Finite Group Databasebeta

← #13247308

Subgroups — #13247308

Order 1792.

538 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2312C211yesmaximal, characteristic
2#11729296C212yesmaximal
3C7⋊#1304C214yesmaximal
4C22×C7×C22⋊C4C2211yescharacteristic
5C7×#388—24no—
6C7×#380C412yescomplemented
7C7×#379C2212yes—
8C4×C7×C22⋊C4C412yescomplemented
9#90019C2214yes—
10#89321—24no—
11#8750—71nomaximal, Sylow
12C25×C7C2311yescharacteristic
13C23×C4×C7C2×C411yescharacteristic
14C23×C4×C7C2×C411yescharacteristic
15C2×C7×C4⋊C4—22no—
16C2×C7×C22⋊C4—24no—
17C2×C7×C22⋊C4—22no—
18C2×C7×C22⋊C4—22no—
19C2×C7×C22⋊C4C2×C414yes—
20C2×C7×C22⋊C4D414yes—
21C2×C42×C7—22no—
22C2×C22⋊Dic7—24no—
23C2×C22⋊Dic7—24no—
24C2×C22⋊Dic7—24no—
25C7×#95—22no—
26C7×#95—22no—
27#2312D711yescharacteristic
28#1819—72no—
29#1304—74no—
30C24×C7—42no—
31C24×C7—81no—
32C24×C7—81no—
33C24×C7C22×C411yescharacteristic
34C24×C7C2×D412yes—
35C24×C7#4114yes—
36C22×C4×C7—22no—
37C22×C4×C7—22no—
38C22×C4×C7—24no—
39C22×C4×C7—81no—
40C22×C4×C7—81no—
41C22×C4×C7C22⋊C412yes—
42C22×C4×C7C22⋊C412yes—
43C22×Dic7—44no—
44C7×C4⋊C4—82no—
45C7×C22⋊C4—28no—
46C7×C22⋊C4—44no—
47C7×C22⋊C4—82no—
48C7×C22⋊C4—82no—
49C42×C7—82no—
50C22⋊Dic7—48no—
51C22⋊Dic7—84no—
52C22⋊Dic7—84no—
53C22×C22⋊C4C2×D711yescharacteristic
54#388—24no—
55#380Dic712yescomplemented
56#379—74no—
57#379C2×D712yes—
58C4×C22⋊C4Dic712yescomplemented
59#342—144no—
60C23×C7—22no—
61C23×C7—24no—
62C23×C7—41no—
63C23×C7—41no—
64C23×C7—44no—
65C23×C7—44no—
66C23×C7—42no—
67C23×C7—42no—
68C23×C7—44no—
69C23×C7—88no—
70C23×C7C2×C22⋊C411yescharacteristic
71C23×C7#11712yes—
72C23×C7#12314yes—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—42no—
76C2×C4×C7—42no—
77C2×C4×C7—44no—
78C2×C4×C7—44no—
79C2×C4×C7—44no—
80C2×C4×C7—82no—
81C2×C4×C7—82no—
82C2×Dic7—84no—
83C2×Dic7—84no—
84C2×Dic7—84no—
85C25C22×D711yescharacteristic
86C23×C4C2×Dic711yescharacteristic
87C23×C4C2×Dic711yescharacteristic
88C2×C4⋊C4—22no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—144no—
93C2×C22⋊C4—144no—
94C2×C22⋊C4—144no—
95C2×C22⋊C4C2×Dic714yes—
96C2×C22⋊C4C22⋊D714yes—
97C2×C42—22no—
98#95—22no—
99#95—22no—
100C22×C7—21no—
101C22×C7—21no—
102C22×C7—24no—
103C22×C7—22no—
104C22×C7—24no—
105C22×C7—24no—
106C22×C7—22no—
107C22×C7—22no—
108C22×C7—44no—
109C22×C7—84no—
110C22×C7—88no—
111C22×C7#38111yescharacteristic
112C22×C7#41012yes—
113C4×C7—42no—
114C4×C7—42no—
115C4×C7—84no—
116Dic7—164no—
117C24—42no—
118C24—81no—
119C24—81no—
120C24#115714yes—
121C24C22×Dic711yescharacteristic
122C24C2×C22⋊D712yes—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—24no—
126C22×C4—81no—
127C22×C4—81no—
128C22×C4—284no—
129C22×C4C22⋊Dic712yes—
130C22×C4C22⋊Dic712yes—
131C4⋊C4—82no—
132C22⋊C4—28no—
133C22⋊C4—44no—
134C22⋊C4—82no—
135C22⋊C4—82no—
136C22⋊C4—288no—
137C22⋊C4—564no—
138C22⋊C4—564no—
139C42—82no—
140C2×C7—22no—
141C2×C7—44no—
142C2×C7—42no—
143C2×C7#182011yescharacteristic
144C2×C7#182811yescharacteristic
145C2×C7#182811yescharacteristic
146C23—22no—
147C23—24no—
148C23—41no—
149C23—41no—
150C23—42no—
151C23—42no—
152C23—44no—
153C23—44no—
154C23—44no—
155C23—88no—
156C23#648912yes—
157C23#649214yes—
158C23C2×C22⋊Dic711yescharacteristic
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—44no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—564no—
169C2×C4—564no—
170C2×C4—564no—
171C7#875011yescharacteristic, complemented, Sylow
172C22—21no—
173C22—21no—
174C22—24no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—24no—
180C22—44no—
181C22—84no—
182C22—88no—
183C22#8998311yescharacteristic
184C22#9001212yes—
185C4—42no—
186C4—42no—
187C4—84no—
188C4—1124no—
189C2—22no—
190C2—44no—
191C2—42no—
192C2#1172934811yescharacteristic
193C2#1172940711yescharacteristic
194C2#1172940711yescharacteristic
subgroupG/H
centerC22#89983
commutatorC24×C7C22×C4
FrattiniC25C22×D7
FittingC7×#2312C2
socleC22×C7#381
prime radical, cosocleC25×C7C23
hypercenter#2312D7
hypocenterC7#8750
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C25 ◁ #2312

Complements

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

NH ≅ G/Ntype
6. C7×#380C4semidirect
8. C4×C7×C22⋊C4C4semidirect
55. #380Dic7semidirect
58. C4×C22⋊C4Dic7semidirect
171. C7#8750semidirect

Maximal subgroups

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

subgroupindex
C7×#23122
#117292962
C7⋊#13042
#87507