Finite Group Databasebeta

← #20290227

Subgroups — #20290227

Order 6784.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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#16811626C211yesmaximal, characteristic, complemented
2C2×#13133284C211yesmaximal, characteristic, complemented
3#16811482C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811168C211yesmaximal, characteristic
6C2×#11719940—21no—
7C22×#88324—21no—
8#13133284—21no—
9#13133284C2212yes—
10#13133284C2212yes—
11C4⋊#88324—21no—
12C2×#11719947—21no—
13#13133216C2211yescharacteristic
14#13133213C2211yescharacteristic
15#13133203—21no—
16#13133191C411yescharacteristic
17#13133189—21no—
18D4×Dic53C411yescharacteristic
19#13133179C411yescharacteristic
20#13133177—21no—
21C2×C4×Dic53C2211yescharacteristic
22#121×C53C411yescharacteristic
23C2×D4×C53—21no—
24C2×D4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29C22×Dic53—21no—
30C22×Dic53D411yescharacteristic
31C22×Dic53D411yescharacteristic
32#11719947—42no—
33#11719947—42no—
34C2×D212C2×C411yescharacteristic
35C2×C4×D53—21no—
36C2×Dic106—21no—
37#11719940—42no—
38C2×#88324—22no—
39C2×#88324—22no—
40C2×#88324—42no—
41C4⋊C4×C53C2×C411yescharacteristic
42C22⋊C4×C53—21no—
43C22⋊Dic53—21no—
44#11719921—21no—
45#11719921—21no—
46C4⋊Dic53C2×C411yescharacteristic
47#11719919—21no—
48C4×Dic53C2×C411yescharacteristic
49C4×Dic53D411yescharacteristic
50C4×Dic53D411yescharacteristic
51C4×Dic53C2311yescharacteristic
52C23×C53—21no—
53C23×C53C22⋊C411yescharacteristic
54C22×D53—21no—
55D4×C53—41no—
56D4×C53—42no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53C22⋊C411yescharacteristic
60C2×C4×C53C22×C411yescharacteristic
61C22⋊D53—42no—
62C22⋊D53—41no—
63C22⋊D53—41no—
64C22⋊D53—41no—
65C22⋊D53—41no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—42no—
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53D812yes—
73C2×Dic53SD1612yes—
74C2×Dic53C2×D411yescharacteristic
75C2×Dic53C2×D411yescharacteristic
76D212—41no—
77C4×D53—42no—
78Dic106—41no—
79Dic106—41no—
80#88324—42no—
81#88324—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53#10212yes—
88C22×C53C2×C22⋊C411yescharacteristic
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—41no—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—41no—
95Dic53—21no—
96Dic53—41no—
97Dic53#10212yes—
98Dic53C2×D811yescharacteristic
99Dic53C2×SD1611yescharacteristic
100Dic53AGL(1,ℤ/8)11yescharacteristic
101Dic53#13711yescharacteristic
102#1547—531nomaximal, Sylow
103C2×C53—22no—
104C2×C53—81no—
105C2×C53#41311yescharacteristic
106C2×C53C2×#10211yescharacteristic
107C2×C53#41711yescharacteristic
108D53—81no—
109#532—531no—
110#423—531no—
111C2×C4⋊C8—531no—
112#332—532no—
113#331—532no—
114C53#154711yescharacteristic, complemented, Sylow
115C2×#41—1061no—
116C2×M16—1061no—
117C22×C8—1061no—
118C4⋊Q8—531no—
119#127—531no—
120#124—1061no—
121#121—531no—
122#121—1061no—
123#121#615711yescharacteristic
124C4×D4—531no—
125C4×D4—531no—
126C4×D4—1061no—
127C2×C42—531no—
128C4⋊C8—532no—
129C4⋊C8—532no—
130C4⋊C8—1061no—
131C4⋊C8—1061no—
132#41—2122no—
133#41—2122no—
134C2×Q8—1061no—
135C2×D4—21no—
136C2×D4—531no—
137C2×D4—1061no—
138C2×D4—1061no—
139C2×D4—1061no—
140C2×D4C2×#615711yescharacteristic
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4C2×#615711yescharacteristic
146M16—2122no—
147C2×C8—1062no—
148C2×C8—1062no—
149C2×C8—2122no—
150C4⋊C4—531no—
151C4⋊C4—1061no—
152C4⋊C4C2×#615711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C42—531no—
158C42—531no—
159C42—531no—
160C42—531no—
161C23—21no—
162C23—1061no—
163C23C22⋊#615711yescharacteristic
164Q8—2121no—
165Q8—2121no—
166D4—41no—
167D4—42no—
168D4—2121no—
169D4—2121no—
170D4—2121no—
171D4—2121no—
172D4—2121no—
173D4—2122no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—532no—
177C2×C4—532no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4C22⋊#615711yescharacteristic
188C2×C4C22×#615711yescharacteristic
189C8—2122no—
190C8—2122no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—2121no—
197C22—2121no—
198C22—2121no—
199C22#1313327912yes—
200C22C2×C22⋊#615711yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—531no—
205C4—532no—
206C4—531no—
207C4—531no—
208C4—531no—
209C4—1061no—
210C4—2121no—
211C2—22no—
212C2—81no—
213C2—4241no—
214C2#1681161011yescharacteristic
215C2C2×#1313327911yescharacteristic
216C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1547
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16811626C2semidirect
2. C2×#13133284C2semidirect
3. #16811482C2semidirect
4. #16811481C2semidirect
114. C53#1547semidirect