Finite Group Databasebeta

← #13284847

Subgroups — #13284847

Order 1792.

1334 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 411 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
1C22×C7×#125C211yesmaximal, characteristic
2C2×#89708C212yesmaximal
3#11729323C214yesmaximal
4#11729319C218yesmaximal
5C2×C7×#125—28no—
6C2×C7×#125C418yescomplemented
7C22×C7×C4⋊C4C2214yes—
8C22×C7×C4⋊C4C2212yes—
9C22×C42×C7C2211yescharacteristic
10C22×C4⋊Dic7C2214yes—
11C2×C7⋊#95C2214yes—
12C2×C7⋊#95C2214yes—
13C2×C7⋊#95C22116yes—
14#89708—28no—
15#14664—71nomaximal, Sylow
16C23×C4×C7C2314yes—
17C23×C4×C7C2312yes—
18C23×C4×C7C2311yescharacteristic
19C23×Dic7C2318yes—
20C7×#125—232no—
21C2×C7×C4⋊C4—28no—
22C2×C7×C4⋊C4—24no—
23C2×C7×C4⋊C4—24no—
24C2×C7×C4⋊C4—28no—
25C2×C7×C4⋊C4C2×C4116yes—
26C2×C7×C4⋊C4C2×C418yes—
27C2×C42×C7C2×C414yes—
28C2×C42×C7D418yes—
29C2×C4⋊Dic7—28no—
30C2×C4⋊Dic7—48no—
31C7⋊#95—28no—
32C7⋊#95—28no—
33C7⋊#95—416no—
34C22×#125D711yescharacteristic
35C2×#382—72no—
36#1445—74no—
37#1441—78no—
38C24×C7C2411yescharacteristic
39C22×C4×C7—24no—
40C22×C4×C7—24no—
41C22×C4×C7—24no—
42C22×C4×C7—24no—
43C22×C4×C7—24no—
44C22×C4×C7—28no—
45C22×C4×C7—28no—
46C22×C4×C7C22×C418yes—
47C22×C4×C7C22×C412yes—
48C22×C4×C7C22×C414yes—
49C22×C4×C7C2×D414yes—
50C22×C4×C7C2×D414yes—
51C22×C4×C7C2×D414yes—
52C22×Dic7—28no—
53C22×Dic7—28no—
54C22×Dic7—216no—
55C22×Dic7—216no—
56C22×Dic7—28no—
57C7×C4⋊C4—216no—
58C7×C4⋊C4—232no—
59C42×C7C22⋊C4116yes—
60C4⋊Dic7—416no—
61C2×#125—28no—
62C2×#125Dic718yescomplemented
63C22×C4⋊C4—74no—
64C22×C4⋊C4C2×D714yes—
65C22×C4⋊C4C2×D712yes—
66C22×C42C2×D711yescharacteristic
67#382—148no—
68C2×#95—74no—
69C2×#95—74no—
70C2×#95—716no—
71C23×C7C23×C411yescharacteristic
72C23×C7C22×D412yes—
73C23×C7H(5,𝔽2)14yes—
74C23×C72−1+414yes—
75C23×C72−1+414yes—
76C2×C4×C7—24no—
77C2×C4×C7—28no—
78C2×C4×C7—24no—
79C2×C4×C7—28no—
80C2×C4×C7—24no—
81C2×C4×C7—28no—
82C2×C4×C7—28no—
83C2×C4×C7—216no—
84C2×C4×C7—44no—
85C2×C4×C7—44no—
86C2×C4×C7—48no—
87C2×C4×C7C2×C22⋊C414yes—
88C2×C4×C7C2×C22⋊C418yes—
89C2×Dic7—48no—
90C2×Dic7—48no—
91C2×Dic7—48no—
92C2×Dic7—416no—
93C2×Dic7—416no—
94C23×C4—78no—
95C23×C4C22×D714yes—
96C23×C4C22×D712yes—
97C23×C4C22×D711yescharacteristic
98#125—232no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—148no—
104C2×C4⋊C4—288no—
105C2×C4⋊C4C2×Dic7116yes—
106C2×C4⋊C4C2×Dic718yes—
107C2×C42C2×Dic714yes—
108C2×C42C22⋊D718yes—
109#95—148no—
110#95—148no—
111#95—2816no—
112C22×C7C22×C22⋊C411yescharacteristic
113C22×C7#51912yes—
114C22×C7#52014yes—
115C22×C7#53512yes—
116C22×C7#53614yes—
117C22×C7#53612yes—
118C22×C7#53714yes—
119C22×C7#56214yes—
120C22×C7#56318yes—
121C22×C7#56414yes—
122C4×C7—24no—
123C4×C7—28no—
124C4×C7—48no—
125Dic7—88no—
126C24C23×D711yescharacteristic
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—28no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—1416no—
135C22×C4—148no—
136C22×C4—1416no—
137C22×C4—148no—
138C22×C4—148no—
139C22×C4C22×Dic718yes—
140C22×C4C22×Dic712yes—
141C22×C4C22×Dic714yes—
142C22×C4C2×C22⋊D714yes—
143C22×C4C2×C22⋊D714yes—
144C22×C4C2×C22⋊D714yes—
145C4⋊C4—216no—
146C4⋊C4—232no—
147C4⋊C4—2816no—
148C42C22⋊Dic7116yes—
149C2×C7#231011yescharacteristic
150C2×C7#231112yes—
151C2×C7#238214yes—
152C2×C7#282014yes—
153C2×C7#282214yes—
154C23#655114yes—
155C23#655314yes—
156C23#655414yes—
157C23C23×Dic711yescharacteristic
158C23C22×C22⋊D712yes—
159C2×C4—24no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—24no—
164C2×C4—28no—
165C2×C4—216no—
166C2×C4—28no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—48no—
170C2×C4—2816no—
171C2×C4—2816no—
172C2×C4—288no—
173C2×C4—288no—
174C2×C4—288no—
175C2×C4C2×C22⋊Dic714yes—
176C2×C4C2×C22⋊Dic718yes—
177C7#1466411yescharacteristic, complemented, Sylow
178C22C7⋊#56414yes—
179C22#9039018yes—
180C22#9039114yes—
181C22#9050312yes—
182C22#9050714yes—
183C22#9051612yes—
184C22#9051814yes—
185C22#9052112yes—
186C22#9052414yes—
187C22C22×C22⋊Dic711yescharacteristic
188C4—24no—
189C4—28no—
190C4—48no—
191C4—568no—
192C2#1174047214yes—
193C2#1174050014yes—
194C2#1174051614yes—
195C2#1174150711yescharacteristic
196C2#1174153212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC22×C7×#125C2
socle, prime radical, cosocleC24×C7C24
hypercenterC22×#125D7
hypocenterC7#14664
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
62. C2×#125Dic7semidirect
177. C7#14664semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1252
C2×#897082
#117293232
#117293192
#146647