Finite Group Databasebeta

← #11729713

Subgroups — #11729713

Order 896.

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

normal subgroups only

This group has 216 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#89867C212yesmaximal, complemented
3#89834C211yesmaximal, characteristic
4#89343C212yesmaximal, complemented
5#89337C211yesmaximal, characteristic, complemented
6#6551C2211yescharacteristic
7C7×C4⋊Q8C2212yes—
8C7×#126—22no—
9C7×#125—21no—
10C7×#124C2211yescharacteristic
11C7×#123—21no—
12C7×C22⋊Q8—22no—
13#6505—22no—
14#6495—21no—
15#6410C2212yes—
16#6410C2211yescharacteristic
17#6379—22no—
18#6379—21no—
19#2203—71nomaximal, Sylow
20C2×C7×Q8D412yes—
21C2×C7×Q8C2311yescharacteristic
22C2×C7×D4D411yescharacteristic
23C22×C4×C7—21no—
24#1161—41no—
25Q8×D7—41no—
26#1157—21no—
27#1157—22no—
28C7×C4⋊C4—21no—
29C7×C4⋊C4—22no—
30C7×C4⋊C4—21no—
31C7×C4⋊C4—22no—
32C7×C4⋊C4—22no—
33C7×C22⋊C4—22no—
34C7×C22⋊C4—21no—
35C7×C22⋊C4—22no—
36C42×C7D411yescharacteristic
37C42×C7D412yes—
38#1144—42no—
39#1143—42no—
40#1142—41no—
41#1141—41no—
42#1136—21no—
43#1136—22no—
44#563D711yescharacteristic, complemented
45#456—72no—
46#454—71no—
47#356—72no—
48#355—71no—
49C23×C7—21no—
50C7×Q8—41no—
51C7×Q8—42no—
52C7×D4—41no—
53C2×C4×C7—21no—
54C2×C4×C7—22no—
55C2×C4×C7—22no—
56C2×C4×C7—81no—
57C2×C4×C7C2×D412yes—
58C2×C4×C7C2×D411yescharacteristic
59C22⋊D7—41no—
60D28—21no—
61D28—22no—
62C4×D7—41no—
63C4×D7—42no—
64Dic14—21no—
65Dic14—22no—
66C7⋊C8—41no—
67C7⋊C8—42no—
682−1+4—71no—
69#137—142no—
70AGL(1,ℤ/8)—141no—
71C4⋊Q8C2×D712yes—
72#126—22no—
73#125—21no—
74#124C2×D711yescharacteristic
75#123—21no—
76C22⋊Q8—22no—
77#104—141no—
78#104—142no—
79#101—72no—
80#101—71no—
81C22×C7—42no—
82C22×C7—41no—
83C22×C7#12011yescharacteristic
84C2×D7—41no—
85C4×C7—21no—
86C4×C7—22no—
87C4×C7—41no—
88C4×C7—42no—
89C4×C7—42no—
90Dic7—41no—
91#41—141no—
92#41—142no—
93#41—281no—
94C2×Q8—281no—
95C2×Q8C22⋊D712yes—
96C2×Q8C22×D711yescharacteristic
97C2×D4C22⋊D711yescharacteristic
98C22×C4—21no—
99Q16—282no—
100SD16—281no—
101SD16—282no—
102D8—281no—
103M16—141no—
104M16—142no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—21no—
112C22⋊C4—22no—
113C42C22⋊D711yescharacteristic
114C42C22⋊D712yes—
115C2×C7—21no—
116C2×C7—81no—
117C2×C7B(4,𝔽2)11yescharacteristic
118D7—81no—
119C23—21no—
120Q8—41no—
121Q8—42no—
122Q8—141no—
123Q8—142no—
124D4—41no—
125D4—141no—
126D4—142no—
127D4—281no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—81no—
132C2×C4—281no—
133C2×C4—282no—
134C2×C4C2×C22⋊D712yes—
135C2×C4C2×C22⋊D711yescharacteristic
136C8—281no—
137C8—282no—
138C7#220311yescharacteristic, complemented, Sylow
139C22—42no—
140C22—41no—
141C22—281no—
142C22#651611yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—281no—
149C2—21no—
150C2—81no—
151C2—561no—
152C2#9001611yescharacteristic
subgroupG/H
centerC2#90016
commutator, cosocleC2×C7×Q8C23
FrattiniC2×Q8C22×D7
FittingC7×#563C2
socleC2×C7B(4,𝔽2)
prime radical#89834C2
hypercenter#563D7
hypocenterC7#2203
LCSG ▷ C2×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C7×#563C2semidirect
2. #89867C2semidirect
4. #89343C2semidirect
5. #89337C2semidirect
44. #563D7semidirect
138. C7#2203semidirect

Maximal subgroups

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

subgroupindex
C7×#5632
#898672
#898342
#893432
#893372
#22037