Finite Group Databasebeta

← #20275249

Subgroups — #20275249

Order 6784.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811100C211yesmaximal, characteristic, complemented
3#16810640C211yesmaximal, characteristic
4C2×#13133135C211yesmaximal, characteristic, complemented
5#16810380C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#421C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133215C412yescomplemented
11#13133204C412yescomplemented
12#13133198—22no—
13#13133196—22no—
14C4×Dic106—22no—
15#13133182—22no—
16#13133176C2211yescharacteristic
17#13133176C2211yescharacteristic
18#13133173—22no—
19#13133144—22no—
20#13133135—22no—
21C2×#41×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23#117×C53C2211yescharacteristic
24#104×C53—22no—
25#41×C53—21no—
26#41×C53—21no—
27#41×C53—21no—
28#41×C53—21no—
29C2×Q8×C53C2×C411yescharacteristic
30C2×D4×C53—21no—
31C2×D4×C53C2×C411yescharacteristic
32C22×C4×C53—21no—
33C22×C4×C53C2311yescharacteristic
34C22×Dic53D412yes—
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106—21no—
38C2×Dic106—21no—
39C2×Dic106C2×C412yes—
40C2×Dic106D412yes—
41M16×C53—21no—
42M16×C53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C42×C53D412yes—
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53—24no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52#11719919—24no—
53#11719919—24no—
54C4×Dic53—24no—
55C4×Dic53C2×C412yes—
56C4×Dic53D412yes—
57#11719917—21no—
58#11719917—21no—
59C2×#88322—21no—
60C23×C53—21no—
61C23×C53C2×D411yescharacteristic
62Q8×C53—21no—
63Q8×C53—21no—
64D4×C53—21no—
65D4×C53—21no—
66D4×C53—41no—
67D4×C53—41no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53C22×C411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53#4111yescharacteristic
77C2×C4×C53#4111yescharacteristic
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53C22⋊C414yes—
83Dic106—22no—
84Dic106—22no—
85Dic106—22no—
86Dic106—22no—
87C8×C53—41no—
88#88322—41no—
89C22×C53—41no—
90C22×C53—42no—
91C22×C53—41no—
92C22×C53C2×C22⋊C411yescharacteristic
93C22×C53#12011yescharacteristic
94C22×C53#12311yescharacteristic
95C4×C53—42no—
96C4×C53—41no—
97C4×C53C4×D411yescharacteristic
98C4×C53C4×D411yescharacteristic
99C4×C53#12111yescharacteristic
100C4×C53#12111yescharacteristic
101Dic53—24no—
102Dic53—42no—
103Dic53—42no—
104#1912—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—21no—
107C2×C53—21no—
108C2×C53—81no—
109C2×C53#38611yescharacteristic
110#536—531no—
111#519—531no—
112#421—531no—
113#421D5311yescharacteristic
114C2×#101—531no—
115H(3,ℤ/4)—531no—
116H(3,ℤ/4)—531no—
117C53#191211yescharacteristic, complemented, Sylow
118C2×#41C2×D5311yescharacteristic
119C22×Q8—531no—
120C2×M16—531no—
121C2×M16C2×D5311yescharacteristic
122C4⋊Q8—532no—
123#124—532no—
124#123—1062no—
125C22⋊Q8—1062no—
126C4×Q8—1062no—
127C4×Q8—1062no—
128#117—531no—
129#117—531no—
130#117—1062no—
131#117C2×D5311yescharacteristic
132#104—22no—
133#104—1062no—
134#101—1062no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—532no—
140C2×Q8—532no—
141C2×Q8—1061no—
142C2×Q8—1061no—
143C2×Q8—1061no—
144C2×Q8—1061no—
145C2×Q8C4×D5311yescharacteristic
146C2×D4—21no—
147C2×D4C4×D5311yescharacteristic
148C22×C4—21no—
149C22×C4—532no—
150C22×C4C22×D5311yescharacteristic
151M16—21no—
152M16—21no—
153M16—1061no—
154M16—1061no—
155C2×C8—21no—
156C2×C8—1061no—
157C4⋊C4—21no—
158C4⋊C4—1064no—
159C4⋊C4—1064no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C22⋊C4—21no—
163C22⋊C4—1064no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C42—532no—
167C42—532no—
168C42—1064no—
169C42C22⋊D5312yes—
170C23—21no—
171C23D4×D5311yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1062no—
175Q8—1062no—
176Q8—1062no—
177Q8—1062no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—534no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4C2×C4×D5311yescharacteristic
192C2×C4#1171994511yescharacteristic
193C2×C4D4×D5311yescharacteristic
194C2×C4D4×D5311yescharacteristic
195C2×C4#1171994711yescharacteristic
196C2×C4C2×C22⋊D5311yescharacteristic
197C8—41no—
198C8—2121no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D5311yescharacteristic
203C22#1313318511yescharacteristic
204C22#1313319911yescharacteristic
205C4—42no—
206C4—41no—
207C4—1064no—
208C4—2122no—
209C4—2122no—
210C4#1313317711yescharacteristic
211C4C4×C22⋊D5311yescharacteristic
212C4#1313318811yescharacteristic
213C4#1313319011yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#1681049911yescharacteristic
subgroupG/H
centerC2#16810499
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#421C2
socleC2×C53#386
hypercenter#421D53
hypocenterC53#1912
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #16811100C2semidirect
4. C2×#13133135C2semidirect
5. #16810380C2semidirect
6. #16810379C2semidirect
10. #13133215C4semidirect
11. #13133204C4semidirect
117. C53#1912semidirect