Finite Group Databasebeta

← #20285847

Subgroups — #20285847

Order 6784.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811180C211yesmaximal, characteristic, complemented
2#16811026C211yesmaximal, characteristic, complemented
3#16811019C211yesmaximal, characteristic, complemented
4#16810969C211yesmaximal, characteristic, complemented
5#16810969C211yesmaximal, characteristic, complemented
6#16810944C214yesmaximal, complemented
7#16810923C214yesmaximal, complemented
8#16810676C211yesmaximal, characteristic, complemented
9C53×#532C211yesmaximal, characteristic
10#13133217C2211yescharacteristic
11#13133217C2211yescharacteristic
12#13133209C2212yes—
13#13133196—21no—
14#13133196—21no—
15C4×Dic106—21no—
16C4×Dic106—21no—
17#13133176C2212yes—
18#13133150C2214yes—
19#13133148C2214yes—
20C4⋊#88322C2212yes—
21#13133138C2214yes—
22#13133138C2214yes—
23C22⋊#88322C2212yes—
24C4×#88322C2211yescharacteristic
25C4×#88322C2211yescharacteristic
26C2×#41×C53—21no—
27C4⋊Q8×C53C2211yescharacteristic
28#127×C53C2211yescharacteristic
29#124×C53—21no—
30#121×C53—21no—
31#121×C53C2212yes—
32C4×D4×C53—21no—
33C4×D4×C53C2212yes—
34C2×C42×C53C2211yescharacteristic
35#41×C53—44no—
36C2×Q8×C53—21no—
37C2×D4×C53—22no—
38C2×D4×C53—21no—
39C2×D4×C53—21no—
40C2×D4×C53C2312yes—
41C22×C4×C53—22no—
42C22×C4×C53D411yescharacteristic
43C22×C4×C53D411yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×Dic106—21no—
46C2×Dic106—21no—
47C4⋊C4×C53—21no—
48C4⋊C4×C53C2312yes—
49C22⋊C4×C53—22no—
50C22⋊C4×C53—22no—
51C42×C53D411yescharacteristic
52C42×C53D411yescharacteristic
53C42×C53C2311yescharacteristic
54C42×C53C2311yescharacteristic
55C22⋊Dic53—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2312yes—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63C2×#88322C2312yes—
64C2×#88322C2312yes—
65C23×C53—22no—
66C23×C53C2×D411yescharacteristic
67Q8×C53—42no—
68D4×C53—24no—
69D4×C53—44no—
70D4×C53—42no—
71D4×C53—42no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—44no—
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2411yescharacteristic
83C2×Dic53—22no—
84C2×Dic53—22no—
85Dic106—42no—
86Dic106—42no—
87#88322—22no—
88#88322—22no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53C22×D411yescharacteristic
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—42no—
100C4×C53#13512yes—
101C4×C53AGL(1,ℤ/8)12yes—
102C4×C532−1+412yes—
103Dic53—42no—
104Dic53—42no—
105#3228—531nomaximal, Sylow
106C2×C53—41no—
107C2×C53—82no—
108C2×C53#53611yescharacteristic
109C2×C53C2×#13511yescharacteristic
110C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
111#533—531no—
112#532D5311yescharacteristic
113#500—531no—
114#498—531no—
115#486—531no—
116#486—531no—
117#482—534no—
118#478—534no—
119#431—531no—
120C53#322811yescharacteristic, complemented, Sylow
121C2×#41—21no—
122C4⋊Q8—531no—
123C4⋊Q8—531no—
124C4⋊Q8C2×D5311yescharacteristic
125#127C2×D5311yescharacteristic
126#125—532no—
127#124—21no—
128C22⋊Q8—1061no—
129C22⋊Q8—1061no—
130#121—21no—
131#121C2×D5312yes—
132C4×Q8—1061no—
133C4×Q8—1061no—
134C4×D4—21no—
135C4×D4C2×D5312yes—
136#117—532no—
137C2×C42C2×D5311yescharacteristic
138#107—534no—
139#106—534no—
140C4⋊C8—532no—
141#102—534no—
142#102—534no—
143C22⋊C8—532no—
144C4×C8—531no—
145C4×C8—531no—
146#41—44no—
147C2×Q8—21no—
148C2×Q8—1061no—
149C2×Q8—1061no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D5312yes—
154C22×C4—22no—
155C22×C4C22⋊D5311yescharacteristic
156C22×C4C22⋊D5311yescharacteristic
157C22×C4C22×D5311yescharacteristic
158C2×C8—532no—
159C2×C8—532no—
160C4⋊C4—21no—
161C4⋊C4—532no—
162C4⋊C4—532no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4C22×D5312yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C22⋊C4—1062no—
171C42—1062no—
172C42C22⋊D5311yescharacteristic
173C42C22⋊D5311yescharacteristic
174C42C22×D5311yescharacteristic
175C42C22×D5311yescharacteristic
176C23—22no—
177C23C2×C22⋊D5311yescharacteristic
178Q8—42no—
179Q8—2122no—
180Q8—2122no—
181D4—24no—
182D4—44no—
183D4—42no—
184D4—42no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4C2×C22⋊D5311yescharacteristic
193C2×C4C2×C22⋊D5311yescharacteristic
194C2×C4C2×C22⋊D5311yescharacteristic
195C2×C4C2×C22⋊D5311yescharacteristic
196C2×C4C2×C22⋊D5311yescharacteristic
197C2×C4C23×D5311yescharacteristic
198C8—1062no—
199C8—1062no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22C22×C22⋊D5311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—2122no—
212C4—2122no—
213C4#1313323712yes—
214C4#1313324312yes—
215C4#1313326412yes—
216C2—41no—
217C2—82no—
218C2#1681119411yescharacteristic
219C2C2×#1313323711yescharacteristic
220C2C2×#1313324311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#532C2
socleC22×C53C22×D4
hypercenter#532D53
hypocenterC53#3228
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #16811180C2semidirect
2. #16811026C2semidirect
3. #16811019C2semidirect
4. #16810969C2semidirect
5. #16810969C2semidirect
6. #16810944C2semidirect
7. #16810923C2semidirect
8. #16810676C2semidirect
120. C53#3228semidirect