Finite Group Databasebeta

← #20284659

Subgroups — #20284659

Order 6784.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133176C211yesmaximal, characteristic
2#16810961C214yesmaximal, complemented
3#16810956C212yesmaximal, complemented
4#16810951C212yesmaximal, complemented
5#16810663C212yesmaximal, complemented
6#16810632C212yesmaximal, complemented
7#16810587C211yesmaximal, characteristic
8C2×C22⋊Q8×C53C211yesmaximal, characteristic
9C2×#11719917C2212yes—
10#13133176—21no—
11#13133176—21no—
12#13133176C2212yes—
13#13133176C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C22⋊Dic53—21no—
17C2×C4×Dic53—21no—
18#13133150C2214yescomplemented
19#13133148C2214yescomplemented
20#13133141C2214yescomplemented
21#13133141C2214yescomplemented
22C22⋊#88322C2212yes—
23C22⋊#88322C2212yes—
24C22×Q8×C53C2211yescharacteristic
25C23×C4×C53C2211yescharacteristic
26C22⋊Q8×C53—21no—
27C22⋊Q8×C53—21no—
28C22⋊Q8×C53C2212yes—
29C22⋊Q8×C53C2212yes—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53—21no—
33C24×C53D411yescharacteristic
34C2×Q8×C53—21no—
35C2×Q8×C53—24no—
36C2×Q8×C53C2312yes—
37C22×C4×C53—24no—
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53D411yescharacteristic
41C22×C4×C53D411yescharacteristic
42C22×C4×C53D411yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×Dic53—21no—
47C22×Dic53—21no—
48C4⋊C4×C53—22no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53C2312yes—
52C22⋊C4×C53—22no—
53C22⋊C4×C53—22no—
54C22⋊Dic53—22no—
55C22⋊Dic53—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4⋊Dic53C2312yes—
59C4⋊Dic53C2312yes—
60C4×Dic53—22no—
61C4×Dic53—22no—
62#11719917—24no—
63C2×#88322C2314yes—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—41no—
67C23×C53C2×D411yescharacteristic
68C23×C53C2×D411yescharacteristic
69C23×C53C2×D411yescharacteristic
70Q8×C53—24no—
71Q8×C53—42no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53#4114yes—
85C2×C4×C53C2411yescharacteristic
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90#88322—24no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—44no—
99C22×C53#12312yes—
100C22×C53#13712yes—
101C22×C53C22×D411yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53#12312yes—
107C4×C53C2×#4112yes—
108Dic53—42no—
109Dic53—42no—
110#3095—531nomaximal, Sylow
111C2×C53—22no—
112C2×C53—22no—
113C2×C53—41no—
114C2×C53C2×#12311yescharacteristic
115C2×C53C2×#13711yescharacteristic
116C2×C53#57511yescharacteristic
117C2×C22⋊Q8D5311yescharacteristic
118C2×#117—531no—
119#485—534no—
120#484—532no—
121#483—532no—
122#428—532no—
123#419—532no—
124#407—531no—
125C53#309511yescharacteristic, complemented, Sylow
126C22×Q8C2×D5311yescharacteristic
127C23×C4C2×D5311yescharacteristic
128C2×M16—532no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8C2×D5312yes—
132C22⋊Q8C2×D5312yes—
133#117—532no—
134#117—532no—
135#117—1061no—
136#117—1061no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4C2×D5311yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—1061no—
143C2×C42—1061no—
144#107—534no—
145#106—534no—
146#103—534no—
147#103—534no—
148C22⋊C8—532no—
149C22⋊C8—532no—
150C24C22⋊D5311yescharacteristic
151C2×Q8—21no—
152C2×Q8—24no—
153C2×Q8C22×D5312yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—1061no—
158C22×C4—1061no—
159C22×C4C22⋊D5311yescharacteristic
160C22×C4C22⋊D5311yescharacteristic
161C22×C4C22⋊D5311yescharacteristic
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165M16—1064no—
166C2×C8—534no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—532no—
171C4⋊C4—532no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C4⋊C4C22×D5312yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C42—1062no—
180C42—1062no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×C22⋊D5311yescharacteristic
185C23C2×C22⋊D5311yescharacteristic
186C23C2×C22⋊D5311yescharacteristic
187Q8—24no—
188Q8—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4#1171994714yes—
203C2×C4C2×C22⋊D5311yescharacteristic
204C2×C4C2×C22⋊D5311yescharacteristic
205C2×C4C2×C22⋊D5311yescharacteristic
206C2×C4C23×D5311yescharacteristic
207C8—1064no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1313319812yes—
217C22#1313324812yes—
218C22C22×C22⋊D5311yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—2122no—
224C4—2122no—
225C4#1313319812yes—
226C4C2×#1171994712yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1313319811yescharacteristic
231C2C2×#1313324811yescharacteristic
232C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C22⋊Q8×C53C2
socleC22×C53C22×D4
hypercenterC2×C22⋊Q8D53
hypocenterC53#3095
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810961C2semidirect
3. #16810956C2semidirect
4. #16810951C2semidirect
5. #16810663C2semidirect
6. #16810632C2semidirect
18. #13133150C22semidirect
19. #13133148C22semidirect
20. #13133141C22semidirect
21. #13133141C22semidirect
125. C53#3095semidirect