Finite Group Databasebeta

← #11729712

Subgroups — #11729712

Order 896.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C7×#563C211yesmaximal, characteristic, complemented
2#89833C211yesmaximal, characteristic
3#89825C212yesmaximal
4#89336C211yesmaximal, characteristic, complemented
5#89334C212yesmaximal, complemented
6#6548C2211yescharacteristic
7C7×C4⋊Q8—21no—
8C7×#126—22no—
9C7×#125—21no—
10C7×#124C2211yescharacteristic
11C7×#123C2212yes—
12C7×C22⋊Q8—22no—
13#6505—21no—
14#6492—22no—
15#6408C2212yes—
16#6407C2211yescharacteristic
17#6380—22no—
18#6379—21no—
19#2201—71nomaximal, Sylow
20C2×C7×Q8—21no—
21C2×C7×Q8D411yescharacteristic
22C2×C7×D4C2311yescharacteristic
23C22×C4×C7D412yes—
24C2×C22⋊D7—22no—
25#1159—41no—
26D4×D7—41no—
27#1157—21no—
28C7×C4⋊C4—21no—
29C7×C4⋊C4—22no—
30C7×C4⋊C4—22no—
31C7×C4⋊C4—21no—
32C7×C4⋊C4—22no—
33C7×C22⋊C4—22no—
34C7×C22⋊C4—22no—
35C7×C22⋊C4D412yes—
36C42×C7—21no—
37C42×C7D411yescharacteristic
38C22⋊Dic7—22no—
39#1144—41no—
40#1143—41no—
41(C2×D7)⋊C4—42no—
42Dic7⋊C4—42no—
43#1136—21no—
44#563D711yescharacteristic, complemented
45#458—72no—
46#455—71no—
47#354—71no—
48#352—72no—
49C23×C7C2×D412yes—
50C22×D7—22no—
51C7×Q8—41no—
52C7×Q8—81no—
53C7×D4—41no—
54C2×C4×C7—21no—
55C2×C4×C7—22no—
56C2×C4×C7—21no—
57C2×C4×C7—22no—
58C2×C4×C7—42no—
59C2×C4×C7C2×D411yescharacteristic
60C22⋊D7—42no—
61C22⋊D7—41no—
62C2×Dic7—22no—
63C2×Dic7—42no—
64D28—21no—
65C4×D7—41no—
66Dic14—21no—
67C7⋊C8—41no—
68H(5,𝔽2)—71no—
69#137—141no—
70C4⋊Q8—21no—
71#126—22no—
72#125—21no—
73#124C2×D711yescharacteristic
74#123—142no—
75#123C2×D712yes—
76C22⋊Q8—22no—
77#104—141no—
78#100—71no—
79#99—72no—
80#99—142no—
81C22×C7—22no—
82C22×C7—42no—
83C22×C7#12011yescharacteristic
84C2×D7—41no—
85C2×D7—82no—
86C4×C7—21no—
87C4×C7—42no—
88C4×C7—41no—
89C4×C7—42no—
90C4×C7—41no—
91Dic7—41no—
92Dic7—82no—
93#41—141no—
94#41—281no—
95C2×Q8—21no—
96C2×Q8C22⋊D711yescharacteristic
97C2×D4—142no—
98C2×D4—281no—
99C2×D4C22×D711yescharacteristic
100C22×C4C22⋊D712yes—
101Q16—281no—
102SD16—281no—
103M16—141no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—282no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—142no—
113C22⋊C4—282no—
114C22⋊C4C22⋊D712yes—
115C42—21no—
116C42C22⋊D711yescharacteristic
117C2×C7—21no—
118C2×C7—42no—
119C2×C7B(4,𝔽2)11yescharacteristic
120D7—81no—
121C23—142no—
122C23C2×C22⋊D712yes—
123Q8—41no—
124Q8—81no—
125Q8—141no—
126D4—41no—
127D4—141no—
128D4—282no—
129D4—281no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—42no—
135C2×C4—142no—
136C2×C4—281no—
137C2×C4—282no—
138C2×C4C2×C22⋊D711yescharacteristic
139C8—281no—
140C7#220111yescharacteristic, complemented, Sylow
141C22—22no—
142C22—42no—
143C22—281no—
144C22—562no—
145C22#651611yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—281no—
152C4—562no—
153C2—21no—
154C2—42no—
155C2—561no—
156C2#9001611yescharacteristic
subgroupG/H
centerC2#90016
commutator, cosocleC2×C7×D4C23
FrattiniC2×D4C22×D7
FittingC7×#563C2
socleC2×C7B(4,𝔽2)
prime radical#6548C22
hypercenter#563D7
hypocenterC7#2201
LCSG ▷ C2×C7×D4 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C7×#563C2semidirect
4. #89336C2semidirect
5. #89334C2semidirect
44. #563D7semidirect
140. C7#2201semidirect

Maximal subgroups

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

subgroupindex
C7×#5632
#898332
#898252
#893362
#893342
#22017