Finite Group Databasebeta

← #20274847

Subgroups — #20274847

Order 6784.

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

normal subgroups only

This group has 224 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
1C2×#13133127C211yesmaximal, characteristic
2#16810492C211yesmaximal, characteristic
3#16810488C211yesmaximal, characteristic
4#16810386C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810385C211yesmaximal, characteristic
7C53×#340C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C2×C4⋊Dic53C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×#11719919—21no—
12C2×#11719919—21no—
13C2×C4×Dic53C2211yescharacteristic
14#13133127C414yescomplemented
15#13133124—21no—
16#13133124—21no—
17C22×C8×C53C2211yescharacteristic
18C2×C4⋊C4×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20C22×C4×C53—21no—
21C22×C4×C53—21no—
22C22×C4×C53C2311yescharacteristic
23C22×Dic53—21no—
24C22×Dic53—21no—
25C22×Dic53D411yescharacteristic
26C22×Dic53D411yescharacteristic
27C2×C8×C53—21no—
28C2×C8×C53—21no—
29C2×C8×C53C2×C412yes—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊Dic53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53—21no—
37C4⋊Dic53—21no—
38#11719919—42no—
39#11719919—42no—
40C4×Dic53C2×C412yes—
41C4×Dic53Q812yes—
42C2×#88322—21no—
43C2×#88322—21no—
44C2×#88322C2×C412yes—
45C23×C53C2×D411yescharacteristic
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53—41no—
49C2×C4×C53—41no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53C22×C411yescharacteristic
53C2×C4×C53C2×Q811yescharacteristic
54C2×C4×C53#4111yescharacteristic
55C2×C4×C53#4111yescharacteristic
56C2×C4×C53#4111yescharacteristic
57C2×C4×C53#4111yescharacteristic
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68C2×Dic53C4⋊C412yes—
69C2×Dic53C4⋊C412yes—
70C8×C53—22no—
71#88322—22no—
72C22×C53C2×C4⋊C411yescharacteristic
73C22×C53#12411yescharacteristic
74C22×C53#12411yescharacteristic
75C22×C53AGL(1,ℤ/8)11yescharacteristic
76C22×C53AGL(1,ℤ/8)11yescharacteristic
77C22×C53#13711yescharacteristic
78C22×C53#13711yescharacteristic
79C4×C53—81no—
80C4×C53—81no—
81C4×C53#11711yescharacteristic
82C4×C53#11711yescharacteristic
83C4×C53#12511yescharacteristic
84C4×C53#12511yescharacteristic
85Dic53—22no—
86Dic53—22no—
87Dic53—81no—
88Dic53—81no—
89#1836—531nomaximal, Sylow
90C2×C53#38211yescharacteristic
91C2×C53#42811yescharacteristic
92C2×C53#42811yescharacteristic
93C2×C53#48911yescharacteristic
94C2×C53#48911yescharacteristic
95C2×C53#49011yescharacteristic
96C2×C53#49111yescharacteristic
97C2×#97—531no—
98#384—531no—
99#382—531no—
100#340—531no—
101#340—531no—
102#340—531no—
103#340D5311yescharacteristic
104C53#183611yescharacteristic, complemented, Sylow
105C22×C8—531no—
106C22×C8C2×D5311yescharacteristic
107C2×C4⋊C4—531no—
108C2×C4⋊C4—531no—
109C2×C4⋊C4—1061no—
110C2×C4⋊C4—1061no—
111C2×C4⋊C4C2×D5311yescharacteristic
112C2×C4⋊C4C2×D5311yescharacteristic
113C2×C42—531no—
114#97—534no—
115#95—1061no—
116#95—1061no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—531no—
120C22×C4—531no—
121C22×C4—1061no—
122C22×C4—1061no—
123C22×C4C22×D5311yescharacteristic
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—532no—
127C2×C8—1061no—
128C2×C8—1061no—
129C2×C8C4×D5312yes—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C4⋊C4—1061no—
137C4⋊C4—1061no—
138C4⋊C4—2122no—
139C4⋊C4—2122no—
140C42—532no—
141C42—532no—
142C23D4×D5311yescharacteristic
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—532no—
150C2×C4—532no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—1061no—
154C2×C4—1061no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4—2121no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2121no—
161C2×C4C2×C4×D5311yescharacteristic
162C2×C4#1171994511yescharacteristic
163C2×C4#1171994511yescharacteristic
164C2×C4#1171994711yescharacteristic
165C2×C4#1171994711yescharacteristic
166C2×C4Q8×D5311yescharacteristic
167C8—22no—
168C8—1062no—
169C22C4⋊C4×D5311yescharacteristic
170C22#1313320311yescharacteristic
171C22#1313320311yescharacteristic
172C22#1313324111yescharacteristic
173C22#1313324411yescharacteristic
174C22#1313324611yescharacteristic
175C22#1313324911yescharacteristic
176C4—81no—
177C4—81no—
178C4—1062no—
179C4—1062no—
180C4—4241no—
181C4—4241no—
182C4#1313317311yescharacteristic
183C4#1313317411yescharacteristic
184C4#1313320711yescharacteristic
185C4#1313320711yescharacteristic
186C2#1681048711yescharacteristic
187C2#1681066211yescharacteristic
188C2#1681066411yescharacteristic
189C2#1681098111yescharacteristic
190C2#1681098411yescharacteristic
191C2#1681098611yescharacteristic
192C2#1681099011yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#1836
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
14. #13133127C4semidirect
104. C53#1836semidirect