Finite Group Databasebeta

← #20273965

Subgroups — #20273965

Order 6784.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811288C211yesmaximal, characteristic
2#16811026C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic
4#16810834C211yesmaximal, characteristic, complemented
5#16810353C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic
7C53×#332C211yesmaximal, characteristic, complemented
8C2×Q8×D53—21no—
9C2×Dic212—21no—
10#13133217C2211yescharacteristic
11#13133216—21no—
12#13133195—22no—
13#13133193—21no—
14C4×Dic106C2211yescharacteristic
15C4×D212C2211yescharacteristic
16#13133179—21no—
17#13133151—21no—
18#13133150—22no—
19#13133142—21no—
20#13133139—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26Q8×D53—42no—
27Q8×D53—42no—
28C2×D212D411yescharacteristic
29C2×C4×D53—22no—
30C2×C4×D53—21no—
31C2×Dic106—22no—
32C2×Dic106—21no—
33C2×Dic106D411yescharacteristic
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D412yes—
38C42×C53C2311yescharacteristic
39#11719921—22no—
40#11719921—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D411yescharacteristic
45#11719919—22no—
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49Dic212—42no—
50C22×D53—21no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—41no—
61D212—21no—
62D212Q1612yescomplemented
63C4×D53—44no—
64C4×D53—41no—
65C4×D53—41no—
66Dic106—21no—
67Dic106—44no—
68Dic106—41no—
69Dic106—41no—
70Dic106D812yes—
71C8×C53—41no—
72C8×C53—41no—
73#88322—22no—
74C22×C53#12011yescharacteristic
75C2×D53—22no—
76C2×D53—41no—
77C2×D53—41no—
78C4×C53—21no—
79C4×C53—42no—
80C4×C53C2×D811yescharacteristic
81C4×C53C2×Q1611yescharacteristic
82C4×C53AGL(1,ℤ/8)11yescharacteristic
83C4×C53#13711yescharacteristic
84Dic53—22no—
85Dic53—41no—
86Dic53—41no—
87Dic53—81no—
88#1649—531nomaximal, Sylow
89C2×C53#44711yescharacteristic
90C2×C53#45111yescharacteristic
91C2×C53#45611yescharacteristic
92D53—42no—
93D4×Q8—531no—
94#500—531no—
95#474—531no—
96#462—531no—
97#332D5311yescharacteristic, complemented
98Q8⋊C8—531no—
99D4⋊C8—531no—
100C53#164911yescharacteristic, complemented, Sylow
101C22×Q8—1061no—
102C2×Q16—1061no—
103C4⋊Q8—531no—
104C4⋊Q8—1061no—
105C4⋊Q8C2×D5311yescharacteristic
106C22⋊Q8—1061no—
107C22⋊Q8—1062no—
108C4×Q8—531no—
109C4×D4—531no—
110C4×D4—1061no—
111#107—1061no—
112#107—1062no—
113C4⋊C8C2×D5311yescharacteristic
114C4⋊C8C2×D5311yescharacteristic
115#103—1061no—
116#102—1061no—
117C4×C8—531no—
118C2×Q8—21no—
119C2×Q8—531no—
120C2×Q8—1062no—
121C2×Q8—1061no—
122C2×Q8—2122no—
123C2×Q8—2122no—
124C2×D4—531no—
125C22×C4—1062no—
126C22×C4—1061no—
127Q16—2122no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—1061no—
131C4⋊C4—21no—
132C4⋊C4—531no—
133C4⋊C4—531no—
134C4⋊C4—1062no—
135C4⋊C4—1061no—
136C4⋊C4—1061no—
137C4⋊C4—1061no—
138C4⋊C4D21212yes—
139C22⋊C4—1062no—
140C22⋊C4—1061no—
141C42—1061no—
142C42C22×D5311yescharacteristic
143C23—1061no—
144Q8—41no—
145Q8—41no—
146Q8—532no—
147Q8—1061no—
148Q8—2124no—
149Q8—2121no—
150Q8—2121no—
151D4—532no—
152D4—1061no—
153C2×C4—22no—
154C2×C4—1061no—
155C2×C4—1061no—
156C2×C4—1061no—
157C2×C4—2124no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2121no—
161C2×C4C2×D21211yescharacteristic
162C2×C4D4×D5311yescharacteristic
163C2×C4D4×D5311yescharacteristic
164C8—41no—
165C8—41no—
166C8—1062no—
167C22—1062no—
168C22—2121no—
169C22—2121no—
170C22#1313318711yescharacteristic
171C4—21no—
172C4—42no—
173C4—1062no—
174C4—4241no—
175C4—2121no—
176C4—2121no—
177C4D8×D5311yescharacteristic
178C4Q16×D5311yescharacteristic
179C4#1313324411yescharacteristic
180C4#1313324611yescharacteristic
181C2—2122no—
182C2#1681075811yescharacteristic
183C2#1681078011yescharacteristic
184C2#1681080811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1649
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #16811026C2semidirect
4. #16810834C2semidirect
5. #16810353C2semidirect
7. C53×#332C2semidirect
62. D212Q16semidirect
97. #332D53semidirect
100. C53#1649semidirect