Finite Group Databasebeta

← #20290226

Subgroups — #20290226

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#16811628C211yesmaximal, characteristic, complemented
2D53⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3#16811482C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811170C211yesmaximal, characteristic
6C2×#11719936—21no—
7C2×D53⋊C8—21no—
8#13133284—21no—
9#13133284C2212yes—
10#13133284C2212yes—
11C4⋊#88324—21no—
12C2×#11719945—21no—
13#13133216C2211yescharacteristic
14#13133213C2211yescharacteristic
15#13133203—21no—
16#13133192C411yescharacteristic
17#13133190—21no—
18#13133188C411yescharacteristic
19C4×D212C411yescharacteristic
20C4×C22⋊D53—21no—
21C42×D53C2211yescharacteristic
22C4×D4×C53C411yescharacteristic
23C2×D4×C53C2×C411yescharacteristic
24C22×C4×C53—21no—
25C2×C22⋊D53—21no—
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28#11719945—42no—
29#11719945—42no—
30C2×D212—21no—
31C2×D212C2×C411yescharacteristic
32C2×C4×D53—21no—
33C2×C4×D53C2×C411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×Dic106—21no—
37C2×#88324—22no—
38C2×#88324—22no—
39#11719936—42no—
40D53⋊C8—42no—
41C4⋊C4×C53C2×C411yescharacteristic
42C22⋊C4×C53—21no—
43C42×C53C2×C411yescharacteristic
44C22⋊Dic53—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53C2×C411yescharacteristic
48#11719919—21no—
49C4×Dic53D411yescharacteristic
50C4×Dic53D411yescharacteristic
51C4×Dic53C2311yescharacteristic
52C23×C53—21no—
53C22×D53—21no—
54C22×D53C22⋊C411yescharacteristic
55D4×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—42no—
58C2×C4×C53C22⋊C411yescharacteristic
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×Dic53C22⋊C411yescharacteristic
68C2×Dic53C2×D411yescharacteristic
69C2×Dic53C2×D411yescharacteristic
70D212—41no—
71D212—42no—
72C4×D53—21no—
73C4×D53—21no—
74C4×D53—41no—
75C4×D53—42no—
76C4×D53D812yes—
77C4×D53SD1612yes—
78Dic106—41no—
79Dic106—41no—
80#88324—42no—
81#88324—42no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53C2×C22⋊C411yescharacteristic
86C2×D53—21no—
87C2×D53—21no—
88C2×D53—41no—
89C2×D53—41no—
90C2×D53—41no—
91C2×D53#10212yes—
92C4×C53—21no—
93C4×C53—21no—
94C4×C53—41no—
95C4×C53#10212yes—
96Dic53—21no—
97Dic53—41no—
98Dic53C2×D811yescharacteristic
99Dic53C2×SD1611yescharacteristic
100Dic53AGL(1,ℤ/8)11yescharacteristic
101Dic53#13711yescharacteristic
102#1547—531nomaximal, Sylow
103C2×C53—81no—
104C2×C53#41311yescharacteristic
105C2×C53C2×#10211yescharacteristic
106C2×C53#41711yescharacteristic
107D53—22no—
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—531no—
123#121—1061no—
124C4×D4—531no—
125C4×D4—1061no—
126C4×D4#615711yescharacteristic
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—531no—
136C2×D4—1061no—
137C2×D4—1061no—
138C2×D4—1061no—
139C2×D4—1061no—
140C2×D4C2×#615711yescharacteristic
141C22×C4—21no—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—531no—
145C22×C4—1061no—
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—
160C42C2×#615711yescharacteristic
161C23—21no—
162C23—531no—
163C23—1061no—
164Q8—2121no—
165Q8—2121no—
166D4—41no—
167D4—2121no—
168D4—2122no—
169D4—2121no—
170D4—2121no—
171D4—2121no—
172D4—2121no—
173D4—2122no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—531no—
177C2×C4—531no—
178C2×C4—532no—
179C2×C4—532no—
180C2×C4—531no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—2121no—
185C2×C4—2122no—
186C2×C4C22⋊#615711yescharacteristic
187C2×C4C22⋊#615711yescharacteristic
188C2×C4C22×#615711yescharacteristic
189C8—2122no—
190C8—2122no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—532no—
195C22—1061no—
196C22—1061no—
197C22—2121no—
198C22—2121no—
199C22—2121no—
200C22C2×C22⋊#615711yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—531no—
205C4—531no—
206C4—531no—
207C4—531no—
208C4—1061no—
209C4—2121no—
210C4#1313327912yes—
211C2—81no—
212C2—1062no—
213C2—4241no—
214C2#1681160911yescharacteristic
215C2C2×#1313327911yescharacteristic
216C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1547
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #16811628C2semidirect
2. D53⋊(C4⋊C8)C2semidirect
3. #16811482C2semidirect
4. #16811481C2semidirect
114. C53#1547semidirect