
224C4—281no—
225C4#644712yes—
226C4C2×#115711yescharacteristic
227C4C2×D4×D711yescharacteristic
228C2—22no—
229C2—82no—
230C2—561no—
231C2C2×#644711yescharacteristic
232C2#9046612yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#418C2
socleC22×C7C22×D4
prime radical#90496C2
hypercenter#418D7
hypocenterC7#3043
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×#6472C2semidirect
4. #90207C2semidirect
5. C7×#418C2semidirect
6. #89891C2semidirect
7. #89774C2semidirect
8. #89539C2semidirect
9. #89534C2semidirect
24. C2×Dic28C22semidirect
25. C2×#1132C22semidirect
26. #6450C22semidirect
30. Dic7⋊C8C22semidirect
31. #6384C22semidirect
32. #6383C22semidirect
49. C2×Dic14D4semidirect
75. #418D7semidirect
212. C7#3043semidirect

Maximal subgroups

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

subgroupindex
#904962
C22×#11422
C2×#64722
#902072
C7×#4182
#898912
#897742
#895392
#895342
#30437