Finite Group Databasebeta

← #20275286

Subgroups — #20275286

Order 6784.

230 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 97 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×#13133150C211yesmaximal, characteristic
2#16810521C211yesmaximal, characteristic
3#16810385C211yesmaximal, characteristic
4#16810385C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810378C211yesmaximal, characteristic
7C4×C4⋊C4×C53C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12C2×#11719919—21no—
13#13133150C414yescomplemented
14#13133124—21no—
15C2×C4⋊C4×C53C2211yescharacteristic
16C2×C4⋊C4×C53C2211yescharacteristic
17C2×C42×C53—21no—
18C2×C42×C53C2211yescharacteristic
19#95×C53—21no—
20C22×C4×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53D411yescharacteristic
23C22×C4×C53D411yescharacteristic
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53C2×C412yes—
30C4⋊C4×C53Q812yes—
31C42×C53—21no—
32C42×C53—21no—
33C42×C53—42no—
34C4⋊Dic53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53C2×C412yes—
38#11719919—42no—
39C2×#88322—21no—
40C2×#88322—21no—
41C2×#88322—21no—
42C2×#88322—21no—
43C2×#88322—21no—
44C2×#88322C2×C412yes—
45C23×C53C2×D411yescharacteristic
46C2×C4×C53—21no—
47C2×C4×C53—21no—
48C2×C4×C53—22no—
49C2×C4×C53—22no—
50C2×C4×C53—21no—
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×C53D812yes—
58C2×C4×C53C22×C411yescharacteristic
59C2×C4×C53C2×Q811yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×C4×C53#4111yescharacteristic
64C2×Dic53—22no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70#88322—22no—
71#88322—41no—
72C22×C53C4×D411yescharacteristic
73C22×C53C22⋊Q811yescharacteristic
74C22×C53#12311yescharacteristic
75C22×C53C2×D811yescharacteristic
76C22×C53#13511yescharacteristic
77C22×C53#13711yescharacteristic
78C22×C53#13711yescharacteristic
79C4×C53—22no—
80C4×C53—42no—
81C4×C53—42no—
82C4×C53—41no—
83C4×C53#11711yescharacteristic
84C4×C53C4×Q811yescharacteristic
85C4×C53#12511yescharacteristic
86C4×C53#12611yescharacteristic
87Dic53—42no—
88Dic53—81no—
89#1918—531nomaximal, Sylow
90C2×C53#38711yescharacteristic
91C2×C53C4×D811yescharacteristic
92C2×C53#44111yescharacteristic
93C2×C53#47411yescharacteristic
94C2×C53#47911yescharacteristic
95C2×C53#48011yescharacteristic
96C2×C53#48511yescharacteristic
97C2×#107—531no—
98#389—531no—
99C4×C4⋊C4D5311yescharacteristic
100#340—531no—
101#340—531no—
102#340—531no—
103#336—531no—
104C53#191811yescharacteristic, complemented, Sylow
105C22×C8—531no—
106C22×C8—531no—
107C2×C4⋊C4—531no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—1061no—
110C2×C4⋊C4C2×D5311yescharacteristic
111C2×C4⋊C4C2×D5311yescharacteristic
112C2×C42—21no—
113C2×C42C2×D5311yescharacteristic
114#107—534no—
115#95—21no—
116#95—1061no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—1061no—
120C22×C4—1061no—
121C22×C4C22⋊D5311yescharacteristic
122C22×C4C22⋊D5311yescharacteristic
123C22×C4C22×D5311yescharacteristic
124C2×C8—532no—
125C2×C8—1061no—
126C2×C8—1061no—
127C2×C8—1061no—
128C2×C8—1061no—
129C2×C8—1061no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—532no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C4⋊C4—2122no—
137C4⋊C4Dic10612yes—
138C4⋊C4C4×D5312yes—
139C42—21no—
140C42—21no—
141C42—42no—
142C23C2×C22⋊D5311yescharacteristic
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—1062no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4#1171992212yes—
161C2×C4C2×Dic10611yescharacteristic
162C2×C4C2×C4×D5311yescharacteristic
163C2×C4#1171994511yescharacteristic
164C2×C4#1171994511yescharacteristic
165C2×C4#1171994511yescharacteristic
166C2×C4#1171994511yescharacteristic
167C8—1062no—
168C8—2121no—
169C22C4×C22⋊D5311yescharacteristic
170C22#1313319611yescharacteristic
171C22#1313320011yescharacteristic
172C22C2×#1171992211yescharacteristic
173C22#1313323711yescharacteristic
174C22#1313324711yescharacteristic
175C22#1313324811yescharacteristic
176C4—22no—
177C4—42no—
178C4—42no—
179C4—41no—
180C4—4241no—
181C4—2122no—
182C4#1313317411yescharacteristic
183C4C4×Dic10611yescharacteristic
184C4#1313320911yescharacteristic
185C4#1313321111yescharacteristic
186C2#1681050711yescharacteristic
187C2C4×#1171992211yescharacteristic
188C2#1681072811yescharacteristic
189C2#1681090311yescharacteristic
190C2#1681093011yescharacteristic
191C2#1681093611yescharacteristic
192C2#1681096411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC4×C4⋊C4×C53C2
socleC23×C53C2×D4
hypercenterC4×C4⋊C4D53
hypocenterC53#1918
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
13. #13133150C4semidirect
104. C53#1918semidirect