Finite Group Databasebeta

← #45664702

Subgroups — #45664702

Order 32896.

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

normal subgroups only

This group has 332 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, complemented
2#30657287C211yesmaximal, characteristic
3#30656755C211yesmaximal, characteristic, complemented
4#30656753C211yesmaximal, characteristic, complemented
5#30656710C211yesmaximal, characteristic, complemented
6#30656497C211yesmaximal, characteristic, complemented
7#339×C257C211yesmaximal, characteristic, complemented
8C2×#16861618C2211yescharacteristic
9C2×#16861618C2211yescharacteristic
10#22551600—22no—
11#22551598—22no—
12C2×#16861602C2211yescharacteristic
13M16×D257—22no—
14#22551592C411yescharacteristic, complemented
15#22551589C411yescharacteristic, complemented
16#22551580C411yescharacteristic, complemented
17#22551580C411yescharacteristic, complemented
18#22551566—22no—
19C4×D1028—22no—
20#22551551C2211yescharacteristic
21#22551520—22no—
22#22551518—22no—
23#22551506—22no—
24C2×M16×C257C2211yescharacteristic
25#117×C257C2211yescharacteristic
26C2×C42×C257C2211yescharacteristic
27#16861618—21no—
28#16861618—21no—
29#16861618—21no—
30#16861618—21no—
31#16861618—21no—
32#16861618—21no—
33#16861618—21no—
34#16861618—21no—
35C2×Dic514C2×C411yescharacteristic
36C2×Dic514C2×C411yescharacteristic
37C2×D1028—22no—
38C2×D1028C2×C411yescharacteristic
39C2×D1028C2×C411yescharacteristic
40C2×C22⋊D257—21no—
41C2×C22⋊D257D412yes—
42C2×C4×D257—21no—
43C2×C4×D257D412yes—
44#16861602—21no—
45#16861602D412yescomplemented
46#16861601—24no—
47C2×#14523339—21no—
48C8×D257—24no—
49C4⋊Dic257—22no—
50C4⋊Dic257—21no—
51C22⋊Dic257—21no—
52#16861595—22no—
53#16861595—22no—
54C4×Dic257—21no—
55C22×C4×C257—21no—
56C22×C4×C257C2311yescharacteristic
57M16×C257—21no—
58M16×C257D412yescomplemented
59C2×C8×C257—21no—
60C4⋊C4×C257—21no—
61C22⋊C4×C257—21no—
62C42×C257—21no—
63C42×C257—21no—
64C42×C257C2×C411yescharacteristic
65C42×C257C2×C411yescharacteristic
66C22×D257—21no—
67C22×D257C22⋊C412yes—
68Dic514—21no—
69Dic514—21no—
70Dic514—21no—
71Dic514—21no—
72D1028—21no—
73D1028—21no—
74D1028—21no—
75D1028—21no—
76D1028—44no—
77C22⋊D257—22no—
78C22⋊D257—22no—
79C22⋊D257—41no—
80C22⋊D257—41no—
81C2×Dic257—21no—
82C2×Dic257—41no—
83C2×Dic257C22⋊C412yes—
84C4×D257—22no—
85C4×D257—22no—
86C4×D257—41no—
87C4×D257—41no—
88#14523339—22no—
89C23×C257C2×D411yescharacteristic
90C2×C4×C257—21no—
91C2×C4×C257—21no—
92C2×C4×C257—42no—
93C2×C4×C257—41no—
94C2×C4×C257C22×C411yescharacteristic
95C2×C4×C257C2×D411yescharacteristic
96C2×C4×C257C2×D411yescharacteristic
97C2×C4×C257C2×D411yescharacteristic
98C2×C4×C257#4111yescharacteristic
99C2×C4×C257#4111yescharacteristic
100C8×C257—22no—
101C22×C257—41no—
102C22×C257C2×C22⋊C411yescharacteristic
103C22×C257#12111yescharacteristic
104C22×C257#12111yescharacteristic
105C2×D257—22no—
106C2×D257—42no—
107C2×D257—42no—
108C2×D257—41no—
109C4×C257—22no—
110C4×C257—41no—
111C4×C257—81no—
112C4×C257C4×D411yescharacteristic
113C4×C257C4×D411yescharacteristic
114C4×C257#12011yescharacteristic
115C4×C257#12311yescharacteristic
116Dic257—22no—
117Dic257—41no—
118Dic257—81no—
119C2×C257—21no—
120C2×C257—21no—
121C2×C257—21no—
122C2×C257#38611yescharacteristic
123D257—42no—
124D257—81no—
125C257#188411yescharacteristic, complemented, Sylow
126#1884—2571nomaximal, Sylow
127#568—2571no—
128#532—2571no—
129#421—2571no—
130#421—2571no—
131#410—2571no—
132#339—2571no—
133#339D25711yescharacteristic, complemented
134C2×#41—2571no—
135C2×#41—2571no—
136#131—5142no—
137#131—5142no—
138C2×M16—2571no—
139C2×M16—5142no—
140C2×M16C2×D25711yescharacteristic
141C4⋊Q8—2571no—
142#127—2571no—
143#124—2571no—
144#124—2571no—
145#121—5142no—
146C4×D4—5142no—
147#117—2571no—
148#117C2×D25711yescharacteristic
149C2×C42C2×D25711yescharacteristic
150#104—5142no—
151#104—5142no—
152#99—5142no—
153#41—5141no—
154#41—5141no—
155#41—5141no—
156#41—5141no—
157#41—5141no—
158#41—5141no—
159#41—5141no—
160#41—5141no—
161C2×Q8—2571no—
162C2×Q8—2571no—
163C2×D4—2572no—
164C2×D4—2571no—
165C2×D4—2571no—
166C2×D4—5142no—
167C2×D4—5141no—
168C22×C4—2572no—
169C22×C4—21no—
170C22×C4—5141no—
171C22×C4C22×D25711yescharacteristic
172M16—2572no—
173M16—21no—
174M16—5144no—
175M16—5141no—
176M16D102812yescomplemented
177C2×C8—21no—
178C2×C8—5144no—
179C2×C8—5141no—
180C4⋊C4—21no—
181C4⋊C4—5142no—
182C4⋊C4—5141no—
183C22⋊C4—21no—
184C22⋊C4—5142no—
185C22⋊C4—5142no—
186C22⋊C4—5141no—
187C42—21no—
188C42—21no—
189C42—5141no—
190C42C4×D25711yescharacteristic
191C42C4×D25711yescharacteristic
192C23—2572no—
193C23—5141no—
194C23D4×D25711yescharacteristic
195Q8—5141no—
196Q8—5141no—
197Q8—5141no—
198Q8—5141no—
199D4—5142no—
200D4—5142no—
201D4—5141no—
202D4—5141no—
203D4—5141no—
204D4—5141no—
205D4—10284no—
206D4—10281no—
207D4—10281no—
208C2×C4—2572no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—5142no—
212C2×C4—5142no—
213C2×C4—5141no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—10281no—
217C2×C4—10281no—
218C2×C4—10281no—
219C2×C4C2×C4×D25711yescharacteristic
220C2×C4D4×D25711yescharacteristic
221C2×C4D4×D25711yescharacteristic
222C2×C4C2×D102811yescharacteristic
223C2×C4#1686161811yescharacteristic
224C2×C4#1686161911yescharacteristic
225C8—22no—
226C8—5142no—
227C22—5142no—
228C22—41no—
229C22—10282no—
230C22—10282no—
231C22—10281no—
232C22C22⋊C4×D25711yescharacteristic
233C22#2255156511yescharacteristic
234C22#2255156711yescharacteristic
235C4—22no—
236C4—5142no—
237C4—41no—
238C4—10281no—
239C4—81no—
240C4—20561no—
241C4#2255155411yescharacteristic
242C4C4×D102811yescharacteristic
243C4#2255156211yescharacteristic
244C4#2255157711yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—10282no—
249C2—20561no—
250C2#3065661511yescharacteristic
subgroupG/H
centerC4#22551562
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#339×C257C2
socleC2×C257#386
prime radical#30657287C2
cosocleC22×C4×C257C23
hypercenter#339D257
hypocenterC257#1884
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #30657503C2semidirect
3. #30656755C2semidirect
4. #30656753C2semidirect
5. #30656710C2semidirect
6. #30656497C2semidirect
7. #339×C257C2semidirect
14. #22551592C4semidirect
15. #22551589C4semidirect
16. #22551580C4semidirect
17. #22551580C4semidirect
45. #16861602D4semidirect
58. M16×C257D4semidirect
125. C257#1884semidirect
133. #339D257semidirect
176. M16D1028semidirect

Maximal subgroups

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

subgroupindex
#306575032
#306572872
#306567552
#306567532
#306567102
#306564972
#339×C2572
#1884257