Finite Group Databasebeta

← #45667586

Subgroups — #45667586

Order 32896.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22551554C211yesmaximal, characteristic
2C2×#22551550C211yesmaximal, characteristic, complemented
3#30656672C211yesmaximal, characteristic, complemented
4#30656667C211yesmaximal, characteristic, complemented
5#30656666C212yesmaximal, complemented
6#30656664C211yesmaximal, characteristic, complemented
7#30656655C212yesmaximal
8#30656588C212yesmaximal, complemented
9#30656584C212yesmaximal, complemented
10#30656584C211yesmaximal, characteristic, complemented
11#400×C257C211yesmaximal, characteristic, complemented
12C22×D1028C2211yescharacteristic
13C22×C4×D257C2211yescharacteristic
14C22×C4×D257C2211yescharacteristic
15#22551554—24no—
16#22551550—24no—
17C2×C4⋊Dic257C2211yescharacteristic
18C2×#16861597C2212yes—
19C2×#16861597C2212yes—
20C2×#16861595C2212yes—
21C2×#16861595C2212yes—
22C2×#16861595C2212yes—
23C2×#16861595C2212yes—
24C2×#16861595C2211yescharacteristic
25C2×#16861595C2211yescharacteristic
26C2×C4×Dic257C2212yes—
27C2×C4×Dic257C2212yes—
28C2×C4×Dic257C2211yescharacteristic
29#22551499C2212yes—
30#22551499C2211yescharacteristic
31#22551499C2212yes—
32C2×C4⋊C4×C257C2211yescharacteristic
33C2×C4⋊C4×C257C2211yescharacteristic
34#95×C257C2211yescharacteristic
35#95×C257C2212yes—
36#95×C257C2212yes—
37C23×D257C2311yescharacteristic
38C23×D257C2311yescharacteristic
39C2×D1028—22no—
40C2×D1028—22no—
41C2×D1028—22no—
42C22×Dic257C2311yescharacteristic
43C22×Dic257C2312yes—
44C22×Dic257C2312yes—
45C22×Dic257C2311yescharacteristic
46C2×C4×D257—24no—
47C2×C4×D257—22no—
48C2×C4×D257—24no—
49C2×C4×D257D414yes—
50C4⋊Dic257—22no—
51#16861597—24no—
52#16861597—24no—
53#16861595—22no—
54#16861595—24no—
55#16861595—24no—
56#16861595—24no—
57#16861595—24no—
58#16861595—22no—
59C4×Dic257—22no—
60C4×Dic257—24no—
61C4×Dic257—24no—
62C22×C4×C257C2311yescharacteristic
63C22×C4×C257C2312yes—
64C22×C4×C257C2312yes—
65C22×C4×C257C2311yescharacteristic
66C22×C4×C257C2311yescharacteristic
67C4⋊C4×C257—22no—
68C4⋊C4×C257—22no—
69C22×D257—21no—
70C22×D257—22no—
71C22×D257—22no—
72C22×D257—22no—
73C22×D257—22no—
74C22×D257—21no—
75C22×D257—21no—
76C22×D257—21no—
77C22×D257C2×D412yes—
78C22×D257#4112yes—
79D1028—28no—
80C2×Dic257—22no—
81C2×Dic257—22no—
82C2×Dic257—21no—
83C2×Dic257—22no—
84C2×Dic257—22no—
85C2×Dic257—22no—
86C2×Dic257—22no—
87C2×Dic257C2×D412yes—
88C2×Dic257#4114yes—
89C2×Dic257#4114yes—
90C4×D257—28no—
91C4×D257—44no—
92C23×C257C2411yescharacteristic
93C2×C4×C257—21no—
94C2×C4×C257—22no—
95C2×C4×C257—21no—
96C2×C4×C257—22no—
97C2×C4×C257—21no—
98C2×C4×C257—22no—
99C2×C4×C257—22no—
100C2×C4×C257—22no—
101C2×C4×C257—22no—
102C2×C4×C257—22no—
103C2×C4×C257—22no—
104C2×C4×C257C2×D412yes—
105C2×C4×C257#4112yes—
106C22×C257C22×D411yescharacteristic
107C22×C257C2×#4112yes—
108C22×C257C2×#4111yescharacteristic
109C22×C257C2×#4112yes—
110C22×C257C2×#4111yescharacteristic
111C2×D257—22no—
112C2×D257—22no—
113C2×D257—22no—
114C2×D257—24no—
115C2×D257—22no—
116C2×D257—41no—
117C2×D257—42no—
118C2×D257—42no—
119C2×D257—42no—
120C2×D257—41no—
121C4×C257—22no—
122C4×C257—22no—
123C4×C257—41no—
124C4×C257—42no—
125C4×C257—42no—
126Dic257—24no—
127Dic257—42no—
128Dic257—41no—
129Dic257#12314yes—
130C2×C257C2×#12311yescharacteristic
131C2×C257#52511yescharacteristic
132C2×C257#52912yes—
133C2×C257#52912yes—
134C2×C257#53211yescharacteristic
135D257—42no—
136D257—42no—
137C257#239511yescharacteristic, complemented, Sylow
138#2395—2571nomaximal, Sylow
139C2×C4×D4—2571no—
140C2×#117—2571no—
141#400D25711yescharacteristic, complemented
142#399—2571no—
143#398—2571no—
144#397—2571no—
145#397—2572no—
146#394—2572no—
147C4×C4⋊C4—2572no—
148C4×C22⋊C4—2572no—
149C4×C22⋊C4—2571no—
150C22×D4—2571no—
151C23×C4—2571no—
152C23×C4—2571no—
153C4×D4—5144no—
154#117—5144no—
155C2×C4⋊C4—2571no—
156C2×C4⋊C4—2572no—
157C2×C4⋊C4—2572no—
158C2×C4⋊C4C2×D25711yescharacteristic
159C2×C4⋊C4C2×D25711yescharacteristic
160C2×C22⋊C4—2572no—
161C2×C22⋊C4—2571no—
162C2×C22⋊C4—2571no—
163C2×C22⋊C4—2572no—
164C2×C22⋊C4—2572no—
165C2×C22⋊C4—2572no—
166C2×C42—2572no—
167C2×C42—2571no—
168C2×C42—2572no—
169#95—2572no—
170#95—2571no—
171#95—2572no—
172#95C2×D25712yes—
173#95C2×D25711yescharacteristic
174#95C2×D25712yes—
175C24—2571no—
176C24—2571no—
177C2×D4—5142no—
178C2×D4—5142no—
179C2×D4—5142no—
180C22×C4—2571no—
181C22×C4—2572no—
182C22×C4—2574no—
183C22×C4—2571no—
184C22×C4—2572no—
185C22×C4—5144no—
186C22×C4—5142no—
187C22×C4—5144no—
188C22×C4C22×D25711yescharacteristic
189C22×C4C22×D25712yes—
190C22×C4C22×D25712yes—
191C22×C4C22×D25711yescharacteristic
192C22×C4C22×D25711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—5144no—
196C4⋊C4—5142no—
197C4⋊C4—5144no—
198C22⋊C4—5144no—
199C22⋊C4—5144no—
200C22⋊C4—5144no—
201C22⋊C4—5142no—
202C22⋊C4—5144no—
203C22⋊C4—5142no—
204C42—5144no—
205C42—5142no—
206C42—5144no—
207C23—2572no—
208C23—2572no—
209C23—5141no—
210C23—5142no—
211C23—5142no—
212C23—5142no—
213C23—5142no—
214C23—5141no—
215C23—5141no—
216C23—5141no—
217C23C23×D25711yescharacteristic
218D4—5148no—
219C2×C4—2572no—
220C2×C4—2574no—
221C2×C4—2574no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—5148no—
234C2×C4—5142no—
235C2×C4—5142no—
236C2×C4—5141no—
237C2×C4—5142no—
238C2×C4—5142no—
239C2×C4—5142no—
240C2×C4—5142no—
241C2×C4—10284no—
242C2×C4D4×D25712yes—
243C2×C4#1686161912yes—
244C22—5142no—
245C22—5142no—
246C22—5142no—
247C22—5142no—
248C22—5144no—
249C22—10281no—
250C22—10281no—
251C22—10282no—
252C22—10282no—
253C22—10282no—
254C22C2×D4×D25711yescharacteristic
255C22#41×D25712yes—
256C22#41×D25712yes—
257C22#41×D25711yescharacteristic
258C22C2×#1686161911yescharacteristic
259C4—2574no—
260C4—22no—
261C4—22no—
262C4—5144no—
263C4—42no—
264C4—41no—
265C4—42no—
266C4—10281no—
267C4—10282no—
268C2—10282no—
269C2—10282no—
270C2#123×D25711yescharacteristic
271C2#3065725011yescharacteristic
272C2#3065727012yes—
273C2#3065727212yes—
274C2#3065728611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#400×C257C2
prime radicalC22×D1028C22
hypercenter#400D257
hypocenterC257#2395
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#22551550C2semidirect
3. #30656672C2semidirect
4. #30656667C2semidirect
5. #30656666C2semidirect
6. #30656664C2semidirect
8. #30656588C2semidirect
9. #30656584C2semidirect
10. #30656584C2semidirect
11. #400×C257C2semidirect
137. C257#2395semidirect
141. #400D257semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#225515542
C2×#225515502
#306566722
#306566672
#306566662
#306566642
#306566552
#306565882
#306565842
#400×C2572
#2395257