Finite Group Databasebeta

← #20275164

Subgroups — #20275164

Order 6784.

358 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13133176C211yesmaximal, characteristic, complemented
2C2×#13133144C212yesmaximal, complemented
3#16810587C211yesmaximal, characteristic, complemented
4#16810382C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C2×#11719917C2212yes—
7#13133176—24no—
8C2×C4⋊Dic53—21no—
9C2×C22⋊Dic53—21no—
10C2×C4×Dic53C2212yes—
11#13133144—24no—
12C22⋊#88322—22no—
13C2×#41×C53C2211yescharacteristic
14C23×C4×C53C2211yescharacteristic
15#123×C53—21no—
16C22⋊Q8×C53C412yescomplemented
17#121×C53C412yescomplemented
18#120×C53—21no—
19C4×D4×C53—22no—
20#117×C53C2211yescharacteristic
21C24×C53D411yescharacteristic
22#41×C53—22no—
23#41×C53—22no—
24C2×Q8×C53C2×C411yescharacteristic
25C2×D4×C53—22no—
26C2×D4×C53—21no—
27C2×D4×C53C2×C411yescharacteristic
28C22×C4×C53—22no—
29C22×C4×C53—22no—
30C22×C4×C53—21no—
31C22×C4×C53C2×C412yes—
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D412yes—
34C22×C4×C53C2311yescharacteristic
35C22×Dic53—22no—
36C4⋊C4×C53—22no—
37C4⋊C4×C53C2×C412yes—
38C22⋊C4×C53—22no—
39C22⋊C4×C53—21no—
40C22⋊C4×C53—22no—
41C42×C53—21no—
42C22⋊Dic53—24no—
43C4⋊Dic53—24no—
44C4×Dic53—22no—
45C4×Dic53D414yescomplemented
46#11719917—22no—
47#11719917—22no—
48C2×#88322—22no—
49C23×C53—21no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53—42no—
53C23×C53C22⋊C412yes—
54C23×C53C2×D411yescharacteristic
55Q8×C53—22no—
56D4×C53—22no—
57D4×C53—44no—
58D4×C53—41no—
59D4×C53—41no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—24no—
63C2×C4×C53—24no—
64C2×C4×C53—21no—
65C2×C4×C53—22no—
66C2×C4×C53—42no—
67C2×C4×C53C22⋊C412yes—
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53#4112yes—
72C2×Dic53—24no—
73C2×Dic53—44no—
74#88322—42no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53—41no—
82C22×C53—42no—
83C22×C53C2×C22⋊C411yescharacteristic
84C22×C53#12011yescharacteristic
85C22×C53#12311yescharacteristic
86C4×C53—22no—
87C4×C53—41no—
88C4×C53—42no—
89C4×C53C4×D412yes—
90C4×C53#12112yes—
91Dic53—44no—
92#1897—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—21no—
95C2×C53—42no—
96C2×C53—81no—
97C2×C53#38611yescharacteristic
98C2×C53#42112yes—
99#525D5311yescharacteristic
100C2×#117—531no—
101C2×#104—532no—
102#407—531no—
103#339—532no—
104C53#189711yescharacteristic, complemented, Sylow
105C2×#41C2×D5311yescharacteristic
106C23×C4C2×D5311yescharacteristic
107C2×M16—532no—
108#123—21no—
109C22⋊Q8Dic5312yescomplemented
110#121Dic5312yescomplemented
111#120—21no—
112C4×D4—22no—
113#117—1064no—
114#117C2×D5311yescharacteristic
115C2×C4⋊C4—1061no—
116C2×C22⋊C4—1061no—
117C2×C42—532no—
118#104—1064no—
119C22⋊C8—1062no—
120C24C22⋊D5311yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic5311yescharacteristic
124C2×D4—21no—
125C2×D4—22no—
126C2×D4C2×Dic5311yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—1062no—
131C22×C4C2×Dic5312yes—
132C22×C4C22⋊D5312yes—
133C22×C4C22⋊D5311yescharacteristic
134C22×C4C22×D5311yescharacteristic
135M16—1062no—
136M16—1062no—
137C2×C8—1062no—
138C4⋊C4—22no—
139C4⋊C4—1064no—
140C4⋊C4C2×Dic5312yes—
141C22⋊C4—21no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—1064no—
145C42—21no—
146C42—534no—
147C42—1062no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊Dic5312yes—
153C23C2×C22⋊D5311yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—1064no—
167C2×C4—2124no—
168C2×C4C22⋊Dic5312yes—
169C2×C4D4×D5312yes—
170C2×C4#1171994712yes—
171C2×C4C22×Dic5311yescharacteristic
172C2×C4C2×C22⋊D5311yescharacteristic
173C8—2122no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—42no—
182C22C2×C22⋊Dic5311yescharacteristic
183C22#1313318511yescharacteristic
184C22#1313319811yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—2124no—
189C4D4×Dic5312yes—
190C4#1313318912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#1681050311yescharacteristic
196C2#1681064012yes—
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#13133176C2semidirect
2. C2×#13133144C2semidirect
3. #16810587C2semidirect
4. #16810382C2semidirect
16. C22⋊Q8×C53C4semidirect
17. #121×C53C4semidirect
45. C4×Dic53D4semidirect
104. C53#1897semidirect
109. C22⋊Q8Dic53semidirect
110. #121Dic53semidirect