Finite Group Databasebeta

← #20276384

Subgroups — #20276384

Order 6784.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13133132C212yesmaximal, complemented
2#16810418C212yesmaximal, complemented
3#16810416C212yesmaximal, complemented
4C53×#535C211yesmaximal, characteristic
5C2×C22⋊Dic53—22no—
6#13133132—22no—
7#13133132C2214yes—
8C2×#41×C53C411yescharacteristic, complemented
9C22×D4×C53C2211yescharacteristic
10#127×C53C411yescharacteristic, complemented
11#127×C53C2211yescharacteristic
12#124×C53C411yescharacteristic, complemented
13#124×C53C2211yescharacteristic
14#121×C53—22no—
15#120×C53—22no—
16#117×C53C411yescharacteristic, complemented
17C24×C53D412yes—
18#41×C53—42no—
19C2×Q8×C53C2×C411yescharacteristic, complemented
20C2×D4×C53—21no—
21C2×D4×C53—22no—
22C2×D4×C53—22no—
23C2×D4×C53—42no—
24C2×D4×C53C2×C411yescharacteristic
25C2×D4×C53C2×C411yescharacteristic, complemented
26C2×D4×C53D412yes—
27C2×D4×C53C2311yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2×C411yescharacteristic
30C22×Dic53—42no—
31C4⋊C4×C53—21no—
32C22⋊C4×C53—21no—
33C22⋊C4×C53—22no—
34C22⋊C4×C53—22no—
35C42×C53C2×C411yescharacteristic, complemented
36C42×C53C2×C411yescharacteristic, complemented
37C22⋊Dic53—24no—
38C22⋊Dic53—42no—
39C23×C53—22no—
40C23×C53—22no—
41C23×C53—21no—
42C23×C53—82no—
43C23×C53C22⋊C412yes—
44C23×C53C22⋊C411yescharacteristic
45C23×C53C2×D412yes—
46Q8×C53—41no—
47D4×C53—41no—
48D4×C53—42no—
49D4×C53—41no—
50D4×C53—41no—
51D4×C53—42no—
52D4×C53—82no—
53D4×C53—82no—
54C2×C4×C53—22no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—42no—
59C2×C4×C53C22⋊C411yescharacteristic
60C2×C4×C53C22×C411yescharacteristic
61C2×Dic53—44no—
62C2×Dic53—84no—
63C22×C53—22no—
64C22×C53—22no—
65C22×C53—41no—
66C22×C53—42no—
67C22×C53—42no—
68C22×C53—44no—
69C22×C53—42no—
70C22×C53—41no—
71C22×C53—82no—
72C22×C53#9912yes—
73C22×C53C2×C22⋊C411yescharacteristic
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—41no—
77C4×C53—41no—
78C4×C53—41no—
79Dic53—84no—
80#2127—531nomaximal, Sylow
81C2×C53—21no—
82C2×C53—22no—
83C2×C53—42no—
84C2×C53—42no—
85C2×C53—81no—
86C2×C53C2×#9911yescharacteristic
87#535D5311yescharacteristic
88C2×#99—532no—
89#354—532no—
90#353—532no—
91C53#212711yescharacteristic, complemented, Sylow
92C2×#41Dic5311yescharacteristic, complemented
93C22×D4C2×D5311yescharacteristic
94#127Dic5311yescharacteristic, complemented
95#127C2×D5311yescharacteristic
96#124Dic5311yescharacteristic, complemented
97#124C2×D5311yescharacteristic
98#121—22no—
99#120—22no—
100#117Dic5311yescharacteristic, complemented
101C2×C22⋊C4—1062no—
102#99—534no—
103#99—1062no—
104C24C22⋊D5312yes—
105#41—42no—
106C2×Q8C2×Dic5311yescharacteristic, complemented
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4C2×Dic5311yescharacteristic
112C2×D4C2×Dic5311yescharacteristic, complemented
113C2×D4C22⋊D5312yes—
114C2×D4C22×D5311yescharacteristic
115C22×C4—21no—
116C22×C4—2122no—
117C22×C4C2×Dic5311yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—21no—
122C22⋊C4—1064no—
123C22⋊C4—2122no—
124C42C2×Dic5311yescharacteristic, complemented
125C42C2×Dic5311yescharacteristic, complemented
126C23—22no—
127C23—22no—
128C23—21no—
129C23—82no—
130C23C22⋊Dic5312yes—
131C23C22⋊Dic5311yescharacteristic
132C23C2×C22⋊D5312yes—
133Q8—41no—
134D4—41no—
135D4—42no—
136D4—41no—
137D4—41no—
138D4—42no—
139D4—82no—
140D4—82no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—4244no—
147C2×C4—2124no—
148C2×C4C22⋊Dic5311yescharacteristic
149C2×C4C22×Dic5311yescharacteristic
150C22—22no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—41no—
158C22—82no—
159C22#1313313212yes—
160C22C2×C22⋊Dic5311yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—4244no—
167C2—21no—
168C2—22no—
169C2—42no—
170C2—42no—
171C2—81no—
172C2C2×#1313313211yescharacteristic
subgroupG/H
centerC2C2×#13133132
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#535C2
socleC2×C53C2×#99
hypercenter#535D53
hypocenterC53#2127
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#13133132C2semidirect
2. #16810418C2semidirect
3. #16810416C2semidirect
8. C2×#41×C53C4semidirect
10. #127×C53C4semidirect
12. #124×C53C4semidirect
16. #117×C53C4semidirect
19. C2×Q8×C53C2×C4semidirect
25. C2×D4×C53C2×C4semidirect
35. C42×C53C2×C4semidirect
36. C42×C53C2×C4semidirect
91. C53#2127semidirect
92. C2×#41Dic53semidirect
94. #127Dic53semidirect
96. #124Dic53semidirect
100. #117Dic53semidirect
106. C2×Q8C2×Dic53semidirect
112. C2×D4C2×Dic53semidirect
124. C42C2×Dic53semidirect
125. C42C2×Dic53semidirect