Finite Group Databasebeta

← #20277428

Subgroups — #20277428

Order 6784.

612 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q8×Dic53C211yesmaximal, characteristic, complemented
2C2×D4×Dic53C211yesmaximal, characteristic
3#16810530C211yesmaximal, characteristic, complemented
4#16810526C211yesmaximal, characteristic
5#16810516C214yesmaximal, complemented
6#16810498C214yesmaximal, complemented
7#16810493C211yesmaximal, characteristic, complemented
8#16810469C211yesmaximal, characteristic, complemented
9C2×#124×C53C211yesmaximal, characteristic
10C23×Dic53C2212yes—
11Q8×Dic53—24no—
12D4×Dic53—24no—
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2214yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2214yes—
19C2×C4×Dic53C2212yes—
20C2×C4×Dic53C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2214yes—
24#13133124C2212yes—
25C22×Q8×C53C2211yescharacteristic
26C22×D4×C53C2211yescharacteristic
27#124×C53C418yescomplemented
28C2×C22⋊C4×C53C2214yes—
29C2×C42×C53C2211yescharacteristic
30C24×C53C2312yes—
31C2×Q8×C53—22no—
32C2×Q8×C53—22no—
33C2×Q8×C53C2×C414yescomplemented
34C2×D4×C53—22no—
35C2×D4×C53—22no—
36C2×D4×C53C2×C414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×Dic53—24no—
41C22×Dic53—44no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22⋊C4×C53C2×C4116yescomplemented
48C42×C53C2×C414yescomplemented
49C22⋊Dic53—24no—
50C22⋊Dic53—44no—
51C4⋊Dic53—22no—
52C4⋊Dic53—22no—
53C4⋊Dic53—24no—
54C4⋊Dic53—44no—
55C4×Dic53—22no—
56C4×Dic53—24no—
57C4×Dic53D414yes—
58C23×C53—22no—
59C23×C53—22no—
60C23×C53—24no—
61C23×C53—24no—
62C23×C53C22×C414yes—
63C23×C53C2411yescharacteristic
64Q8×C53—28no—
65D4×C53—28no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53C22×C414yes—
71C2×C4×C53C22×C414yes—
72C2×C4×C53C22×C412yes—
73C2×C4×C53C2×D412yes—
74C2×C4×C53#4112yes—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—24no—
81C2×Dic53—24no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—48no—
85C2×Dic53C2×D412yes—
86C2×Dic53C2×D412yes—
87C22×C53—24no—
88C22×C53—28no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—44no—
92C22×C53C23×C411yescharacteristic
93C22×C53C22×D411yescharacteristic
94C22×C53C2×#4111yescharacteristic
95C22×C53H(5,𝔽2)12yes—
96C22×C532−1+412yes—
97C4×C53—24no—
98C4×C53—24no—
99C4×C53C4×D414yes—
100Dic53—22no—
101Dic53—22no—
102Dic53—42no—
103Dic53—42no—
104Dic53—42no—
105#2326—531nomaximal, Sylow
106C2×C53—44no—
107C2×C53C2×C4×D411yescharacteristic
108C2×C53#52012yes—
109C2×C53#53511yescharacteristic
110C2×C53#53611yescharacteristic
111C2×C53#54212yes—
112C2×#124D5311yescharacteristic
113C2×C4×Q8—531no—
114C2×C4×D4—531no—
115#391—531no—
116#390—531no—
117#388—534no—
118#385—534no—
119#384—531no—
120#376—531no—
121C53#232611yescharacteristic, complemented, Sylow
122C22×Q8C2×D5311yescharacteristic
123C22×D4C2×D5311yescharacteristic
124C23×C4—532no—
125#124Dic5318yescomplemented
126C4×Q8—1064no—
127C4×D4—1064no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—534no—
132C2×C22⋊C4—532no—
133C2×C22⋊C4—534no—
134C2×C22⋊C4C2×D5314yes—
135C2×C42—531no—
136C2×C42—532no—
137C2×C42—531no—
138C2×C42C2×D5311yescharacteristic
139#95—532no—
140#95—532no—
141#95—534no—
142C24C22×D5312yes—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C2×Dic5314yescomplemented
146C2×D4—22no—
147C2×D4—22no—
148C2×D4C2×Dic5314yes—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—1064no—
155C22×C4—2124no—
156C22×C4C22×D5312yes—
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C4⋊C4—1064no—
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—2124no—
163C22⋊C4—1064no—
164C22⋊C4—2124no—
165C22⋊C4C2×Dic53116yescomplemented
166C42—534no—
167C42—1064no—
168C42—1062no—
169C42C2×Dic5314yescomplemented
170C23—22no—
171C23—22no—
172C23—24no—
173C23—24no—
174C23C22×Dic5314yes—
175C23C23×D5311yescharacteristic
176Q8—28no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—532no—
183C2×C4—532no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1064no—
191C2×C4—1064no—
192C2×C4—1062no—
193C2×C4—2128no—
194C2×C4D4×D5312yes—
195C2×C4#1171994712yes—
196C2×C4C22×Dic5314yes—
197C2×C4C22×Dic5314yes—
198C2×C4C22×Dic5312yes—
199C22—24no—
200C22—28no—
201C22—42no—
202C22—42no—
203C22—44no—
204C22C23×Dic5311yescharacteristic
205C22C2×D4×D5311yescharacteristic
206C22C2×#1171994711yescharacteristic
207C22#1313326312yes—
208C22#1313326412yes—
209C4—24no—
210C4—24no—
211C4—1062no—
212C4—1062no—
213C4—2122no—
214C4—2122no—
215C4—2122no—
216C4D4×Dic5314yes—
217C2—44no—
218C2C2×D4×Dic5311yescharacteristic
219C2#1681110612yes—
220C2#1681119111yescharacteristic
221C2#1681119311yescharacteristic
222C2#1681124212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC2×#124×C53C2
socleC23×C53C24
hypercenterC2×#124D53
hypocenterC53#2326
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic53C2semidirect
3. #16810530C2semidirect
5. #16810516C2semidirect
6. #16810498C2semidirect
7. #16810493C2semidirect
8. #16810469C2semidirect
27. #124×C53C4semidirect
33. C2×Q8×C53C2×C4semidirect
47. C22⋊C4×C53C2×C4semidirect
48. C42×C53C2×C4semidirect
121. C53#2326semidirect
125. #124Dic53semidirect
145. C2×Q8C2×Dic53semidirect
165. C22⋊C4C2×Dic53semidirect
169. C42C2×Dic53semidirect