Finite Group Databasebeta

← #20273258

Subgroups — #20273258

Order 6784.

286 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 81 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#16810587C211yesmaximal, characteristic, complemented
2#16810483C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C2×#11719917—21no—
7C2×C4⋊Dic53—21no—
8C2×C22⋊Dic53—21no—
9C22⋊#88322—21no—
10C22⋊#88322C2212yes—
11#13133124C2212yes—
12C2×#41×C53C411yescharacteristic, complemented
13C23×C4×C53C2211yescharacteristic
14#123×C53—21no—
15C22⋊Q8×C53C411yescharacteristic, complemented
16C22⋊Q8×C53C2211yescharacteristic
17#121×C53C411yescharacteristic, complemented
18#121×C53C2211yescharacteristic
19#120×C53—21no—
20C4×D4×C53—21no—
21C4×D4×C53—21no—
22#117×C53C411yescharacteristic, complemented
23C24×C53D411yescharacteristic
24#41×C53—42no—
25C2×Q8×C53C2×C411yescharacteristic
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53C2×C411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53—42no—
32C22×C4×C53C2×C411yescharacteristic
33C22×C4×C53C2×C411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53—22no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53C2×C411yescharacteristic
42C4⋊C4×C53C2×C411yescharacteristic
43C22⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C42×C53—21no—
49C22⋊Dic53—42no—
50C4⋊Dic53—42no—
51#11719917—42no—
52C2×#88322—22no—
53C23×C53—21no—
54C23×C53—21no—
55C23×C53—21no—
56C23×C53—41no—
57C23×C53—41no—
58C23×C53C22⋊C411yescharacteristic
59C23×C53C22⋊C411yescharacteristic
60C23×C53C2×D411yescharacteristic
61Q8×C53—41no—
62D4×C53—41no—
63D4×C53—42no—
64D4×C53—42no—
65D4×C53—41no—
66D4×C53—41no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—42no—
77C2×C4×C53C22⋊C411yescharacteristic
78C2×C4×C53C22⋊C411yescharacteristic
79C2×C4×C53C22×C411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84#88322—42no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53C2×C22⋊C411yescharacteristic
99C22×C53AGL(1,ℤ/8)11yescharacteristic
100C22×C53#13711yescharacteristic
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53—41no—
106C4×C53—41no—
107C4×C53#9912yes—
108Dic53—82no—
109#1513—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53—81no—
115C2×C53C2×#9911yescharacteristic
116C2×C53#41711yescharacteristic
117C2×C53#42111yescharacteristic
118#525D5311yescharacteristic
119#407—531no—
120#380—531no—
121#328—532no—
122#327—532no—
123C53#151311yescharacteristic, complemented, Sylow
124C2×#41Dic5311yescharacteristic, complemented
125C23×C4C2×D5311yescharacteristic
126C2×M16—1061no—
127#123—21no—
128C22⋊Q8Dic5311yescharacteristic, complemented
129C22⋊Q8C2×D5311yescharacteristic
130#121Dic5311yescharacteristic, complemented
131#121C2×D5311yescharacteristic
132#120—21no—
133C4×D4—21no—
134C4×D4—21no—
135#117Dic5311yescharacteristic, complemented
136C2×C4⋊C4—1061no—
137C2×C22⋊C4—1061no—
138C22⋊C8—532no—
139C22⋊C8—1061no—
140#95—532no—
141C24C22⋊D5311yescharacteristic
142#41—42no—
143C2×Q8C2×Dic5311yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×Dic5311yescharacteristic
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—1062no—
151C22×C4C2×Dic5311yescharacteristic
152C22×C4C2×Dic5311yescharacteristic
153C22×C4C22⋊D5311yescharacteristic
154C22×C4C22⋊D5311yescharacteristic
155C22×C4C22⋊D5311yescharacteristic
156C22×C4C22×D5311yescharacteristic
157M16—2122no—
158C2×C8—1062no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2122no—
162C4⋊C4C2×Dic5311yescharacteristic
163C4⋊C4C2×Dic5311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—2122no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23C22⋊Dic5311yescharacteristic
177C23C22⋊Dic5311yescharacteristic
178C23C2×C22⋊D5311yescharacteristic
179Q8—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—41no—
184D4—41no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4C22⋊Dic5311yescharacteristic
199C2×C4C22⋊Dic5311yescharacteristic
200C2×C4C22×Dic5311yescharacteristic
201C2×C4C2×C22⋊D5311yescharacteristic
202C8—2122no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22C2×C22⋊Dic5311yescharacteristic
217C22#1313324211yescharacteristic
218C22#1313324711yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—4242no—
226C4#1313313212yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—81no—
232C2C2×#1313313211yescharacteristic
233C2#1681062411yescharacteristic
234C2#1681064011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1513
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #16810587C2semidirect
2. #16810483C2semidirect
3. #16810360C2semidirect
4. #16810357C2semidirect
12. C2×#41×C53C4semidirect
15. C22⋊Q8×C53C4semidirect
17. #121×C53C4semidirect
22. #117×C53C4semidirect
123. C53#1513semidirect
124. C2×#41Dic53semidirect
128. C22⋊Q8Dic53semidirect
130. #121Dic53semidirect
135. #117Dic53semidirect