Finite Group Databasebeta

← #20274683

Subgroups — #20274683

Order 6784.

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

normal subgroups only

This group has 588 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
1C2×#13133263C211yesmaximal, characteristic
2#16810641C212yesmaximal, complemented
3#16810627C212yesmaximal, complemented
4#16810607C211yesmaximal, characteristic, complemented
5C53×#410C211yesmaximal, characteristic, complemented
6#13133263—24no—
7#13133263—24no—
8C2×#11719949—21no—
9C2×#11719945C412yescomplemented
10C2×#11719945C2211yescharacteristic
11C22×D212C412yescomplemented
12C22×D212C2211yescharacteristic
13C2×D4×D53—22no—
14C2×D4×D53—21no—
15C2×#11719917C2212yescomplemented
16#13133145—24no—
17#13133137—24no—
18#13133136—21no—
19#13133134—21no—
20C2×#41×C53C2211yescharacteristic
21#117×C53C2212yescomplemented
22#99×C53—22no—
23C23×D53—22no—
24C23×D53—21no—
25#41×C53—22no—
26#41×C53—22no—
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53D412yes—
29C2×D4×C53D411yescharacteristic
30C22×C4×C53C2×C412yes—
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—22no—
33C2×C22⋊D53—21no—
34#11719949—28no—
35D4×D53—216no—
36D4×D53—28no—
37#11719945—22no—
38#11719945—24no—
39#11719945—24no—
40#11719945D414yes—
41C2×D212—22no—
42C2×D212—21no—
43C2×D212—22no—
44C2×D212—24no—
45C2×D212—24no—
46C2×D212C2×C411yescharacteristic
47C2×D212C2×C412yes—
48C2×D212D414yes—
49C2×C4×D53—22no—
50C2×C4×D53—21no—
51C2×Dic106C2×C411yescharacteristic
52C4⋊C4×C53—22no—
53C22⋊C4×C53—22no—
54C42×C53—22no—
55#11719917—22no—
56#11719917—22no—
57C2×#88322—22no—
58C23×C53C22⋊C412yes—
59C23×C53C2×D411yescharacteristic
60C22×D53—22no—
61C22×D53—21no—
62C22×D53—28no—
63C22×D53—24no—
64C22×D53—82no—
65C22×D53—84no—
66Q8×C53—22no—
67D4×C53—24no—
68D4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53C22⋊C412yes—
73C2×C4×C53C22×C411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×D412yes—
77C22⋊D53—28no—
78C22⋊D53—24no—
79C2×Dic53—21no—
80D212—28no—
81D212—21no—
82D212—22no—
83D212—24no—
84D212C22⋊C412yes—
85D212C22⋊C414yes—
86C4×D53—28no—
87C4×D53—24no—
88Dic106—21no—
89Dic106C22⋊C412yes—
90#88322—42no—
91C22×C53—22no—
92C22×C53—41no—
93C22×C53—42no—
94C22×C53C2×C22⋊C411yescharacteristic
95C22×C53#12012yes—
96C2×D53—24no—
97C2×D53—22no—
98C2×D53—44no—
99C2×D53—42no—
100C2×D53—84no—
101C2×D53—88no—
102C4×C53—22no—
103C4×C53—82no—
104C4×C53C2×C22⋊C411yescharacteristic
105C4×C53C2×C22⋊C411yescharacteristic
106C4×C53#12012yes—
107Dic53—22no—
108#1795—531nomaximal, Sylow
109C2×C53—22no—
110C2×C53—21no—
111C2×C53—42no—
112C2×C53#37911yescharacteristic
113D53—44no—
114D53—42no—
115C2×H(5,𝔽2)—531no—
116#421—532no—
117#418—532no—
118#413—531no—
119#410D5311yescharacteristic, complemented
120C53#179511yescharacteristic, complemented, Sylow
121H(5,𝔽2)—1064no—
122H(5,𝔽2)—1064no—
123C2×#41—532no—
124C2×#41—531no—
125C2×#41—1061no—
126C2×#41C2×D5311yescharacteristic
127C22×D4—532no—
128C22×D4—531no—
129C22×D4—1061no—
130C22×D4—1062no—
131C2×M16—532no—
132#117C2×D5312yescomplemented
133#104—1064no—
134#102—1064no—
135#101—1061no—
136#100—1061no—
137#99—22no—
138C24—1061no—
139C24—1062no—
140#41—22no—
141#41—22no—
142#41—534no—
143#41—1064no—
144#41—1064no—
145#41—1068no—
146#41—1062no—
147C2×Q8—531no—
148C2×Q8D21211yescharacteristic
149C2×D4—532no—
150C2×D4—531no—
151C2×D4—534no—
152C2×D4—1064no—
153C2×D4—1064no—
154C2×D4—10616no—
155C2×D4—1068no—
156C2×D4—1061no—
157C2×D4—1062no—
158C2×D4—1061no—
159C2×D4—1062no—
160C2×D4—1062no—
161C2×D4D21211yescharacteristic
162C2×D4C22⋊D5312yes—
163C22×C4—1061no—
164C22×C4—1062no—
165C22×C4C4×D5312yes—
166C22×C4C22×D5311yescharacteristic
167M16—1062no—
168M16—1062no—
169C2×C8—1062no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—1068no—
174C23—1064no—
175C23—1061no—
176C23—1062no—
177C23—4242no—
178C23—4244no—
179C23#1171992112yes—
180C23C2×C22⋊D5311yescharacteristic
181Q8—22no—
182Q8—532no—
183Q8—1061no—
184D4—24no—
185D4—22no—
186D4—534no—
187D4—532no—
188D4—1068no—
189D4—1068no—
190D4—1064no—
191D4—1061no—
192D4—1062no—
193D4—1064no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—1068no—
198C2×C4—1061no—
199C2×C4—1064no—
200C2×C4#1171992112yes—
201C2×C4C2×C4×D5311yescharacteristic
202C2×C4C2×D21211yescharacteristic
203C2×C4D4×D5312yes—
204C2×C4D4×D5312yes—
205C8—2122no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—1064no—
210C22—1062no—
211C22—4244no—
212C22—4248no—
213C22—2124no—
214C22—2122no—
215C22C2×#1171992111yescharacteristic
216C22#1313318512yes—
217C4—22no—
218C4—82no—
219C4—1062no—
220C4C22⋊C4×D5311yescharacteristic
221C4C22⋊C4×D5311yescharacteristic
222C4#1313318712yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—2124no—
227C2—2122no—
228C2#1681047811yescharacteristic
subgroupG/H
centerC2#16810478
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#410C2
socleC2×C53#379
hypercenter#410D53
hypocenterC53#1795
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16810641C2semidirect
3. #16810627C2semidirect
4. #16810607C2semidirect
5. C53×#410C2semidirect
9. C2×#11719945C4semidirect
11. C22×D212C4semidirect
15. C2×#11719917C22semidirect
21. #117×C53C22semidirect
119. #410D53semidirect
120. C53#1795semidirect
132. #117C2×D53semidirect