Finite Group Databasebeta

← #20275099

Subgroups — #20275099

Order 6784.

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

normal subgroups only

This group has 332 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#16810640C212yesmaximal, complemented
3#16810597C211yesmaximal, characteristic, complemented
4#16810383C212yesmaximal, complemented
5C53×#532C211yesmaximal, characteristic
6#13133224—24no—
7C2×#11719917—22no—
8C2×#11719917C2212yes—
9#13133176C2212yes—
10#13133144—24no—
11#13133132—22no—
12C2×#41×C53C2211yescharacteristic
13C2×#41×C53C2211yescharacteristic
14C4⋊Q8×C53C411yescharacteristic, complemented
15#127×C53C411yescharacteristic, complemented
16#124×C53C411yescharacteristic, complemented
17#124×C53C411yescharacteristic, complemented
18#121×C53—22no—
19C4×D4×C53—22no—
20C2×C42×C53C2211yescharacteristic
21#41×C53—22no—
22#41×C53—22no—
23#41×C53—22no—
24#41×C53—22no—
25C2×Q8×C53C2×C411yescharacteristic
26C2×Q8×C53C2×C411yescharacteristic
27C2×D4×C53—22no—
28C2×D4×C53—21no—
29C2×D4×C53C2×C411yescharacteristic
30C2×D4×C53C2×C411yescharacteristic
31C2×D4×C53D412yes—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53D412yes—
35C22×C4×C53C2311yescharacteristic
36C4⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C42×C53—21no—
40C42×C53C2×C411yescharacteristic
41C42×C53C2×C411yescharacteristic
42C22⋊Dic53—22no—
43C4⋊Dic53—22no—
44C4×Dic53—22no—
45#11719917—24no—
46#11719917—22no—
47#11719917D414yescomplemented
48C2×#88322—24no—
49C2×#88322—22no—
50C23×C53—21no—
51C23×C53C22⋊C412yes—
52C23×C53C2×D411yescharacteristic
53Q8×C53—22no—
54Q8×C53—22no—
55D4×C53—24no—
56D4×C53—22no—
57D4×C53—22no—
58D4×C53—44no—
59D4×C53—42no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—42no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C22⋊C412yes—
69C2×C4×C53C22×C411yescharacteristic
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53#4112yes—
73C2×Dic53—42no—
74#88322—24no—
75C22×C53—22no—
76C22×C53—41no—
77C22×C53—42no—
78C22×C53—42no—
79C22×C53—41no—
80C22×C53C2×C22⋊C411yescharacteristic
81C22×C53#12112yes—
82C4×C53—22no—
83C4×C53—22no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53C4×D412yes—
87C4×C53#12011yescharacteristic
88C4×C53#12311yescharacteristic
89Dic53—82no—
90#1884—531nomaximal, Sylow
91C2×C53—22no—
92C2×C53—21no—
93C2×C53—42no—
94C2×C53—81no—
95C2×C53#38611yescharacteristic
96#568—531no—
97#532D5311yescharacteristic
98#421—532no—
99#410—531no—
100#339—532no—
101C53#188411yescharacteristic, complemented, Sylow
102C2×#41C2×D5311yescharacteristic
103C2×#41C2×D5311yescharacteristic
104#131—1064no—
105C2×M16—532no—
106C2×M16—1062no—
107C4⋊Q8Dic5311yescharacteristic, complemented
108#127Dic5311yescharacteristic, complemented
109#124Dic5311yescharacteristic, complemented
110#124Dic5311yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—532no—
114C2×C42C2×D5311yescharacteristic
115#104—1064no—
116#99—1062no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic5311yescharacteristic
122C2×Q8C2×Dic5311yescharacteristic
123C2×D4—21no—
124C2×D4—22no—
125C2×D4C2×Dic5311yescharacteristic
126C2×D4C2×Dic5311yescharacteristic
127C2×D4C22⋊D5312yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D5312yes—
131C22×C4C22×D5311yescharacteristic
132M16—534no—
133M16—1064no—
134M16—1062no—
135C2×C8—1064no—
136C2×C8—1062no—
137C4⋊C4—22no—
138C4⋊C4—1062no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—1062no—
142C42—21no—
143C42—1062no—
144C42C2×Dic5311yescharacteristic
145C42C2×Dic5311yescharacteristic
146C23—21no—
147C23C22⋊Dic5312yes—
148C23C2×C22⋊D5311yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—2122no—
165C2×C4C22⋊Dic5312yes—
166C2×C4D4×D5312yes—
167C2×C4#1171994712yes—
168C2×C4C22×Dic5311yescharacteristic
169C2×C4C2×C22⋊D5311yescharacteristic
170C8—1064no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic5311yescharacteristic
177C22#1313318912yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—4242no—
183C4D4×Dic5312yes—
184C4#1313318511yescharacteristic
185C4#1313319811yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#1681050311yescharacteristic
subgroupG/H
centerC4#13133185
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#532C2
socleC2×C53#386
hypercenter#532D53
hypocenterC53#1884
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #16811390C2semidirect
2. #16810640C2semidirect
3. #16810597C2semidirect
4. #16810383C2semidirect
14. C4⋊Q8×C53C4semidirect
15. #127×C53C4semidirect
16. #124×C53C4semidirect
17. #124×C53C4semidirect
47. #11719917D4semidirect
101. C53#1884semidirect
107. C4⋊Q8Dic53semidirect
108. #127Dic53semidirect
109. #124Dic53semidirect
110. #124Dic53semidirect