Finite Group Databasebeta

← #11739086

Subgroups — #11739086

Order 896.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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
1D7×#125C211yesmaximal, characteristic, complemented
2#90359C211yesmaximal, characteristic
3C7×#488C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic, complemented
5#89828C211yesmaximal, characteristic, complemented
6#89575C212yesmaximal, complemented
7#89569C212yesmaximal, complemented
8#89543C212yesmaximal, complemented
9#89536C212yesmaximal, complemented
10#89469C211yesmaximal, characteristic, complemented
11C4×#1131C211yesmaximal, characteristic, complemented
12C2×#1161C2211yescharacteristic
13C2×#1159C2211yescharacteristic
14C7×#125C2211yescharacteristic
15C7×#124C2211yescharacteristic
16#6507C2211yescharacteristic
17#6503C2211yescharacteristic
18C2×#1131C2212yes—
19D7×C4⋊C4—22no—
20D7×C4⋊C4C2212yes—
21#6453C2212yes—
22#6452—22no—
23#6447—22no—
24(C22⋊D7)⋊C4—22no—
25#6438C2211yescharacteristic
26C42×D7C2211yescharacteristic
27#6433C2211yescharacteristic
28C7×#103C2212yes—
29C7×#102C2212yes—
30C4×C7×C8C2211yescharacteristic
31#6409C2212yes—
32#6406C2212yes—
33#6395C2212yes—
34#6390C2212yes—
35#6389C2212yes—
36#6384C2212yes—
37#6383C2212yes—
38C4×C7⋊C8C2211yescharacteristic
39#3142—71nomaximal, Sylow
40C2×C7×Q8C2311yescharacteristic
41C2×C7×D4C2311yescharacteristic
42C2×C22⋊D7—21no—
43C2×C22⋊D7—22no—
44C22×Dic7—21no—
45#1161—24no—
46#1159—24no—
47C2×D28—21no—
48C2×D28C2311yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—21no—
51C2×C4×D7D412yes—
52C2×C4×D7C2311yescharacteristic
53C2×Dic14C2311yescharacteristic
54C2×C7×C8C2312yes—
55C7×C4⋊C4—22no—
56C7×C4⋊C4C2312yes—
57C7×C22⋊C4—22no—
58C42×C7C2311yescharacteristic
59(C2×D7)⋊C4—22no—
60C4⋊Dic7—22no—
61C4⋊Dic7C2312yes—
62Dic7⋊C4—22no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66C4×Dic7D412yes—
67C4×Dic7C2311yescharacteristic
68C2×C7⋊C8C2312yes—
69#1131—24no—
70#532—71no—
71C2×#125—71no—
72#491—72no—
73#490—72no—
74#488—71no—
75#488—71no—
76#488—71no—
77#488D711yescharacteristic, complemented
78#417—72no—
79#417—72no—
80C4×M16—71no—
81C23×C7—21no—
82C22×D7—21no—
83C22×D7C2×D411yescharacteristic
84C7×Q8—22no—
85C7×D4—22no—
86C2×C4×C7—22no—
87C2×C4×C7—21no—
88C2×C4×C7C2×D412yes—
89C2×C4×C7C2411yescharacteristic
90C22⋊D7—44no—
91C22⋊D7—42no—
92C2×Dic7—22no—
93C2×Dic7—21no—
94C2×Dic7—42no—
95C2×Dic7C2×D412yes—
96C2×Dic7C2×D411yescharacteristic
97D28—22no—
98D28—42no—
99C4×D7—42no—
100C4×D7—44no—
101C4×D7—42no—
102C4×D7#4114yes—
103Dic14—22no—
104C7×C8—22no—
105C7⋊C8—22no—
106C2×#41—71no—
107C2×#41—71no—
108C2×M16—72no—
109C4⋊Q8—71no—
110#127—71no—
111#125—72no—
112#125—71no—
113#125—142no—
114#125C2×D711yescharacteristic
115#124—71no—
116#124C2×D711yescharacteristic
117#121—142no—
118C4×D4—142no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—142no—
121C2×C42—71no—
122#103—72no—
123#103—72no—
124#103—72no—
125#103C2×D712yes—
126#102—72no—
127#102—72no—
128#102—72no—
129#102C2×D712yes—
130#97—72no—
131C4×C8—71no—
132C4×C8C2×D711yescharacteristic
133C22×C7—41no—
134C22×C7—42no—
135C22×C7C22×D411yescharacteristic
136C2×D7—22no—
137C2×D7—41no—
138C2×D7—42no—
139C2×D7#12412yes—
140C4×C7—22no—
141C4×C7—41no—
142C4×C7—42no—
143C4×C7C2×#4112yes—
144Dic7—22no—
145Dic7—41no—
146Dic7—42no—
147Dic7#12412yes—
148#41—144no—
149#41—144no—
150C2×Q8—71no—
151C2×Q8C22×D711yescharacteristic
152C2×D4—71no—
153C2×D4—141no—
154C2×D4—142no—
155C2×D4—141no—
156C2×D4C22×D711yescharacteristic
157C22×C4—71no—
158C22×C4—72no—
159C22×C4—141no—
160C22×C4—142no—
161C22×C4—141no—
162M16—144no—
163C2×C8—72no—
164C2×C8C22×D712yes—
165C4⋊C4—22no—
166C4⋊C4—72no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4C22×D712yes—
173C22⋊C4—22no—
174C22⋊C4—142no—
175C42—72no—
176C42—71no—
177C42C22×D711yescharacteristic
178C2×C7—81no—
179C2×C7C2×#12411yescharacteristic
180C2×C7#57512yes—
181D7—22no—
182D7—81no—
183C23—21no—
184C23—71no—
185C23—141no—
186Q8—22no—
187Q8—142no—
188D4—22no—
189D4—142no—
190D4—282no—
191D4—284no—
192D4—282no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—71no—
196C2×C4—74no—
197C2×C4—72no—
198C2×C4—141no—
199C2×C4—142no—
200C2×C4—282no—
201C2×C4—282no—
202C2×C4—284no—
203C2×C4—282no—
204C2×C4D4×D712yes—
205C2×C4C23×D711yescharacteristic
206C8—22no—
207C8—142no—
208C7#314211yescharacteristic, complemented, Sylow
209C22—42no—
210C22—41no—
211C22—72no—
212C22—142no—
213C22—282no—
214C22—281no—
215C22C2×D4×D711yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—72no—
220C4—142no—
221C4—282no—
222C4—281no—
223C4D7×#4112yes—
224C2—81no—
225C2—142no—
226C2—561no—
227C2D7×#12411yescharacteristic
228C2#9045912yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#488C2
socleC22×C7C22×D4
prime radical#90359C2
hypercenter#488D7
hypocenterC7#3142
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. D7×#125C2semidirect
3. C7×#488C2semidirect
4. #89840C2semidirect
5. #89828C2semidirect
6. #89575C2semidirect
7. #89569C2semidirect
8. #89543C2semidirect
9. #89536C2semidirect
10. #89469C2semidirect
11. C4×#1131C2semidirect
77. #488D7semidirect
208. C7#3142semidirect

Maximal subgroups

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

subgroupindex
D7×#1252
#903592
C7×#4882
#898402
#898282
#895752
#895692
#895432
#895362
#894692
C4×#11312
#31427