Finite Group Databasebeta

← #20276406

Subgroups — #20276406

Order 6784.

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

normal subgroups only

This group has 262 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#16811187C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic, complemented
3#16810419C214yesmaximal, complemented
4C53×#413C211yesmaximal, characteristic, complemented
5C22×D212C411yescharacteristic
6C2×#11719917—21no—
7#13133213C412yes—
8#13133206C2212yes—
9#13133191—21no—
10#13133191—21no—
11#13133185—21no—
12#13133185—21no—
13#13133176C411yescharacteristic
14#13133136C2212yescomplemented
15#13133134—21no—
16C2×#41×C53C2211yescharacteristic
17C2×M16×C53—21no—
18#101×C53C2212yescomplemented
19#100×C53—21no—
20C23×D53—21no—
21#41×C53—41no—
22#41×C53—41no—
23C2×Q8×C53C2311yescharacteristic
24C2×D4×C53C2×C411yescharacteristic
25C2×D4×C53D411yescharacteristic
26C2×D4×C53D411yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32C2×D212—21no—
33C2×D212—41no—
34C2×D212—41no—
35C2×D212C2×C412yes—
36M16×C53—22no—
37M16×C53—41no—
38C2×C8×C53—41no—
39C22⋊Dic53—21no—
40#11719921—22no—
41#11719921—22no—
42C4⋊Dic53—21no—
43C4×Dic53C2×C412yes—
44#11719917—22no—
45#11719917—41no—
46C2×#88322—41no—
47C23×C53C22⋊C411yescharacteristic
48C23×C53C22⋊C411yescharacteristic
49C23×C53C22⋊C411yescharacteristic
50C22×D53—22no—
51C22×D53—42no—
52C22×D53—82no—
53Q8×C53—41no—
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22×C411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C22⋊D53—82no—
68C22⋊D53—82no—
69C2×Dic53—22no—
70D212—42no—
71D212—41no—
72D212—41no—
73C8×C53—42no—
74#88322—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—84no—
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53#9911yescharacteristic
90C4×C53#9911yescharacteristic
91Dic53—42no—
92#2134—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—41no—
95C2×C53—41no—
96C2×C53—41no—
97C2×C53C2×#9911yescharacteristic
98D53—82no—
99#535—531no—
100#413—531no—
101#413D5311yescharacteristic, complemented
102#355—534no—
103C53#213411yescharacteristic, complemented, Sylow
104C2×#41C2×D5311yescharacteristic
105C22×D4—531no—
106C2×M16—21no—
107C2×M16—1061no—
108#127—532no—
109#124—532no—
110#121—1061no—
111#121—1061no—
112#120—1061no—
113#120—1061no—
114#117—531no—
115#101—532no—
116#101C2×D5312yescomplemented
117#100—21no—
118#100—1061no—
119C24—1061no—
120#41—41no—
121#41—41no—
122C2×Q8C22×D5311yescharacteristic
123C2×D4—532no—
124C2×D4—1062no—
125C2×D4—1062no—
126C2×D4—1061no—
127C2×D4—2121no—
128C2×D4—2121no—
129C2×D4C4×D5311yescharacteristic
130C2×D4D21211yescharacteristic
131C2×D4C22⋊D5311yescharacteristic
132C22×C4C4×D5311yescharacteristic
133C22×C4D21211yescharacteristic
134C22×C4C22⋊D5311yescharacteristic
135M16—22no—
136M16—41no—
137M16—1062no—
138M16—2121no—
139C2×C8—41no—
140C2×C8—2121no—
141C4⋊C4—1061no—
142C22⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—532no—
146C23—1062no—
147C23—4242no—
148C23—2122no—
149C23#1171992111yescharacteristic
150C23#1171992111yescharacteristic
151C23#1171992111yescharacteristic
152Q8—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—4242no—
157D4—4242no—
158D4—2122no—
159D4—2121no—
160D4—2121no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—1062no—
168C2×C4#1171992111yescharacteristic
169C2×C4C2×C4×D5311yescharacteristic
170C2×C4C2×D21211yescharacteristic
171C2×C4C2×C22⋊D5311yescharacteristic
172C8—42no—
173C8—2122no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—4244no—
181C22—2122no—
182C22—2122no—
183C22—2122no—
184C22C2×#1171992111yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—2122no—
189C4#1313313111yescharacteristic
190C4#1313313111yescharacteristic
191C2—21no—
192C2—41no—
193C2—41no—
194C2—41no—
195C2—4242no—
196C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#413C2
socleC2×C53C2×#99
hypercenter#413D53
hypocenterC53#2134
LCSG ▷ C2×C4×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
2. #16810607C2semidirect
3. #16810419C2semidirect
4. C53×#413C2semidirect
14. #13133136C22semidirect
18. #101×C53C22semidirect
101. #413D53semidirect
103. C53#2134semidirect
116. #101C2×D53semidirect