Finite Group Databasebeta

← #20291007

Subgroups — #20291007

Order 6784.

250 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 52 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#16811610C212yesmaximal, complemented
2#16811534C212yesmaximal, complemented
3#16811531C212yesmaximal, complemented
4#16811171C211yesmaximal, characteristic
5C2×#11719940—22no—
6#13133278C2214yes—
7#13133277—22no—
8C2×#11719947C411yescharacteristic
9C2×#11719947C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133204C411yescharacteristic
12#13133204C2211yescharacteristic
13#13133188—22no—
14D4×Dic53—22no—
15C42×D53C411yescharacteristic
16#127×C53C411yescharacteristic
17C2×D4×C53—22no—
18C2×D4×C53C2×C411yescharacteristic
19C2×D4×C53C2×C411yescharacteristic
20C2×C22⋊D53—21no—
21C2×C22⋊D53D412yes—
22C22×Dic53—21no—
23C22×Dic53D412yes—
24#11719947—41no—
25#11719947—41no—
26#11719947—41no—
27#11719947—41no—
28C2×C4×D53—21no—
29C2×C4×D53C2×C411yescharacteristic
30C2×Dic106C2×C411yescharacteristic
31C2×Dic106C2311yescharacteristic
32#11719940—24no—
33#11719940—42no—
34C2×#88324—42no—
35C42×C53C2×C411yescharacteristic
36C22⋊Dic53—22no—
37C22⋊Dic53—22no—
38C4⋊Dic53—22no—
39C4×Dic53—21no—
40C4×Dic53C2×C411yescharacteristic
41C23×C53—21no—
42C23×C53C22⋊C412yes—
43C22×D53C22⋊C411yescharacteristic
44D4×C53—41no—
45D4×C53—41no—
46D4×C53—44no—
47C2×C4×C53—21no—
48C2×C4×C53C22×C411yescharacteristic
49C22⋊D53—42no—
50C22⋊D53—42no—
51C2×Dic53—22no—
52C2×Dic53—22no—
53C2×Dic53—21no—
54C2×Dic53—21no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53C22⋊C411yescharacteristic
58C2×Dic53C2×D412yes—
59C4×D53—21no—
60C4×D53—21no—
61C4×D53—42no—
62Dic106—41no—
63Dic106—41no—
64#88324—44no—
65C22×C53—22no—
66C22×C53—42no—
67C22×C53—42no—
68C22×C53—41no—
69C22×C53C2×C22⋊C411yescharacteristic
70C2×D53—21no—
71C2×D53—41no—
72C4×C53—22no—
73C4×C53—21no—
74Dic53—22no—
75Dic53—41no—
76Dic53—41no—
77Dic53#9911yescharacteristic
78Dic53#9911yescharacteristic
79#2132—531nomaximal, Sylow
80C2×C53—21no—
81C2×C53—42no—
82C2×C53—81no—
83C2×C53C2×#9911yescharacteristic
84D53—41no—
85#532—531no—
86#413—532no—
87#356—532no—
88#355—532no—
89C53#213211yescharacteristic, complemented, Sylow
90C2×#41—531no—
91C2×#41—531no—
92C2×M16—1062no—
93C4⋊Q8—531no—
94#127#615711yescharacteristic
95#124—531no—
96#124—531no—
97#121—1062no—
98C4×D4—1062no—
99C2×C42—531no—
100#101—534no—
101#100—1062no—
102#41—2121no—
103#41—2121no—
104#41—2121no—
105#41—2121no—
106C2×Q8—531no—
107C2×Q8—531no—
108C2×D4—22no—
109C2×D4—532no—
110C2×D4—1061no—
111C2×D4C2×#615711yescharacteristic
112C2×D4C2×#615711yescharacteristic
113C22×C4—532no—
114C22×C4—531no—
115C22×C4—1061no—
116C22×C4—1061no—
117M16—1064no—
118M16—2122no—
119C2×C8—2122no—
120C4⋊C4—1062no—
121C22⋊C4—1062no—
122C22⋊C4—1062no—
123C42—531no—
124C42—1061no—
125C42C2×#615711yescharacteristic
126C23—21no—
127C23—531no—
128C23C22⋊#615712yes—
129Q8—2121no—
130Q8—2121no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—2122no—
135D4—2122no—
136C2×C4—21no—
137C2×C4—531no—
138C2×C4—532no—
139C2×C4—1061no—
140C2×C4—1061no—
141C2×C4—1062no—
142C2×C4—1062no—
143C2×C4—1061no—
144C2×C4—1061no—
145C2×C4—2122no—
146C2×C4—2121no—
147C2×C4—2121no—
148C2×C4C22×#615711yescharacteristic
149C8—2124no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—1061no—
155C22—2121no—
156C22C2×C22⋊#615711yescharacteristic
157C4—22no—
158C4—21no—
159C4—531no—
160C4—531no—
161C4—1062no—
162C4—2121no—
163C4—2121no—
164C2—21no—
165C2—42no—
166C2—81no—
167C2—2121no—
168C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#127×C53C4
socleC2×C53C2×#99
hypercenter#127#6157
hypocenterC53#2132
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #16811610C2semidirect
2. #16811534C2semidirect
3. #16811531C2semidirect
89. C53#2132semidirect