Finite Group Databasebeta

← #45667582

Subgroups — #45667582

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×#22551552C211yesmaximal, characteristic
2C2×#22551550C211yesmaximal, characteristic, complemented
3#30656677C211yesmaximal, characteristic, complemented
4#30656673C211yesmaximal, characteristic, complemented
5#30656668C211yesmaximal, characteristic, complemented
6#30656666C212yesmaximal, complemented
7#30656652C212yesmaximal
8#30656588C212yesmaximal, complemented
9C22⋊C4×Dic257C212yesmaximal, complemented
10#30656584C211yesmaximal, characteristic, complemented
11#397×C257C211yesmaximal, characteristic, complemented
12C22×C22⋊D257C2211yescharacteristic
13C23×Dic257C2211yescharacteristic
14C22×C4×D257C2211yescharacteristic
15#22551552—24no—
16#22551550—24no—
17C2×C4⋊Dic257C2211yescharacteristic
18C2×#16861597C2212yes—
19C2×#16861597C2212yes—
20C2×#16861597C2211yescharacteristic
21C2×C22⋊Dic257C2212yes—
22C2×#16861595C2212yes—
23C2×#16861595C2212yes—
24C2×#16861595C2211yescharacteristic
25C2×C4×Dic257C2212yes—
26C2×C4×Dic257C2212yes—
27C2×C4×Dic257C2211yescharacteristic
28#22551499C2211yescharacteristic
29#22551499C2212yes—
30#22551499C2212yes—
31#22551499C2212yes—
32C2×C4⋊C4×C257C2211yescharacteristic
33C2×C22⋊C4×C257C2212yes—
34C2×C22⋊C4×C257C2211yescharacteristic
35#95×C257C2211yescharacteristic
36#95×C257C2212yes—
37C23×D257C2311yescharacteristic
38C2×C22⋊D257—22no—
39C2×C22⋊D257—22no—
40C2×C22⋊D257—22no—
41C22×Dic257—22no—
42C22×Dic257—24no—
43C22×Dic257C2311yescharacteristic
44C22×Dic257C2312yes—
45C22×Dic257C2311yescharacteristic
46C22×Dic257C2312yes—
47C22×Dic257C2311yescharacteristic
48C2×C4×D257—24no—
49C2×C4×D257D414yes—
50C4⋊Dic257—22no—
51#16861597—24no—
52#16861597—24no—
53#16861597—22no—
54C22⋊Dic257—24no—
55#16861595—22no—
56#16861595—24no—
57#16861595—24no—
58C4×Dic257—22no—
59C4×Dic257—24no—
60C4×Dic257—24no—
61C24×C257C2311yescharacteristic
62C22×C4×C257C2312yes—
63C22×C4×C257C2311yescharacteristic
64C22×C4×C257C2312yes—
65C22×C4×C257C2311yescharacteristic
66C4⋊C4×C257—22no—
67C22⋊C4×C257—24no—
68C22⋊C4×C257—22no—
69C22×D257—22no—
70C22×D257—22no—
71C22×D257—21no—
72C22×D257—21no—
73C22×D257C2×D412yes—
74C22⋊D257—28no—
75C2×Dic257—21no—
76C2×Dic257—22no—
77C2×Dic257—22no—
78C2×Dic257—21no—
79C2×Dic257—22no—
80C2×Dic257—21no—
81C2×Dic257—22no—
82C2×Dic257—22no—
83C2×Dic257—22no—
84C2×Dic257—44no—
85C2×Dic257C2×D412yes—
86C2×Dic257#4114yes—
87C2×Dic257#4112yes—
88C2×Dic257#4114yes—
89C4×D257—28no—
90C23×C257—21no—
91C23×C257—22no—
92C23×C257—22no—
93C23×C257—21no—
94C23×C257#4112yes—
95C23×C257C2411yescharacteristic
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×C257—22no—
105C2×C4×C257C2×D412yes—
106C22×C257—22no—
107C22×C257—24no—
108C22×C257—41no—
109C22×C257—42no—
110C22×C257—41no—
111C22×C257C22×D411yescharacteristic
112C22×C257C2×#4112yes—
113C22×C257C2×#4111yescharacteristic
114C22×C257C2×#4112yes—
115C22×C257C2×#4111yescharacteristic
116C2×D257—22no—
117C2×D257—22no—
118C2×D257—22no—
119C2×D257—42no—
120C2×D257—42no—
121C4×C257—22no—
122C4×C257—42no—
123C4×C257—42no—
124C4×C257—41no—
125Dic257—22no—
126Dic257—24no—
127Dic257—41no—
128Dic257—42no—
129Dic257#12314yes—
130C2×C257—42no—
131C2×C257C2×#12311yescharacteristic
132C2×C257#52511yescharacteristic
133C2×C257#52912yes—
134C2×C257#52912yes—
135C2×C257#53211yescharacteristic
136D257—42no—
137C257#239511yescharacteristic, complemented, Sylow
138#2395—2571nomaximal, Sylow
139C2×C4×D4—2571no—
140C2×#117—2571no—
141#400—2571no—
142#399—2571no—
143#398—2571no—
144#397—2572no—
145#397D25711yescharacteristic, complemented
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—2571no—
157C2×C4⋊C4—2572no—
158C2×C4⋊C4—2572no—
159C2×C4⋊C4C2×D25711yescharacteristic
160C2×C22⋊C4—2572no—
161C2×C22⋊C4—2572no—
162C2×C22⋊C4—2571no—
163C2×C22⋊C4—2572no—
164C2×C22⋊C4C2×D25711yescharacteristic
165C2×C22⋊C4C2×D25712yes—
166C2×C42—2572no—
167C2×C42—2571no—
168C2×C42—2572no—
169#95—2572no—
170#95—2572no—
171#95—2571no—
172#95—2572no—
173#95C2×D25711yescharacteristic
174#95C2×D25712yes—
175C24—2571no—
176C24C22×D25711yescharacteristic
177C2×D4—5142no—
178C2×D4—5142no—
179C2×D4—5142no—
180C22×C4—2571no—
181C22×C4—2572no—
182C22×C4—2571no—
183C22×C4—2574no—
184C22×C4—2571no—
185C22×C4—2572no—
186C22×C4—5144no—
187C22×C4—5142no—
188C22×C4—5144no—
189C22×C4C22×D25712yes—
190C22×C4C22×D25712yes—
191C22×C4C22×D25711yescharacteristic
192C22×C4C22×D25711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—5144no—
195C4⋊C4—5144no—
196C4⋊C4—5142no—
197C4⋊C4—5142no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—5144no—
201C22⋊C4—5144no—
202C22⋊C4—5144no—
203C22⋊C4—5142no—
204C42—5144no—
205C42—5142no—
206C42—5144no—
207C23—2572no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—5142no—
213C23—5142no—
214C23—5141no—
215C23—5141no—
216C23#1686161712yes—
217C23C23×D25711yescharacteristic
218D4—5148no—
219C2×C4—2572no—
220C2×C4—2574no—
221C2×C4—2574no—
222C2×C4—2572no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—5148no—
233C2×C4—5141no—
234C2×C4—5142no—
235C2×C4—5142no—
236C2×C4—5141no—
237C2×C4—5142no—
238C2×C4—5141no—
239C2×C4—5142no—
240C2×C4—5142no—
241C2×C4—5142no—
242C2×C4—10284no—
243C2×C4D4×D25712yes—
244C22—22no—
245C22—24no—
246C22—5142no—
247C22—5142no—
248C22—5142no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—10282no—
253C22—10282no—
254C22C2×D4×D25711yescharacteristic
255C22#41×D25712yes—
256C22#41×D25712yes—
257C22#41×D25711yescharacteristic
258C22C2×#1686161711yescharacteristic
259C4—2574no—
260C4—22no—
261C4—5142no—
262C4—5144no—
263C4—41no—
264C4—42no—
265C4—42no—
266C4—10282no—
267C4—10281no—
268C2—42no—
269C2—10282no—
270C2#123×D25711yescharacteristic
271C2#3065724711yescharacteristic
272C2#3065726412yes—
273C2#3065727012yes—
274C2#3065728611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutator, socle, cosocleC23×C257C24
Fitting#397×C257C2
prime radicalC22×C22⋊D257C22
hypercenter#397D257
hypocenterC257#2395
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#22551550C2semidirect
3. #30656677C2semidirect
4. #30656673C2semidirect
5. #30656668C2semidirect
6. #30656666C2semidirect
8. #30656588C2semidirect
9. C22⋊C4×Dic257C2semidirect
10. #30656584C2semidirect
11. #397×C257C2semidirect
137. C257#2395semidirect
145. #397D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515522
C2×#225515502
#306566772
#306566732
#306566682
#306566662
#306566522
#306565882
C22⋊C4×Dic2572
#306565842
#397×C2572
#2395257