Finite Group Databasebeta

← #13054483

Subgroups — #13054483

Order 1344.

300 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 88 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×C3×#1159C211yesmaximal, characteristic
2D7×#208C212yesmaximal, complemented
3#10620672C211yesmaximal, characteristic, complemented
4C7×#833C211yesmaximal, characteristic, complemented
5#10620496C212yesmaximal, complemented
6#89517—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×C3×C22⋊D7C412yes—
9C22×C3×Dic7C412yes—
10C3×#1159—22no—
11C3×#1159—22no—
12C2×C3×C4×D7C2211yescharacteristic
13C2×C3×Dic14C2211yescharacteristic
14#66627C2212yes—
15C7×#208C2212yes—
16#66557—22no—
17D7×C3⋊C8—22no—
18C2×#1159D311yescharacteristic
19D7×M16—32no—
20C7×#100—31no—
21#6407—31no—
22#6398—32no—
23#5237—71nomaximal, Hall
24C23×C3×C7C2×C412yes—
25C22×C3×D7C2×C411yescharacteristic
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28C3×C22⋊D7—24no—
29C2×C3×Dic7—22no—
30C2×C3×Dic7—22no—
31C2×C3×Dic7C2×C412yes—
32C2×C3×Dic7C2×C411yescharacteristic
33C3×C4×D7D412yes—
34C3×C4×D7D412yes—
35C3×Dic14—22no—
36C3⋊(C7⋊C8)—22no—
37C7×C3⋊C8—22no—
38C2×C7×D4C2×D311yescharacteristic
39C2×C22⋊D7Dic312yes—
40C22×Dic7Dic312yes—
41#1159—22no—
42#1159—22no—
43C2×C4×D7C2×D311yescharacteristic
44C2×Dic14C2×D311yescharacteristic
45C7×M16—32no—
46#1136—32no—
47#1131—62no—
48C8×D7—62no—
49C2×C3×#41—71no—
50C2×#208—72no—
51#836—72no—
52#833—71no—
53#833D711yescharacteristic, complemented
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C22×C411yescharacteristic
57C2×C3×D7—41no—
58C2×C3×D7C22⋊C412yes—
59C3×C4×C7C2×D412yes—
60C3×Dic7—22no—
61C3×Dic7C22⋊C411yescharacteristic
62C3×Dic7C22⋊C411yescharacteristic
63#413—211noSylow
64C23×C7C2×Dic312yes—
65C22×D7C2×Dic311yescharacteristic
66C7×D4—22no—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—24no—
69C2×Dic7—22no—
70C2×Dic7—22no—
71C2×Dic7C2×Dic312yes—
72C2×Dic7C2×Dic311yescharacteristic
73C4×D7#6712yes—
74C4×D7#6712yes—
75Dic14—22no—
76C7×C8—62no—
77C7⋊C8—62no—
78C3×#41—142no—
79C3×#41—142no—
80C2×C3×Q8—71no—
81C2×C3×D4—72no—
82C2×C3×D4C2×D711yescharacteristic
83C22×C3×C4—71no—
84C22×C3×C4—72no—
85#208—72no—
86#208—142no—
87#208C2×D712yes—
88C2×C3⋊C8—142no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7C2×C22⋊C411yescharacteristic
92C3×D7—22no—
93C2×#41—71no—
94C2×M16—212no—
95#101—212no—
96#100—31no—
97#100—211no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C22×Dic311yescharacteristic
101C2×D7—41no—
102C2×D7#21712yes—
103C4×C7C2×#6712yes—
104Dic7—22no—
105Dic7#21711yescharacteristic
106Dic7#21711yescharacteristic
107C23×C3—71no—
108C23×C3C4×D712yes—
109C3×Q8—142no—
110C3×D4—22no—
111C3×D4—144no—
112C2×C3×C4—71no—
113C2×C3×C4—72no—
114C2×C3×C4—72no—
115C2×C3×C4—72no—
116C2×C3×C4—142no—
117C2×C3×C4—142no—
118C2×C3×C4C22×D711yescharacteristic
119C3⋊C8—22no—
120C3⋊C8—142no—
121C3×C7#41311yescharacteristic, complemented, Hall
122#41—142no—
123#41—142no—
124C2×Q8—71no—
125C2×D4—72no—
126C2×D4D3×D711yescharacteristic
127C22×C4—71no—
128C22×C4—72no—
129M16—32no—
130M16—212no—
131M16—422no—
132C2×C8—422no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7C2×#21711yescharacteristic
136D7—22no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3—72no—
140C22×C3—281no—
141C22×C3C2×C4×D711yescharacteristic
142C3×C4—71no—
143C3×C4—71no—
144C3×C4—142no—
145C3×C4D4×D712yes—
146C23—71no—
147C23Dic3×D712yes—
148Q8—142no—
149D4—22no—
150D4—144no—
151C2×C4—71no—
152C2×C4—72no—
153C2×C4—72no—
154C2×C4—72no—
155C2×C4—142no—
156C2×C4—142no—
157C2×C4C2×D3×D711yescharacteristic
158C8—62no—
159C8—422no—
160C7#523711yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3—142no—
164C2×C3D7×C22⋊C411yescharacteristic
165C22—22no—
166C22—42no—
167C22—72no—
168C22—281no—
169C22C2×Dic3×D711yescharacteristic
170C4—71no—
171C4—71no—
172C4—142no—
173C4D7×#6712yes—
174C3#8951711yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—142no—
178C2D7×#21711yescharacteristic
subgroupG/H
centerC2D7×#217
commutatorC22×C3×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×C3×#1159C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#413
LCSG ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D7×#208C2semidirect
3. #10620672C2semidirect
4. C7×#833C2semidirect
5. #10620496C2semidirect
53. #833D7semidirect
121. C3×C7#413semidirect
160. C7#5237semidirect
174. C3#89517semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
D7×#2082
#106206722
C7×#8332
#106204962
#895173
#52377