Finite Group Databasebeta

← #20274998

Subgroups — #20274998

Order 6784.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#16811098C211yesmaximal, characteristic
2#16810503C211yesmaximal, characteristic
3#16810498C212yesmaximal, complemented
4#16810389C212yesmaximal, complemented
5C2×#99×C53C211yesmaximal, characteristic
6C23×Dic53C2211yescharacteristic
7D4×Dic53—22no—
8D4×Dic53—22no—
9#13133176—21no—
10C2×C4⋊Dic53—22no—
11C2×C22⋊Dic53—21no—
12C2×C22⋊Dic53—21no—
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2212yes—
15#13133124—21no—
16#13133124—22no—
17C22×D4×C53C2211yescharacteristic
18C2×C22⋊C4×C53C2212yes—
19#99×C53C414yescomplemented
20C24×C53D411yescharacteristic
21C24×C53C2311yescharacteristic
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53C2×C412yes—
25C2×D4×C53Q812yes—
26C22×C4×C53—22no—
27C22×C4×C53D411yescharacteristic
28C22×Dic53—21no—
29C22×Dic53—22no—
30C22×Dic53—21no—
31C22×Dic53—21no—
32C22×Dic53—21no—
33C22×Dic53—42no—
34C22×Dic53D412yes—
35C22×Dic53D412yes—
36C22⋊C4×C53—22no—
37C22⋊C4×C53C2×C414yescomplemented
38C22⋊Dic53—22no—
39C22⋊Dic53—22no—
40C22⋊Dic53—22no—
41C22⋊Dic53—22no—
42C22⋊Dic53—42no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—44no—
46C4×Dic53—22no—
47C23×C53—21no—
48C23×C53—21no—
49C23×C53—41no—
50C23×C53—41no—
51C23×C53—41no—
52C23×C53—41no—
53C23×C53C4⋊C412yescomplemented
54C23×C53C22×C411yescharacteristic
55C23×C53C2×D411yescharacteristic
56C23×C53C2×D412yes—
57C23×C53C2×Q811yescharacteristic
58C23×C53#4112yes—
59D4×C53—24no—
60D4×C53—24no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—24no—
64C2×C4×C53—42no—
65C2×C4×C53—42no—
66C2×C4×C53C4⋊C412yescomplemented
67C2×Dic53—24no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—22no—
71C2×Dic53—42no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—42no—
75C2×Dic53—42no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—81no—
80C2×Dic53—81no—
81C22×C53—22no—
82C22×C53—21no—
83C22×C53—22no—
84C22×C53—21no—
85C22×C53—22no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—42no—
89C22×C53—42no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—81no—
93C22×C53C2×C4⋊C411yescharacteristic
94C22×C53C4×D412yes—
95C22×C53#12011yescharacteristic
96C22×C53C22⋊Q812yes—
97C22×C53#12311yescharacteristic
98C4×C53—22no—
99C4×C53—44no—
100Dic53—42no—
101Dic53—42no—
102Dic53—42no—
103Dic53—81no—
104#1867—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—22no—
107C2×C53—21no—
108C2×C53—22no—
109C2×C53—42no—
110C2×C53#38511yescharacteristic
111C2×C53B(4,𝔽2)11yescharacteristic
112C2×C53#45811yescharacteristic
113#518—531no—
114C2×#99D5311yescharacteristic
115#386—531no—
116#385—532no—
117#342—532no—
118C53#186711yescharacteristic, complemented, Sylow
119C22×D4C2×D5311yescharacteristic
120C23×C4—531no—
121C4×D4—1062no—
122C4×D4—1062no—
123#117—1061no—
124C2×C4⋊C4—1062no—
125C2×C22⋊C4—532no—
126C2×C22⋊C4—531no—
127C2×C22⋊C4—1061no—
128C2×C22⋊C4—1061no—
129C2×C22⋊C4C2×D5312yes—
130#99Dic5314yescomplemented
131#95—1062no—
132#95—1061no—
133C24D21211yescharacteristic
134C24C22×D5311yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4Dic10612yes—
138C2×D4C2×Dic5312yes—
139C22×C4—22no—
140C22×C4—532no—
141C22×C4—532no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4—1062no—
146C22×C4—1061no—
147C22×C4—2122no—
148C22×C4D21211yescharacteristic
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—2124no—
152C22⋊C4—22no—
153C22⋊C4—1062no—
154C22⋊C4—1062no—
155C22⋊C4—1062no—
156C22⋊C4—1062no—
157C22⋊C4—2122no—
158C22⋊C4C2×Dic5314yescomplemented
159C42—1062no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23C4⋊Dic5312yescomplemented
167C23C2×Dic10611yescharacteristic
168C23C2×D21211yescharacteristic
169C23D4×D5312yes—
170C23#1171994712yes—
171C23C22×Dic5311yescharacteristic
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—1062no—
180C2×C4—1064no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—4241no—
184C2×C4—4241no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2122no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4C4⋊Dic5312yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×C4⋊Dic5311yescharacteristic
207C22D4×Dic5312yes—
208C22#1313318711yescharacteristic
209C22#1313319412yes—
210C22#1313320111yescharacteristic
211C4—22no—
212C4—44no—
213C4—4241no—
214C4—2122no—
215C4—2122no—
216C4—2122no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#1681049811yescharacteristic
223C2#1681081111yescharacteristic
224C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC22×C53#120
hypercenterC2×#99D53
hypocenterC53#1867
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #16810498C2semidirect
4. #16810389C2semidirect
19. #99×C53C4semidirect
37. C22⋊C4×C53C2×C4semidirect
53. C23×C53C4⋊C4semidirect
66. C2×C4×C53C4⋊C4semidirect
118. C53#1867semidirect
130. #99Dic53semidirect
158. C22⋊C4C2×Dic53semidirect
166. C23C4⋊Dic53semidirect
193. C2×C4C4⋊Dic53semidirect