Finite Group Databasebeta

← #11736768

Subgroups — #11736768

Order 896.

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

normal subgroups only

This group has 362 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
2C2×#6432C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89623C212yesmaximal, complemented
5#89618C212yesmaximal, complemented
6#89507C212yesmaximal
7#89494C212yesmaximal, complemented
8#89486C212yesmaximal, complemented
9#89474C212yesmaximal, complemented
10C22×Dic14—21no—
11C2×C42×C7C2211yescharacteristic
12#6493C2211yescharacteristic
13#6492—21no—
14C4×C22⋊D7—21no—
15C2×C4⋊Dic7—21no—
16C2×C4⋊Dic7C2212yes—
17#6487C2211yescharacteristic
18#6439—21no—
19#6438C2211yescharacteristic
20C4×D28C2211yescharacteristic
21#6435—21no—
22#6433C2211yescharacteristic
23#6432—21no—
24#6432—21no—
25#6432C2212yes—
26#6432C2212yes—
27C4×Dic14C2211yescharacteristic
28C7×C4⋊C8C2212yes—
29C7×C22⋊C8C2212yes—
30C7×#97C2212yes—
31#6395C2214yes—
32#6392C2214yes—
33#6391C2214yes—
34#6390C2214yes—
35#3239—71nomaximal, Sylow
36C22×C4×C7D411yescharacteristic
37C22×C4×C7D411yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—21no—
40C22×Dic7—22no—
41C2×D28C2311yescharacteristic
42C2×C4×D7—21no—
43C2×Dic14—22no—
44C2×Dic14—22no—
45C2×Dic14—44no—
46C2×Dic14C2311yescharacteristic
47C2×C7×C8C2314yes—
48C42×C7D411yescharacteristic
49C42×C7D411yescharacteristic
50C42×C7C2311yescharacteristic
51C42×C7C2311yescharacteristic
52C22⋊Dic7—21no—
53(C2×D7)⋊C4—21no—
54(C2×D7)⋊C4—21no—
55(C2×D7)⋊C4—21no—
56(C2×D7)⋊C4—21no—
57C4⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7C2311yescharacteristic
61C4⋊Dic7C2314yes—
62C4⋊Dic7C2311yescharacteristic
63Dic7⋊C4—21no—
64Dic7⋊C4—21no—
65Dic7⋊C4—21no—
66Dic7⋊C4—21no—
67C4×Dic7—21no—
68C2×C4⋊Q8—71no—
69#529—71no—
70#501—72no—
71#489—72no—
72#484—72no—
73#481—72no—
74#475—72no—
75#474—72no—
76#432D711yescharacteristic, complemented
77C23×C7C2×D411yescharacteristic
78C22×D7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2411yescharacteristic
88C22⋊D7—42no—
89C2×Dic7—21no—
90C2×Dic7—21no—
91C2×Dic7—21no—
92C2×Dic7—24no—
93C2×Dic7—44no—
94D28—22no—
95C4×D7—42no—
96Dic14—22no—
97Dic14—44no—
98Dic14—44no—
99C7×C8—24no—
100C22×Q8—141no—
101C4⋊Q8—72no—
102C4⋊Q8—72no—
103C4⋊Q8—141no—
104C4⋊Q8—141no—
105#126—141no—
106#125—71no—
107#124—71no—
108#123—141no—
109C22⋊Q8—71no—
110#121—71no—
111C4×Q8—71no—
112C4×D4—71no—
113C4×D4—141no—
114#117—141no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—141no—
117C2×C42C2×D711yescharacteristic
118#107—74no—
119#106—74no—
120C4⋊C8C2×D712yes—
121#103—74no—
122#102—74no—
123C22⋊C8C2×D712yes—
124#97C2×D712yes—
125C22×C7—21no—
126C22×C7—21no—
127C22×C7#13712yes—
128C22×C7C22×D411yescharacteristic
129C2×D7—41no—
130C2×D7—41no—
131C2×D7—41no—
132C4×C7—21no—
133C4×C7—21no—
134C4×C7—21no—
135C4×C7—21no—
136C4×C7AGL(1,ℤ/8)12yes—
137C4×C72−1+412yes—
138Dic7—41no—
139Dic7—44no—
140Dic7—41no—
141Dic7—41no—
142C2×Q8—71no—
143C2×Q8—142no—
144C2×Q8—142no—
145C2×Q8—284no—
146C2×D4—71no—
147C2×D4—141no—
148C22×C4—142no—
149C22×C4—141no—
150C22×C4D2811yescharacteristic
151C22×C4D2811yescharacteristic
152C22×C4C22×D711yescharacteristic
153C2×C8C22×D714yes—
154C4⋊C4—74no—
155C4⋊C4—71no—
156C4⋊C4—71no—
157C4⋊C4—141no—
158C4⋊C4—141no—
159C4⋊C4—141no—
160C4⋊C4—141no—
161C4⋊C4—142no—
162C4⋊C4—142no—
163C4⋊C4—142no—
164C22⋊C4—141no—
165C22⋊C4—141no—
166C22⋊C4—141no—
167C22⋊C4—141no—
168C22⋊C4—141no—
169C42—141no—
170C42D2811yescharacteristic
171C42D2811yescharacteristic
172C42C22×D711yescharacteristic
173C42C22×D711yescharacteristic
174C2×C7—22no—
175C2×C7#53611yescharacteristic
176C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C7C2×#13711yescharacteristic
178D7—81no—
179C23—141no—
180C23C2×D2811yescharacteristic
181Q8—142no—
182Q8—284no—
183Q8—284no—
184D4—142no—
185D4—282no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—141no—
190C2×C4—141no—
191C2×C4—141no—
192C2×C4—144no—
193C2×C4—284no—
194C2×C4—282no—
195C2×C4C2×D2811yescharacteristic
196C2×C4C2×D2811yescharacteristic
197C2×C4C2×D2811yescharacteristic
198C2×C4C2×D2811yescharacteristic
199C2×C4C2×D2811yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—24no—
202C7#323911yescharacteristic, complemented, Sylow
203C22—21no—
204C22—21no—
205C22—281no—
206C22—281no—
207C22—281no—
208C22#647212yes—
209C22C22×D2811yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—284no—
215C4—281no—
216C4—281no—
217C4—281no—
218C4#647112yes—
219C4#655412yes—
220C2—22no—
221C2—561no—
222C2#9021711yescharacteristic
223C2C2×#647111yescharacteristic
224C2C2×#647211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#432D7
hypocenterC7#3239
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. C2×#6432C2semidirect
3. C7×#432C2semidirect
4. #89623C2semidirect
5. #89618C2semidirect
7. #89494C2semidirect
8. #89486C2semidirect
9. #89474C2semidirect
76. #432D7semidirect
202. C7#3239semidirect

Maximal subgroups

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

subgroupindex
#901702
C2×#64322
C7×#4322
#896232
#896182
#895072
#894942
#894862
#894742
#32397