Finite Group Databasebeta

← #20285938

Subgroups — #20285938

Order 6784.

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

normal subgroups only

This group has 362 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
1C2×#13133217C211yesmaximal, characteristic, complemented
2#16811157C211yesmaximal, characteristic
3#16811032C212yesmaximal, complemented
4#16810983C212yesmaximal, complemented
5#16810954C212yesmaximal, complemented
6#16810937C212yesmaximal
7#16810919C212yesmaximal, complemented
8#16810914C212yesmaximal, complemented
9C53×#432C211yesmaximal, characteristic, complemented
10C22×Dic106—21no—
11#13133217—21no—
12#13133217—21no—
13#13133217C2212yes—
14#13133217C2212yes—
15#13133211—21no—
16#13133209C2211yescharacteristic
17#13133205C2211yescharacteristic
18#13133200—21no—
19#13133196C2211yescharacteristic
20#13133191C2211yescharacteristic
21C4×Dic106C2211yescharacteristic
22C4×D212C2211yescharacteristic
23C4×C22⋊D53—21no—
24#13133174—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2212yes—
27#13133151C2214yes—
28#13133149C2214yes—
29#13133142C2214yes—
30#13133139C2214yes—
31C2×C42×C53C2211yescharacteristic
32C4⋊C8×C53C2212yes—
33C22⋊C8×C53C2212yes—
34#97×C53C2212yes—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—21no—
39C22×Dic53—22no—
40C2×D212C2311yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—22no—
43C2×Dic106—22no—
44C2×Dic106—44no—
45C2×Dic106C2311yescharacteristic
46C2×C8×C53C2314yes—
47C42×C53D411yescharacteristic
48C42×C53D411yescharacteristic
49C42×C53C2311yescharacteristic
50C42×C53C2311yescharacteristic
51C22⋊Dic53—21no—
52#11719921—21no—
53#11719921—21no—
54#11719921—21no—
55#11719921—21no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53C2314yes—
60C4⋊Dic53C2311yescharacteristic
61C4⋊Dic53C2311yescharacteristic
62#11719919—21no—
63#11719919—21no—
64#11719919—21no—
65#11719919—21no—
66C4×Dic53—21no—
67C23×C53C2×D411yescharacteristic
68C22×D53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—42no—
79C2×Dic53—24no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—44no—
84D212—22no—
85C4×D53—42no—
86Dic106—22no—
87Dic106—44no—
88Dic106—44no—
89C8×C53—24no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53AGL(1,ℤ/8)12yes—
93C22×C53C22×D411yescharacteristic
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—41no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53#13712yes—
102C4×C532−1+412yes—
103Dic53—41no—
104Dic53—44no—
105Dic53—41no—
106Dic53—41no—
107#3238—531nomaximal, Sylow
108C2×C53—22no—
109C2×C53#53611yescharacteristic
110C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C53C2×#13711yescharacteristic
112D53—81no—
113C2×C4⋊Q8—531no—
114#529—531no—
115#501—532no—
116#489—532no—
117#483—532no—
118#480—532no—
119#477—532no—
120#476—532no—
121#432D5311yescharacteristic, complemented
122C53#323811yescharacteristic, complemented, Sylow
123C22×Q8—1061no—
124C4⋊Q8—532no—
125C4⋊Q8—532no—
126C4⋊Q8—1061no—
127C4⋊Q8—1061no—
128#126—1061no—
129#125—531no—
130#124—531no—
131#123—1061no—
132C22⋊Q8—531no—
133#121—531no—
134C4×Q8—531no—
135C4×D4—531no—
136C4×D4—1061no—
137#117—1061no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1061no—
140C2×C42C2×D5311yescharacteristic
141#107—534no—
142#106—534no—
143C4⋊C8C2×D5312yes—
144#103—534no—
145#102—534no—
146C22⋊C8C2×D5312yes—
147#97C2×D5312yes—
148C2×Q8—531no—
149C2×Q8—1062no—
150C2×Q8—1062no—
151C2×Q8—2124no—
152C2×D4—531no—
153C2×D4—1061no—
154C22×C4—1062no—
155C22×C4—1061no—
156C22×C4D21211yescharacteristic
157C22×C4D21211yescharacteristic
158C22×C4C22×D5311yescharacteristic
159C2×C8C22×D5314yes—
160C4⋊C4—534no—
161C4⋊C4—531no—
162C4⋊C4—531no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C4⋊C4—1061no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C22⋊C4—1061no—
174C22⋊C4—1061no—
175C42—1061no—
176C42D21211yescharacteristic
177C42D21211yescharacteristic
178C42C22×D5311yescharacteristic
179C42C22×D5311yescharacteristic
180C23—1061no—
181C23C2×D21211yescharacteristic
182Q8—1062no—
183Q8—2124no—
184Q8—2124no—
185D4—1062no—
186D4—2122no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1064no—
194C2×C4—2124no—
195C2×C4—2122no—
196C2×C4C2×D21211yescharacteristic
197C2×C4C2×D21211yescharacteristic
198C2×C4C2×D21211yescharacteristic
199C2×C4C2×D21211yescharacteristic
200C2×C4C2×D21211yescharacteristic
201C2×C4C23×D5311yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—2121no—
206C22—2121no—
207C22—2121no—
208C22#1313324512yes—
209C22C22×D21211yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—2124no—
215C4—2121no—
216C4—2121no—
217C4—2121no—
218C4#1313325012yes—
219C4#1313326412yes—
220C2—22no—
221C2—4241no—
222C2#1681119911yescharacteristic
223C2C2×#1313324511yescharacteristic
224C2C2×#1313325011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3238
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. C2×#13133217C2semidirect
3. #16811032C2semidirect
4. #16810983C2semidirect
5. #16810954C2semidirect
7. #16810919C2semidirect
8. #16810914C2semidirect
9. C53×#432C2semidirect
121. #432D53semidirect
122. C53#3238semidirect