Finite Group Databasebeta

← #11740500

Subgroups — #11740500

Order 896.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×C7×#125C211yesmaximal, characteristic
2C2×#6433C211yesmaximal, characteristic
3#89758C212yesmaximal
4#89756C214yesmaximal
5#89755C212yesmaximal
6#89754C212yesmaximal
7#89752C212yesmaximal
8#89708C211yesmaximal, characteristic
9C7×#125—24no—
10C2×C7×C4⋊C4C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C4⋊C4C2212yes—
13C2×C7×C4⋊C4C2212yes—
14C2×C42×C7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2214yes—
21#6433—24no—
22C7⋊#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2214yes—
25C7⋊#95C2214yes—
26C7⋊#95C2214yes—
27#2822—71nomaximal, Sylow
28C22×C4×C7C2312yes—
29C22×C4×C7C2312yes—
30C22×C4×C7C2311yescharacteristic
31C22×C4×C7C2311yescharacteristic
32C22×C4×C7C2311yescharacteristic
33C22×Dic7C2314yes—
34C22×Dic7C2312yes—
35C22×Dic7C2312yes—
36C7×C4⋊C4—22no—
37C7×C4⋊C4—22no—
38C7×C4⋊C4—24no—
39C7×C4⋊C4—24no—
40C42×C7D414yes—
41C4⋊Dic7—22no—
42C4⋊Dic7—22no—
43C4⋊Dic7—42no—
44Dic7⋊C4—24no—
45Dic7⋊C4—24no—
46Dic7⋊C4—44no—
47C2×#125—71no—
48C2×#125D711yescharacteristic
49#400—72no—
50#400—72no—
51#398—72no—
52#398—72no—
53#398—74no—
54#382—71no—
55C23×C7C2411yescharacteristic
56C2×C4×C7—21no—
57C2×C4×C7—22no—
58C2×C4×C7—22no—
59C2×C4×C7—21no—
60C2×C4×C7—22no—
61C2×C4×C7—22no—
62C2×C4×C7—22no—
63C2×C4×C7—24no—
64C2×C4×C7—22no—
65C2×C4×C7C2×D412yes—
66C2×C4×C7C2×D412yes—
67C2×C4×C7C2×D412yes—
68C2×Dic7—22no—
69C2×Dic7—22no—
70C2×Dic7—22no—
71C2×Dic7—24no—
72C2×Dic7—24no—
73C2×Dic7—24no—
74C2×Dic7—22no—
75C2×Dic7—24no—
76#125—24no—
77#125—144no—
78C2×C4⋊C4—72no—
79C2×C4⋊C4—72no—
80C2×C4⋊C4—71no—
81C2×C4⋊C4—71no—
82C2×C4⋊C4—74no—
83C2×C4⋊C4—72no—
84C2×C4⋊C4C2×D712yes—
85C2×C4⋊C4C2×D711yescharacteristic
86C2×C4⋊C4C2×D712yes—
87C2×C4⋊C4C2×D711yescharacteristic
88C2×C42C2×D711yescharacteristic
89#95—72no—
90#95—72no—
91#95—74no—
92#95—74no—
93#95—74no—
94C22×C7C22×D411yescharacteristic
95C22×C7H(5,𝔽2)11yescharacteristic
96C22×C72−1+412yes—
97C22×C72−1+412yes—
98C22×C72−1+411yescharacteristic
99C4×C7—22no—
100C4×C7—22no—
101C4×C7—22no—
102C4×C7—42no—
103C4×C7—42no—
104Dic7—42no—
105Dic7—42no—
106Dic7—44no—
107C22×C4—74no—
108C22×C4—72no—
109C22×C4—72no—
110C22×C4C22×D712yes—
111C22×C4C22×D712yes—
112C22×C4C22×D711yescharacteristic
113C22×C4C22×D711yescharacteristic
114C22×C4C22×D711yescharacteristic
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—144no—
120C4⋊C4—144no—
121C4⋊C4—142no—
122C4⋊C4—142no—
123C4⋊C4—284no—
124C4⋊C4—282no—
125C42C22⋊D714yes—
126C2×C7#53611yescharacteristic
127C2×C7#53611yescharacteristic
128C2×C7#53711yescharacteristic
129C2×C7#56312yes—
130C2×C7#56412yes—
131C23C23×D711yescharacteristic
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—144no—
142C2×C4—144no—
143C2×C4—144no—
144C2×C4—142no—
145C2×C4—144no—
146C2×C4—142no—
147C2×C4—142no—
148C2×C4—142no—
149C2×C4C2×C22⋊D712yes—
150C2×C4C2×C22⋊D712yes—
151C2×C4C2×C22⋊D712yes—
152C7#282211yescharacteristic, complemented, Sylow
153C22#655112yes—
154C22#655311yescharacteristic
155C22#655411yescharacteristic
156C22#655412yes—
157C22C22×C22⋊D711yescharacteristic
158C4—22no—
159C4—22no—
160C4—22no—
161C4—42no—
162C4—42no—
163C4—284no—
164C4—282no—
165C4—282no—
166C2C7⋊#56412yes—
167C2#9039012yes—
168C2#9050711yescharacteristic
169C2#9051611yescharacteristic
170C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC2×C7×#125C2
hypercenterC2×#125D7
hypocenterC7#2822
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
152. C7#2822semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1252
C2×#64332
#897582
#897562
#897552
#897542
#897522
#897082
#28227