Finite Group Databasebeta

← #20290892

Subgroups — #20290892

Order 6784.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#11719936C211yesmaximal, characteristic
2C4×C4⋊#6157C211yesmaximal, characteristic
3#16811493C212yesmaximal
4#16811493C212yesmaximal
5#16811179C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7C2×C4⋊#6157—21no—
8C2×C4×#6157C2212yes—
9#13133272—21no—
10C4×#88324—21no—
11(C2×#6157)⋊C4—21no—
12#13133216C411yescharacteristic, complemented
13#13133216C411yescharacteristic, complemented
14#13133209C411yescharacteristic, complemented
15#13133195—21no—
16#13133195—21no—
17#13133182—21no—
18#13133182—21no—
19#13133175C2211yescharacteristic
20#13133175C2211yescharacteristic
21C42×D53C2211yescharacteristic
22#125×C53C411yescharacteristic, complemented
23C22×#6157—22no—
24C2×C4×D53D411yescharacteristic
25C2×C4×D53D411yescharacteristic
26C2×C4×D53C2311yescharacteristic
27C2×Dic106—21no—
28C2×Dic106—21no—
29C2×#88324—22no—
30C4⋊#6157—24no—
31C4×#6157—24no—
32#11719936—22no—
33#11719936Q814yescomplemented
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53C2×C411yescharacteristic
37C4⋊C4×C53C2×C411yescharacteristic
38C42×C53C2×C411yescharacteristic
39#11719921—21no—
40#11719921—21no—
41C4⋊Dic53C2×C411yescharacteristic
42C4⋊Dic53C2×C411yescharacteristic
43#11719919—21no—
44#11719919—21no—
45#11719919—21no—
46#11719919—21no—
47C4×Dic53—21no—
48C4×Dic53—21no—
49C4×Dic53C2×C411yescharacteristic
50C4×Dic53D411yescharacteristic
51C4×Dic53D411yescharacteristic
52C22×D53C2×D411yescharacteristic
53C2×#6157—24no—
54C2×#6157—42no—
55C2×#6157—42no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22⋊C411yescharacteristic
60C2×C4×C53C22×C411yescharacteristic
61C2×Dic53—21no—
62C2×Dic53—21no—
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C22⋊C411yescharacteristic
65C2×Dic53C2×D411yescharacteristic
66C4×D53—22no—
67C4×D53—41no—
68C4×D53C2×Q812yes—
69C4×D53#4112yes—
70Dic106—42no—
71Dic106—42no—
72#88324—24no—
73C22×C53C2×C22⋊C411yescharacteristic
74C2×D53—21no—
75C2×D53—21no—
76C2×D53#12411yescharacteristic
77C2×D53C4⋊Q811yescharacteristic
78#6157—44no—
79C4×C53—21no—
80C4×C53—21no—
81C4×C53—41no—
82C4×C53—41no—
83C4×C53C4×Q812yes—
84Dic53—21no—
85Dic53—41no—
86Dic53—41no—
87Dic53#10412yes—
88Dic53C22⋊Q812yes—
89#1986—531nomaximal, Sylow
90C2×C53#39111yescharacteristic
91C2×C53C2×#10411yescharacteristic
92C2×C53#42111yescharacteristic
93D53—21no—
94D53—21no—
95#533—531no—
96C4×M16—531no—
97C4×C4⋊C4—531no—
98#339—532no—
99#339—532no—
100C53#198611yescharacteristic, complemented, Sylow
101C2×M16—532no—
102C4⋊Q8—531no—
103C4⋊Q8—531no—
104#125—531no—
105#125#615711yescharacteristic, complemented
106C22⋊Q8—1061no—
107C22⋊Q8—1061no—
108C4×Q8—1061no—
109C4×Q8—1061no—
110#117—531no—
111#117—531no—
112C2×C4⋊C4—1061no—
113C2×C42—531no—
114C2×C42—532no—
115#97—1061no—
116C4×C8—1061no—
117#95—1061no—
118C2×Q8—1061no—
119C2×Q8—1061no—
120C22×C4—531no—
121C22×C4—531no—
122C22×C4—531no—
123C22×C4—1062no—
124M16—534no—
125M16—1062no—
126C2×C8—1062no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—531no—
130C4⋊C4—531no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C4⋊C4—1064no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C4⋊C4C2×#615711yescharacteristic
137C4⋊C4C2×#615711yescharacteristic
138C22⋊C4—1061no—
139C22⋊C4—1061no—
140C42—531no—
141C42—531no—
142C42—531no—
143C42—1061no—
144C42—1061no—
145C42—1064no—
146C42C2×#615711yescharacteristic
147C23—531no—
148Q8—2122no—
149Q8—2122no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—531no—
153C2×C4—531no—
154C2×C4—532no—
155C2×C4—532no—
156C2×C4—531no—
157C2×C4—1062no—
158C2×C4—1064no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—2121no—
162C2×C4—2122no—
163C2×C4—2122no—
164C2×C4C22⋊#615711yescharacteristic
165C2×C4C22⋊#615711yescharacteristic
166C2×C4C22×#615711yescharacteristic
167C8—1064no—
168C22—531no—
169C22—531no—
170C22—1061no—
171C22—1061no—
172C22C2×C22⋊#615711yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—532no—
178C4—532no—
179C4—1061no—
180C4—2124no—
181C4—2121no—
182C4—2121no—
183C4Q8×#615712yes—
184C2—1061no—
185C2—1061no—
186C2(C2×#6157)⋊Q811yescharacteristic
187C2D53⋊#10411yescharacteristic
188C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#125×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#125#6157
hypocenterC53#1986
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
12. #13133216C4semidirect
13. #13133216C4semidirect
14. #13133209C4semidirect
22. #125×C53C4semidirect
33. #11719936Q8semidirect
100. C53#1986semidirect
105. #125#6157semidirect