Finite Group Databasebeta

← C22×#66730 #13062373

Subgroups — C22×#66730 #13062373

Order 1344.

1780 conjugacy classes of subgroups; only the 619 normal subgroups are shown.

normal subgroups only

This group has 1780 conjugacy classes of subgroups — too many to draw. Only its 619 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×C3×C7×D4C211yesmaximal, characteristic, complemented
2C22×#4268C212yesmaximal, complemented
3C23×Dic21C212yesmaximal, complemented
4C2×#66730C2124yesmaximal, direct product factor
5C22×C4×D21C211yesmaximal, characteristic, complemented
6C22×Dic42C211yesmaximal, characteristic, complemented
7C22×#1159—31nomaximal, Hall
8C24×C3×C7C2212yes—
9C23×D21C2211yescharacteristic
10C2×C3×C7×D4C22112yescomplemented
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268C22124yescomplemented
13C22×Dic21C2212yes—
14C22×Dic21C2211yescharacteristic
15C22×Dic21C22124yescomplemented
16#66730C22164yesdirect product factor
17C2×C4×D21C22112yescomplemented
18C2×Dic42C22112yescomplemented
19C22×C7×D4D311yescharacteristic, complemented
20C22×C22⋊D7—32no—
21C23×Dic7—32no—
22C2×#1159—324no—
23C22×C4×D7—31no—
24C22×Dic14—31no—
25C22×#237—71nomaximal, Hall
26C23×C3×C7—28no—
27C23×C3×C7C23112yes—
28C23×C3×C7C2311yescharacteristic
29C22×D21—24no—
30C22×D21C2316yes—
31C3×C7×D4C23116yescomplemented
32C2×C3×C4×C7C2316yes—
33#4268C23132yescomplemented
34C2×Dic21C23112yes—
35C2×Dic21C2316yes—
36C2×Dic21C23132yescomplemented
37C4×D21C23116yescomplemented
38Dic42C23116yescomplemented
39C24×C7C2×D312yes—
40C23×D7—31no—
41C2×C7×D4C2×D3112yescomplemented
42C22×C4×C7C2×D311yescharacteristic
43C2×C22⋊D7—324no—
44C22×Dic7—32no—
45C22×Dic7—31no—
46C22×Dic7—324no—
47#1159—364no—
48C2×C4×D7—312no—
49C2×Dic14—312no—
50C22×C3×D4D711yescharacteristic, complemented
51C22×#67—72no—
52C23×Dic3—72no—
53C2×#237—724no—
54C22×C4×D3—71no—
55C22×Dic6—71no—
56C22×C3×C7—224no—
57C22×C3×C7#4114yescomplemented
58C22×C3×C7C2418yes—
59C22×C3×C7C2413yes—
60C2×D21—212no—
61C2×D21C2414yes—
62C3×C4×C7C2414yes—
63Dic21C2418yes—
64Dic21C2414yes—
65C22×#41—211noSylow
66C23×C7—28no—
67C23×C7C22×D3112yes—
68C23×C7C22×D311yescharacteristic
69C22×D7—36no—
70C22×D7—64no—
71C7×D4C22×D3116yescomplemented
72C2×C4×C7C22×D316yes—
73C22⋊D7—332no—
74C2×Dic7—312no—
75C2×Dic7—36no—
76C2×Dic7—332no—
77C4×D7—316no—
78Dic14—316no—
79C24×C3C2×D712yes—
80C23×D3—71no—
81C2×C3×D4C2×D7112yescomplemented
82C22×C3×C4C2×D711yescharacteristic
83C2×#67—724no—
84C22×Dic3—72no—
85C22×Dic3—71no—
86C22×Dic3—724no—
87#237—764no—
88C2×C4×D3—712no—
89C2×Dic6—712no—
90C2×C3×C7—28no—
91C2×C3×C7C2×#4116yescomplemented
92C2×C3×C7C2511yescharacteristic
93D21—24no—
94C2×#41—2124no—
95C22×Q8—211no—
96C22×D4—212no—
97C22×D4D2111yescharacteristic, complemented
98C23×C4—212no—
99C23×C4—211no—
100C22×C7—224no—
101C22×C7#23714yescomplemented
102C22×C7C23×D318yes—
103C22×C7C23×D313yes—
104C2×D7—34no—
105C2×D7—612no—
106C4×C7C23×D314yes—
107Dic7—38no—
108Dic7—34no—
109C23×C3—28no—
110C23×C3C22×D7112yes—
111C23×C3C22×D711yescharacteristic
112C22×D3—76no—
113C22×D3—144no—
114C3×D4C22×D7116yescomplemented
115C2×C3×C4C22×D716yes—
116#67—732no—
117C2×Dic3—712no—
118C2×Dic3—76no—
119C2×Dic3—732no—
120C4×D3—716no—
121Dic6—716no—
122C3×C7C22×#4111yescharacteristic, complemented, Hall
123C24—211no—
124C24C2×D2112yes—
125#41—2164no—
126C2×Q8—2112no—
127C2×D4—2124no—
128C2×D4C2×D21112yescomplemented
129C22×C4—2124no—
130C22×C4—211no—
131C22×C4—212no—
132C22×C4—2112no—
133C22×C4C2×D2111yescharacteristic
134C2×C7—28no—
135C2×C7C2×#23716yescomplemented
136C2×C7C24×D311yescharacteristic
137D7—64no—
138C22×C3—224no—
139C22×C3#115914yescomplemented
140C22×C3C23×D718yes—
141C22×C3C23×D713yes—
142C2×D3—74no—
143C2×D3—1412no—
144C3×C4C23×D714yes—
145Dic3—78no—
146Dic3—74no—
147C23—28no—
148C23—216no—
149C23—424no—
150C23C22×D21112yes—
151C23C22×D2111yescharacteristic
152Q8—2116no—
153D4—2132no—
154D4C22×D21116yescomplemented
155C2×C4—216no—
156C2×C4—2112no—
157C2×C4—2132no—
158C2×C4—2116no—
159C2×C4C22×D2116yes—
160C7C22×#23711yescharacteristic, complemented, Sylow
161C2×C3—28no—
162C2×C3C2×#115916yescomplemented
163C2×C3C24×D711yescharacteristic
164D3—144no—
165C22—224no—
166C22—214no—
167C22—4212no—
168C22#6673014yesdirect product factor
169C22C23×D2118yes—
170C22C23×D2113yes—
171C4—214no—
172C4—218no—
173C4C23×D2114yes—
174C3C22×#115911yescharacteristic, complemented, Sylow
175C2—28no—
176C2—424no—
177C2C2×#6673016yesdirect product factor
178C2C24×D2111yescharacteristic
subgroupG/H
centerC23C22×D21
commutator, cosocleC2×C3×C7C25
FrattiniC2C24×D21
FittingC22×C3×C7×D4C2
socleC23×C3×C7C23
hypercenterC22×D4D21
hypocenterC3×C7C22×#41
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C7×D4C2semidirect
2. C22×#4268C2semidirect
3. C23×Dic21C2semidirect
4. C2×#66730C2direct (N × H)
5. C22×C4×D21C2semidirect
6. C22×Dic42C2semidirect
10. C2×C3×C7×D4C22semidirect
12. C2×#4268C22semidirect
15. C22×Dic21C22semidirect
16. #66730C22direct (N × H)
17. C2×C4×D21C22semidirect
18. C2×Dic42C22semidirect
19. C22×C7×D4D3semidirect
31. C3×C7×D4C23semidirect
33. #4268C23semidirect
36. C2×Dic21C23semidirect
37. C4×D21C23semidirect
38. Dic42C23semidirect
41. C2×C7×D4C2×D3semidirect
50. C22×C3×D4D7semidirect
57. C22×C3×C7#41semidirect
71. C7×D4C22×D3semidirect
81. C2×C3×D4C2×D7semidirect
91. C2×C3×C7C2×#41semidirect
97. C22×D4D21semidirect
101. C22×C7#237semidirect
114. C3×D4C22×D7semidirect
122. C3×C7C22×#41semidirect
128. C2×D4C2×D21semidirect
135. C2×C7C2×#237semidirect
139. C22×C3#1159semidirect
154. D4C22×D21semidirect
160. C7C22×#237semidirect
162. C2×C3C2×#1159semidirect
168. C22#66730direct (N × H)
174. C3C22×#1159semidirect
177. C2C2×#66730direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C7×D42
C22×#42682
C23×Dic212
C2×#667302
C22×C4×D212
C22×Dic422
C22×#11593
C22×#2377