Finite Group Databasebeta

← #20291009

Subgroups — #20291009

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#16811533C212yesmaximal, complemented
3#16811532C212yesmaximal, complemented
4#16811172C211yesmaximal, characteristic
5C2×#11719940—22no—
6#13133278C2214yes—
7#13133277—22no—
8C2×#11719949C411yescharacteristic
9C2×#11719947C2211yescharacteristic
10#13133215C2211yescharacteristic
11#13133213C411yescharacteristic
12#13133205C2211yescharacteristic
13#13133190—22no—
14#13133177—22no—
15C42×D53C411yescharacteristic
16#124×C53C411yescharacteristic
17C2×Q8×C53C2×C411yescharacteristic
18C2×D4×C53C2×C411yescharacteristic
19C2×C22⋊D53—22no—
20C2×C22⋊D53D412yes—
21C22×Dic53D412yes—
22#11719949—41no—
23#11719949—41no—
24#11719947—41no—
25#11719947—41no—
26C2×D212—21no—
27C2×D212C2×C411yescharacteristic
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53C2×C411yescharacteristic
31C2×Dic106C2311yescharacteristic
32#11719940—24no—
33#11719940—42no—
34C2×#88324—42no—
35C22⋊C4×C53—22no—
36C42×C53C2×C411yescharacteristic
37#11719921—22no—
38#11719919—22no—
39C4×Dic53—21no—
40C4×Dic53C2×C411yescharacteristic
41C23×C53C22⋊C412yes—
42C22×D53—21no—
43C22×D53C22⋊C411yescharacteristic
44Q8×C53—41no—
45D4×C53—41no—
46C2×C4×C53—21no—
47C2×C4×C53—21no—
48C2×C4×C53C22×C411yescharacteristic
49C22⋊D53—42no—
50C22⋊D53—44no—
51C2×Dic53—21no—
52C2×Dic53—22no—
53C2×Dic53—22no—
54C2×Dic53C22⋊C411yescharacteristic
55C2×Dic53C2×D412yes—
56D212—41no—
57D212—42no—
58C4×D53—21no—
59C4×D53—21no—
60C4×D53—42no—
61C4×D53—41no—
62C4×D53—41no—
63Dic106—41no—
64#88324—44no—
65C22×C53—22no—
66C22×C53—42no—
67C22×C53C2×C22⋊C411yescharacteristic
68C2×D53—21no—
69C2×D53—41no—
70C2×D53—42no—
71C2×D53—41no—
72C4×C53—21no—
73C4×C53—41no—
74C4×C53—41no—
75Dic53—22no—
76Dic53—22no—
77Dic53#9911yescharacteristic
78Dic53#9911yescharacteristic
79#2132—531nomaximal, Sylow
80C2×C53—21no—
81C2×C53—42no—
82C2×C53C2×#9911yescharacteristic
83D53—41no—
84D53—81no—
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—531no—
95#124—531no—
96#124#615711yescharacteristic
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×Q8C2×#615711yescharacteristic
108C2×D4—532no—
109C2×D4—531no—
110C2×D4—1061no—
111C2×D4—1062no—
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—22no—
122C22⋊C4—1062no—
123C42—531no—
124C42—1061no—
125C42C2×#615711yescharacteristic
126C23—531no—
127C23—1061no—
128C23C22⋊#615712yes—
129Q8—41no—
130Q8—2121no—
131D4—41no—
132D4—2121no—
133D4—2122no—
134D4—2122no—
135D4—2124no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—531no—
139C2×C4—532no—
140C2×C4—1061no—
141C2×C4—1061no—
142C2×C4—1062no—
143C2×C4—1062no—
144C2×C4—1061no—
145C2×C4—2121no—
146C2×C4—2121no—
147C2×C4—2122no—
148C2×C4C22×#615711yescharacteristic
149C8—2124no—
150C22—22no—
151C22—42no—
152C22—1061no—
153C22—2122no—
154C22—2121no—
155C22—2121no—
156C22C2×C22⋊#615711yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—531no—
161C4—531no—
162C4—1062no—
163C4—1062no—
164C2—21no—
165C2—42no—
166C2—4241no—
167C2—2121no—
168C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC2×C53C2×#99
hypercenter#124#6157
hypocenterC53#2132
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811610C2semidirect
2. #16811533C2semidirect
3. #16811532C2semidirect
89. C53#2132semidirect