Finite Group Databasebeta

← #45664839

Subgroups — #45664839

Order 32896.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#30657503C211yesmaximal, characteristic
2#30657311C211yesmaximal, characteristic
3#30656746C211yesmaximal, characteristic, complemented
4#30656710C211yesmaximal, characteristic
5#30656494C211yesmaximal, characteristic, complemented
6#30656493C211yesmaximal, characteristic, complemented
7#419×C257C211yesmaximal, characteristic, complemented
8C2×#16861618C2211yescharacteristic
9#22551600—22no—
10#22551598—22no—
11C2×#16861602C2211yescharacteristic
12M16×D257—22no—
13#22551590C412yescomplemented
14#22551581C412yescomplemented
15#22551575—22no—
16#22551568—22no—
17#22551551C2211yescharacteristic
18#22551551C2211yescharacteristic
19#22551516—22no—
20#22551506—22no—
21C22×Q8×C257C2211yescharacteristic
22C2×M16×C257C2211yescharacteristic
23#117×C257C2211yescharacteristic
24#103×C257—22no—
25#16861618—21no—
26#16861618—21no—
27#16861618—21no—
28#16861618—21no—
29C2×Dic514C2×C411yescharacteristic
30C2×D1028C2×C411yescharacteristic
31C2×C22⋊D257D412yes—
32C2×C4×D257D412yes—
33#16861602—21no—
34#16861602—21no—
35#16861601—24no—
36C2×#14523339D412yes—
37C8×D257—24no—
38C4⋊Dic257—21no—
39C4⋊Dic257—21no—
40#16861597—24no—
41C22⋊Dic257—21no—
42C22⋊Dic257—21no—
43#16861595—24no—
44C4×Dic257—21no—
45C4×Dic257C2×C412yes—
46C2×Q8×C257—21no—
47C2×Q8×C257—21no—
48C2×Q8×C257—21no—
49C2×Q8×C257—21no—
50C2×Q8×C257—21no—
51C2×Q8×C257C2×C412yes—
52C22×C4×C257—21no—
53C22×C4×C257C2311yescharacteristic
54M16×C257—21no—
55M16×C257—21no—
56C2×C8×C257D412yes—
57C4⋊C4×C257—21no—
58C22⋊C4×C257—21no—
59C42×C257—21no—
60C22×D257C22⋊C412yes—
61Dic514—21no—
62Dic514—21no—
63D1028—21no—
64D1028—21no—
65C22⋊D257—22no—
66C22⋊D257—22no—
67C2×Dic257—22no—
68C2×Dic257—41no—
69C2×Dic257C22⋊C412yes—
70C4×D257—22no—
71C4×D257—22no—
72#14523339—22no—
73C23×C257C2×D411yescharacteristic
74Q8×C257—22no—
75Q8×C257—22no—
76Q8×C257—41no—
77Q8×C257—41no—
78C2×C4×C257—22no—
79C2×C4×C257—41no—
80C2×C4×C257—41no—
81C2×C4×C257—41no—
82C2×C4×C257C22×C411yescharacteristic
83C2×C4×C257C2×D411yescharacteristic
84C2×C4×C257C2×D411yescharacteristic
85C2×C4×C257C2×D411yescharacteristic
86C2×C4×C257#4111yescharacteristic
87C2×C4×C257#4111yescharacteristic
88C8×C257—22no—
89C22×C257—41no—
90C22×C257C2×C22⋊C411yescharacteristic
91C22×C257#12111yescharacteristic
92C22×C257#12111yescharacteristic
93C2×D257—22no—
94C2×D257—42no—
95C4×C257—42no—
96C4×C257—81no—
97C4×C257C4×D411yescharacteristic
98C4×C257C4×D411yescharacteristic
99C4×C257#12011yescharacteristic
100C4×C257#12311yescharacteristic
101Dic257—22no—
102Dic257—42no—
103Dic257—81no—
104C2×C257—21no—
105C2×C257—21no—
106C2×C257—21no—
107C2×C257#38611yescharacteristic
108D257—42no—
109C257#190911yescharacteristic, complemented, Sylow
110#1909—2571nomaximal, Sylow
111#568—2571no—
112#536—2571no—
113#419—2571no—
114#419D25711yescharacteristic, complemented
115#410—2571no—
116H(3,ℤ/4)—2571no—
117H(3,ℤ/4)—2571no—
118C2×#41—2571no—
119C22×Q8C2×D25711yescharacteristic
120#131—5142no—
121#131—5142no—
122C2×M16—2571no—
123C2×M16—5142no—
124C2×M16C2×D25711yescharacteristic
125C4⋊Q8—2572no—
126#124—2572no—
127#123—5142no—
128C22⋊Q8—5142no—
129#117—2571no—
130#117—2571no—
131#117C2×D25711yescharacteristic
132#103—22no—
133#103—5142no—
134#99—5142no—
135#41—5141no—
136#41—5141no—
137#41—5141no—
138#41—5141no—
139C2×Q8—2571no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8C4×D25712yes—
146C2×D4—2572no—
147C2×D4—2571no—
148C22×C4—2572no—
149C22×C4—21no—
150C22×C4C22×D25711yescharacteristic
151M16—21no—
152M16—21no—
153M16—5144no—
154M16—5141no—
155M16—5141no—
156C2×C8—2572no—
157C2×C8—5144no—
158C2×C8C22⋊D25712yes—
159C4⋊C4—21no—
160C4⋊C4—5144no—
161C4⋊C4—5141no—
162C4⋊C4—5141no—
163C22⋊C4—21no—
164C22⋊C4—5144no—
165C22⋊C4—5141no—
166C22⋊C4—5141no—
167C42—2572no—
168C42—21no—
169C42—5141no—
170C23—2572no—
171C23D4×D25711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—5141no—
175Q8—5141no—
176Q8—41no—
177Q8—41no—
178D4—5142no—
179D4—5142no—
180D4—5141no—
181D4—5141no—
182C2×C4—2572no—
183C2×C4—22no—
184C2×C4—5142no—
185C2×C4—5142no—
186C2×C4—5142no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—10281no—
191C2×C4C2×C4×D25711yescharacteristic
192C2×C4D4×D25711yescharacteristic
193C2×C4D4×D25711yescharacteristic
194C2×C4C2×C22⋊D25711yescharacteristic
195C2×C4#1686161711yescharacteristic
196C2×C4#1686161811yescharacteristic
197C8—22no—
198C8—5142no—
199C22—5142no—
200C22—41no—
201C22—10282no—
202C22C22⋊C4×D25711yescharacteristic
203C22#2255156311yescharacteristic
204C22#2255156511yescharacteristic
205C4—5142no—
206C4—42no—
207C4—10282no—
208C4—81no—
209C4—20561no—
210C4#2255155211yescharacteristic
211C4C4×C22⋊D25711yescharacteristic
212C4#2255156011yescharacteristic
213C4#2255157411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—10282no—
218C2#3065661311yescharacteristic
subgroupG/H
centerC2#30656613
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#419×C257C2
socleC2×C257#386
prime radicalC2×#16861618C22
cosocleC22×C4×C257C23
hypercenter#419D257
hypocenterC257#1909
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #30656746C2semidirect
5. #30656494C2semidirect
6. #30656493C2semidirect
7. #419×C257C2semidirect
13. #22551590C4semidirect
14. #22551581C4semidirect
109. C257#1909semidirect
114. #419D257semidirect

Maximal subgroups

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

subgroupindex
#306575032
#306573112
#306567462
#306567102
#306564942
#306564932
#419×C2572
#1909257