Finite Group Databasebeta

← #13054888

Subgroups — #13054888

Order 1344.

468 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C23×C7×Dic3C211yesmaximal, characteristic
2C2×#66636C211yesmaximal, characteristic
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic
4C2×Dic3⋊Dic7C212yesmaximal, complemented
5#10620529C212yesmaximal, complemented
6#89984—31nomaximal, Hall
7C24×C3×C7C2211yescharacteristic
8C22×Dic21C2212yes—
9C22×C7×Dic3—24no—
10C22×C7×Dic3C414yescomplemented
11C22×C7×Dic3C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C3×Dic7C2212yes—
14#66636—22no—
15C3×C22⋊Dic7—22no—
16Dic3⋊Dic7—24no—
17C23×C4×C7—31no—
18C2×C22⋊Dic7—31no—
19C2×C22⋊Dic7D311yescharacteristic
20C2×C4⋊Dic7—32no—
21C7⋊#95—32no—
22#4942—71nomaximal, Hall
23C23×C3×C7—22no—
24C23×C3×C7—22no—
25C23×C3×C7C2×C412yes—
26C23×C3×C7D412yes—
27C23×C3×C7Q812yes—
28C23×C3×C7C2311yescharacteristic
29C2×Dic21—22no—
30C2×Dic21—22no—
31C2×Dic21—22no—
32C2×C7×Dic3—22no—
33C2×C7×Dic3—28no—
34C2×C7×Dic3C2×C412yes—
35C2×C7×Dic3C2×C412yes—
36C2×C7×Dic3D414yes—
37C2×C3×Dic7—22no—
38C2×C3×Dic7—22no—
39C2×C3×Dic7—22no—
40C24×C7C2×D311yescharacteristic
41C22×C4×C7—31no—
42C22×C4×C7—34no—
43C22×C4×C7—31no—
44C22×C4×C7—64no—
45C22×Dic7—32no—
46C22×Dic7C2×D312yes—
47C22⋊Dic7—22no—
48C22⋊Dic7—62no—
49C4⋊Dic7—64no—
50C23×Dic3D711yescharacteristic
51C2×C3×C22⋊C4—71no—
52C2×#217—71no—
53C2×Dic3⋊C4—72no—
54C3⋊#95—72no—
55C22×C3×C7—22no—
56C22×C3×C7—22no—
57C22×C3×C7—24no—
58C22×C3×C7—24no—
59C22×C3×C7C4⋊C414yescomplemented
60C22×C3×C7C22×C411yescharacteristic
61C22×C3×C7C2×D411yescharacteristic
62C22×C3×C7C2×D412yes—
63C22×C3×C7C2×Q811yescharacteristic
64C22×C3×C7#4112yes—
65Dic21—42no—
66C3×Dic7—42no—
67C7×Dic3—22no—
68C7×Dic3C22⋊C414yescomplemented
69#380—211noSylow
70C23×C7—22no—
71C23×C7—22no—
72C23×C7Dic612yes—
73C23×C7C4×D312yes—
74C23×C7#6712yes—
75C23×C7C22×D311yescharacteristic
76C2×C4×C7—32no—
77C2×C4×C7—34no—
78C2×C4×C7—32no—
79C2×C4×C7—68no—
80C2×C4×C7—62no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—62no—
85C2×Dic7—62no—
86C2×Dic7—62no—
87C24×C3C2×D711yescharacteristic
88C22×C3×C4—72no—
89C22×Dic3—24no—
90C22×Dic3—72no—
91C22×Dic3Dic714yescomplemented
92C22×Dic3C2×D711yescharacteristic
93C22×Dic3C2×D711yescharacteristic
94C3×C22⋊C4—142no—
95#217—142no—
96Dic3⋊C4—144no—
97C2×C3×C7—24no—
98C2×C3×C7C2×C22⋊C411yescharacteristic
99C2×C3×C7C2×C4⋊C411yescharacteristic
100C2×C3×C7#11711yescharacteristic
101C2×C3×C7#12112yes—
102C2×C3×C7C22⋊Q812yes—
103C23×C4—31no—
104C2×C4⋊C4—212no—
105C2×C22⋊C4—71no—
106C2×C22⋊C4—211no—
107#95—212no—
108C22×C7—22no—
109C22×C7—24no—
110C22×C7—22no—
111C22×C7—24no—
112C22×C7Dic3⋊C414yescomplemented
113C22×C7C2×Dic611yescharacteristic
114C22×C7C2×C4×D311yescharacteristic
115C22×C7D3×D412yes—
116C22×C7#23712yes—
117C22×C7C2×#6711yescharacteristic
118C4×C7—34no—
119C4×C7—62no—
120Dic7—42no—
121Dic7—122no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3Dic1412yes—
125C23×C3D2812yes—
126C23×C3C2×Dic712yes—
127C23×C3C22×D711yescharacteristic
128C2×C3×C4—142no—
129C2×C3×C4—142no—
130C2×C3×C4—142no—
131C2×Dic3—22no—
132C2×Dic3—28no—
133C2×Dic3—142no—
134C2×Dic3—142no—
135C2×Dic3—142no—
136C2×Dic3C2×Dic712yes—
137C2×Dic3C2×Dic712yes—
138C2×Dic3C22⋊D714yes—
139C3×C7#38011yescharacteristic, complemented, Hall
140C24D3×D711yescharacteristic
141C22×C4—31no—
142C22×C4—34no—
143C22×C4—31no—
144C22×C4—64no—
145C22×C4—72no—
146C22×C4—212no—
147C4⋊C4—424no—
148C22⋊C4—142no—
149C22⋊C4—422no—
150C2×C7—24no—
151C2×C7#87711yescharacteristic
152C2×C7#87812yes—
153C2×C7D3×C22⋊C411yescharacteristic
154C2×C7C2×Dic3⋊C411yescharacteristic
155C2×C7#93912yes—
156C22×C3—24no—
157C22×C3—22no—
158C22×C3—24no—
159C22×C3—22no—
160C22×C3C4⋊Dic714yescomplemented
161C22×C3C2×Dic1411yescharacteristic
162C22×C3C2×D2811yescharacteristic
163C22×C3#115712yes—
164C22×C3C22×Dic711yescharacteristic
165C22×C3C2×C22⋊D712yes—
166C3×C4—282no—
167Dic3—22no—
168Dic3—282no—
169Dic3C22⋊Dic714yescomplemented
170C23—22no—
171C23—22no—
172C23D3×Dic712yes—
173C23#424612yes—
174C23#424812yes—
175C23C2×D3×D711yescharacteristic
176C2×C4—32no—
177C2×C4—34no—
178C2×C4—32no—
179C2×C4—68no—
180C2×C4—62no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—422no—
185C2×C4—422no—
186C2×C4—422no—
187C7#494211yescharacteristic, complemented, Sylow
188C2×C3—24no—
189C2×C3#648712yes—
190C2×C3C2×C4⋊Dic711yescharacteristic
191C2×C3#648911yescharacteristic
192C2×C3#649312yes—
193C2×C3C2×C22⋊Dic711yescharacteristic
194C22—22no—
195C22—22no—
196C22—24no—
197C22—24no—
198C22Dic3⋊Dic714yescomplemented
199C22#6668312yes—
200C22C2×D3×Dic711yescharacteristic
201C22C2×#424611yescharacteristic
202C22C2×#424811yescharacteristic
203C22D3×C22⋊D712yes—
204C4—34no—
205C4—62no—
206C4—282no—
207C4—842no—
208C3#8998411yescharacteristic, complemented, Sylow
209C2—24no—
210C2#1062109111yescharacteristic
211C2C2×Dic3⋊Dic711yescharacteristic
212C2D3×C22⋊Dic711yescharacteristic
213C2#1062113112yes—
214C2#1062113712yes—
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
Fitting, prime radicalC24×C3×C7C22
socle, cosocleC23×C3×C7C23
hypercenterC24D3×D7
hypocenterC3×C7#380
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic7C2semidirect
5. #10620529C2semidirect
10. C22×C7×Dic3C4semidirect
59. C22×C3×C7C4⋊C4semidirect
68. C7×Dic3C22⋊C4semidirect
91. C22×Dic3Dic7semidirect
112. C22×C7Dic3⋊C4semidirect
139. C3×C7#380semidirect
160. C22×C3C4⋊Dic7semidirect
169. Dic3C22⋊Dic7semidirect
187. C7#4942semidirect
198. C22Dic3⋊Dic7semidirect
208. C3#89984semidirect

Maximal subgroups

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

subgroupindex
C23×C7×Dic32
C2×#666362
C2×C3×C22⋊Dic72
C2×Dic3⋊Dic72
#106205292
#899843
#49427