Finite Group Databasebeta

← #13054890

Subgroups — #13054890

Order 1344.

468 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 148 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×Dic21C211yesmaximal, characteristic
2C2×C7×#217C211yesmaximal, characteristic
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic
4C2×Dic7⋊Dic3C212yesmaximal, complemented
5#10620529C212yesmaximal, complemented
6#89718—31nomaximal, Hall
7C24×C3×C7C2211yescharacteristic
8C22×Dic21—22no—
9C22×Dic21—24no—
10C22×Dic21C2212yes—
11C22×C7×Dic3C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C3×Dic7C2212yes—
14C7×#217—22no—
15C3×C22⋊Dic7C414yescomplemented
16Dic7⋊Dic3—24no—
17C23×Dic7—31no—
18C2×C7×C22⋊C4—31no—
19C2×C22⋊Dic7D311yescharacteristic
20C2×Dic7⋊C4—32no—
21C7⋊#95—32no—
22#4950—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—28no—
30C2×Dic21—22no—
31C2×Dic21—22no—
32C2×Dic21D414yes—
33C2×C7×Dic3—22no—
34C2×C7×Dic3—21no—
35C2×C7×Dic3—21no—
36C2×C7×Dic3—22no—
37C2×C3×Dic7—22no—
38C2×C3×Dic7—22no—
39C2×C3×Dic7C2×C414yescomplemented
40C24×C7C2×D311yescharacteristic
41C22×C4×C7—31no—
42C22×C4×C7—31no—
43C22×Dic7—32no—
44C22×Dic7—64no—
45C22×Dic7—62no—
46C22×Dic7C2×D312yes—
47C7×C22⋊C4—62no—
48C22⋊Dic7Dic314yescomplemented
49Dic7⋊C4—64no—
50C23×Dic3—71no—
51C2×C3×C22⋊C4—71no—
52C2×#217D711yescharacteristic
53C2×C4⋊Dic3—72no—
54C3⋊#95—72no—
55C22×C3×C7—24no—
56C22×C3×C7—22no—
57C22×C3×C7—24no—
58C22×C3×C7—22no—
59C22×C3×C7C4⋊C414yescomplemented
60C22×C3×C7C22×C411yescharacteristic
61C22×C3×C7C2×D412yes—
62C22×C3×C7C2×D411yescharacteristic
63C22×C3×C7C2×Q811yescharacteristic
64C22×C3×C7#4112yes—
65Dic21—24no—
66C3×Dic7—24no—
67C7×Dic3—41no—
68C7×Dic3—41no—
69#385—211noSylow
70C23×C7—22no—
71C23×C7—22no—
72C23×C7Dic612yes—
73C23×C7D1212yes—
74C23×C7C2×Dic312yes—
75C23×C7C22×D311yescharacteristic
76C2×C4×C7—61no—
77C2×C4×C7—61no—
78C2×C4×C7—62no—
79C2×C4×C7—62no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—34no—
83C2×Dic7—68no—
84C2×Dic7—62no—
85C2×Dic7—62no—
86C2×Dic7C2×Dic314yescomplemented
87C24×C3C2×D711yescharacteristic
88C22×C3×C4—72no—
89C22×Dic3—72no—
90C22×Dic3—144no—
91C22×Dic3—142no—
92C22×Dic3C2×D711yescharacteristic
93C22×Dic3C2×D711yescharacteristic
94C3×C22⋊C4—74no—
95#217—22no—
96C4⋊Dic3—144no—
97C2×C3×C7—24no—
98C2×C3×C7C2×C4⋊C411yescharacteristic
99C2×C3×C7C4×D412yes—
100C2×C3×C7#12011yescharacteristic
101C2×C3×C7C22⋊Q812yes—
102C2×C3×C7#12311yescharacteristic
103C23×C4—211no—
104C2×C4⋊C4—212no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—71no—
107#95—212no—
108C22×C7—24no—
109C22×C7—22no—
110C22×C7—22no—
111C22×C7—24no—
112C22×C7C4⋊Dic314yescomplemented
113C22×C7C2×Dic611yescharacteristic
114C22×C7C2×D1211yescharacteristic
115C22×C7D3×D412yes—
116C22×C7#23712yes—
117C22×C7C22×Dic311yescharacteristic
118C4×C7—121no—
119C4×C7—121no—
120Dic7—24no—
121Dic7—64no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3Dic1412yes—
125C23×C3C4×D712yes—
126C23×C3C22⋊D712yes—
127C23×C3C22×D711yescharacteristic
128C2×C3×C4—74no—
129C2×C3×C4—142no—
130C2×C3×C4—142no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—74no—
136C2×Dic3—148no—
137C2×Dic3—142no—
138C2×Dic3—142no—
139C3×C7#38511yescharacteristic, complemented, Hall
140C24D3×D711yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—72no—
144C22×C4—212no—
145C22×C4—424no—
146C22×C4—422no—
147C4⋊C4—424no—
148C22⋊C4—62no—
149C22⋊C4—74no—
150C2×C7—24no—
151C2×C7#87812yes—
152C2×C7#88211yescharacteristic
153C2×C7#88611yescharacteristic
154C2×C7C2×C4⋊Dic311yescharacteristic
155C2×C7D4×Dic312yes—
156C22×C3—24no—
157C22×C3—24no—
158C22×C3—22no—
159C22×C3—22no—
160C22×C3Dic7⋊C414yescomplemented
161C22×C3C2×Dic1411yescharacteristic
162C22×C3C2×C4×D711yescharacteristic
163C22×C3D4×D712yes—
164C22×C3#115912yes—
165C22×C3C2×C22⋊D711yescharacteristic
166C3×C4—144no—
167Dic3—41no—
168Dic3—41no—
169Dic3—144no—
170C23—22no—
171C23—22no—
172C23Dic3×D712yes—
173C23#424712yes—
174C23#424812yes—
175C23C2×D3×D711yescharacteristic
176C2×C4—62no—
177C2×C4—61no—
178C2×C4—61no—
179C2×C4—62no—
180C2×C4—74no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—214no—
184C2×C4—428no—
185C2×C4—422no—
186C2×C4—422no—
187C7#495011yescharacteristic, complemented, Sylow
188C2×C3—24no—
189C2×C3#644112yes—
190C2×C3(C22⋊D7)⋊C412yes—
191C2×C3C2×Dic7⋊C411yescharacteristic
192C2×C3#649811yescharacteristic
193C2×C3#650011yescharacteristic
194C22—22no—
195C22—24no—
196C22—22no—
197C22—24no—
198C22Dic7⋊Dic314yescomplemented
199C22C2×Dic3×D711yescharacteristic
200C22#6668412yes—
201C22C2×#424711yescharacteristic
202C22C2×#424811yescharacteristic
203C22#6669412yes—
204C4—121no—
205C4—121no—
206C4—144no—
207C4—424no—
208C3#8971811yescharacteristic, complemented, Sylow
209C2—24no—
210C2C2×Dic7⋊Dic311yescharacteristic
211C2#1062110511yescharacteristic
212C2(C22⋊D7)⋊Dic312yes—
213C2#1062113511yescharacteristic
214C2#1062113812yes—
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
Fitting, prime radicalC24×C3×C7C22
socle, cosocleC23×C3×C7C23
hypercenterC24D3×D7
hypocenterC3×C7#385
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×Dic7⋊Dic3C2semidirect
5. #10620529C2semidirect
15. C3×C22⋊Dic7C4semidirect
39. C2×C3×Dic7C2×C4semidirect
48. C22⋊Dic7Dic3semidirect
59. C22×C3×C7C4⋊C4semidirect
86. C2×Dic7C2×Dic3semidirect
112. C22×C7C4⋊Dic3semidirect
139. C3×C7#385semidirect
160. C22×C3Dic7⋊C4semidirect
187. C7#4950semidirect
198. C22Dic7⋊Dic3semidirect
208. C3#89718semidirect

Maximal subgroups

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

subgroupindex
C23×Dic212
C2×C7×#2172
C2×C3×C22⋊Dic72
C2×Dic7⋊Dic32
#106205292
#897183
#49507