Finite Group Databasebeta

← #20276217

Subgroups — #20276217

Order 6784.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#13133148C211yesmaximal, characteristic
2C2×C4⋊#88322C211yesmaximal, characteristic
3#16810521C211yesmaximal, characteristic
4#16810385C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810385C211yesmaximal, characteristic
7C53×#389C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133148—22no—
14C4⋊#88322—22no—
15#13133124—21no—
16C2×C4⋊C4×C53—21no—
17C2×C4⋊C4×C53C2211yescharacteristic
18C2×C4⋊C4×C53C2211yescharacteristic
19C2×C42×C53C2211yescharacteristic
20#95×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53—21no—
23C22×C4×C53D411yescharacteristic
24C22×C4×C53D411yescharacteristic
25C22×C4×C53C2311yescharacteristic
26C22×Dic53—21no—
27C22×Dic53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—42no—
32C4⋊C4×C53Q812yes—
33C42×C53—21no—
34C42×C53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53—21no—
38C4⋊Dic53Q812yes—
39#11719919—42no—
40C2×#88322—21no—
41C2×#88322—21no—
42C2×#88322—21no—
43C2×#88322—21no—
44C2×#88322—21no—
45C2×#88322D412yes—
46C23×C53C2×D411yescharacteristic
47C2×C4×C53—21no—
48C2×C4×C53—21no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53SD1612yes—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×Q811yescharacteristic
61C2×C4×C53C2×Q811yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×Dic53—22no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71#88322—22no—
72#88322—41no—
73C22×C53#12111yescharacteristic
74C22×C53C22⋊Q811yescharacteristic
75C22×C53C22⋊Q811yescharacteristic
76C22×C53C2×SD1611yescharacteristic
77C22×C53#13511yescharacteristic
78C22×C53AGL(1,ℤ/8)11yescharacteristic
79C22×C53#13711yescharacteristic
80C4×C53—22no—
81C4×C53—41no—
82C4×C53—42no—
83C4×C53—81no—
84C4×C53#12311yescharacteristic
85C4×C53#12511yescharacteristic
86C4×C53#12511yescharacteristic
87C4×C53C4⋊Q811yescharacteristic
88Dic53—42no—
89Dic53—81no—
90#2088—531nomaximal, Sylow
91C2×C53#39811yescharacteristic
92C2×C53#46511yescharacteristic
93C2×C53#46811yescharacteristic
94C2×C53#47511yescharacteristic
95C2×C53#47611yescharacteristic
96C2×C53#47811yescharacteristic
97C2×C53#47911yescharacteristic
98C2×#106—531no—
99C2×C4⋊C8—531no—
100#389—531no—
101#389D5311yescharacteristic
102#340—531no—
103#340—531no—
104#340—531no—
105C53#208811yescharacteristic, complemented, Sylow
106C22×C8—531no—
107C22×C8—531no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—531no—
110C2×C4⋊C4—531no—
111C2×C4⋊C4—1061no—
112C2×C4⋊C4C2×D5311yescharacteristic
113C2×C4⋊C4C2×D5311yescharacteristic
114C2×C42C2×D5311yescharacteristic
115#106—1062no—
116C4⋊C8—1062no—
117#95—21no—
118#95—1061no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—1061no—
122C22×C4—1061no—
123C22×C4C22⋊D5311yescharacteristic
124C22×C4C22⋊D5311yescharacteristic
125C22×C4C22×D5311yescharacteristic
126C2×C8—532no—
127C2×C8—1061no—
128C2×C8—1061no—
129C2×C8—1061no—
130C2×C8—1061no—
131C2×C8—1061no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—42no—
136C4⋊C4—532no—
137C4⋊C4—1061no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4—2122no—
141C4⋊C4Dic10612yes—
142C42—21no—
143C42—21no—
144C23C2×C22⋊D5311yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—1062no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4#1171992412yes—
163C2×C4C2×Dic10611yescharacteristic
164C2×C4#1171994511yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4#1171994711yescharacteristic
167C2×C4Q8×D5311yescharacteristic
168C2×C4#1171994911yescharacteristic
169C8—1062no—
170C8—2121no—
171C22#1313318911yescharacteristic
172C22#1313319311yescharacteristic
173C22#1313319611yescharacteristic
174C22C2×#1171992411yescharacteristic
175C22#1313323711yescharacteristic
176C22#1313324311yescharacteristic
177C22#1313324711yescharacteristic
178C4—22no—
179C4—41no—
180C4—42no—
181C4—81no—
182C4—4241no—
183C4—2122no—
184C4#1313320211yescharacteristic
185C4#1313320711yescharacteristic
186C4#1313320811yescharacteristic
187C4#1313321611yescharacteristic
188C2#1681055611yescharacteristic
189C2#1681085411yescharacteristic
190C2#1681086711yescharacteristic
191C2#1681090811yescharacteristic
192C2#1681091011yescharacteristic
193C2#1681092311yescharacteristic
194C2#1681092711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#389C2
socleC23×C53C2×D4
hypercenter#389D53
hypocenterC53#2088
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
105. C53#2088semidirect