Finite Group Databasebeta

← #20275134

Subgroups — #20275134

Order 6784.

342 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 93 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#16811390C211yesmaximal, characteristic, complemented
2#16810607C211yesmaximal, characteristic, complemented
3#16810587C212yesmaximal, complemented
4#16810393C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6#13133224—24no—
7C2×#11719917—22no—
8C2×#11719917C2212yes—
9C2×#11719917C2212yes—
10#13133136—21no—
11#13133134—21no—
12C22⋊#88322—24no—
13C2×#41×C53C2211yescharacteristic
14C23×C4×C53C2211yescharacteristic
15#123×C53C412yes—
16C22⋊Q8×C53—21no—
17#121×C53—21no—
18#120×C53C412yes—
19C4×D4×C53—22no—
20#117×C53C2211yescharacteristic
21C24×C53C2×C411yescharacteristic
22#41×C53—22no—
23#41×C53—22no—
24C2×Q8×C53D411yescharacteristic
25C2×D4×C53—22no—
26C2×D4×C53C2×C412yes—
27C2×D4×C53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53—22no—
31C22×C4×C53—22no—
32C22×C4×C53C2×C411yescharacteristic
33C22×C4×C53D412yes—
34C22×C4×C53C2311yescharacteristic
35C4⋊C4×C53—22no—
36C4⋊C4×C53—21no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53C2×C412yes—
40C42×C53—21no—
41#11719917—24no—
42#11719917—22no—
43#11719917—22no—
44#11719917—22no—
45#11719917—22no—
46C2×#88322—24no—
47C2×#88322—22no—
48C2×#88322D414yescomplemented
49C23×C53—21no—
50C23×C53—22no—
51C23×C53—42no—
52C23×C53C22⋊C412yes—
53C23×C53C22×C411yescharacteristic
54Q8×C53—22no—
55D4×C53—24no—
56D4×C53—22no—
57D4×C53—44no—
58C2×C4×C53—22no—
59C2×C4×C53—22no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—41no—
64C2×C4×C53—42no—
65C2×C4×C53—41no—
66C2×C4×C53C22⋊C412yes—
67C2×C4×C53C2×D412yes—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53#4112yes—
71#88322—24no—
72#88322—42no—
73C22×C53—22no—
74C22×C53—24no—
75C22×C53—22no—
76C22×C53—41no—
77C22×C53—42no—
78C22×C53—41no—
79C22×C53—42no—
80C22×C53—41no—
81C22×C53C2×C22⋊C411yescharacteristic
82C22×C53C4×D412yes—
83C4×C53—22no—
84C4×C53—42no—
85C4×C53—41no—
86C4×C53#12011yescharacteristic
87C4×C53#12112yes—
88C4×C53#12311yescharacteristic
89#1891—531nomaximal, Sylow
90C2×C53—21no—
91C2×C53—22no—
92C2×C53—42no—
93C2×C53—42no—
94C2×C53#38611yescharacteristic
95#568—531no—
96#525D5311yescharacteristic
97#413—531no—
98#407—532no—
99#344—532no—
100C53#189111yescharacteristic, complemented, Sylow
101C2×#41C2×D5311yescharacteristic
102C23×C4C2×D5311yescharacteristic
103#131—1064no—
104C2×M16—532no—
105C2×M16—532no—
106C2×M16—1062no—
107#123Dic5312yes—
108C22⋊Q8—21no—
109#121—21no—
110#120Dic5312yes—
111C4×D4—22no—
112#117C2×D5311yescharacteristic
113#101—1061no—
114#100—1061no—
115C22⋊C8—1064no—
116C24C2×Dic5311yescharacteristic
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D5311yescharacteristic
120C2×D4—22no—
121C2×D4C2×Dic5312yes—
122C2×D4C22⋊D5311yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C2×Dic5311yescharacteristic
128C22×C4C22⋊D5312yes—
129C22×C4C22×D5311yescharacteristic
130M16—1064no—
131M16—1062no—
132M16—1062no—
133M16—1062no—
134M16—1062no—
135C2×C8—534no—
136C2×C8—1064no—
137C2×C8—1062no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4C2×Dic5312yes—
143C42—21no—
144C23—21no—
145C23—22no—
146C23—42no—
147C23C22⋊Dic5312yes—
148C23C22×Dic5311yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊Dic5312yes—
162C2×C4D4×D5312yes—
163C2×C4#1171994712yes—
164C2×C4C2×C22⋊D5311yescharacteristic
165C2×C4C2×C22⋊D5311yescharacteristic
166C8—1064no—
167C8—2122no—
168C22—22no—
169C22—24no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic5311yescharacteristic
177C22D4×Dic5312yes—
178C4—22no—
179C4—42no—
180C4—41no—
181C4#1313318511yescharacteristic
182C4#1313318912yes—
183C4#1313319811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—42no—
188C2#1681050311yescharacteristic
subgroupG/H
centerC4#13133185
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC2×C53#386
hypercenter#525D53
hypocenterC53#1891
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #16811390C2semidirect
2. #16810607C2semidirect
3. #16810587C2semidirect
4. #16810393C2semidirect
48. C2×#88322D4semidirect
100. C53#1891semidirect