Finite Group Databasebeta

← #11728749

Subgroups — #11728749

Order 896.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#90413C211yesmaximal, characteristic, complemented
2C7×#456C211yesmaximal, characteristic
3#90018C212yesmaximal
4#89343C211yesmaximal, characteristic, complemented
5#89273C212yesmaximal, complemented
6C7×2−1+4C2211yescharacteristic
7C7×#137—21no—
8C7×C4⋊Q8C2211yescharacteristic
9#6514—22no—
10#6507—21no—
11#6507C2212yes—
12#6451—22no—
13#6450—22no—
14C4×Dic14—21no—
15C7×#104—21no—
16C7×#101C2211yescharacteristic
17#6411—22no—
18#6410C2212yes—
19#2205—71nomaximal, Sylow
20C7×#41—22no—
21C7×#41—21no—
22C7×#41—41no—
23C2×C7×Q8—41no—
24C2×C7×Q8D411yescharacteristic
25C2×C7×Q8C2311yescharacteristic
26C2×Dic14—21no—
27C2×Dic14—21no—
28C2×Dic14D412yes—
29C7×Q16—41no—
30C7×SD16—41no—
31C7×M16—21no—
32C7×C4⋊C4—22no—
33C42×C7D411yescharacteristic
34#1144—42no—
35#1142—42no—
36C4⋊Dic7—21no—
37Dic7⋊C4—22no—
38Dic7⋊C4—22no—
39Dic7⋊C4—22no—
40C4×Dic7—22no—
41C4×Dic7—21no—
42C4×Dic7D412yes—
43#1136—22no—
44Q82—71no—
45#456—72no—
46#456D711yescharacteristic
47#356—71no—
48#356—72no—
49C7×Q8—22no—
50C7×Q8—21no—
51C7×Q8—41no—
52C7×Q8—41no—
53C7×D4—22no—
54C7×D4—21no—
55C7×D4—41no—
56C2×C4×C7—21no—
57C2×C4×C7—21no—
58C2×C4×C7—42no—
59C2×C4×C7—41no—
60C2×C4×C7C2×D412yes—
61C2×C4×C7C2×D411yescharacteristic
62C2×Dic7—22no—
63C2×Dic7—22no—
64Dic14—24no—
65Dic14—42no—
66Dic14—81no—
67C7×C8—41no—
68C7⋊C8—42no—
692−1+4C2×D711yescharacteristic
70#137—21no—
71#137—142no—
72C4⋊Q8—72no—
73C4⋊Q8—142no—
74C4⋊Q8—141no—
75C4⋊Q8C2×D711yescharacteristic
76C4×Q8—142no—
77C4×Q8—141no—
78#104—21no—
79#104—142no—
80#101—72no—
81#101C2×D711yescharacteristic
82C22×C7—41no—
83C22×C7#12011yescharacteristic
84C4×C7—22no—
85C4×C7—22no—
86C4×C7—21no—
87C4×C7—41no—
88C4×C7—41no—
89Dic7—24no—
90Dic7—42no—
91#41—22no—
92#41—21no—
93#41—41no—
94C2×Q8—41no—
95C2×Q8—72no—
96C2×Q8—141no—
97C2×Q8—141no—
98C2×Q8C22⋊D711yescharacteristic
99C2×Q8C22×D711yescharacteristic
100Q16—41no—
101Q16—282no—
102SD16—41no—
103SD16—282no—
104M16—21no—
105M16—142no—
106C4⋊C4—22no—
107C4⋊C4—142no—
108C4⋊C4—141no—
109C4⋊C4—142no—
110C4⋊C4—142no—
111C42—72no—
112C42—142no—
113C42—141no—
114C42C22⋊D711yescharacteristic
115C2×C7—21no—
116C2×C7—81no—
117C2×C7B(4,𝔽2)11yescharacteristic
118Q8—21no—
119Q8—22no—
120Q8—41no—
121Q8—41no—
122Q8—144no—
123Q8—282no—
124Q8—561no—
125D4—21no—
126D4—22no—
127D4—41no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—142no—
133C2×C4—142no—
134C2×C4D4×D712yes—
135C2×C4C2×C22⋊D711yescharacteristic
136C8—41no—
137C8—282no—
138C7#220511yescharacteristic, complemented, Sylow
139C22—41no—
140C22#650011yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—144no—
147C4—282no—
148C2—21no—
149C2—81no—
150C2#8980411yescharacteristic
subgroupG/H
centerC2#89804
commutator, cosocleC2×C7×Q8C23
FrattiniC2×Q8C22×D7
FittingC7×#456C2
socleC2×C7B(4,𝔽2)
prime radicalC7×2−1+4C22
hypercenter#456D7
hypocenterC7#2205
LCSG ▷ C2×C7×Q8 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #90413C2semidirect
4. #89343C2semidirect
5. #89273C2semidirect
138. C7#2205semidirect

Maximal subgroups

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

subgroupindex
#904132
C7×#4562
#900182
#893432
#892732
#22057