Finite Group Databasebeta

← #20275907

Subgroups — #20275907

Order 6784.

368 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133192C211yesmaximal, characteristic
2C2×#13133148C211yesmaximal, characteristic, complemented
3C2×#13133139C211yesmaximal, characteristic, complemented
4C2×#13133137C211yesmaximal, characteristic, complemented
5C2×#13133129C211yesmaximal, characteristic, complemented
6#16810480C211yesmaximal, characteristic, complemented
7C53×#340C211yesmaximal, characteristic, complemented
8C22×D212—21no—
9C22×D212C2211yescharacteristic
10C22×C4×D53C2211yescharacteristic
11C22×#88322C2211yescharacteristic
12#13133192—22no—
13#13133192—22no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719921—21no—
16C2×#11719921—21no—
17#13133148—22no—
18#13133139—22no—
19#13133137—22no—
20#13133129—22no—
21#13133124—21no—
22C22×C8×C53C2211yescharacteristic
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C23×D53—21no—
26C23×D53D411yescharacteristic
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53D411yescharacteristic
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—24no—
37C2×D212—21no—
38C2×D212—42no—
39C2×D212—42no—
40C2×D212D412yes—
41C2×C4×D53—21no—
42C2×C4×D53—21no—
43C2×C4×D53—21no—
44C2×C4×D53—21no—
45C2×C4×D53—21no—
46C2×C4×D53—21no—
47C2×C8×C53—21no—
48C2×C8×C53—21no—
49C2×C8×C53—21no—
50C4⋊C4×C53—21no—
51C4⋊C4×C53—21no—
52C4⋊C4×C53—21no—
53C4⋊C4×C53D412yes—
54#11719921—24no—
55#11719921—42no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C2×#88322—21no—
59C2×#88322—21no—
60C2×#88322D412yescomplemented
61C23×C53C2×D411yescharacteristic
62C22×D53—21no—
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—22no—
69C22×D53—41no—
70C22×D53—41no—
71C22×D53—41no—
72C22×D53—41no—
73C22×D53—41no—
74C22×D53—41no—
75C22×D53SD1612yes—
76C2×C4×C53—22no—
77C2×C4×C53—41no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53#4111yescharacteristic
86C2×C4×C53#4111yescharacteristic
87C2×C4×C53#4111yescharacteristic
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—41no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94D212—22no—
95D212—22no—
96D212—48no—
97D212—41no—
98D212—41no—
99C4×D53—41no—
100C4×D53—41no—
101C4×D53—41no—
102C4×D53—41no—
103C8×C53—41no—
104#88322—22no—
105C22×C53#12011yescharacteristic
106C22×C53#12111yescharacteristic
107C22×C53#12311yescharacteristic
108C22×C53C2×SD1611yescharacteristic
109C22×C53#13511yescharacteristic
110C22×C53AGL(1,ℤ/8)11yescharacteristic
111C22×C53AGL(1,ℤ/8)11yescharacteristic
112C2×D53—22no—
113C2×D53—22no—
114C2×D53—22no—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—41no—
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—42no—
122C2×D53—81no—
123C2×D53—81no—
124C2×D53—81no—
125C2×D53—81no—
126C4×C53—42no—
127C4×C53—81no—
128C4×C53#12111yescharacteristic
129C4×C53#12111yescharacteristic
130C4×C53#12311yescharacteristic
131C4×C53#12411yescharacteristic
132Dic53—41no—
133Dic53—81no—
134#2034—531nomaximal, Sylow
135C2×C53#39411yescharacteristic
136C2×C53#44911yescharacteristic
137C2×C53#45011yescharacteristic
138C2×C53#46511yescharacteristic
139C2×C53#46911yescharacteristic
140C2×C53#48111yescharacteristic
141C2×C53#48211yescharacteristic
142D53—42no—
143D53—82no—
144C2×#121—531no—
145C2×#106—531no—
146C2×#102—531no—
147C2×#102—531no—
148C2×C22⋊C8—531no—
149#380—531no—
150#340D5311yescharacteristic, complemented
151C53#203411yescharacteristic, complemented, Sylow
152C22×D4—531no—
153C22×D4—1061no—
154C23×C4—531no—
155C22×C8—531no—
156C22×C8C2×D5311yescharacteristic
157#121—1062no—
158#121—1062no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4C2×D5311yescharacteristic
161C2×C4⋊C4C2×D5311yescharacteristic
162C2×C22⋊C4—1061no—
163C2×C22⋊C4—1061no—
164#106—1062no—
165#102—1062no—
166#102—1062no—
167C22⋊C8—1062no—
168#95—1061no—
169C24—531no—
170C24—1061no—
171C2×D4—532no—
172C2×D4—1064no—
173C2×D4—1061no—
174C2×D4—1061no—
175C2×D4—1061no—
176C2×D4—1061no—
177C2×D4—1061no—
178C2×D4—2122no—
179C2×D4—2122no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—531no—
183C22×C4—1061no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—1061no—
190C22×C4C22×D5311yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—532no—
195C2×C8—1061no—
196C2×C8—1061no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—1061no—
201C4⋊C4—1061no—
202C4⋊C4D21212yes—
203C22⋊C4—1064no—
204C22⋊C4—2122no—
205C23—532no—
206C23—1061no—
207C23—1061no—
208C23—1061no—
209C23—1061no—
210C23—1061no—
211C23—1061no—
212C23—1062no—
213C23—2121no—
214C23—2121no—
215C23—2121no—
216C23—2121no—
217C23—2121no—
218C23—2121no—
219C23D4×D5311yescharacteristic
220D4—1062no—
221D4—1062no—
222D4—2128no—
223D4—2121no—
224D4—2121no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2121no—
240C2×C4—2121no—
241C2×C4C2×D21211yescharacteristic
242C2×C4#1171994511yescharacteristic
243C2×C4#1171994511yescharacteristic
244C2×C4D4×D5311yescharacteristic
245C2×C4D4×D5311yescharacteristic
246C2×C4#1171994911yescharacteristic
247C8—41no—
248C8—1062no—
249C22—1062no—
250C22—1062no—
251C22—1062no—
252C22—4241no—
253C22—4241no—
254C22—4241no—
255C22—4241no—
256C22—2122no—
257C22—2122no—
258C22—2122no—
259C22—2121no—
260C22—2121no—
261C22—2121no—
262C22—2121no—
263C22#1313318711yescharacteristic
264C22#1313319011yescharacteristic
265C22#1313320211yescharacteristic
266C22SD16×D5311yescharacteristic
267C22#1313324011yescharacteristic
268C22#1313324111yescharacteristic
269C22#1313324411yescharacteristic
270C4—42no—
271C4—81no—
272C4—4241no—
273C4—2121no—
274C4#1313319011yescharacteristic
275C4#1313319211yescharacteristic
276C4#1313320211yescharacteristic
277C4#1313320511yescharacteristic
278C2—4242no—
279C2—2122no—
280C2#1681054111yescharacteristic
281C2#1681077111yescharacteristic
282C2#1681077511yescharacteristic
283C2#1681085511yescharacteristic
284C2#1681087511yescharacteristic
285C2#1681094311yescharacteristic
286C2#1681095011yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#2034
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#13133148C2semidirect
3. C2×#13133139C2semidirect
4. C2×#13133137C2semidirect
5. C2×#13133129C2semidirect
6. #16810480C2semidirect
7. C53×#340C2semidirect
60. C2×#88322D4semidirect
150. #340D53semidirect
151. C53#2034semidirect