Finite Group Databasebeta

← #20276378

Subgroups — #20276378

Order 6784.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811169C211yesmaximal, characteristic
2#16810597C211yesmaximal, characteristic, complemented
3#16810417C212yesmaximal, complemented
4#16810415C212yesmaximal, complemented
5C53×#410C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7#13133215C411yescharacteristic, complemented
8#13133213C2211yescharacteristic
9#13133206C411yescharacteristic, complemented
10#13133204C2211yescharacteristic
11#13133189—21no—
12#13133189—21no—
13C4×C22⋊D53—21no—
14C4×C22⋊D53—21no—
15#13133176—21no—
16C2×C4×Dic53C411yescharacteristic, complemented
17#13133132—21no—
18#13133132C2212yes—
19C2×#41×C53C2211yescharacteristic
20#117×C53—21no—
21#99×C53—21no—
22#99×C53C2212yes—
23#41×C53—41no—
24#41×C53—41no—
25C2×Q8×C53C2×C411yescharacteristic
26C2×D4×C53D411yescharacteristic
27C2×D4×C53D411yescharacteristic
28C2×D4×C53C2311yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53D411yescharacteristic
31C22×C4×C53D411yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C22×Dic53—21no—
36#11719945—41no—
37#11719945—41no—
38C2×D212C2×C411yescharacteristic
39C2×C4×D53—21no—
40C2×Dic106C2×C411yescharacteristic
41C4⋊C4×C53—41no—
42C22⋊C4×C53—22no—
43C42×C53—41no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C22⋊Dic53—22no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53—41no—
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53C4×Dic53—41no—
54C4×Dic53C2×C411yescharacteristic
55C4×Dic53C2×C411yescharacteristic
56C23×C53C22⋊C411yescharacteristic
57C23×C53C2×D411yescharacteristic
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60Q8×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63D4×C53—41no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—42no—
71C2×C4×C53C22⋊C411yescharacteristic
72C2×C4×C53C22⋊C411yescharacteristic
73C2×C4×C53C22⋊C411yescharacteristic
74C2×C4×C53C22×C411yescharacteristic
75C22⋊D53—42no—
76C22⋊D53—42no—
77C22⋊D53—42no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83D212—41no—
84C4×D53—41no—
85C4×D53—41no—
86Dic106—41no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53C2×C22⋊C411yescharacteristic
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—41no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—82no—
101C4×C53#9911yescharacteristic
102C4×C53#9911yescharacteristic
103Dic53—22no—
104Dic53—41no—
105Dic53—41no—
106Dic53—82no—
107#2126—531nomaximal, Sylow
108C2×C53—21no—
109C2×C53—41no—
110C2×C53—41no—
111C2×C53—41no—
112C2×C53C2×#9911yescharacteristic
113D53—81no—
114#532—531no—
115#410—531no—
116#410D5311yescharacteristic, complemented
117#354—532no—
118#353—532no—
119C53#212611yescharacteristic, complemented, Sylow
120C2×#41—531no—
121C2×#41C2×D5311yescharacteristic
122C4⋊Q8—531no—
123#127—531no—
124#124—531no—
125#124—531no—
126#121—1061no—
127#121—1061no—
128C4×D4—1061no—
129C4×D4—1061no—
130#117—21no—
131#117—1061no—
132C2×C42—531no—
133#99—21no—
134#99—532no—
135#99—1061no—
136#99C2×D5312yes—
137#41—41no—
138#41—41no—
139#41—2121no—
140#41—2121no—
141C2×Q8—531no—
142C2×Q8C4×D5311yescharacteristic
143C2×D4—531no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4D21211yescharacteristic
148C2×D4C22⋊D5311yescharacteristic
149C2×D4C22×D5311yescharacteristic
150C22×C4—1061no—
151C22×C4—1061no—
152C22×C4C4×D5311yescharacteristic
153C22×C4D21211yescharacteristic
154C22×C4C22⋊D5311yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—2121no—
159C22⋊C4—22no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1062no—
165C42—41no—
166C42—531no—
167C42—531no—
168C42—1061no—
169C42—2121no—
170C23—1061no—
171C23#1171992111yescharacteristic
172C23C2×D21211yescharacteristic
173C23C2×C22⋊D5311yescharacteristic
174Q8—41no—
175Q8—2121no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—2121no—
180D4—2122no—
181D4—2122no—
182D4—2122no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4#1171992111yescharacteristic
198C2×C4#1171992111yescharacteristic
199C2×C4#1171992111yescharacteristic
200C2×C4C2×C4×D5311yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—2121no—
208C22—2121no—
209C22—2121no—
210C22C2×#1171992111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—1062no—
216C4—4242no—
217C4—2121no—
218C4—2121no—
219C4#1313313111yescharacteristic
220C4#1313313111yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—4241no—
226C2C2×#1313313111yescharacteristic
subgroupG/H
centerC4#13133131
commutatorC2×C4×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#410C2
socleC2×C53C2×#99
hypercenter#410D53
hypocenterC53#2126
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16810597C2semidirect
3. #16810417C2semidirect
4. #16810415C2semidirect
5. C53×#410C2semidirect
6. C2×#11719945C4semidirect
7. #13133215C4semidirect
9. #13133206C4semidirect
16. C2×C4×Dic53C4semidirect
116. #410D53semidirect
119. C53#2126semidirect