Finite Group Databasebeta

← #20290977

Subgroups — #20290977

Order 6784.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#16811609C211yesmaximal, characteristic
2#16811602C211yesmaximal, characteristic
3#16811525C214yesmaximal, complemented
4#16811197C211yesmaximal, characteristic
5D53⋊M16—21no—
6#13133296—21no—
7#13133278—21no—
8#13133277C2212yes—
9#13133276—21no—
10#13133275C2212yes—
11C2×#11719945C2211yescharacteristic
12#13133215—21no—
13#13133206—21no—
14#13133200C412yescomplemented
15#13133200C2212yes—
16#13133193—21no—
17#13133193—21no—
18#13133176C411yescharacteristic, complemented
19C22×Q8×C53C411yescharacteristic, complemented
20C2×Q8×C53—21no—
21C2×Q8×C53—21no—
22C2×Q8×C53—41no—
23C2×Q8×C53—41no—
24C22×C4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C412yes—
26C2×C22⋊D53C2×C411yescharacteristic
27C2×C22⋊D53C2311yescharacteristic
28#11719945—41no—
29#11719945—41no—
30C2×D212D411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53D411yescharacteristic
33C2×Dic106D411yescharacteristic
34C22⋊#6157—22no—
35#11719940—22no—
36C4⋊#6157—41no—
37C4×#6157—41no—
38#11719936—41no—
39D53⋊C8—41no—
40C22⋊Dic53C2×C412yes—
41#11719921—22no—
42#11719921—22no—
43C4⋊Dic53—21no—
44#11719919—22no—
45#11719919—22no—
46C4×Dic53—21no—
47C23×C53C22×C411yescharacteristic
48C22×D53C22⋊C411yescharacteristic
49C22×D53C2×D411yescharacteristic
50C2×#6157—42no—
51Q8×C53—42no—
52Q8×C53—41no—
53Q8×C53—41no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—22no—
57C2×C4×C53—42no—
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22⋊C411yescharacteristic
60C22⋊D53—41no—
61C22⋊D53—41no—
62C2×Dic53—22no—
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C2×D411yescharacteristic
65D212—41no—
66C4×D53—21no—
67C4×D53—21no—
68C4×D53—21no—
69C4×D53—21no—
70Dic106—41no—
71#88324—42no—
72C22×C53—21no—
73C22×C53—41no—
74C22×C53C2×C22⋊C411yescharacteristic
75C2×D53—21no—
76C2×D53—21no—
77C2×D53—41no—
78C2×D53—41no—
79#6157—82no—
80C4×C53—21no—
81C4×C53—42no—
82C4×C53#9911yescharacteristic
83C4×C53#9911yescharacteristic
84Dic53—21no—
85Dic53—21no—
86Dic53—42no—
87#2123—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53—41no—
90C2×C53C2×#9911yescharacteristic
91D53—41no—
92D53—41no—
93#536—531no—
94#413—531no—
95#410—531no—
96#352—534no—
97C53#212311yescharacteristic, complemented, Sylow
98C2×#41—531no—
99C22×Q8#615711yescharacteristic, complemented
100C2×M16—1061no—
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—1061no—
110#100—532no—
111#99—532no—
112#99—1061no—
113#41—2121no—
114#41—2121no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—531no—
120C2×D4—531no—
121C2×D4—531no—
122C2×D4—531no—
123C22×C4—531no—
124C22×C4—531no—
125C22×C4C2×#615712yes—
126C22×C4C2×#615711yescharacteristic
127M16—1062no—
128M16—2121no—
129C2×C8—2121no—
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—531no—
141C23—531no—
142C23C22×#615711yescharacteristic
143Q8—42no—
144Q8—41no—
145Q8—41no—
146Q8—2121no—
147D4—2121no—
148D4—2121no—
149D4—2121no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—531no—
155C2×C4—531no—
156C2×C4—1061no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—1061no—
160C2×C4—1062no—
161C2×C4—2122no—
162C2×C4C22⋊#615711yescharacteristic
163C2×C4C22⋊#615711yescharacteristic
164C8—2122no—
165C22—21no—
166C22—41no—
167C22—1061no—
168C22—1061no—
169C22—2121no—
170C22—2121no—
171C22C2×C22⋊#615711yescharacteristic
172C4—21no—
173C4—42no—
174C4—1061no—
175C4—1061no—
176C4—4242no—
177C4—2122no—
178C4#1313327511yescharacteristic
179C4#1313327511yescharacteristic
180C2—21no—
181C2—41no—
182C2—2121no—
183C2—2121no—
184C2C2×#1313327511yescharacteristic
subgroupG/H
centerC2C2×#13133275
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC22×Q8×C53C4
socleC2×C53C2×#99
hypercenterC22×Q8#6157
hypocenterC53#2123
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #16811525C2semidirect
14. #13133200C4semidirect
18. #13133176C4semidirect
19. C22×Q8×C53C4semidirect
97. C53#2123semidirect
99. C22×Q8#6157semidirect