Finite Group Databasebeta

← #20275163

Subgroups — #20275163

Order 6784.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#16811134C211yesmaximal, characteristic
2C2×#13133145C212yesmaximal, complemented
3#16810587C211yesmaximal, characteristic, complemented
4#16810383C212yesmaximal, complemented
5C2×#117×C53C211yesmaximal, characteristic, complemented
6C2×#11719945C2211yescharacteristic
7C2×#11719917C2212yes—
8#13133200—21no—
9#13133196C412yescomplemented
10#13133191C412yescomplemented
11#13133186—21no—
12C4×C22⋊D53—22no—
13#13133176C2211yescharacteristic
14#13133145—24no—
15C22⋊#88322—22no—
16C23×C4×C53C2211yescharacteristic
17#117×C53—24no—
18C2×C4⋊C4×C53—21no—
19C2×C22⋊C4×C53—21no—
20C2×C42×C53C2212yes—
21C24×C53D411yescharacteristic
22C22×C4×C53—22no—
23C22×C4×C53—22no—
24C22×C4×C53—22no—
25C22×C4×C53C2×C412yes—
26C22×C4×C53D411yescharacteristic
27C22×C4×C53D412yes—
28C22×C4×C53C2311yescharacteristic
29C2×C22⋊D53—22no—
30C2×C22⋊D53—21no—
31#11719945—22no—
32#11719945—22no—
33C2×D212C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106C2×C411yescharacteristic
36C4⋊C4×C53—24no—
37C22⋊C4×C53—24no—
38C42×C53—22no—
39C42×C53D414yescomplemented
40C22⋊Dic53—22no—
41C22⋊Dic53—21no—
42#11719921—22no—
43C4⋊Dic53C2×C412yes—
44#11719919—22no—
45C4×Dic53—21no—
46#11719917—22no—
47#11719917—22no—
48C2×#88322—22no—
49C23×C53—21no—
50C23×C53—21no—
51C23×C53—42no—
52C23×C53C22⋊C412yes—
53C23×C53C2×D411yescharacteristic
54C22×D53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—24no—
58C2×C4×C53—24no—
59C2×C4×C53—24no—
60C2×C4×C53—44no—
61C2×C4×C53C22⋊C412yes—
62C2×C4×C53C22×C411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D412yes—
65C2×C4×C53#4112yes—
66C22⋊D53—44no—
67C22⋊D53—41no—
68C22⋊D53—41no—
69C2×Dic53—21no—
70C2×Dic53—22no—
71D212—22no—
72C4×D53—42no—
73Dic106—22no—
74#88322—42no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53C2×C22⋊C411yescharacteristic
82C22×C53#12011yescharacteristic
83C22×C53#12311yescharacteristic
84C2×D53—41no—
85C2×D53—42no—
86C4×C53—22no—
87C4×C53—44no—
88C4×C53C4×D412yes—
89C4×C53#12112yes—
90Dic53—41no—
91Dic53—42no—
92#1897—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—21no—
95C2×C53—42no—
96C2×C53#38611yescharacteristic
97C2×C53#42112yes—
98D53—81no—
99#525—531no—
100C2×#117D5311yescharacteristic, complemented
101C2×#104—532no—
102#407—531no—
103#339—532no—
104C53#189711yescharacteristic, complemented, Sylow
105C2×#41—531no—
106C23×C4C2×D5311yescharacteristic
107C2×M16—532no—
108#123—1061no—
109C22⋊Q8—532no—
110#121—532no—
111#120—1061no—
112C4×D4—1062no—
113#117—24no—
114#117—531no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D5312yes—
118#104—1064no—
119C22⋊C8—1062no—
120C24C22⋊D5311yescharacteristic
121#41—1062no—
122#41—1062no—
123C2×Q8—531no—
124C2×D4—531no—
125C2×D4—1061no—
126C2×D4—1062no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—1061no—
131C22×C4C4×D5312yes—
132C22×C4D21212yes—
133C22×C4C22⋊D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135M16—1062no—
136M16—1062no—
137C2×C8—1062no—
138C4⋊C4—24no—
139C4⋊C4—532no—
140C4⋊C4—1062no—
141C22⋊C4—24no—
142C22⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—22no—
146C42—1061no—
147C42C22⋊D5314yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—1061no—
152C23#1171992112yes—
153C23C2×C22⋊D5311yescharacteristic
154Q8—1062no—
155D4—1062no—
156D4—2124no—
157D4—2121no—
158D4—2121no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—1062no—
166C2×C4—1061no—
167C2×C4—2122no—
168C2×C4#1171992112yes—
169C2×C4C2×C4×D5311yescharacteristic
170C2×C4C2×D21211yescharacteristic
171C2×C4#1171994512yes—
172C2×C4C2×C22⋊D5312yes—
173C8—2122no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—2121no—
181C22—2122no—
182C22C2×#1171992111yescharacteristic
183C22#1313318611yescharacteristic
184C22#1313320011yescharacteristic
185C4—22no—
186C4—44no—
187C4—2121no—
188C4—2122no—
189C4C4×C22⋊D5312yes—
190C4#1313319112yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—4241no—
195C2#1681050211yescharacteristic
196C2#1681064112yes—
subgroupG/H
centerC2×C4C2×D212
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#117×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#117D53
hypocenterC53#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#13133145C2semidirect
3. #16810587C2semidirect
4. #16810383C2semidirect
5. C2×#117×C53C2semidirect
9. #13133196C4semidirect
10. #13133191C4semidirect
39. C42×C53D4semidirect
100. C2×#117D53semidirect
104. C53#1897semidirect
147. C42C22⋊D53semidirect