Finite Group Databasebeta

← #20276368

Subgroups — #20276368

Order 6784.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811194C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic
3#16810412C214yesmaximal, complemented
4C53×#410C211yesmaximal, characteristic
5C22×Dic106C411yescharacteristic, complemented
6C2×#11719917—21no—
7#13133215—21no—
8#13133204—21no—
9#13133198C412yescomplemented
10#13133198C2212yes—
11#13133196—21no—
12#13133196—21no—
13#13133176C411yescharacteristic, complemented
14#13133136—21no—
15#13133134C2212yes—
16C2×#41×C53C2211yescharacteristic
17#117×C53—21no—
18#99×C53—21no—
19#99×C53C2212yes—
20#41×C53—41no—
21#41×C53—41no—
22C2×Q8×C53D411yescharacteristic
23C2×D4×C53C2×C411yescharacteristic
24C2×D4×C53D411yescharacteristic
25C2×D4×C53C2311yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×Dic53C2×C412yes—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—41no—
33C2×Dic106—41no—
34C4⋊C4×C53—41no—
35C22⋊C4×C53—22no—
36C42×C53—41no—
37C22⋊Dic53—22no—
38C22⋊Dic53—22no—
39C22⋊Dic53C2×C412yes—
40C4⋊Dic53—21no—
41#11719919—22no—
42#11719919—22no—
43C4×Dic53—21no—
44#11719917—22no—
45#11719917—41no—
46C2×#88322—41no—
47C23×C53C22⋊C411yescharacteristic
48C23×C53C22×C411yescharacteristic
49C23×C53C2×D411yescharacteristic
50Q8×C53—41no—
51D4×C53—41no—
52D4×C53—41no—
53D4×C53—41no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—42no—
61C2×C4×C53C22⋊C411yescharacteristic
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×Dic53—22no—
66C2×Dic53—22no—
67C2×Dic53—42no—
68Dic106—42no—
69Dic106—41no—
70Dic106—41no—
71#88322—42no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53—21no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53C2×C22⋊C411yescharacteristic
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—21no—
82C4×C53—82no—
83C4×C53#9911yescharacteristic
84C4×C53#9911yescharacteristic
85Dic53—42no—
86Dic53—42no—
87#2123—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—41no—
90C2×C53—41no—
91C2×C53—41no—
92C2×C53C2×#9911yescharacteristic
93#536—531no—
94#413—531no—
95#410D5311yescharacteristic
96#352—534no—
97C53#212311yescharacteristic, complemented, Sylow
98C2×#41C2×D5311yescharacteristic
99C22×Q8—531no—
100C2×M16—1061no—
101C4⋊Q8—1061no—
102#124—1061no—
103#123—532no—
104#123—532no—
105C22⋊Q8—1061no—
106C22⋊Q8—1061no—
107#117—21no—
108#117—531no—
109#101—1061no—
110#100—532no—
111#99—21no—
112#99C2×D5312yes—
113#41—41no—
114#41—41no—
115C2×Q8—1061no—
116C2×Q8—1061no—
117C2×Q8—2121no—
118C2×Q8—2121no—
119C2×Q8D21211yescharacteristic
120C2×D4C4×D5311yescharacteristic
121C2×D4C22⋊D5311yescharacteristic
122C2×D4C22×D5311yescharacteristic
123C22×C4—532no—
124C22×C4C4×D5311yescharacteristic
125C22×C4D21211yescharacteristic
126C22×C4C22⋊D5311yescharacteristic
127M16—1062no—
128M16—2121no—
129C2×C8—2121no—
130C4⋊C4—41no—
131C4⋊C4—1062no—
132C4⋊C4—1062no—
133C4⋊C4—1061no—
134C22⋊C4—22no—
135C22⋊C4—532no—
136C22⋊C4—1062no—
137C22⋊C4—1062no—
138C42—41no—
139C42—1061no—
140C23#1171992111yescharacteristic
141C23C2×C4×D5311yescharacteristic
142C23C2×C22⋊D5311yescharacteristic
143Q8—41no—
144Q8—2122no—
145Q8—2121no—
146Q8—2121no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—1062no—
158C2×C4—1062no—
159C2×C4—2122no—
160C2×C4#1171992111yescharacteristic
161C2×C4#1171992111yescharacteristic
162C2×C4#1171992111yescharacteristic
163C2×C4C2×D21211yescharacteristic
164C8—2122no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1171992111yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—2122no—
177C4—2122no—
178C4#1313313111yescharacteristic
179C4#1313313111yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#410C2
socleC2×C53C2×#99
hypercenter#410D53
hypocenterC53#2123
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #16810412C2semidirect
5. C22×Dic106C4semidirect
9. #13133198C4semidirect
13. #13133176C4semidirect
97. C53#2123semidirect