Finite Group Databasebeta

← #20276386

Subgroups — #20276386

Order 6784.

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

normal subgroups only

This group has 270 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#16810597C211yesmaximal, characteristic, complemented
3#16810415C214yesmaximal, complemented
4C53×#410C211yesmaximal, characteristic, complemented
5C22×D212C411yescharacteristic, complemented
6#13133213C2212yes—
7#13133206C412yescomplemented
8#13133191—21no—
9#13133191—21no—
10#13133185—21no—
11#13133185—21no—
12#13133176—21no—
13#13133176C411yescharacteristic, complemented
14#13133132—21no—
15#13133132C2212yescomplemented
16C2×#41×C53C2211yescharacteristic
17#117×C53—21no—
18#99×C53—21no—
19#99×C53C2212yescomplemented
20C23×D53—21no—
21#41×C53—41no—
22#41×C53—41no—
23C2×Q8×C53C2×C411yescharacteristic
24C2×D4×C53D411yescharacteristic
25C2×D4×C53D411yescharacteristic
26C2×D4×C53C2311yescharacteristic
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—
36C4⋊C4×C53—41no—
37C22⋊C4×C53—22no—
38C42×C53—41no—
39C22⋊Dic53—21no—
40C22⋊Dic53—22no—
41#11719921—22no—
42#11719921—22no—
43C4⋊Dic53—21no—
44C4⋊Dic53—41no—
45C4×Dic53—41no—
46C4×Dic53C2×C412yes—
47C23×C53C22⋊C411yescharacteristic
48C23×C53C2×D411yescharacteristic
49C23×C53C2×D411yescharacteristic
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×C53—42no—
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22×C411yescharacteristic
68C22⋊D53—82no—
69C22⋊D53—82no—
70C2×Dic53—22no—
71C2×Dic53—42no—
72D212—42no—
73D212—41no—
74D212—41no—
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—82no—
90C4×C53#9911yescharacteristic
91C4×C53#9911yescharacteristic
92Dic53—42no—
93Dic53—82no—
94#2128—531nomaximal, Sylow
95C2×C53—21no—
96C2×C53—41no—
97C2×C53—41no—
98C2×C53—41no—
99C2×C53C2×#9911yescharacteristic
100D53—82no—
101#535—531no—
102#410—531no—
103#410D5311yescharacteristic, complemented
104#353—534no—
105C53#212811yescharacteristic, complemented, Sylow
106C2×#41C2×D5311yescharacteristic
107C22×D4—531no—
108#127—532no—
109#124—532no—
110#121—1061no—
111#121—1061no—
112#120—1061no—
113#120—1061no—
114#117—21no—
115#117—531no—
116#117—1061no—
117#99—21no—
118#99—532no—
119#99—1061no—
120#99C2×D5312yescomplemented
121C24—1061no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D5311yescharacteristic
125C2×D4—532no—
126C2×D4—1062no—
127C2×D4—1062no—
128C2×D4—1061no—
129C2×D4—2121no—
130C2×D4—2121no—
131C2×D4D21211yescharacteristic
132C2×D4C22⋊D5311yescharacteristic
133C2×D4C22×D5311yescharacteristic
134C22×C4C4×D5311yescharacteristic
135C22×C4D21211yescharacteristic
136C22×C4C22⋊D5311yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—1061no—
139C4⋊C4—2121no—
140C22⋊C4—22no—
141C22⋊C4—1061no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—41no—
146C42—532no—
147C42—2121no—
148C23—1062no—
149C23—4242no—
150C23—2122no—
151C23#1171992111yescharacteristic
152C23C2×D21211yescharacteristic
153C23C2×C22⋊D5311yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—4242no—
159D4—4242no—
160D4—2122no—
161D4—2121no—
162D4—2121no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—1062no—
171C2×C4—2122no—
172C2×C4#1171992111yescharacteristic
173C2×C4#1171992111yescharacteristic
174C2×C4#1171992111yescharacteristic
175C2×C4C2×C4×D5311yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—4244no—
183C22—2122no—
184C22—2122no—
185C22—2122no—
186C22C2×#1171992111yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—4242no—
192C4—2122no—
193C4#1313313111yescharacteristic
194C4#1313313111yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—4242no—
200C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#410C2
socleC2×C53C2×#99
hypercenter#410D53
hypocenterC53#2128
LCSG ▷ C2×C4×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
2. #16810597C2semidirect
3. #16810415C2semidirect
4. C53×#410C2semidirect
5. C22×D212C4semidirect
7. #13133206C4semidirect
13. #13133176C4semidirect
15. #13133132C22semidirect
19. #99×C53C22semidirect
103. #410D53semidirect
105. C53#2128semidirect
120. #99C2×D53semidirect