Finite Group Databasebeta

← #20274669

Subgroups — #20274669

Order 6784.

620 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133144C212yesmaximal, complemented
2#16810624C212yesmaximal, complemented
3C2×#13133134C211yesmaximal, characteristic, complemented
4#16810526C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C53C211yesmaximal, characteristic
6C2×#11719917C2212yescomplemented
7C2×C4⋊Dic53C2211yescharacteristic, complemented
8C2×C22⋊Dic53—22no—
9C2×C4×Dic53C2211yescharacteristic, complemented
10#13133144—24no—
11#13133141—22no—
12#13133138—22no—
13#13133134—22no—
14H(5,𝔽2)×C53—24no—
15H(5,𝔽2)×C53—22no—
16H(5,𝔽2)×C53C414yescomplemented
17C2×#41×C53—22no—
18C2×#41×C53C2212yes—
19C22×D4×C53—22no—
20C22×D4×C53—22no—
21C22×D4×C53C2211yescharacteristic
22C24×C53—22no—
23C24×C53D412yes—
24#41×C53—28no—
25#41×C53—28no—
26#41×C53—22no—
27#41×C53—22no—
28#41×C53C2×C414yes—
29#41×C53D414yes—
30C2×Q8×C53D412yes—
31C2×D4×C53—24no—
32C2×D4×C53—28no—
33C2×D4×C53—28no—
34C2×D4×C53—28no—
35C2×D4×C53—22no—
36C2×D4×C53—28no—
37C2×D4×C53—22no—
38C2×D4×C53—22no—
39C2×D4×C53C2×C412yes—
40C2×D4×C53D412yes—
41C2×D4×C53D412yes—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53C2311yescharacteristic
45C22×Dic53—21no—
46C22×Dic53—21no—
47C22⋊Dic53—44no—
48C4⋊Dic53—22no—
49C4×Dic53—22no—
50#11719917—22no—
51#11719917—22no—
52C2×#88322—22no—
53C23×C53—28no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—22no—
57C23×C53—42no—
58C23×C53—42no—
59C23×C53—82no—
60C23×C53—82no—
61C23×C53C22⋊C414yes—
62C23×C53C2×D411yescharacteristic
63Q8×C53—22no—
64Q8×C53C22⋊C414yescomplemented
65D4×C53—216no—
66D4×C53—24no—
67D4×C53—24no—
68D4×C53—24no—
69D4×C53—24no—
70D4×C53—22no—
71D4×C53C22⋊C414yescomplemented
72C2×C4×C53—28no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—24no—
76C2×C4×C53C22×C411yescharacteristic
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×D411yescharacteristic
80C2×Dic53—22no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53—42no—
84C2×Dic53—41no—
85#88322—42no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—24no—
89C22×C53—24no—
90C22×C53—44no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—82no—
96C22×C53—88no—
97C22×C53C2×C22⋊C411yescharacteristic
98C22×C53#12011yescharacteristic
99C22×C53#12011yescharacteristic
100C4×C53—24no—
101C4×C53C2×C22⋊C412yes—
102C4×C53#12012yes—
103Dic53—42no—
104Dic53—81no—
105#1792—531nomaximal, Sylow
106C2×C53—21no—
107C2×C53—21no—
108C2×C53—44no—
109C2×C53—44no—
110C2×C53#37911yescharacteristic
111C2×C53#45311yescharacteristic
112C2×C53#45511yescharacteristic
113C2×H(5,𝔽2)D5311yescharacteristic
114C2×#104—532no—
115#417—532no—
116C2×#100—531no—
117#390—531no—
118C53#179211yescharacteristic, complemented, Sylow
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)Dic5314yescomplemented
122C2×#41—22no—
123C2×#41C2×D5312yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D5311yescharacteristic
127C2×M16—532no—
128C2×C4⋊C4—531no—
129C2×C22⋊C4—1062no—
130C2×C42—531no—
131#104—1064no—
132#103—1062no—
133#102—1062no—
134#100—1062no—
135C24—22no—
136C24C22⋊D5312yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×Dic5314yes—
142#41C22⋊D5314yes—
143C2×Q8C22⋊D5312yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—22no—
148C2×D4—28no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×Dic5312yes—
153C2×D4C22⋊D5312yes—
154C2×D4C22⋊D5312yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—1061no—
158C22×C4—1061no—
159C22×C4C22×D5311yescharacteristic
160M16—1062no—
161M16—1062no—
162C2×C8—1062no—
163C4⋊C4—1062no—
164C22⋊C4—2124no—
165C42—1062no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊Dic5314yes—
175C23C2×C22⋊D5311yescharacteristic
176Q8—22no—
177Q8C22⋊Dic5314yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊Dic5314yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—1062no—
190C2×C4—2122no—
191C2×C4—2121no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4C22×Dic5311yescharacteristic
195C2×C4C2×C22⋊D5312yes—
196C2×C4C2×C22⋊D5312yes—
197C2×C4C2×C22⋊D5311yescharacteristic
198C8—2122no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊Dic5311yescharacteristic
211C22#1313318611yescharacteristic
212C22#1313318611yescharacteristic
213C4—24no—
214C4—4241no—
215C4—2122no—
216C4C2×C22⋊Dic5312yes—
217C4#1313318612yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#1681047911yescharacteristic
223C2#1681079111yescharacteristic
224C2#1681080111yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×H(5,𝔽2)×C53C2
socleC22×C53#120
hypercenterC2×H(5,𝔽2)D53
hypocenterC53#1792
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#13133144C2semidirect
2. #16810624C2semidirect
3. C2×#13133134C2semidirect
4. #16810526C2semidirect
6. C2×#11719917C22semidirect
7. C2×C4⋊Dic53C22semidirect
9. C2×C4×Dic53C22semidirect
16. H(5,𝔽2)×C53C4semidirect
64. Q8×C53C22⋊C4semidirect
71. D4×C53C22⋊C4semidirect
118. C53#1792semidirect
121. H(5,𝔽2)Dic53semidirect
177. Q8C22⋊Dic53semidirect
184. D4C22⋊Dic53semidirect