Finite Group Databasebeta

← #13054494

Subgroups — #13054494

Order 1344.

300 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 72 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×#66730C211yesmaximal, characteristic
2#10620854C212yesmaximal, complemented
3C7×#833C211yesmaximal, characteristic, complemented
4C3×#6407C211yesmaximal, characteristic, complemented
5#10620498C212yesmaximal, complemented
6#89517—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×#4268C412yes—
9C22×Dic21C412yes—
10#66730—22no—
11#66730—22no—
12C2×C4×D21C2211yescharacteristic
13C2×Dic42C2211yescharacteristic
14C7×#208C2212yes—
15C3×#1136C2212yes—
16#66559—22no—
17D21⋊C8—22no—
18C2×#1159—31no—
19D7×M16—32no—
20C7×#100—31no—
21#6407D311yescharacteristic, complemented
22#6398—32no—
23#4745—71nomaximal, Hall
24C23×C3×C7C2×C412yes—
25C22×D21C2×C411yescharacteristic
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—24no—
29C2×Dic21—22no—
30C2×Dic21—22no—
31C2×Dic21C2×C412yes—
32C2×Dic21C2×C411yescharacteristic
33C4×D21D412yes—
34C4×D21D412yes—
35Dic42—22no—
36C3×C7⋊C8—22no—
37C7×C3⋊C8—22no—
38C2×C7×D4C2×D311yescharacteristic
39C2×C22⋊D7—32no—
40C22×Dic7—32no—
41#1159—62no—
42#1159—62no—
43C2×C4×D7—31no—
44C2×Dic14—31no—
45C7×M16—32no—
46#1136C2×D312yes—
47#1131—62no—
48C8×D7—62no—
49C2×#237—71no—
50D3×M16—72no—
51C3×#100—71no—
52#833D711yescharacteristic, complemented
53#824—72no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C22×C411yescharacteristic
57C2×D21—41no—
58C2×D21C22⋊C412yes—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61Dic21C22⋊C411yescharacteristic
62Dic21C22⋊C411yescharacteristic
63#413—211noSylow
64C23×C7C4×D312yes—
65C22×D7—31no—
66C7×D4—22no—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—64no—
69C2×Dic7—31no—
70C2×Dic7—32no—
71C2×Dic7—62no—
72C2×Dic7—62no—
73C4×D7—32no—
74C4×D7—32no—
75Dic14—62no—
76C7×C8—62no—
77C7⋊C8—22no—
78C2×C3×D4C2×D711yescharacteristic
79C2×#67—72no—
80C22×Dic3—72no—
81#237—142no—
82#237—142no—
83C2×C4×D3—71no—
84C2×Dic6—71no—
85C3×M16—72no—
86#208C2×D712yes—
87ΓL(1,𝔽25)—142no—
88D3×C8—142no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7C2×C22⋊C411yescharacteristic
92D21—22no—
93C2×#41—211no—
94C2×M16—212no—
95#101—212no—
96#100—31no—
97#100—71no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×C4×D311yescharacteristic
101C2×D7—32no—
102C2×D7—121no—
103C4×C7D3×D412yes—
104Dic7—31no—
105Dic7—31no—
106Dic7—62no—
107C23×C3C4×D712yes—
108C22×D3—71no—
109C3×D4—22no—
110C2×C3×C4C22×D711yescharacteristic
111#67—144no—
112C2×Dic3—71no—
113C2×Dic3—72no—
114C2×Dic3—142no—
115C2×Dic3—142no—
116C4×D3—72no—
117C4×D3—72no—
118Dic6—142no—
119C3×C8—142no—
120C3⋊C8—22no—
121C3×C7#41311yescharacteristic, complemented, Hall
122#41—422no—
123#41—422no—
124C2×Q8—211no—
125C2×D4—212no—
126C2×D4D3×D711yescharacteristic
127C22×C4—211no—
128C22×C4—212no—
129M16—32no—
130M16—72no—
131M16—422no—
132C2×C8—422no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7D3×C22⋊C411yescharacteristic
136D7—62no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3C2×C4×D711yescharacteristic
140C2×D3—72no—
141C2×D3—281no—
142C3×C4D4×D712yes—
143Dic3—71no—
144Dic3—71no—
145Dic3—142no—
146C23—211no—
147C23D21⋊C412yes—
148Q8—422no—
149D4—22no—
150D4—424no—
151C2×C4—211no—
152C2×C4—212no—
153C2×C4—212no—
154C2×C4—212no—
155C2×C4—422no—
156C2×C4—422no—
157C2×C4C2×D3×D711yescharacteristic
158C8—62no—
159C8—142no—
160C7#474511yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3D7×C22⋊C411yescharacteristic
164D3—142no—
165C22—22no—
166C22—42no—
167C22—212no—
168C22—841no—
169C22C2×D21⋊C411yescharacteristic
170C4—211no—
171C4—211no—
172C4—422no—
173C4#6669412yes—
174C3#8951711yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—422no—
178C2#1062112311yescharacteristic
subgroupG/H
centerC2#10621123
commutatorC22×C3×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×#66730C2
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. #10620854C2semidirect
3. C7×#833C2semidirect
4. C3×#6407C2semidirect
5. #10620498C2semidirect
21. #6407D3semidirect
52. #833D7semidirect
121. C3×C7#413semidirect
160. C7#4745semidirect
174. C3#89517semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
#106208542
C7×#8332
C3×#64072
#106204982
#895173
#47457