Finite Group Databasebeta

← #11739161

Subgroups — #11739161

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
1#90407C211yesmaximal, characteristic
2D7×#125C211yesmaximal, characteristic, complemented
3C7×#490C211yesmaximal, characteristic, complemented
4#89843C211yesmaximal, characteristic, complemented
5#89841C211yesmaximal, characteristic, complemented
6#89543C214yesmaximal, complemented
7#89540C214yesmaximal, complemented
8#89479C211yesmaximal, characteristic, complemented
9#89475C211yesmaximal, characteristic, complemented
10C2×#1159C2212yes—
11C7×#127C2211yescharacteristic
12C7×#125C2211yescharacteristic
13#6501—22no—
14#6499C2212yes—
15D4×Dic7—22no—
16C2×#1131C2212yes—
17D7×C4⋊C4—22no—
18D7×C4⋊C4C2212yes—
19#6453C2212yes—
20#6452—22no—
21C42×D7C2211yescharacteristic
22#6433C2211yescharacteristic
23#6432C2211yescharacteristic
24C7×#102C2214yes—
25C7×#97C2211yescharacteristic
26#6406C2214yes—
27#6390C2214yes—
28C8×Dic7C2212yes—
29#6384C2214yes—
30#6377C2211yescharacteristic
31#3142—71nomaximal, Sylow
32C2×C7×D4—22no—
33C2×C7×D4C2312yes—
34C2×C22⋊D7—22no—
35C22×Dic7—22no—
36#1159—28no—
37C2×C4×D7—22no—
38C2×C4×D7D412yes—
39C2×C4×D7C2311yescharacteristic
40C2×Dic14C2312yes—
41C2×C7×C8C2312yes—
42C7×C4⋊C4—22no—
43C7×C4⋊C4C2312yes—
44C42×C7C2311yescharacteristic
45C22⋊Dic7—24no—
46C4⋊Dic7—22no—
47C4⋊Dic7—22no—
48C4⋊Dic7C2312yes—
49Dic7⋊C4—22no—
50Dic7⋊C4—22no—
51Dic7⋊C4—22no—
52C4×Dic7D412yes—
53C4×Dic7C2311yescharacteristic
54C2×C7⋊C8C2312yes—
55#1131—24no—
56#532—71no—
57C2×#125—71no—
58#491—71no—
59#491—71no—
60#490—71no—
61#490D711yescharacteristic, complemented
62#488—74no—
63#417—74no—
64C4×M16—71no—
65C23×C7—22no—
66C22×D7C2×D411yescharacteristic
67C7×D4—24no—
68C7×D4—44no—
69C2×C4×C7—22no—
70C2×C4×C7C2×D412yes—
71C2×C4×C7C2411yescharacteristic
72C22⋊D7—44no—
73C2×Dic7—22no—
74C2×Dic7—22no—
75C2×Dic7—44no—
76C2×Dic7C2×D412yes—
77C2×Dic7C2×D411yescharacteristic
78C4×D7—42no—
79C4×D7—44no—
80C4×D7#4114yes—
81Dic14—24no—
82C7×C8—22no—
83C7⋊C8—22no—
84C2×#41—72no—
85C2×M16—72no—
86C4⋊Q8—71no—
87#127C2×D711yescharacteristic
88#125—72no—
89#125—71no—
90#125—142no—
91#125C2×D711yescharacteristic
92#124—72no—
93#121—142no—
94C4×D4—142no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—142no—
97C2×C42—71no—
98#103—74no—
99#103—74no—
100#102—74no—
101#102C2×D714yes—
102#97—71no—
103#97C2×D711yescharacteristic
104C4×C8—72no—
105C22×C7—42no—
106C22×C7—44no—
107C22×C7C22×D411yescharacteristic
108C2×D7—22no—
109C2×D7#12412yes—
110C4×C7—22no—
111C4×C7—42no—
112C4×C7C2×#4112yes—
113Dic7—22no—
114Dic7—42no—
115Dic7—42no—
116Dic7#12412yes—
117#41—148no—
118C2×Q8—72no—
119C2×D4—22no—
120C2×D4—142no—
121C2×D4C22×D712yes—
122C22×C4—71no—
123C22×C4—72no—
124C22×C4—142no—
125C22×C4—142no—
126M16—144no—
127C2×C8—72no—
128C2×C8C22×D712yes—
129C4⋊C4—22no—
130C4⋊C4—72no—
131C4⋊C4—142no—
132C4⋊C4—142no—
133C4⋊C4—142no—
134C4⋊C4—142no—
135C4⋊C4—142no—
136C4⋊C4C22×D712yes—
137C22⋊C4—144no—
138C42—72no—
139C42—71no—
140C42C22×D711yescharacteristic
141C2×C7—82no—
142C2×C7C2×#12411yescharacteristic
143C2×C7#57512yes—
144D7—22no—
145C23—22no—
146C23—71no—
147Q8—144no—
148D4—24no—
149D4—44no—
150D4—284no—
151C2×C4—22no—
152C2×C4—71no—
153C2×C4—74no—
154C2×C4—72no—
155C2×C4—142no—
156C2×C4—142no—
157C2×C4—282no—
158C2×C4—284no—
159C2×C4—284no—
160C2×C4D4×D712yes—
161C2×C4C23×D711yescharacteristic
162C8—22no—
163C8—142no—
164C7#314211yescharacteristic, complemented, Sylow
165C22—44no—
166C22—42no—
167C22—72no—
168C22—142no—
169C22C2×D4×D711yescharacteristic
170C4—22no—
171C4—42no—
172C4—72no—
173C4—142no—
174C4—282no—
175C4—282no—
176C4D7×#4112yes—
177C2—82no—
178C2—142no—
179C2D7×#12411yescharacteristic
180C2#9046412yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#490C2
socleC22×C7C22×D4
prime radical#90407C2
hypercenter#490D7
hypocenterC7#3142
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. D7×#125C2semidirect
3. C7×#490C2semidirect
4. #89843C2semidirect
5. #89841C2semidirect
6. #89543C2semidirect
7. #89540C2semidirect
8. #89479C2semidirect
9. #89475C2semidirect
61. #490D7semidirect
164. C7#3142semidirect

Maximal subgroups

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

subgroupindex
#904072
D7×#1252
C7×#4902
#898432
#898412
#895432
#895402
#894792
#894752
#31427