Finite Group Databasebeta

← #20287605

Subgroups — #20287605

Order 6784.

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

normal subgroups only

This group has 324 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#16811352C211yesmaximal, characteristic
2#16811348C211yesmaximal, characteristic
3#16811026C211yesmaximal, characteristic
4#16811023C212yesmaximal
5#16810933C212yesmaximal
6#16810928C212yesmaximal
7#16810909C211yesmaximal, characteristic
8#16810747C211yesmaximal, characteristic
9C4×#11719925C211yesmaximal, characteristic
10#16810715C212yesmaximal
11C53×#475C211yesmaximal, characteristic
12C2×#11719925C2211yescharacteristic
13#13133217—21no—
14#13133217C2211yescharacteristic
15#13133215—21no—
16#13133208—22no—
17#13133208C2212yes—
18#13133208C2212yes—
19#13133207—22no—
20Q8×Dic53—21no—
21Q8×Dic53C2212yes—
22C4×Dic106—21no—
23C4×Dic106C2211yescharacteristic
24C4×Dic106C2211yescharacteristic
25#13133182C2212yes—
26#13133150C2211yescharacteristic
27#13133150C2212yes—
28#13133150C2212yes—
29#13133149C2212yes—
30#13133146C2212yes—
31#13133142C2212yes—
32#13133141C2211yescharacteristic
33#13133140C2211yescharacteristic
34C4×#88322C2211yescharacteristic
35C8×Dic53C2212yes—
36C4⋊Q8×C53C2211yescharacteristic
37C4×Q8×C53C2211yescharacteristic
38#106×C53C2212yes—
39C4⋊C8×C53C2211yescharacteristic
40#103×C53C2212yes—
41C2×Q8×C53—21no—
42C2×Q8×C53C2311yescharacteristic
43C2×Dic106—21no—
44C2×Dic106D411yescharacteristic
45C2×Dic106C2311yescharacteristic
46C2×C8×C53C2312yes—
47C4⋊C4×C53—21no—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C4⋊C4×C53C2311yescharacteristic
51C42×C53—21no—
52C42×C53C2311yescharacteristic
53#11719925Q814yescomplemented
54C4⋊Dic53—21no—
55C4⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53D411yescharacteristic
59C4⋊Dic53C2311yescharacteristic
60C4⋊Dic53C2312yes—
61#11719919—21no—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65#11719919D412yes—
66C4×Dic53—21no—
67C4×Dic53—22no—
68C4×Dic53—22no—
69C4×Dic53C2312yes—
70C2×#88322C2311yescharacteristic
71C2×#88322C2311yescharacteristic
72Q8×C53—21no—
73Q8×C53—42no—
74Q8×C53C2×Q812yes—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53—21no—
84C2×Dic53C2×D412yes—
85C2×Dic53C2×D412yes—
86Dic106—22no—
87Dic106—21no—
88Dic106—42no—
89Dic106C2×Q812yes—
90C8×C53—22no—
91#88322—21no—
92#88322C2×Q812yes—
93C22×C53C22×D411yescharacteristic
94C4×C53—22no—
95C4×C53—21no—
96C4×C53—42no—
97C4×C53—41no—
98C4×C53#13512yes—
99C4×C53C22×Q811yescharacteristic
100C4×C532−1+411yescharacteristic
101Dic53—22no—
102Dic53—22no—
103Dic53—22no—
104Dic53—42no—
105Dic53—41no—
106#3383—531nomaximal, Sylow
107C2×C53D4×Q811yescharacteristic
108C2×C53C2×#13511yescharacteristic
109C2×C53#57811yescharacteristic
110#557—531no—
111#557—531no—
112#500—531no—
113#499—532no—
114#479—532no—
115#479—532no—
116#475—531no—
117#475D5311yescharacteristic
118C8×Q8—531no—
119C4×Q16—531no—
120C4×Q16—532no—
121C53#338311yescharacteristic, complemented, Sylow
122C2×Q16—531no—
123C4⋊Q8—531no—
124C4⋊Q8—1061no—
125C4⋊Q8—1061no—
126C4⋊Q8C2×D5311yescharacteristic
127#125—532no—
128#125—532no—
129#125—1062no—
130#125—1062no—
131C4×Q8—531no—
132C4×Q8—532no—
133C4×Q8—532no—
134C4×Q8—531no—
135C4×Q8—1061no—
136C4×Q8—1061no—
137C4×Q8C2×D5311yescharacteristic
138#107—531no—
139#107—532no—
140#107—532no—
141#106—532no—
142#106C2×D5312yes—
143C4⋊C8—532no—
144C4⋊C8C2×D5311yescharacteristic
145#103—531no—
146#103—532no—
147#103—531no—
148#103C2×D5312yes—
149C4×C8—532no—
150C4×C8—531no—
151C2×Q8—21no—
152C2×Q8—531no—
153C2×Q8—531no—
154C2×Q8—1061no—
155C2×Q8C22×D5311yescharacteristic
156Q16—534no—
157C2×C8—531no—
158C2×C8—531no—
159C2×C8C22×D5312yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—531no—
163C4⋊C4—532no—
164C4⋊C4—531no—
165C4⋊C4—532no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C4⋊C4—1062no—
171C4⋊C4—1062no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4C22×D5311yescharacteristic
175C4⋊C4C22×D5312yes—
176C42—21no—
177C42—532no—
178C42—1062no—
179C42—1062no—
180C42—1061no—
181C42C22×D5311yescharacteristic
182Q8—21no—
183Q8—42no—
184Q8—532no—
185Q8—1062no—
186Q8—1061no—
187Q8—2122no—
188Q8Q8×D5312yes—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—532no—
193C2×C4—532no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1062no—
197C2×C4D4×D5311yescharacteristic
198C2×C4D4×D5311yescharacteristic
199C2×C4C23×D5311yescharacteristic
200C8—22no—
201C8—532no—
202C8—1061no—
203C22C2×D4×D5311yescharacteristic
204C4—22no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—1062no—
209C4—1062no—
210C4—1062no—
211C4—2122no—
212C4—2121no—
213C4#1313323812yes—
214C4C2×Q8×D5311yescharacteristic
215C4#1313326411yescharacteristic
216C2#1681128611yescharacteristic
217C2C2×#1313323811yescharacteristic
218C2#1681143911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3383
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
53. #11719925Q8semidirect
121. C53#3383semidirect