Finite Group Databasebeta

← #20276370

Subgroups — #20276370

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#16810597C211yesmaximal, characteristic
3#16810414C214yesmaximal, complemented
4C53×#413C211yesmaximal, characteristic
5C22×Dic106C411yescharacteristic, complemented
6#13133215—21no—
7#13133204—21no—
8#13133198C412yescomplemented
9#13133198C2212yes—
10#13133196—21no—
11#13133196—21no—
12#13133176—21no—
13#13133176C411yescharacteristic, complemented
14#13133132—21no—
15#13133132C2212yes—
16C2×#41×C53C2211yescharacteristic
17C2×M16×C53—21no—
18#101×C53—21no—
19#100×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—
34M16×C53—22no—
35M16×C53—41no—
36C2×C8×C53—41no—
37C22⋊Dic53—22no—
38C22⋊Dic53—22no—
39C22⋊Dic53—22no—
40C22⋊Dic53C2×C412yes—
41C4⋊Dic53—21no—
42C4⋊Dic53—41no—
43#11719919—22no—
44#11719919—22no—
45C4×Dic53—21no—
46C4×Dic53—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×C53C22⋊C411yescharacteristic
61C2×C4×C53C22⋊C411yescharacteristic
62C2×C4×C53C22⋊C411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×Dic53—22no—
65C2×Dic53—22no—
66C2×Dic53—42no—
67C2×Dic53—42no—
68Dic106—42no—
69Dic106—41no—
70Dic106—41no—
71C8×C53—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#9911yescharacteristic
83C4×C53#9911yescharacteristic
84Dic53—42no—
85Dic53—42no—
86Dic53—82no—
87#2123—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—41no—
90C2×C53—41no—
91C2×C53—41no—
92C2×C53C2×#9911yescharacteristic
93#536—531no—
94#413D5311yescharacteristic
95#410—531no—
96#352—534no—
97C53#212311yescharacteristic, complemented, Sylow
98C2×#41C2×D5311yescharacteristic
99C22×Q8—531no—
100C2×M16—21no—
101C4⋊Q8—1061no—
102#124—1061no—
103#123—532no—
104#123—532no—
105C22⋊Q8—1061no—
106C22⋊Q8—1061no—
107#117—531no—
108#117—1061no—
109#101—21no—
110#100C2×D5312yes—
111#99—532no—
112#99—1061no—
113#41—41no—
114#41—41no—
115C2×Q8—1061no—
116C2×Q8—1061no—
117C2×Q8—2121no—
118C2×Q8—2121no—
119C2×Q8C22⋊D5311yescharacteristic
120C2×D4C4×D5311yescharacteristic
121C2×D4D21211yescharacteristic
122C2×D4C22×D5311yescharacteristic
123C22×C4—532no—
124C22×C4C4×D5311yescharacteristic
125C22×C4D21211yescharacteristic
126C22×C4C22⋊D5311yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—1062no—
131C4⋊C4—1062no—
132C4⋊C4—1061no—
133C4⋊C4—2121no—
134C22⋊C4—532no—
135C22⋊C4—1062no—
136C22⋊C4—1062no—
137C22⋊C4—1062no—
138C42—1061no—
139C42—2121no—
140C23#1171992111yescharacteristic
141C23C2×C4×D5311yescharacteristic
142C23C2×D21211yescharacteristic
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—1062no—
157C2×C4—1062no—
158C2×C4—2122no—
159C2×C4—2122no—
160C2×C4#1171992111yescharacteristic
161C2×C4#1171992111yescharacteristic
162C2×C4#1171992111yescharacteristic
163C2×C4C2×C22⋊D5311yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×#1171992111yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—4242no—
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×#413C2
socleC2×C53C2×#99
hypercenter#413D53
hypocenterC53#2123
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #16810414C2semidirect
5. C22×Dic106C4semidirect
8. #13133198C4semidirect
13. #13133176C4semidirect
97. C53#2123semidirect