Finite Group Databasebeta

← #20280848

Subgroups — #20280848

Order 6784.

598 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133202C211yesmaximal, characteristic, complemented
2C2×#13133192C211yesmaximal, characteristic, complemented
3#16810549C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810540C212yesmaximal, complemented
6#16810533C212yesmaximal, complemented
7#16810533C211yesmaximal, characteristic, complemented
8#16810524C212yesmaximal, complemented
9#16810512C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×D212C2212yes—
15C22×D212C2211yescharacteristic
16C22×C4×D53C2211yescharacteristic
17#13133202—24no—
18#13133192—24no—
19C2×#11719919C2211yescharacteristic
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×C4×Dic53C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2312yes—
38C23×D53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×D212—24no—
49C2×D212—22no—
50C2×D212—22no—
51C2×D212—24no—
52C2×D212—42no—
53C2×D212—42no—
54C2×D212—42no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—42no—
58C2×C4×D53—42no—
59C2×C4×D53D414yescomplemented
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—42no—
63#11719921—22no—
64#11719921—22no—
65#11719921—24no—
66#11719921—24no—
67#11719921—24no—
68#11719921—22no—
69#11719921—42no—
70#11719921—42no—
71#11719921—42no—
72#11719919—22no—
73C4×Dic53—42no—
74C23×C53C2411yescharacteristic
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53—21no—
79C22×D53—22no—
80C22×D53—21no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—21no—
91C22×D53C2×D412yes—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53#4114yes—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53C2×D412yes—
112D212—44no—
113D212—48no—
114D212—44no—
115D212—48no—
116C4×D53—44no—
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53H(5,𝔽2)11yescharacteristic
120C22×C53H(5,𝔽2)12yes—
121C22×C53H(5,𝔽2)11yescharacteristic
122C2×D53—22no—
123C2×D53—22no—
124C2×D53—22no—
125C2×D53—42no—
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—41no—
131C2×D53—42no—
132C2×D53—41no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C4×C53—22no—
140C4×C53—24no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—41no—
144Dic53—22no—
145Dic53—41no—
146Dic53—42no—
147#2661—531nomaximal, Sylow
148C2×C53#53211yescharacteristic
149C2×C53#53411yescharacteristic
150C2×C53#53511yescharacteristic
151C2×C53#53812yes—
152C2×C53#54012yes—
153D53—42no—
154D53—82no—
155D53—81no—
156C2×#123—531no—
157C2×#121—531no—
158#398D5311yescharacteristic, complemented
159#397—531no—
160#394—531no—
161#394—532no—
162#392—531no—
163#392—532no—
164#390—532no—
165#388—532no—
166C4×C22⋊C4—531no—
167C53#266111yescharacteristic, complemented, Sylow
168C22×D4—532no—
169C22×D4—531no—
170C22×D4—531no—
171C22×D4—532no—
172C23×C4—531no—
173#123—1064no—
174#121—1064no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4C2×D5311yescharacteristic
177C2×C4⋊C4C2×D5311yescharacteristic
178C2×C4⋊C4C2×D5312yes—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—531no—
182C2×C22⋊C4—531no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C42—532no—
189#95—531no—
190#95—532no—
191#95C2×D5311yescharacteristic
192#95C2×D5312yes—
193C24—531no—
194C24—531no—
195C24—532no—
196C2×D4—1064no—
197C2×D4—1062no—
198C2×D4—1062no—
199C2×D4—1064no—
200C2×D4—2122no—
201C2×D4—2122no—
202C2×D4—2122no—
203C2×D4—2122no—
204C2×D4—2122no—
205C2×D4—2122no—
206C22×C4—531no—
207C22×C4—534no—
208C22×C4—532no—
209C22×C4—531no—
210C22×C4—2122no—
211C22×C4C22×D5312yes—
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5312yes—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—1062no—
220C22⋊C4—1064no—
221C22⋊C4—1062no—
222C22⋊C4—1062no—
223C22⋊C4—1062no—
224C22⋊C4—1064no—
225C22⋊C4—1064no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C42—2122no—
230C23—532no—
231C23—1062no—
232C23—1061no—
233C23—1062no—
234C23—1061no—
235C23—1061no—
236C23—1062no—
237C23—1062no—
238C23—1062no—
239C23—1062no—
240C23—1062no—
241C23—1061no—
242C23—1061no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23C23×D5311yescharacteristic
248D4—2124no—
249D4—2128no—
250D4—2124no—
251D4—2128no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—532no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—2124no—
271C2×C4D4×D5312yes—
272C2×C4#1171994914yes—
273C22—1062no—
274C22—1062no—
275C22—1062no—
276C22—2121no—
277C22—2122no—
278C22—2122no—
279C22—2121no—
280C22—2122no—
281C22—2121no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22C2×D4×D5311yescharacteristic
291C22C2×#1171994912yes—
292C22#1313326211yescharacteristic
293C22#1313326312yes—
294C22#1313326311yescharacteristic
295C4—22no—
296C4—24no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—1062no—
301C4—2121no—
302C4—2122no—
303C2—4242no—
304C2—4241no—
305C2—2122no—
306C2#1681117411yescharacteristic
307C2#1681118411yescharacteristic
308C2#1681118911yescharacteristic
309C2#1681121312yes—
310C2#1681122912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2661
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133192C2semidirect
3. #16810549C2semidirect
4. #16810540C2semidirect
5. #16810540C2semidirect
6. #16810533C2semidirect
7. #16810533C2semidirect
8. #16810524C2semidirect
9. #16810512C2semidirect
10. #16810470C2semidirect
11. C53×#398C2semidirect
59. C2×C4×D53D4semidirect
158. #398D53semidirect
167. C53#2661semidirect